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The original proofs of theorems with recently shortened proofs can often be found by appending "OLD" to the theorem name, for example 19.43OLD for 19.43. The "OLD" versions are usually deleted after a year.
Other links Email: Norm Megill. Mailing list: Metamath Google Group Updated 7-Dec-2021 . Contributing: How can I contribute to Metamath? Syndication: RSS feed (courtesy of Dan Getz) Related wikis: Ghilbert site; Ghilbert Google Group.
Recent news items (7-Aug-2021) Version 0.198 of the metamath program fixes a bug in "write source ... /rewrap" that prevented end-of-sentence punctuation from appearing in column 79, causing some rewrapped lines to be shorter than necessary. Because this affects about 2000 lines in set.mm, you should use version 0.198 or later for rewrapping before submitting to GitHub.
(7-May-2021) Mario Carneiro has written a Metamath verifier in Lean.
(5-May-2021) Marnix Klooster has written a Metamath verifier in Zig.
(24-Mar-2021) Metamath was mentioned in a couple of articles about OpenAI: Researchers find that large language models struggle with math and What Is GPT-F?.
(26-Dec-2020) Version 0.194 of the metamath program adds the keyword "htmlexturl" to the $t comment to specify external versions of theorem pages. This keyward has been added to set.mm, and you must update your local copy of set.mm for "verify markup" to pass with the new program version.
(19-Dec-2020) Aleksandr A. Adamov has translated the Wikipedia Metamath page into Russian.
(19-Nov-2020) Eric Schmidt's checkmm.cpp was used as a test case for C'est, "a non-standard version of the C++20 standard library, with enhanced support for compile-time evaluation." See C++20 Compile-time Metamath Proof Verification using C'est.
(10-Nov-2020) Filip Cernatescu has updated the XPuzzle (Android app) to version 1.2. XPuzzle is a puzzle with math formulas derived from the Metamath system. At the bottom of the web page is a link to the Google Play Store, where the app can be found.
(7-Nov-2020) Richard Penner created a cross-reference guide between Frege's logic notation and the notation used by set.mm.
(4-Sep-2020) Version 0.192 of the metamath program adds the qualifier '/extract' to 'write source'. See 'help write source' and also this Google Group post.
(23-Aug-2020) Version 0.188 of the metamath program adds keywords Conclusion, Fact, Introduction, Paragraph, Scolia, Scolion, Subsection, and Table to bibliographic references. See 'help write bibliography' for the complete current list.
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| Date | Label | Description |
|---|---|---|
| Theorem | ||
| 10-Feb-2022 | df-bj-magmahom 33080 | Define the set of magma morphisms between two magmas. (Contributed by BJ, 10-Feb-2022.) |
| ⊢ Magma⟶ = (𝑥 ∈ Mgm, 𝑦 ∈ Mgm ↦ {𝑓 ∈ ((Base‘𝑥) Set⟶ (Base‘𝑦)) ∣ ∀𝑢 ∈ (Base‘𝑥)∀𝑣 ∈ (Base‘𝑥)(𝑓‘(𝑢(+g‘𝑥)𝑣)) = ((𝑓‘𝑢)(+g‘𝑦)(𝑓‘𝑣))}) | ||
| 10-Feb-2022 | df-bj-tophom 33079 | Define the set of continuous functions (morphisms of topological spaces) between two topological spaces. Similar to df-cn 21031 (which is in terms of topologies instead of topological spaces). (Contributed by BJ, 10-Feb-2022.) |
| ⊢ Top⟶ = (𝑥 ∈ TopSp, 𝑦 ∈ TopSp ↦ {𝑓 ∈ ((Base‘𝑥) Set⟶ (Base‘𝑦)) ∣ ∀𝑢 ∈ (TopOpen‘𝑦)(◡𝑓 “ 𝑢) ∈ (TopOpen‘𝑥)}) | ||
| 5-Feb-2022 | dfxlim2 40074 | An alternative definition for the convergence relation in the extended real numbers. This resembles what's found in most textbooks: three distinct definitions for the same symbol (limit of a sequence). (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
| ⊢ Ⅎ𝑘𝐹 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) ⇒ ⊢ (𝜑 → (𝐹~~>*𝐴 ↔ (𝐹 ⇝ 𝐴 ∨ (𝐴 = -∞ ∧ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ≤ 𝑥) ∨ (𝐴 = +∞ ∧ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝑥 ≤ (𝐹‘𝑘))))) | ||
| 5-Feb-2022 | dfxlim2v 40073 | An alternative definition for the convergence relation in the extended real numbers. This resembles what's found in most textbooks: three distinct definitions for the same symbol (limit of a sequence). (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
| ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) ⇒ ⊢ (𝜑 → (𝐹~~>*𝐴 ↔ (𝐹 ⇝ 𝐴 ∨ (𝐴 = -∞ ∧ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ≤ 𝑥) ∨ (𝐴 = +∞ ∧ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝑥 ≤ (𝐹‘𝑘))))) | ||
| 5-Feb-2022 | climxlim2 40072 | A sequence of extended reals, converging w.r.t. the standard topology on the complex numbers is a converging sequence w.r.t. the standard topology on the extended reals. This is non-trivial, because +∞ and -∞ could, in principle, be complex numbers. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
| ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) & ⊢ (𝜑 → 𝐹 ⇝ 𝐴) ⇒ ⊢ (𝜑 → 𝐹~~>*𝐴) | ||
| 5-Feb-2022 | climxlim2lem 40071 | In this lemma for climxlim2 40072 there is the additional assumption that the converging function is complex valued on the whole domain. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
| ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) & ⊢ (𝜑 → 𝐹:𝑍⟶ℂ) & ⊢ (𝜑 → 𝐹 ⇝ 𝐴) ⇒ ⊢ (𝜑 → 𝐹~~>*𝐴) | ||
| 5-Feb-2022 | xlimpnfmpt 40070 | A function converges to plus infinity if it eventually becomes (and stays) larger than any given real number. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
| ⊢ Ⅎ𝑘𝜑 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℝ*) & ⊢ 𝐹 = (𝑘 ∈ 𝑍 ↦ 𝐵) ⇒ ⊢ (𝜑 → (𝐹~~>*+∞ ↔ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝑥 ≤ 𝐵)) | ||
| 5-Feb-2022 | xlimmnfmpt 40069 | A function converges to plus infinity if it eventually becomes (and stays) larger than any given real number. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
| ⊢ Ⅎ𝑘𝜑 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℝ*) & ⊢ 𝐹 = (𝑘 ∈ 𝑍 ↦ 𝐵) ⇒ ⊢ (𝜑 → (𝐹~~>*-∞ ↔ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝐵 ≤ 𝑥)) | ||
| 5-Feb-2022 | xlimpnf 40068 | A function converges to plus infinity if it eventually becomes (and stays) larger than any given real number. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
| ⊢ Ⅎ𝑘𝐹 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) ⇒ ⊢ (𝜑 → (𝐹~~>*+∞ ↔ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝑥 ≤ (𝐹‘𝑘))) | ||
| 5-Feb-2022 | xlimmnf 40067 | A function converges to minus infinity if it eventually becomes (and stays) smaller than any given real number. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
| ⊢ Ⅎ𝑘𝐹 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) ⇒ ⊢ (𝜑 → (𝐹~~>*-∞ ↔ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ≤ 𝑥)) | ||
| 5-Feb-2022 | xlimclim2 40066 | Given a sequence of extended reals, it converges to a real number 𝐴 w.r.t. the standard topology on the reals (see climreeq 39845), if and only if it converges to 𝐴 w.r.t. to the standard topology on the extended reals. In order for the first part of the statement to even make sense, the sequence will of course eventually become (and stay) real: showing this, is the key step of the proof. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
| ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) & ⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝐹~~>*𝐴 ↔ 𝐹 ⇝ 𝐴)) | ||
| 5-Feb-2022 | xlimclim2lem 40065 | Lemma for xlimclim2 40066. Here it is additionally assumed that the sequence will eventually become (and stay) real. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
| ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℝ) ⇒ ⊢ (𝜑 → (𝐹~~>*𝐴 ↔ 𝐹 ⇝ 𝐴)) | ||
| 5-Feb-2022 | xlimpnfv 40064 | A function converges to plus infinity if it eventually becomes (and stays) larger than any given real number. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
| ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) ⇒ ⊢ (𝜑 → (𝐹~~>*+∞ ↔ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝑥 ≤ (𝐹‘𝑘))) | ||
| 5-Feb-2022 | xlimpnfvlem2 40063 | The "if" part of the biconditional in xlimpnf 40068. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
| ⊢ Ⅎ𝑘𝜑 & ⊢ Ⅎ𝑗𝜑 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) & ⊢ (𝜑 → ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝑥 < (𝐹‘𝑘)) ⇒ ⊢ (𝜑 → 𝐹~~>*+∞) | ||
| 5-Feb-2022 | xlimpnfvlem1 40062 | The "only if" part of the biconditional in xlimmnf 40067. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
| ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) & ⊢ (𝜑 → 𝐹~~>*+∞) & ⊢ (𝜑 → 𝑋 ∈ ℝ) ⇒ ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝑋 ≤ (𝐹‘𝑘)) | ||
| 5-Feb-2022 | xlimconst2 40061 | A sequence that eventually becomes constant, converges to its constant value (w.r.t. the standard topology on the extended reals). (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
| ⊢ Ⅎ𝑘𝜑 & ⊢ Ⅎ𝑘𝐹 & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) & ⊢ (𝜑 → 𝑁 ∈ 𝑍) & ⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → (𝐹‘𝑘) = 𝐴) ⇒ ⊢ (𝜑 → 𝐹~~>*𝐴) | ||
| 5-Feb-2022 | xlimmnfv 40060 | A function converges to minus infinity if it eventually becomes (and stays) smaller than any given real number. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
| ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) ⇒ ⊢ (𝜑 → (𝐹~~>*-∞ ↔ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ≤ 𝑥)) | ||
| 5-Feb-2022 | xlimmnfvlem2 40059 | The "if" part of the biconditional in xlimpnf 40068. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
| ⊢ Ⅎ𝑘𝜑 & ⊢ Ⅎ𝑗𝜑 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) & ⊢ (𝜑 → ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) < 𝑥) ⇒ ⊢ (𝜑 → 𝐹~~>*-∞) | ||
| 5-Feb-2022 | xlimmnfvlem1 40058 | The "only if" part of the biconditional in xlimmnf 40067. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
| ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) & ⊢ (𝜑 → 𝐹~~>*-∞) & ⊢ (𝜑 → 𝑋 ∈ ℝ) ⇒ ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ≤ 𝑋) | ||
| 5-Feb-2022 | xlimxrre 40057 | If a sequence ranging over the extended reals converges w.r.t. the standard topology on the complex numbers, then there exists an upper set of the integers over which the function is real-valued (the weaker hypothesis 𝐹 ∈ dom ⇝ is probably not enough, since in principle we could have +∞ ∈ ℂ and -∞ ∈ ℂ). (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
| ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐹~~>*𝐴) ⇒ ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℝ) | ||
| 5-Feb-2022 | cnrefiisp 40056 | A non-real, complex number is an isolated point w.r.t. the union of the reals with any finite set (the extended reals is an example of such a union). (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → ¬ 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ Fin) & ⊢ 𝐶 = (ℝ ∪ 𝐵) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ ℝ+ ∀𝑦 ∈ 𝐶 ((𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴) → 𝑥 ≤ (abs‘(𝑦 − 𝐴)))) | ||
| 5-Feb-2022 | cnrefiisplem 40055 | Lemma for cnrefiisp 40056 (some local definitions are used). (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → ¬ 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ Fin) & ⊢ 𝐶 = (ℝ ∪ 𝐵) & ⊢ 𝐷 = ({(abs‘(ℑ‘𝐴))} ∪ ∪ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑦 − 𝐴))}) & ⊢ 𝑋 = inf(𝐷, ℝ*, < ) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ ℝ+ ∀𝑦 ∈ 𝐶 ((𝑦 ∈ ℂ ∧ 𝑦 ≠ 𝐴) → 𝑥 ≤ (abs‘(𝑦 − 𝐴)))) | ||
| 5-Feb-2022 | fuzxrpmcn 40054 | A function mapping from an upper set of integers to the extended reals is a partial map on the complex numbers. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
| ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) ⇒ ⊢ (𝜑 → 𝐹 ∈ (ℝ* ↑pm ℂ)) | ||
| 5-Feb-2022 | xlimbr 40053 | Express the binary relation "sequence 𝐹 converges to point 𝑃 " w.r.t. the standard topology on the extended reals. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
| ⊢ Ⅎ𝑘𝐹 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) & ⊢ 𝐽 = (ordTop‘ ≤ ) ⇒ ⊢ (𝜑 → (𝐹~~>*𝑃 ↔ (𝑃 ∈ ℝ* ∧ ∀𝑢 ∈ 𝐽 (𝑃 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢))))) | ||
| 5-Feb-2022 | climxlim 40052 | A converging sequence in the reals is a converging sequence in the extended reals. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
| ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ) & ⊢ (𝜑 → 𝐹 ⇝ 𝐴) ⇒ ⊢ (𝜑 → 𝐹~~>*𝐴) | ||
| 5-Feb-2022 | xlimconst 40051 | A constant sequence converges to its value, w.r.t. the standard topology on the extended reals. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
| ⊢ Ⅎ𝑘𝜑 & ⊢ Ⅎ𝑘𝐹 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹 Fn 𝑍) & ⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴) ⇒ ⊢ (𝜑 → 𝐹~~>*𝐴) | ||
| 5-Feb-2022 | xlimclim 40050 | Given a sequence of reals, it converges to a real number 𝐴 w.r.t. the standard topology on the reals, if and only if it converges to 𝐴 w.r.t. to the standard topology on the extended reals (see climreeq 39845). (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
| ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ) & ⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝐹~~>*𝐴 ↔ 𝐹 ⇝ 𝐴)) | ||
| 5-Feb-2022 | rexlimddv2 40049 | Restricted existential elimination rule of natural deduction. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
| ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓) & ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝜓) → 𝜒) ⇒ ⊢ (𝜑 → 𝜒) | ||
| 5-Feb-2022 | xlimcl 40048 | The limit of a sequence of extended real numbers is an extended real number. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
| ⊢ (𝐹~~>*𝐴 → 𝐴 ∈ ℝ*) | ||
| 5-Feb-2022 | xlimres 40047 | A function converges iff its restriction to an upper integers set converges. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
| ⊢ (𝜑 → 𝐹 ∈ (ℝ* ↑pm ℂ)) & ⊢ (𝜑 → 𝑀 ∈ ℤ) ⇒ ⊢ (𝜑 → (𝐹~~>*𝐴 ↔ (𝐹 ↾ (ℤ≥‘𝑀))~~>*𝐴)) | ||
| 5-Feb-2022 | xlimrel 40046 | The limit on extended reals is a relation. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
| ⊢ Rel ~~>* | ||
| 5-Feb-2022 | df-xlim 40045 | Define the convergence relation for extended real sequences. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
| ⊢ ~~>* = (⇝𝑡‘(ordTop‘ ≤ )) | ||
| 5-Feb-2022 | climxrre 39982 | If a sequence ranging over the extended reals converges w.r.t. the standard topology on the complex numbers, then there exists an upper set of the integers over which the function is real-valued (the weaker hypothesis 𝐹 ∈ dom ⇝ is probably not enough, since in principle we could have +∞ ∈ ℂ and -∞ ∈ ℂ). (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
| ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐹 ⇝ 𝐴) ⇒ ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℝ) | ||
| 5-Feb-2022 | climxrrelem 39981 | If a seqence ranging over the extended reals converges w.r.t. the standard topology on the complex numbers, then there exists an upper set of the integers over which the function is real-valued. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
| ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) & ⊢ (𝜑 → 𝐹 ⇝ 𝐴) & ⊢ (𝜑 → 𝐷 ∈ ℝ+) & ⊢ ((𝜑 ∧ +∞ ∈ ℂ) → 𝐷 ≤ (abs‘(+∞ − 𝐴))) & ⊢ ((𝜑 ∧ -∞ ∈ ℂ) → 𝐷 ≤ (abs‘(-∞ − 𝐴))) ⇒ ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℝ) | ||
| 5-Feb-2022 | climrescn 39980 | A sequence converging w.r.t. the standard topology on the complex numbers, eventually becomes a sequence of complex numbers. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
| ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹 Fn 𝑍) & ⊢ (𝜑 → 𝐹 ∈ dom ⇝ ) ⇒ ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℂ) | ||
| 5-Feb-2022 | lmbr3 39979 | Express the binary relation "sequence 𝐹 converges to point 𝑃 " in a metric space using an arbitrary upper set of integers. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
| ⊢ Ⅎ𝑘𝐹 & ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) ⇒ ⊢ (𝜑 → (𝐹(⇝𝑡‘𝐽)𝑃 ↔ (𝐹 ∈ (𝑋 ↑pm ℂ) ∧ 𝑃 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝐽 (𝑃 ∈ 𝑢 → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢))))) | ||
| 5-Feb-2022 | climisp 39978 | If a sequence converges to an isolated point (w.r.t. the standard topology on the complex numbers) then the sequence eventually becomes that point. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
| ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℂ) & ⊢ (𝜑 → 𝐹 ⇝ 𝐴) & ⊢ (𝜑 → 𝑋 ∈ ℝ+) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍 ∧ (𝐹‘𝑘) ≠ 𝐴) → 𝑋 ≤ (abs‘((𝐹‘𝑘) − 𝐴))) ⇒ ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) = 𝐴) | ||
| 5-Feb-2022 | lmbr3v 39977 | Express the binary relation "sequence 𝐹 converges to point 𝑃 " in a metric space using an arbitrary upper set of integers. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
| ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) ⇒ ⊢ (𝜑 → (𝐹(⇝𝑡‘𝐽)𝑃 ↔ (𝐹 ∈ (𝑋 ↑pm ℂ) ∧ 𝑃 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝐽 (𝑃 ∈ 𝑢 → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢))))) | ||
| 5-Feb-2022 | tgioo4 39800 | The standard topology on the reals is a subspace of the complex metric topology. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
| ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂfld) ↾t ℝ) | ||
| 5-Feb-2022 | xrtgioo2 39799 | The topology on the extended reals coincides with the standard topology on the reals, when restricted to ℝ. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
| ⊢ (topGen‘ran (,)) = ((ordTop‘ ≤ ) ↾t ℝ) | ||
| 5-Feb-2022 | iocgtlbd 39798 | An element of a left open right closed interval is larger than its lower bound. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐶 ∈ (𝐴(,]𝐵)) ⇒ ⊢ (𝜑 → 𝐴 < 𝐶) | ||
| 5-Feb-2022 | rpssxr 39711 | The positive reals are a subset of the extended reals. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
| ⊢ ℝ+ ⊆ ℝ* | ||
| 5-Feb-2022 | absimlere 39710 | The absolute value of the imaginary part of a complex number is a lower bound of the distance to any real number. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (abs‘(ℑ‘𝐴)) ≤ (abs‘(𝐵 − 𝐴))) | ||
| 5-Feb-2022 | xrtgcntopre 39709 | The standard topologies on the extended reals and on the complex numbers, coincide when restricted to the reals. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
| ⊢ ((ordTop‘ ≤ ) ↾t ℝ) = ((TopOpen‘ℂfld) ↾t ℝ) | ||
| 5-Feb-2022 | uzsscn2 39708 | An upper set of integers is a subset of the complex numbers. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
| ⊢ 𝑍 = (ℤ≥‘𝑀) ⇒ ⊢ 𝑍 ⊆ ℂ | ||
| 5-Feb-2022 | absimnre 39707 | The absolute value of the imaginary part of a non real, complex number, is strictly positive. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → ¬ 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → (abs‘(ℑ‘𝐴)) ∈ ℝ+) | ||
| 5-Feb-2022 | uzsscn 39706 | An upper set of integers is a subset of the complex numbers. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
| ⊢ (ℤ≥‘𝑀) ⊆ ℂ | ||
| 5-Feb-2022 | xrnpnfmnf 39705 | An extended real that is neither real nor plus infinity, is minus infinity. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → ¬ 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐴 ≠ +∞) ⇒ ⊢ (𝜑 → 𝐴 = -∞) | ||
| 5-Feb-2022 | pnfged 39704 | Plus infinity is an upper bound for extended reals. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ*) ⇒ ⊢ (𝜑 → 𝐴 ≤ +∞) | ||
| 5-Feb-2022 | min2d 39703 | The minimum of two numbers is less than or equal to the second. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → if(𝐴 ≤ 𝐵, 𝐴, 𝐵) ≤ 𝐵) | ||
| 5-Feb-2022 | min1d 39702 | The minimum of two numbers is less than or equal to the first. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → if(𝐴 ≤ 𝐵, 𝐴, 𝐵) ≤ 𝐴) | ||
| 5-Feb-2022 | fconst7 39484 | An alternative way to express a constant function. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝐹 & ⊢ (𝜑 → 𝐹 Fn 𝐴) & ⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵) ⇒ ⊢ (𝜑 → 𝐹 = (𝐴 × {𝐵})) | ||
| 5-Feb-2022 | fpmd 39483 | A total function is a partial function. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
| ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ (𝜑 → 𝐶 ⊆ 𝐴) & ⊢ (𝜑 → 𝐹:𝐶⟶𝐵) ⇒ ⊢ (𝜑 → 𝐹 ∈ (𝐵 ↑pm 𝐴)) | ||
| 5-Feb-2022 | fnssresd 39482 | Restriction of a function to a subclass of its domain. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
| ⊢ (𝜑 → 𝐹 Fn 𝐴) & ⊢ (𝜑 → 𝐵 ⊆ 𝐴) ⇒ ⊢ (𝜑 → (𝐹 ↾ 𝐵) Fn 𝐵) | ||
| 5-Feb-2022 | unfid 39345 | The union of two finite sets is finite. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
| ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐵 ∈ Fin) ⇒ ⊢ (𝜑 → (𝐴 ∪ 𝐵) ∈ Fin) | ||
| 5-Feb-2022 | reximdd 39344 | Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝜓) → 𝜒) & ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜒) | ||
| 5-Feb-2022 | reximddv3 39343 | Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
| ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝜓) → 𝜒) & ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜒) | ||
| 5-Feb-2022 | eliunid 39342 | Membership in indexed union. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
| ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵) → 𝐶 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) | ||
| 5-Feb-2022 | frgrwopreglem5ALT 27186 | Alternate direct proof of frgrwopreglem5 27185, not using frgrwopreglem5a 27175. This proof would be even a little bit shorter than the proof of frgrwopreglem5 27185 without using frgrwopreglem5lem 27184. (Contributed by Alexander van der Vekens, 31-Dec-2017.) (Revised by AV, 3-Jan-2022.) (Proof shortened by AV, 5-Feb-2022.) (New usage is discouraged.) (Proof modification is discouraged.) |
| ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐷 = (VtxDeg‘𝐺) & ⊢ 𝐴 = {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾} & ⊢ 𝐵 = (𝑉 ∖ 𝐴) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ ((𝐺 ∈ FriendGraph ∧ 1 < (#‘𝐴) ∧ 1 < (#‘𝐵)) → ∃𝑎 ∈ 𝐴 ∃𝑥 ∈ 𝐴 ∃𝑏 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ((𝑎 ≠ 𝑥 ∧ 𝑏 ≠ 𝑦) ∧ ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑥} ∈ 𝐸) ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑦, 𝑎} ∈ 𝐸))) | ||
| 5-Feb-2022 | frgrwopreglem5 27185 | Lemma 5 for frgrwopreg 27187. If 𝐴 as well as 𝐵 contain at least two vertices, there is a 4-cycle in a friendship graph. This corresponds to statement 6 in [Huneke] p. 2: "... otherwise, there are two different vertices in A, and they have two common neighbors in B, ...". (Contributed by Alexander van der Vekens, 31-Dec-2017.) (Proof shortened by AV, 5-Feb-2022.) |
| ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐷 = (VtxDeg‘𝐺) & ⊢ 𝐴 = {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾} & ⊢ 𝐵 = (𝑉 ∖ 𝐴) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ ((𝐺 ∈ FriendGraph ∧ 1 < (#‘𝐴) ∧ 1 < (#‘𝐵)) → ∃𝑎 ∈ 𝐴 ∃𝑥 ∈ 𝐴 ∃𝑏 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ((𝑎 ≠ 𝑥 ∧ 𝑏 ≠ 𝑦) ∧ ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑥} ∈ 𝐸) ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑦, 𝑎} ∈ 𝐸))) | ||
| 5-Feb-2022 | frgrwopreglem5lem 27184 | Lemma for frgrwopreglem5 27185. (Contributed by AV, 5-Feb-2022.) |
| ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐷 = (VtxDeg‘𝐺) & ⊢ 𝐴 = {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾} & ⊢ 𝐵 = (𝑉 ∖ 𝐴) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ (((𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ (𝑏 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝐷‘𝑎) = (𝐷‘𝑥) ∧ (𝐷‘𝑎) ≠ (𝐷‘𝑏) ∧ (𝐷‘𝑥) ≠ (𝐷‘𝑦))) | ||
| 4-Feb-2022 | frgrwopreg2 27183 | According to statement 5 in [Huneke] p. 2: "If ... B is a singleton, then that singleton is a universal friend". (Contributed by Alexander van der Vekens, 1-Jan-2018.) (Proof shortened by AV, 4-Feb-2022.) |
| ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐷 = (VtxDeg‘𝐺) & ⊢ 𝐴 = {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾} & ⊢ 𝐵 = (𝑉 ∖ 𝐴) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ ((𝐺 ∈ FriendGraph ∧ (#‘𝐵) = 1) → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸) | ||
| 4-Feb-2022 | frgrwopreg1 27182 | According to statement 5 in [Huneke] p. 2: "If A ... is a singleton, then that singleton is a universal friend". (Contributed by Alexander van der Vekens, 1-Jan-2018.) (Proof shortened by AV, 4-Feb-2022.) |
| ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐷 = (VtxDeg‘𝐺) & ⊢ 𝐴 = {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾} & ⊢ 𝐵 = (𝑉 ∖ 𝐴) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ ((𝐺 ∈ FriendGraph ∧ (#‘𝐴) = 1) → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸) | ||
| 4-Feb-2022 | frgrwopregbsn 27181 | According to statement 5 in [Huneke] p. 2: "If ... B is a singleton, then that singleton is a universal friend". This version of frgrwopreg2 27183 is stricter (claiming that the singleton itself is a universal friend instead of claiming the existence of a universal friend only) and therefore closer to Huneke's statement. This strict variant, however, is not required for the proof of the friendship theorem. (Contributed by AV, 4-Feb-2022.) |
| ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐷 = (VtxDeg‘𝐺) & ⊢ 𝐴 = {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾} & ⊢ 𝐵 = (𝑉 ∖ 𝐴) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝐵 = {𝑋}) → ∀𝑤 ∈ (𝑉 ∖ {𝑋}){𝑋, 𝑤} ∈ 𝐸) | ||
| 4-Feb-2022 | frgrwopregasn 27180 | According to statement 5 in [Huneke] p. 2: "If A ... is a singleton, then that singleton is a universal friend". This version of frgrwopreg1 27182 is stricter (claiming that the singleton itself is a universal friend instead of claiming the existence of a universal friend only) and therefore closer to Huneke's statement. This strict variant, however, is not required for the proof of the friendship theorem. (Contributed by Alexander van der Vekens, 1-Jan-2018.) (Revised by AV, 4-Feb-2022.) |
| ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐷 = (VtxDeg‘𝐺) & ⊢ 𝐴 = {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾} & ⊢ 𝐵 = (𝑉 ∖ 𝐴) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝐴 = {𝑋}) → ∀𝑤 ∈ (𝑉 ∖ {𝑋}){𝑋, 𝑤} ∈ 𝐸) | ||
| 4-Feb-2022 | frgrwopreglem4 27179 | Lemma 4 for frgrwopreg 27187. In a friendship graph each vertex with degree 𝐾 is connected with any vertex with degree other than 𝐾. This corresponds to statement 4 in [Huneke] p. 2: "By the first claim, every vertex in A is adjacent to every vertex in B.". (Contributed by Alexander van der Vekens, 30-Dec-2017.) (Revised by AV, 10-May-2021.) (Proof shortened by AV, 4-Feb-2022.) |
| ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐷 = (VtxDeg‘𝐺) & ⊢ 𝐴 = {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾} & ⊢ 𝐵 = (𝑉 ∖ 𝐴) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ (𝐺 ∈ FriendGraph → ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 {𝑎, 𝑏} ∈ 𝐸) | ||
| 4-Feb-2022 | frgrwopreglem5a 27175 | If a friendship graph has two vertices with the same degree and two other vertices with different degrees, then there is a 4-cycle in the graph. Alternate version of frgrwopreglem5 27185 without a fixed degree and without using the sets 𝐴 and 𝐵. (Contributed by Alexander van der Vekens, 31-Dec-2017.) (Revised by AV, 4-Feb-2022.) |
| ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐷 = (VtxDeg‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ ((𝐺 ∈ FriendGraph ∧ ((𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) ∧ (𝐵 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) ∧ ((𝐷‘𝐴) = (𝐷‘𝑋) ∧ (𝐷‘𝐴) ≠ (𝐷‘𝐵) ∧ (𝐷‘𝑋) ≠ (𝐷‘𝑌))) → (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝑋} ∈ 𝐸) ∧ ({𝑋, 𝑌} ∈ 𝐸 ∧ {𝑌, 𝐴} ∈ 𝐸))) | ||
| 4-Feb-2022 | frgrwopreglem4a 27174 | In a friendship graph any two vertices with different degrees are connected. Alternate version of frgrwopreglem4 27179 without a fixed degree and without using the sets 𝐴 and 𝐵. (Contributed by Alexander van der Vekens, 30-Dec-2017.) (Revised by AV, 4-Feb-2022.) |
| ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐷 = (VtxDeg‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ ((𝐺 ∈ FriendGraph ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ≠ (𝐷‘𝑌)) → {𝑋, 𝑌} ∈ 𝐸) | ||
| 29-Jan-2022 | extwwlkfablem1 27207 | Lemma 1 for extwwlkfab 27223. (Contributed by Alexander van der Vekens, 15-Sep-2018.) (Revised by AV, 27-May-2021.) (Proof shortened by AV, 29-Jan-2022.) |
| ⊢ (((𝐺 ∈ USGraph ∧ 𝑁 ∈ (ℤ≥‘2)) ∧ 𝑊 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝑊‘(𝑁 − 2)) = (𝑊‘0)) → (𝑊‘(𝑁 − 1)) ∈ (𝐺 NeighbVtx (𝑊‘0))) | ||
| 27-Jan-2022 | numclwwlkovf2exlem2 27212 | Lemma 2 for numclwwlkovf2ex 27219: Transformation of a walk and two edges into a walk extended by two vertices/edges. (Contributed by AV, 22-Sep-2018.) (Revised by AV, 27-Jan-2022.) |
| ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ ((((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3)) ∧ ((𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 ∧ {( lastS ‘𝑊), (𝑊‘0)} ∈ 𝐸) ∧ (#‘𝑊) = (𝑁 − 2) ∧ (𝑊‘0) = 𝑋)) ∧ 𝑌 ∈ (𝐺 NeighbVtx 𝑋)) → ∀𝑖 ∈ ((0..^((#‘𝑊) − 1)) ∪ {((#‘𝑊) − 1), (#‘𝑊)}){(((𝑊 ++ ⟨“𝑋”⟩) ++ ⟨“𝑌”⟩)‘𝑖), (((𝑊 ++ ⟨“𝑋”⟩) ++ ⟨“𝑌”⟩)‘(𝑖 + 1))} ∈ 𝐸) | ||
| 26-Jan-2022 | numclwwlkovf2exlem1 27211 | Lemma 1 for numclwwlkovf2ex 27219: Transformation of a special half-open integer range into a union of a smaller half-open integer range and an unordered pair. (Contributed by AV, 22-Sep-2018.) (Revised by AV, 26-Jan-2022.) |
| ⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ (#‘𝑊) = (𝑁 − 2)) → (0..^(((#‘𝑊) + 2) − 1)) = ((0..^((#‘𝑊) − 1)) ∪ {((#‘𝑊) − 1), (#‘𝑊)})) | ||
| 25-Jan-2022 | numclwwlkovf2ex 27219 | Extending a closed walk starting at a fixed vertex by an additional edge (forth and back). (Contributed by AV, 22-Sep-2018.) (Proof shortened by AV, 25-Jan-2022.) |
| ⊢ 𝐹 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣}) & ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3)) ∧ 𝑌 ∈ (𝐺 NeighbVtx 𝑋) ∧ 𝑊 ∈ (𝑋𝐹(𝑁 − 2))) → ((𝑊 ++ ⟨“𝑋”⟩) ++ ⟨“𝑌”⟩) ∈ (𝑁 ClWWalksN 𝐺)) | ||
| 24-Jan-2022 | extwwlkfablem2 27210 | Lemma 2 for extwwlkfab 27223. (Contributed by Alexander van der Vekens, 15-Sep-2018.) (Revised by AV, 28-May-2021.) (Proof shortened by AV, 24-Jan-2022.) |
| ⊢ (((𝐺 ∈ USGraph ∧ 𝑁 ∈ (ℤ≥‘3)) ∧ 𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝑤‘(𝑁 − 2)) = (𝑤‘0)) → (𝑤 substr ⟨0, (𝑁 − 2)⟩) ∈ ((𝑁 − 2) ClWWalksN 𝐺)) | ||
| 24-Jan-2022 | clwwlksnwwlksn 27209 | A word representing a closed walk of length 𝑁 also represents a walk of length 𝑁 − 1. The walk is one edge shorter than the closed walk, because the last edge connecting the last with the first vertex is missing. For example, if ⟨“𝑎𝑏𝑐”⟩ ∈ (3 ClWWalksN 𝐺) represents a closed walk "abca" of length 3, then ⟨“𝑎𝑏𝑐”⟩ ∈ (2 WWalksN 𝐺) represents a walk "abc" (not closed if 𝑎 ≠ 𝑐) of length 2, and ⟨“𝑎𝑏𝑐𝑎”⟩ ∈ (3 WWalksN 𝐺) represents also a closed walk "abca" of length 3. (Contributed by AV, 24-Jan-2022.) |
| ⊢ ((𝑁 ∈ ℕ ∧ 𝑊 ∈ (𝑁 ClWWalksN 𝐺)) → 𝑊 ∈ ((𝑁 − 1) WWalksN 𝐺)) | ||
| 23-Jan-2022 | numclwlk3lem3 27206 | Lemma 3 for numclwwlk3 27243. (Contributed by Alexander van der Vekens, 26-Aug-2018.) (Proof shortened by AV, 23-Jan-2022.) |
| ⊢ ((𝐾 ∈ ℂ ∧ 𝑌 ∈ ℂ ∧ 𝑁 ∈ (ℤ≥‘2)) → (((𝐾↑(𝑁 − 2)) − 𝑌) + (𝐾 · 𝑌)) = (((𝐾 − 1) · 𝑌) + (𝐾↑(𝑁 − 2)))) | ||
| 23-Jan-2022 | clwwlkinwwlk 26905 | If the initial vertex of a walk occurs another time in the walk, the walk starts with a closed walk. Since the walk is expressed as a word over vertices, the closed walk can be expressed as a subword of this word. (Contributed by Alexander van der Vekens, 15-Sep-2018.) (Revised by AV, 23-Jan-2022.) |
| ⊢ (((𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ≥‘𝑁)) ∧ 𝑊 ∈ (𝑀 WWalksN 𝐺) ∧ (𝑊‘𝑁) = (𝑊‘0)) → (𝑊 substr ⟨0, 𝑁⟩) ∈ (𝑁 ClWWalksN 𝐺)) | ||
| 23-Jan-2022 | elfz1uz 12410 | Membership in a 1 based finite set of sequential integers with an upper integer. (Contributed by AV, 23-Jan-2022.) |
| ⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ≥‘𝑁)) → 𝑁 ∈ (1...𝑀)) | ||
| 23-Jan-2022 | elfz1b 12409 | Membership in a 1 based finite set of sequential integers. (Contributed by AV, 30-Oct-2018.) (Proof shortened by AV, 23-Jan-2022.) |
| ⊢ (𝑁 ∈ (1...𝑀) ↔ (𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑁 ≤ 𝑀)) | ||
| 23-Jan-2022 | nrelv 5244 | The universal class is not a relation. (Contributed by Thierry Arnoux, 23-Jan-2022.) |
| ⊢ ¬ Rel V | ||
| 23-Jan-2022 | eqvinc 3330 | A variable introduction law for class equality. (Contributed by NM, 14-Apr-1995.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) (Proof shortened by Thierry Arnoux, 23-Jan-2022.) |
| ⊢ 𝐴 ∈ V ⇒ ⊢ (𝐴 = 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 = 𝐵)) | ||
| 21-Jan-2022 | numclwwlk3 27243 | Statement 12 in [Huneke] p. 2: "Thus f(n) = (k - 1)f(n - 2) + k^(n-2)." - the number of the closed walks v(0) ... v(n-2) v(n-1) v(n) is the sum of the number of the closed walks v(0) ... v(n-2) v(n-1) v(n) with v(n-2) = v(n) (see numclwwlk1 27231) and with v(n-2) =/= v(n) (see numclwwlk2 27240): f(n) = kf(n-2) + k^(n-2) - f(n-2) = (k-1)f(n-2) + k^(n-2). (Contributed by Alexander van der Vekens, 26-Aug-2018.) (Revised by AV, 21-Jan-2022.) |
| ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐹 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣}) ⇒ ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝐺 ∈ FriendGraph ) ∧ (𝑉 ∈ Fin ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))) → (#‘(𝑋𝐹𝑁)) = (((𝐾 − 1) · (#‘(𝑋𝐹(𝑁 − 2)))) + (𝐾↑(𝑁 − 2)))) | ||
| 21-Jan-2022 | numclwlk2lem2f1o 27238 | R is a 1-1 onto function. (Contributed by Alexander van der Vekens, 6-Oct-2018.) (Revised by AV, 21-Jan-2022.) |
| ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝑄 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ ( lastS ‘𝑤) ≠ 𝑣)}) & ⊢ 𝐹 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣}) & ⊢ 𝐻 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) ≠ (𝑤‘0))}) & ⊢ 𝑅 = (𝑥 ∈ (𝑋𝐻(𝑁 + 2)) ↦ (𝑥 substr ⟨0, (𝑁 + 1)⟩)) ⇒ ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → 𝑅:(𝑋𝐻(𝑁 + 2))–1-1-onto→(𝑋𝑄𝑁)) | ||
| 21-Jan-2022 | reuccats1v 13481 | A set of words having the length of a given word increased by 1 contains a unique word with the given word as prefix if there is a unique symbol which extends the given word to be a word of the set. (Contributed by Alexander van der Vekens, 6-Oct-2018.) (Proof shortened by AV, 21-Jan-2022.) |
| ⊢ ((𝑊 ∈ Word 𝑉 ∧ ∀𝑥 ∈ 𝑋 (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1))) → (∃!𝑣 ∈ 𝑉 (𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 → ∃!𝑥 ∈ 𝑋 𝑊 = (𝑥 substr ⟨0, (#‘𝑊)⟩))) | ||
| 21-Jan-2022 | reuccats1 13480 | A set of words having the length of a given word increased by 1 contains a unique word with the given word as prefix if there is a unique symbol which extends the given word to be a word of the set. (Contributed by Alexander van der Vekens, 6-Oct-2018.) (Revised by AV, 21-Jan-2022.) |
| ⊢ Ⅎ𝑣𝑋 ⇒ ⊢ ((𝑊 ∈ Word 𝑉 ∧ ∀𝑥 ∈ 𝑋 (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1))) → (∃!𝑣 ∈ 𝑉 (𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 → ∃!𝑥 ∈ 𝑋 𝑊 = (𝑥 substr ⟨0, (#‘𝑊)⟩))) | ||
| 21-Jan-2022 | reu8nf 3516 | Restricted uniqueness using implicit substitution. This version of reu8 3402 uses a non-freeness hypothesis for 𝑥 and 𝜓 instead of distinct variable conditions. (Contributed by AV, 21-Jan-2022.) |
| ⊢ Ⅎ𝑥𝜓 & ⊢ Ⅎ𝑥𝜒 & ⊢ (𝑥 = 𝑤 → (𝜑 ↔ 𝜒)) & ⊢ (𝑤 = 𝑦 → (𝜒 ↔ 𝜓)) ⇒ ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐴 (𝜑 ∧ ∀𝑦 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦))) | ||
| 19-Jan-2022 | fvmptdf 6296 | Alternate deduction version of fvmpt 6282, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.) (Proof shortened by AV, 19-Jan-2022.) |
| ⊢ (𝜑 → 𝐴 ∈ 𝐷) & ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → ((𝐹‘𝐴) = 𝐵 → 𝜓)) & ⊢ Ⅎ𝑥𝐹 & ⊢ Ⅎ𝑥𝜓 ⇒ ⊢ (𝜑 → (𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) → 𝜓)) | ||
| 19-Jan-2022 | fvmptd3f 6295 | Alternate deduction version of fvmpt 6282 with three non-freeness hypotheses instead of distinct variable conditions. (Contributed by AV, 19-Jan-2022.) |
| ⊢ (𝜑 → 𝐴 ∈ 𝐷) & ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → ((𝐹‘𝐴) = 𝐵 → 𝜓)) & ⊢ Ⅎ𝑥𝐹 & ⊢ Ⅎ𝑥𝜓 & ⊢ Ⅎ𝑥𝜑 ⇒ ⊢ (𝜑 → (𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) → 𝜓)) | ||
| 17-Jan-2022 | br2ndeqg 31673 | Uniqueness condition for the binary relation 2nd. (Contributed by Scott Fenton, 2-Jul-2020.) Revised to remove sethood hypothesis on 𝐶. (Revised by Peter Mazsa, 17-Jan-2022.) |
| ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (⟨𝐴, 𝐵⟩2nd 𝐶 ↔ 𝐶 = 𝐵)) | ||
| 17-Jan-2022 | br1steqg 31672 | Uniqueness condition for the binary relation 1st. (Contributed by Scott Fenton, 2-Jul-2020.) Revised to remove sethood hypothesis on 𝐶. (Revised by Peter Mazsa, 17-Jan-2022.) |
| ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (⟨𝐴, 𝐵⟩1st 𝐶 ↔ 𝐶 = 𝐴)) | ||
| 15-Jan-2022 | tgoldbachgtALTV 41700 | Variant of Thierry Arnoux's tgoldbachgt 30741 using the symbols Odd and GoldbachOdd: The ternary Goldbach conjecture is valid for large odd numbers (i.e. for all odd numbers greater than a fixed 𝑚). This is proven by Helfgott (see section 7.4 in [Helfgott] p. 70) for 𝑚 = 10^27. (Contributed by AV, 2-Aug-2020.) (Revised by AV, 15-Jan-2022.) |
| ⊢ ∃𝑚 ∈ ℕ (𝑚 ≤ (;10↑;27) ∧ ∀𝑛 ∈ Odd (𝑚 < 𝑛 → 𝑛 ∈ GoldbachOdd )) | ||
| 12-Jan-2022 | frrusgrord 27205 | If a nonempty finite friendship graph is k-regular, its order is k(k-1)+1. This corresponds to claim 3 in [Huneke] p. 2: "Next we claim that the number n of vertices in G is exactly k(k-1)+1.". Variant of frrusgrord0 27204, using the definition RegUSGraph (df-rusgr 26454). (Contributed by Alexander van der Vekens, 25-Aug-2018.) (Revised by AV, 26-May-2021.) (Proof shortened by AV, 12-Jan-2022.) |
| ⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → ((𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾) → (#‘𝑉) = ((𝐾 · (𝐾 − 1)) + 1))) | ||
| 12-Jan-2022 | frrusgrord0 27204 | If a nonempty finite friendship graph is k-regular, its order is k(k-1)+1. This corresponds to claim 3 in [Huneke] p. 2: "Next we claim that the number n of vertices in G is exactly k(k-1)+1.". (Contributed by Alexander van der Vekens, 11-Mar-2018.) (Revised by AV, 26-May-2021.) (Proof shortened by AV, 12-Jan-2022.) |
| ⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → (∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾 → (#‘𝑉) = ((𝐾 · (𝐾 − 1)) + 1))) | ||
| 12-Jan-2022 | frrusgrord0lem 27203 | Lemma for frrusgrord0 27204. (Contributed by AV, 12-Jan-2022.) |
| ⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ ∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾) → (𝐾 ∈ ℂ ∧ (#‘𝑉) ∈ ℂ ∧ (#‘𝑉) ≠ 0)) | ||
| 12-Jan-2022 | fusgreghash2wsp 27202 | In a finite k-regular graph with N vertices there are N times "k choose 2" paths with length 2, according to statement 8 in [Huneke] p. 2: "... giving n * ( k 2 ) total paths of length two.", if the direction of traversing the path is not respected. For simple paths of length 2 represented by length 3 strings, however, we have again n*k*(k-1) such paths. (Contributed by Alexander van der Vekens, 11-Mar-2018.) (Revised by AV, 19-May-2021.) (Proof shortened by AV, 12-Jan-2022.) |
| ⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑉 ≠ ∅) → (∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾 → (#‘(2 WSPathsN 𝐺)) = ((#‘𝑉) · (𝐾 · (𝐾 − 1))))) | ||
| 12-Jan-2022 | frgrregorufrg 27190 | If there is a vertex having degree 𝑘 for each nonnegative integer 𝑘 in a friendship graph, then there is a universal friend. This corresponds to claim 2 in [Huneke] p. 2: "Suppose there is a vertex of degree k > 1. ... all vertices have degree k, unless there is a universal friend. ... It follows that G is k-regular, i.e., the degree of every vertex is k". Variant of frgrregorufr 27189 with generalization. (Contributed by Alexander van der Vekens, 6-Sep-2018.) (Revised by AV, 26-May-2021.) (Proof shortened by AV, 12-Jan-2022.) |
| ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ (𝐺 ∈ FriendGraph → ∀𝑘 ∈ ℕ0 (∃𝑎 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑎) = 𝑘 → (𝐺 RegUSGraph 𝑘 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸))) | ||
| 10-Jan-2022 | 2wspmdisj 27201 | The sets of paths of length 2 with a given vertex in the middle are distinct for different vertices in the middle. (Contributed by Alexander van der Vekens, 11-Mar-2018.) (Revised by AV, 18-May-2021.) (Proof shortened by AV, 10-Jan-2022.) |
| ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝑀 = (𝑎 ∈ 𝑉 ↦ {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑎}) ⇒ ⊢ Disj 𝑥 ∈ 𝑉 (𝑀‘𝑥) | ||
| 10-Jan-2022 | fusgreg2wsp 27200 | In a finite simple graph, the set of all paths of length 2 is the union of all the paths of length 2 over the vertices which are in the middle of such a path. (Contributed by Alexander van der Vekens, 10-Mar-2018.) (Revised by AV, 18-May-2021.) (Proof shortened by AV, 10-Jan-2022.) |
| ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝑀 = (𝑎 ∈ 𝑉 ↦ {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑎}) ⇒ ⊢ (𝐺 ∈ FinUSGraph → (2 WSPathsN 𝐺) = ∪ 𝑥 ∈ 𝑉 (𝑀‘𝑥)) | ||
| 10-Jan-2022 | fusgreghash2wspv 27199 | According to statement 7 in [Huneke] p. 2: "For each vertex v, there are exactly ( k 2 ) paths with length two having v in the middle, ..." in a finite k-regular graph. For directed simple paths of length 2 represented by length 3 strings, we have again k*(k-1) such paths, see also comment of frgrhash2wsp 27196. (Contributed by Alexander van der Vekens, 10-Mar-2018.) (Revised by AV, 17-May-2021.) (Proof shortened by AV, 10-Jan-2022.) |
| ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝑀 = (𝑎 ∈ 𝑉 ↦ {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑎}) ⇒ ⊢ (𝐺 ∈ FinUSGraph → ∀𝑣 ∈ 𝑉 (((VtxDeg‘𝐺)‘𝑣) = 𝐾 → (#‘(𝑀‘𝑣)) = (𝐾 · (𝐾 − 1)))) | ||
| 10-Jan-2022 | frgrhash2wsp 27196 | The number of simple paths of length 2 is n*(n-1) in a friendship graph with n vertices. This corresponds to the proof of claim 3 in [Huneke] p. 2: "... the paths of length two in G: by assumption there are ( n 2 ) such paths.". However, Huneke counts undirected paths, so obtains the result ((𝑛C2) = ((𝑛 · (𝑛 − 1)) / 2)), whereas we count directed paths, obtaining twice that number. (Contributed by Alexander van der Vekens, 6-Mar-2018.) (Revised by AV, 10-Jan-2022.) |
| ⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin) → (#‘(2 WSPathsN 𝐺)) = ((#‘𝑉) · ((#‘𝑉) − 1))) | ||
| 10-Jan-2022 | midwwlks2s3 26860 | There is a vertex between the endpoints of a walk of length 2 between two vertices as length 3 string. (Contributed by AV, 10-Jan-2022.) |
| ⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ (𝑊 ∈ (2 WWalksN 𝐺) → ∃𝑏 ∈ 𝑉 (𝑊‘1) = 𝑏) | ||
| 9-Jan-2022 | 2wspiundisj 26856 | All simple paths of length 2 from a fixed vertex to another vertex are disjunct. (Contributed by Alexander van der Vekens, 5-Mar-2018.) (Revised by AV, 14-May-2021.) (Proof shortened by AV, 9-Jan-2022.) |
| ⊢ Disj 𝑎 ∈ 𝑉 ∪ 𝑏 ∈ (𝑉 ∖ {𝑎})(𝑎(2 WSPathsNOn 𝐺)𝑏) | ||
| 9-Jan-2022 | 2wspdisj 26855 | All simple paths of length 2 from a fixed vertex to another vertex are disjunct. (Contributed by Alexander van der Vekens, 4-Mar-2018.) (Revised by AV, 9-Jan-2022.) |
| ⊢ Disj 𝑏 ∈ (𝑉 ∖ {𝐴})(𝐴(2 WSPathsNOn 𝐺)𝑏) | ||
| 9-Jan-2022 | hash2iun1dif1 14556 | The cardinality of a nested disjoint indexed union. (Contributed by AV, 9-Jan-2022.) |
| ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ 𝐵 = (𝐴 ∖ {𝑥}) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝐶 ∈ Fin) & ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 𝐶) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → Disj 𝑦 ∈ 𝐵 𝐶) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (#‘𝐶) = 1) ⇒ ⊢ (𝜑 → (#‘∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 𝐶) = ((#‘𝐴) · ((#‘𝐴) − 1))) | ||
| 9-Jan-2022 | hash2iun 14555 | The cardinality of a nested disjoint indexed union. (Contributed by AV, 9-Jan-2022.) |
| ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝐶 ∈ Fin) & ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 𝐶) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → Disj 𝑦 ∈ 𝐵 𝐶) ⇒ ⊢ (𝜑 → (#‘∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 𝐶) = Σ𝑥 ∈ 𝐴 Σ𝑦 ∈ 𝐵 (#‘𝐶)) | ||
| 9-Jan-2022 | disjiund 4643 | Conditions for a collection of index unions of sets 𝐴(𝑎, 𝑏) for 𝑎 ∈ 𝑉 and 𝑏 ∈ 𝑊 to be disjoint. (Contributed by AV, 9-Jan-2022.) |
| ⊢ (𝑎 = 𝑐 → 𝐴 = 𝐶) & ⊢ (𝑏 = 𝑑 → 𝐶 = 𝐷) & ⊢ (𝑎 = 𝑐 → 𝑊 = 𝑋) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐷) → 𝑎 = 𝑐) ⇒ ⊢ (𝜑 → Disj 𝑎 ∈ 𝑉 ∪ 𝑏 ∈ 𝑊 𝐴) | ||
| 9-Jan-2022 | disjiunb 4642 | Two ways to say that a collection of index unions 𝐶(𝑖, 𝑥) for 𝑖 ∈ 𝐴 and 𝑥 ∈ 𝐵 is disjoint. (Contributed by AV, 9-Jan-2022.) |
| ⊢ (𝑖 = 𝑗 → 𝐵 = 𝐷) & ⊢ (𝑖 = 𝑗 → 𝐶 = 𝐸) ⇒ ⊢ (Disj 𝑖 ∈ 𝐴 ∪ 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐴 (𝑖 = 𝑗 ∨ (∪ 𝑥 ∈ 𝐵 𝐶 ∩ ∪ 𝑥 ∈ 𝐷 𝐸) = ∅)) | ||
| 9-Jan-2022 | disjord 4641 | Conditions for a collection of sets 𝐴(𝑎) for 𝑎 ∈ 𝑉 to be disjoint. (Contributed by AV, 9-Jan-2022.) |
| ⊢ (𝑎 = 𝑏 → 𝐴 = 𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → 𝑎 = 𝑏) ⇒ ⊢ (𝜑 → Disj 𝑎 ∈ 𝑉 𝐴) | ||
| 8-Jan-2022 | fusgr2wsp2nb 27198 | The set of paths of length 2 with a given vertex in the middle for a finite simple graph is the union of all paths of length 2 from one neighbor to another neighbor of this vertex via this vertex. (Contributed by Alexander van der Vekens, 9-Mar-2018.) (Revised by AV, 17-May-2021.) (Proof shortened by AV, 8-Jan-2022.) |
| ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝑀 = (𝑎 ∈ 𝑉 ↦ {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑎}) ⇒ ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → (𝑀‘𝑁) = ∪ 𝑥 ∈ (𝐺 NeighbVtx 𝑁)∪ 𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥}){⟨“𝑥𝑁𝑦”⟩}) | ||
| 8-Jan-2022 | fusgreg2wsplem 27197 | Lemma for fusgreg2wsp 27200 and related theorems. (Contributed by AV, 8-Jan-2022.) |
| ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝑀 = (𝑎 ∈ 𝑉 ↦ {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑎}) ⇒ ⊢ (𝑁 ∈ 𝑉 → (𝑝 ∈ (𝑀‘𝑁) ↔ (𝑝 ∈ (2 WSPathsN 𝐺) ∧ (𝑝‘1) = 𝑁))) | ||
| 8-Jan-2022 | rspcebdv 3314 | Restricted existential specialization, using implicit substitution in both directions. (Contributed by AV, 8-Jan-2022.) |
| ⊢ (𝜑 → 𝐴 ∈ 𝐵) & ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)) & ⊢ ((𝜑 ∧ 𝜓) → 𝑥 = 𝐴) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ 𝐵 𝜓 ↔ 𝜒)) | ||
| 7-Jan-2022 | frgr2wwlkeqm 27195 | If there is a (simple) path of length 2 from one vertex to another vertex and a (simple) path of length 2 from the other vertex back to the first vertex in a friendship graph, then the middle vertex is the same. This is only an observation, which is not required to proof the friendship theorem. (Contributed by Alexander van der Vekens, 20-Feb-2018.) (Revised by AV, 13-May-2021.) (Proof shortened by AV, 7-Jan-2022.) |
| ⊢ ((𝐺 ∈ FriendGraph ∧ 𝐴 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌)) → ((⟨“𝐴𝑃𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐵𝑄𝐴”⟩ ∈ (𝐵(2 WWalksNOn 𝐺)𝐴)) → 𝑄 = 𝑃)) | ||
| 7-Jan-2022 | wwlks2onsym 26851 | There is a walk of length 2 from one vertex to another vertex iff there is a walk of length 2 from the other vertex to the first vertex. (Contributed by AV, 7-Jan-2022.) |
| ⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ ((𝐺 ∈ UMGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ ⟨“𝐶𝐵𝐴”⟩ ∈ (𝐶(2 WWalksNOn 𝐺)𝐴))) | ||
| 7-Jan-2022 | fsumdifsnconst 14523 | The sum of constant terms (𝑘 is not free in 𝐶) over an index set excluding a singleton. (Contributed by AV, 7-Jan-2022.) |
| ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ ℂ) → Σ𝑘 ∈ (𝐴 ∖ {𝐵})𝐶 = (((#‘𝐴) − 1) · 𝐶)) | ||
| 5-Jan-2022 | reximdvva 3019 | Deduction doubly quantifying both antecedent and consequent, based on Theorem 19.22 of [Margaris] p. 90. (Contributed by AV, 5-Jan-2022.) |
| ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜒)) | ||
| 4-Jan-2022 | frgr2wwlk1 27193 | In a friendship graph, there is exactly one walk of length 2 between two different vertices. (Contributed by Alexander van der Vekens, 19-Feb-2018.) (Revised by AV, 13-May-2021.) (Proof shortened by AV, 4-Jan-2022.) |
| ⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ ((𝐺 ∈ FriendGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 ≠ 𝐵) → (#‘(𝐴(2 WWalksNOn 𝐺)𝐵)) = 1) | ||
| 4-Jan-2022 | frgr2wwlkeu 27191 | For two different vertices in a friendship graph, there is exactly one third vertex being the middle vertex of a (simple) path/walk of length 2 between the two vertices. (Contributed by Alexander van der Vekens, 18-Feb-2018.) (Revised by AV, 12-May-2021.) (Proof shortened by AV, 4-Jan-2022.) |
| ⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ ((𝐺 ∈ FriendGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 ≠ 𝐵) → ∃!𝑐 ∈ 𝑉 ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) | ||
| 4-Jan-2022 | frgreu 27132 | Variant of frcond2 27131: Any two (different) vertices in a friendship graph have a unique common neighbor. (Contributed by Alexander van der Vekens, 18-Feb-2018.) (Revised by AV, 12-May-2021.) (Proof shortened by AV, 4-Jan-2022.) |
| ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ (𝐺 ∈ FriendGraph → ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶) → ∃!𝑏({𝐴, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝐶} ∈ 𝐸))) | ||
| 4-Jan-2022 | sqrt2irrlem 14977 | Lemma for sqrt2irr 14979. This is the core of the proof: if 𝐴 / 𝐵 = √(2), then 𝐴 and 𝐵 are even, so 𝐴 / 2 and 𝐵 / 2 are smaller representatives, which is absurd by the method of infinite descent (here implemented by strong induction). This is Metamath 100 proof #1. (Contributed by NM, 20-Aug-2001.) (Revised by Mario Carneiro, 12-Sep-2015.) (Proof shortened by JV, 4-Jan-2022.) |
| ⊢ (𝜑 → 𝐴 ∈ ℤ) & ⊢ (𝜑 → 𝐵 ∈ ℕ) & ⊢ (𝜑 → (√‘2) = (𝐴 / 𝐵)) ⇒ ⊢ (𝜑 → ((𝐴 / 2) ∈ ℤ ∧ (𝐵 / 2) ∈ ℕ)) | ||
| 4-Jan-2022 | s3eq3seq 13684 | Two length 3 words are equal iff the corresponding symbols are equal. (Contributed by AV, 4-Jan-2022.) |
| ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ (𝐷 ∈ 𝑉 ∧ 𝐸 ∈ 𝑉 ∧ 𝐹 ∈ 𝑉)) → (⟨“𝐴𝐵𝐶”⟩ = ⟨“𝐷𝐸𝐹”⟩ ↔ (𝐴 = 𝐷 ∧ 𝐵 = 𝐸 ∧ 𝐶 = 𝐹))) | ||
| 4-Jan-2022 | s3eqs2s1eq 13683 | Two length 3 words are equal iff the corresponding length 2 words and singleton words consisting of their symbols are equal. (Contributed by AV, 4-Jan-2022.) |
| ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ (𝐷 ∈ 𝑉 ∧ 𝐸 ∈ 𝑉 ∧ 𝐹 ∈ 𝑉)) → (⟨“𝐴𝐵𝐶”⟩ = ⟨“𝐷𝐸𝐹”⟩ ↔ (⟨“𝐴𝐵”⟩ = ⟨“𝐷𝐸”⟩ ∧ ⟨“𝐶”⟩ = ⟨“𝐹”⟩))) | ||
| 4-Jan-2022 | s3eq2 13615 | Equality theorem for a length 3 word for the second symbol. (Contributed by AV, 4-Jan-2022.) |
| ⊢ (𝐵 = 𝐷 → ⟨“𝐴𝐵𝐶”⟩ = ⟨“𝐴𝐷𝐶”⟩) | ||
| 4-Jan-2022 | eqeuel 3941 | A condition which implies the existence of a unique element of a class. (Contributed by AV, 4-Jan-2022.) |
| ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑥 = 𝑦)) → ∃!𝑥 𝑥 ∈ 𝐴) | ||
| 3-Jan-2022 | pmtridfv2 29858 | Value at Y of the transposition of 𝑋 and 𝑌 (understood to be the identity when X = Y ). (Contributed by Thierry Arnoux, 3-Jan-2022.) |
| ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → 𝑌 ∈ 𝐴) & ⊢ 𝑇 = if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌})) ⇒ ⊢ (𝜑 → (𝑇‘𝑌) = 𝑋) | ||
| 3-Jan-2022 | pmtridfv1 29857 | Value at X of the transposition of 𝑋 and 𝑌 (understood to be the identity when X = Y ). (Contributed by Thierry Arnoux, 3-Jan-2022.) |
| ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → 𝑌 ∈ 𝐴) & ⊢ 𝑇 = if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌})) ⇒ ⊢ (𝜑 → (𝑇‘𝑋) = 𝑌) | ||
| 3-Jan-2022 | elimampt 29438 | Membership in the image of a mapping. (Contributed by Thierry Arnoux, 3-Jan-2022.) |
| ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) & ⊢ (𝜑 → 𝐶 ∈ 𝑊) & ⊢ (𝜑 → 𝐷 ⊆ 𝐴) ⇒ ⊢ (𝜑 → (𝐶 ∈ (𝐹 “ 𝐷) ↔ ∃𝑥 ∈ 𝐷 𝐶 = 𝐵)) | ||
| 3-Jan-2022 | difrab2 29339 | Difference of two restricted class abstractions. Compare with difrab 3901. (Contributed by Thierry Arnoux, 3-Jan-2022.) |
| ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∖ {𝑥 ∈ 𝐵 ∣ 𝜑}) = {𝑥 ∈ (𝐴 ∖ 𝐵) ∣ 𝜑} | ||
| 3-Jan-2022 | frgrregorufr0 27188 | In a friendship graph there are either no vertices having degree 𝐾, or all vertices have degree 𝐾 for any (nonnegative integer) 𝐾, unless there is a universal friend. This corresponds to claim 2 in [Huneke] p. 2: "... all vertices have degree k, unless there is a universal friend." (Contributed by Alexander van der Vekens, 1-Jan-2018.) (Revised by AV, 11-May-2021.) (Proof shortened by AV, 3-Jan-2022.) |
| ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) & ⊢ 𝐷 = (VtxDeg‘𝐺) ⇒ ⊢ (𝐺 ∈ FriendGraph → (∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾 ∨ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) ≠ 𝐾 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸)) | ||
| 3-Jan-2022 | frgrwopreg 27187 | In a friendship graph there are either no vertices (𝐴 = ∅) or exactly one vertex ((#‘𝐴) = 1) having degree 𝐾, or all (𝐵 = ∅) or all except one vertices ((#‘𝐵) = 1) have degree 𝐾. (Contributed by Alexander van der Vekens, 31-Dec-2017.) (Revised by AV, 10-May-2021.) (Proof shortened by AV, 3-Jan-2022.) |
| ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐷 = (VtxDeg‘𝐺) & ⊢ 𝐴 = {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾} & ⊢ 𝐵 = (𝑉 ∖ 𝐴) ⇒ ⊢ (𝐺 ∈ FriendGraph → (((#‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((#‘𝐵) = 1 ∨ 𝐵 = ∅))) | ||
| 3-Jan-2022 | ssdifsym 3863 | Symmetric class differences for subclasses. (Contributed by AV, 3-Jan-2022.) |
| ⊢ ((𝐴 ⊆ 𝑉 ∧ 𝐵 ⊆ 𝑉) → (𝐵 = (𝑉 ∖ 𝐴) ↔ 𝐴 = (𝑉 ∖ 𝐵))) | ||
| 3-Jan-2022 | ssdifim 3862 | Implication of a class difference with a subclass. (Contributed by AV, 3-Jan-2022.) |
| ⊢ ((𝐴 ⊆ 𝑉 ∧ 𝐵 = (𝑉 ∖ 𝐴)) → 𝐴 = (𝑉 ∖ 𝐵)) | ||
| 2-Jan-2022 | smfliminfmpt 41038 | The inferior limit of a countable set of sigma-measurable functions is sigma-measurable. Proposition 121F (e) of [Fremlin1] p. 39 . 𝐴 can contain 𝑚 as a free variable, in other words it can be thought of as an indexed collection 𝐴(𝑚). 𝐵 can be thought of as a collection with two indexes 𝐵(𝑚, 𝑥). (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ Ⅎ𝑚𝜑 & ⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑛𝜑 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆)) & ⊢ 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐴 ∣ (lim inf‘(𝑚 ∈ 𝑍 ↦ 𝐵)) ∈ ℝ} & ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ (lim inf‘(𝑚 ∈ 𝑍 ↦ 𝐵))) ⇒ ⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆)) | ||
| 2-Jan-2022 | smfliminf 41037 | The inferior limit of a countable set of sigma-measurable functions is sigma-measurable. Proposition 121F (e) of [Fremlin1] p. 39 . (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ Ⅎ𝑚𝐹 & ⊢ Ⅎ𝑥𝐹 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) & ⊢ 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ} & ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ (lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)))) ⇒ ⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆)) | ||
| 2-Jan-2022 | smfliminflem 41036 | The inferior limit of a countable set of sigma-measurable functions is sigma-measurable. Proposition 121F (e) of [Fremlin1] p. 39 . (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) & ⊢ 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ} & ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ (lim inf‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)))) ⇒ ⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆)) | ||
| 2-Jan-2022 | climliminflimsup4 40043 | A sequence of real numbers converges if and only if its superior limit is real and equal to its inferior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ) ⇒ ⊢ (𝜑 → (𝐹 ∈ dom ⇝ ↔ ((lim sup‘𝐹) ∈ ℝ ∧ (lim inf‘𝐹) = (lim sup‘𝐹)))) | ||
| 2-Jan-2022 | climliminflimsup3 40042 | A sequence of real numbers converges if and only if its inferior limit is real and equal to its superior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ) ⇒ ⊢ (𝜑 → (𝐹 ∈ dom ⇝ ↔ ((lim inf‘𝐹) ∈ ℝ ∧ (lim inf‘𝐹) = (lim sup‘𝐹)))) | ||
| 2-Jan-2022 | climliminflimsup2 40041 | A sequence of real numbers converges if and only if its superior limit is real and it is less than or equal to its inferior limit (in such a case, they are actually equal, see liminfgelimsupuz 40020). (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ) ⇒ ⊢ (𝜑 → (𝐹 ∈ dom ⇝ ↔ ((lim sup‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹)))) | ||
| 2-Jan-2022 | climliminflimsup 40040 | A sequence of real numbers converges if and only if its inferior limit is real and it is greater than or equal to its superior limit (in such a case, they are actually equal, see liminfgelimsupuz 40020). (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ) ⇒ ⊢ (𝜑 → (𝐹 ∈ dom ⇝ ↔ ((lim inf‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹)))) | ||
| 2-Jan-2022 | liminflimsupclim 40039 | A sequence of real numbers converges if its inferior limit is real, and it is greater or equal to the superior limit (in such a case, they are actually equal, see liminflelimsupuz 40017). (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ) & ⊢ (𝜑 → (lim inf‘𝐹) ∈ ℝ) & ⊢ (𝜑 → (lim sup‘𝐹) ≤ (lim inf‘𝐹)) ⇒ ⊢ (𝜑 → 𝐹 ∈ dom ⇝ ) | ||
| 2-Jan-2022 | climliminf 40038 | A sequence of real numbers converges if and only if it converges to its inferior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ) ⇒ ⊢ (𝜑 → (𝐹 ∈ dom ⇝ ↔ 𝐹 ⇝ (lim inf‘𝐹))) | ||
| 2-Jan-2022 | liminflt 40037 | Given a sequence of real numbers, there exists an upper part of the sequence that's approximated from above by the inferior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ Ⅎ𝑘𝐹 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ) & ⊢ (𝜑 → (lim inf‘𝐹) ∈ ℝ) & ⊢ (𝜑 → 𝑋 ∈ ℝ+) ⇒ ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(lim inf‘𝐹) < ((𝐹‘𝑘) + 𝑋)) | ||
| 2-Jan-2022 | liminfltlem 40036 | Given a sequence of real numbers, there exists an upper part of the sequence that's approximated from above by the inferior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ) & ⊢ (𝜑 → (lim inf‘𝐹) ∈ ℝ) & ⊢ (𝜑 → 𝑋 ∈ ℝ+) ⇒ ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(lim inf‘𝐹) < ((𝐹‘𝑘) + 𝑋)) | ||
| 2-Jan-2022 | liminfreuz 40035 | Given a function on the reals, its inferior limit is real if and only if two condition holds: 1. there is a real number that is greater than or equal to the function, infinitely often; 2. there is a real number that is smaller than or equal to the function. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ Ⅎ𝑗𝐹 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ) ⇒ ⊢ (𝜑 → ((lim inf‘𝐹) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥 ∧ ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗)))) | ||
| 2-Jan-2022 | liminfreuzlem 40034 | Given a function on the reals, its inferior limit is real if and only if two condition holds: 1. there is a real number that is greater than or equal to the function, infinitely often; 2. there is a real number that is smaller than or equal to the function. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ Ⅎ𝑗𝐹 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ) ⇒ ⊢ (𝜑 → ((lim inf‘𝐹) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥 ∧ ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗)))) | ||
| 2-Jan-2022 | climliminflimsupd 40033 | If a sequence of real numbers converges, its inferior limit and its superior limit are equal. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ) & ⊢ (𝜑 → 𝐹 ∈ dom ⇝ ) ⇒ ⊢ (𝜑 → (lim inf‘𝐹) = (lim sup‘𝐹)) | ||
| 2-Jan-2022 | limsupvaluz4 40032 | Alternate definition of lim inf for a real-valued function, defined on a set of upper integers. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ Ⅎ𝑘𝜑 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (lim sup‘(𝑘 ∈ 𝑍 ↦ 𝐵)) = -𝑒(lim inf‘(𝑘 ∈ 𝑍 ↦ -𝐵))) | ||
| 2-Jan-2022 | liminfvaluz4 40031 | Alternate definition of lim inf for a real-valued function, defined on a set of upper integers. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ Ⅎ𝑘𝜑 & ⊢ Ⅎ𝑘𝐹 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ) ⇒ ⊢ (𝜑 → (lim inf‘𝐹) = -𝑒(lim sup‘(𝑘 ∈ 𝑍 ↦ -(𝐹‘𝑘)))) | ||
| 2-Jan-2022 | limsupvaluz3 40030 | Alternate definition of lim inf for an extended real valued function, defined on a set of upper integers. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ Ⅎ𝑘𝜑 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℝ*) ⇒ ⊢ (𝜑 → (lim sup‘(𝑘 ∈ 𝑍 ↦ 𝐵)) = -𝑒(lim inf‘(𝑘 ∈ 𝑍 ↦ -𝑒𝐵))) | ||
| 2-Jan-2022 | liminflelimsupcex 40029 | A counterexample for liminflelimsup 40008, showing that the second hypothesis is needed. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (lim sup‘∅) < (lim inf‘∅) | ||
| 2-Jan-2022 | liminfvaluz3 40028 | Alternate definition of lim inf for an extended real valued function, defined on a set of upper integers. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ Ⅎ𝑘𝜑 & ⊢ Ⅎ𝑘𝐹 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) ⇒ ⊢ (𝜑 → (lim inf‘𝐹) = -𝑒(lim sup‘(𝑘 ∈ 𝑍 ↦ -𝑒(𝐹‘𝑘)))) | ||
| 2-Jan-2022 | liminfvaluz2 40027 | Alternate definition of lim inf for a real-valued function, defined on a set of upper integers. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ Ⅎ𝑘𝜑 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (lim inf‘(𝑘 ∈ 𝑍 ↦ 𝐵)) = -𝑒(lim sup‘(𝑘 ∈ 𝑍 ↦ -𝐵))) | ||
| 2-Jan-2022 | limsupval4 40026 | Alternate definition of lim inf when the given a function is eventually extended real valued. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝑀 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ (𝑀[,)+∞))) → 𝐵 ∈ ℝ*) ⇒ ⊢ (𝜑 → (lim sup‘(𝑥 ∈ 𝐴 ↦ 𝐵)) = -𝑒(lim inf‘(𝑥 ∈ 𝐴 ↦ -𝑒𝐵))) | ||
| 2-Jan-2022 | liminf0 40025 | The inferior limit of the empty set. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (lim inf‘∅) = +∞ | ||
| 2-Jan-2022 | liminfvaluz 40024 | Alternate definition of lim inf for an extended real valued function, defined on a set of upper integers. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ Ⅎ𝑘𝜑 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℝ*) ⇒ ⊢ (𝜑 → (lim inf‘(𝑘 ∈ 𝑍 ↦ 𝐵)) = -𝑒(lim sup‘(𝑘 ∈ 𝑍 ↦ -𝑒𝐵))) | ||
| 2-Jan-2022 | liminfequzmpt2 40023 | Two functions that are eventually equal to one another have the same superior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ Ⅎ𝑗𝜑 & ⊢ Ⅎ𝑗𝐴 & ⊢ Ⅎ𝑗𝐵 & ⊢ 𝐴 = (ℤ≥‘𝑀) & ⊢ 𝐵 = (ℤ≥‘𝑁) & ⊢ (𝜑 → 𝐾 ∈ 𝐴) & ⊢ (𝜑 → 𝐾 ∈ 𝐵) & ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝐾)) → 𝐶 ∈ 𝑉) ⇒ ⊢ (𝜑 → (lim inf‘(𝑗 ∈ 𝐴 ↦ 𝐶)) = (lim inf‘(𝑗 ∈ 𝐵 ↦ 𝐶))) | ||
| 2-Jan-2022 | liminfval3 40022 | Alternate definition of lim inf when the given function is eventually extended real valued. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝑀 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ (𝑀[,)+∞))) → 𝐵 ∈ ℝ*) ⇒ ⊢ (𝜑 → (lim inf‘(𝑥 ∈ 𝐴 ↦ 𝐵)) = -𝑒(lim sup‘(𝑥 ∈ 𝐴 ↦ -𝑒𝐵))) | ||
| 2-Jan-2022 | liminfval4 40021 | Alternate definition of lim inf when the given function is eventually real valued. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝑀 ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ (𝑀[,)+∞))) → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (lim inf‘(𝑥 ∈ 𝐴 ↦ 𝐵)) = -𝑒(lim sup‘(𝑥 ∈ 𝐴 ↦ -𝐵))) | ||
| 2-Jan-2022 | liminfgelimsupuz 40020 | The inferior limit is greater than or equal to the superior limit if and only if they are equal. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ) ⇒ ⊢ (𝜑 → ((lim sup‘𝐹) ≤ (lim inf‘𝐹) ↔ (lim inf‘𝐹) = (lim sup‘𝐹))) | ||
| 2-Jan-2022 | liminfresuz2 40019 | If the domain of a function is a subset of the integers, the inferior limit doesn't change when the function is restricted to an upper set of integers. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹 ∈ 𝑉) & ⊢ (𝜑 → dom 𝐹 ⊆ ℤ) ⇒ ⊢ (𝜑 → (lim inf‘(𝐹 ↾ 𝑍)) = (lim inf‘𝐹)) | ||
| 2-Jan-2022 | liminfvalxrmpt 40018 | Alternate definition of lim inf when 𝐹 is an extended real valued function. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*) ⇒ ⊢ (𝜑 → (lim inf‘(𝑥 ∈ 𝐴 ↦ 𝐵)) = -𝑒(lim sup‘(𝑥 ∈ 𝐴 ↦ -𝑒𝐵))) | ||
| 2-Jan-2022 | liminflelimsupuz 40017 | The superior limit is greater than or equal to the inferior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) ⇒ ⊢ (𝜑 → (lim inf‘𝐹) ≤ (lim sup‘𝐹)) | ||
| 2-Jan-2022 | liminfresuz 40016 | If the real part of the domain of a function is a subset of the integers, the inferior limit doesn't change when the function is restricted to an upper set of integers. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹 ∈ 𝑉) & ⊢ (𝜑 → dom (𝐹 ↾ ℝ) ⊆ ℤ) ⇒ ⊢ (𝜑 → (lim inf‘(𝐹 ↾ 𝑍)) = (lim inf‘𝐹)) | ||
| 2-Jan-2022 | liminfvalxr 40015 | Alternate definition of lim inf when 𝐹 is an extended real valued function. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ Ⅎ𝑥𝐹 & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*) ⇒ ⊢ (𝜑 → (lim inf‘𝐹) = -𝑒(lim sup‘(𝑥 ∈ 𝐴 ↦ -𝑒(𝐹‘𝑥)))) | ||
| 2-Jan-2022 | liminfgelimsup 40014 | The inferior limit is greater than or equal to the superior limit if and only if they are equal. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝜑 → 𝐹 ∈ 𝑉) & ⊢ (𝜑 → ∀𝑘 ∈ ℝ ∃𝑗 ∈ (𝑘[,)+∞)((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*) ≠ ∅) ⇒ ⊢ (𝜑 → ((lim sup‘𝐹) ≤ (lim inf‘𝐹) ↔ (lim inf‘𝐹) = (lim sup‘𝐹))) | ||
| 2-Jan-2022 | liminfltlimsupex 40013 | An example where the lim inf is strictly smaller than the lim sup. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, 1)) ⇒ ⊢ (lim inf‘𝐹) < (lim sup‘𝐹) | ||
| 2-Jan-2022 | liminfresicompt 40012 | The inferior limit doesn't change when a function is restricted to the upper part of the reals. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝜑 → 𝑀 ∈ ℝ) & ⊢ 𝑍 = (𝑀[,)+∞) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) ⇒ ⊢ (𝜑 → (lim inf‘(𝑥 ∈ (𝐴 ∩ 𝑍) ↦ 𝐵)) = (lim inf‘(𝑥 ∈ 𝐴 ↦ 𝐵))) | ||
| 2-Jan-2022 | liminfresre 40011 | The inferior limit of a function only depends on the real part of its domain. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝜑 → 𝐹 ∈ 𝑉) ⇒ ⊢ (𝜑 → (lim inf‘(𝐹 ↾ ℝ)) = (lim inf‘𝐹)) | ||
| 2-Jan-2022 | limsupgt 40010 | Given a sequence of real numbers, there exists an upper part of the sequence that's appxoximated from below by the superior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ Ⅎ𝑘𝐹 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ) & ⊢ (𝜑 → (lim sup‘𝐹) ∈ ℝ) & ⊢ (𝜑 → 𝑋 ∈ ℝ+) ⇒ ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) − 𝑋) < (lim sup‘𝐹)) | ||
| 2-Jan-2022 | limsupgtlem 40009 | For any positive real, the superior limit of F is larger than any of its values at large enough arguments, up to that positive real. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ) & ⊢ (𝜑 → (lim sup‘𝐹) ∈ ℝ) & ⊢ (𝜑 → 𝑋 ∈ ℝ+) ⇒ ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)((𝐹‘𝑘) − 𝑋) < (lim sup‘𝐹)) | ||
| 2-Jan-2022 | liminflelimsup 40008 | The superior limit is greater than or equal to the inferior limit. The second hypothesis is needed (see liminflelimsupcex 40029 for a counterexample). The inequality can be strict, see liminfltlimsupex 40013. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝜑 → 𝐹 ∈ 𝑉) & ⊢ (𝜑 → ∀𝑘 ∈ ℝ ∃𝑗 ∈ (𝑘[,)+∞)((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*) ≠ ∅) ⇒ ⊢ (𝜑 → (lim inf‘𝐹) ≤ (lim sup‘𝐹)) | ||
| 2-Jan-2022 | liminflelimsuplem 40007 | The superior limit is greater than or equal to the inferior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝜑 → 𝐹 ∈ 𝑉) & ⊢ (𝜑 → ∀𝑘 ∈ ℝ ∃𝑗 ∈ (𝑘[,)+∞)((𝐹 “ (𝑗[,)+∞)) ∩ ℝ*) ≠ ∅) ⇒ ⊢ (𝜑 → (lim inf‘𝐹) ≤ (lim sup‘𝐹)) | ||
| 2-Jan-2022 | liminf10ex 40006 | The inferior limit of a function that alternates between two values. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, 1)) ⇒ ⊢ (lim inf‘𝐹) = 0 | ||
| 2-Jan-2022 | limsup10ex 40005 | The superior limit of a function that alternates between two values. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, 1)) ⇒ ⊢ (lim sup‘𝐹) = 1 | ||
| 2-Jan-2022 | limsup10exlem 40004 | The range of the given function. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ if(2 ∥ 𝑛, 0, 1)) & ⊢ (𝜑 → 𝐾 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝐹 “ (𝐾[,)+∞)) = {0, 1}) | ||
| 2-Jan-2022 | liminfresico 40003 | The inferior limit doesn't change when a function is restricted to an upperset of reals. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝜑 → 𝑀 ∈ ℝ) & ⊢ 𝑍 = (𝑀[,)+∞) & ⊢ (𝜑 → 𝐹 ∈ 𝑉) ⇒ ⊢ (𝜑 → (lim inf‘(𝐹 ↾ 𝑍)) = (lim inf‘𝐹)) | ||
| 2-Jan-2022 | liminfcld 40002 | Closure of the inferior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝜑 → 𝐹 ∈ 𝑉) ⇒ ⊢ (𝜑 → (lim inf‘𝐹) ∈ ℝ*) | ||
| 2-Jan-2022 | climlimsupcex 40001 | Counterexample for climlimsup 39992, showing that the first hypothesis is needed, if the empty set is a complex number (see 0ncn 9954 and its comment) (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ ¬ 𝑀 ∈ ℤ & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ 𝐹 = ∅ ⇒ ⊢ ((∅ ∈ ℂ ∧ ¬ -∞ ∈ ℂ) → (𝐹:𝑍⟶ℝ ∧ 𝐹 ∈ dom ⇝ ∧ ¬ 𝐹 ⇝ (lim sup‘𝐹))) | ||
| 2-Jan-2022 | liminfval2 40000 | The superior limit, relativized to an unbounded set. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ 𝐺 = (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) & ⊢ (𝜑 → 𝐹 ∈ 𝑉) & ⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ (𝜑 → sup(𝐴, ℝ*, < ) = +∞) ⇒ ⊢ (𝜑 → (lim inf‘𝐹) = sup((𝐺 “ 𝐴), ℝ*, < )) | ||
| 2-Jan-2022 | liminfresxr 39999 | The inferior limit of a function only depends on the preimage of the extended real part. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝜑 → 𝐹 ∈ 𝑉) & ⊢ (𝜑 → Fun 𝐹) & ⊢ 𝐴 = (◡𝐹 “ ℝ*) ⇒ ⊢ (𝜑 → (lim inf‘(𝐹 ↾ 𝐴)) = (lim inf‘𝐹)) | ||
| 2-Jan-2022 | limsupresxr 39998 | The superior limit of a function only depends on the restriction of that function to the preimage of the set of extended reals. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝜑 → 𝐹 ∈ 𝑉) & ⊢ (𝜑 → Fun 𝐹) & ⊢ 𝐴 = (◡𝐹 “ ℝ*) ⇒ ⊢ (𝜑 → (lim sup‘(𝐹 ↾ 𝐴)) = (lim sup‘𝐹)) | ||
| 2-Jan-2022 | liminfval5 39997 | The inferior limit of an infinite sequence 𝐹 of extended real numbers. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ Ⅎ𝑘𝜑 & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*) & ⊢ 𝐺 = (𝑘 ∈ ℝ ↦ inf((𝐹 “ (𝑘[,)+∞)), ℝ*, < )) ⇒ ⊢ (𝜑 → (lim inf‘𝐹) = sup(ran 𝐺, ℝ*, < )) | ||
| 2-Jan-2022 | liminfvald 39996 | The inferior limit of a set 𝐹. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝜑 → 𝐹 ∈ 𝑉) & ⊢ 𝐺 = (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) ⇒ ⊢ (𝜑 → (lim inf‘𝐹) = sup(ran 𝐺, ℝ*, < )) | ||
| 2-Jan-2022 | liminfcl 39995 | Closure of the inferior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝐹 ∈ 𝑉 → (lim inf‘𝐹) ∈ ℝ*) | ||
| 2-Jan-2022 | liminfgval 39994 | Value of the inferior limit function. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ 𝐺 = (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) ⇒ ⊢ (𝑀 ∈ ℝ → (𝐺‘𝑀) = inf(((𝐹 “ (𝑀[,)+∞)) ∩ ℝ*), ℝ*, < )) | ||
| 2-Jan-2022 | limsupge 39993 | The defining property of the superior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝜑 → 𝐵 ⊆ ℝ) & ⊢ (𝜑 → 𝐹:𝐵⟶ℝ*) & ⊢ (𝜑 → 𝐴 ∈ ℝ*) ⇒ ⊢ (𝜑 → (𝐴 ≤ (lim sup‘𝐹) ↔ ∀𝑘 ∈ ℝ 𝐴 ≤ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ))) | ||
| 2-Jan-2022 | climlimsup 39992 | A sequence of real numbers converges if and only if it converges to its superior limit. The first hypothesis is needed (see climlimsupcex 40001 for a counterexample) (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ) ⇒ ⊢ (𝜑 → (𝐹 ∈ dom ⇝ ↔ 𝐹 ⇝ (lim sup‘𝐹))) | ||
| 2-Jan-2022 | liminfval 39991 | The inferior limit of a set 𝐹. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ 𝐺 = (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) ⇒ ⊢ (𝐹 ∈ 𝑉 → (lim inf‘𝐹) = sup(ran 𝐺, ℝ*, < )) | ||
| 2-Jan-2022 | liminfgf 39990 | Closure of the inferior limit function. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ 𝐺 = (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) ⇒ ⊢ 𝐺:ℝ⟶ℝ* | ||
| 2-Jan-2022 | limsupcli 39989 | Closure of the superior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ 𝐹 ∈ 𝑉 ⇒ ⊢ (lim sup‘𝐹) ∈ ℝ* | ||
| 2-Jan-2022 | limsupresicompt 39988 | The superior limit doesn't change when a function is restricted to the upper part of the reals. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝑀 ∈ ℝ) & ⊢ 𝑍 = (𝑀[,)+∞) ⇒ ⊢ (𝜑 → (lim sup‘(𝑥 ∈ 𝐴 ↦ 𝐵)) = (lim sup‘(𝑥 ∈ (𝐴 ∩ 𝑍) ↦ 𝐵))) | ||
| 2-Jan-2022 | limsupvald 39987 | The superior limit of a sequence 𝐹 of extended real numbers is the infimum of the set of suprema of all restrictions of 𝐹 to an upperset of reals . (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝜑 → 𝐹 ∈ 𝑉) & ⊢ 𝐺 = (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) ⇒ ⊢ (𝜑 → (lim sup‘𝐹) = inf(ran 𝐺, ℝ*, < )) | ||
| 2-Jan-2022 | liminfgord 39986 | Ordering property of the inferior limit function. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → inf(((𝐹 “ (𝐴[,)+∞)) ∩ ℝ*), ℝ*, < ) ≤ inf(((𝐹 “ (𝐵[,)+∞)) ∩ ℝ*), ℝ*, < )) | ||
| 2-Jan-2022 | limsuplt2 39985 | The defining property of the superior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝜑 → 𝐵 ⊆ ℝ) & ⊢ (𝜑 → 𝐹:𝐵⟶ℝ*) & ⊢ (𝜑 → 𝐴 ∈ ℝ*) ⇒ ⊢ (𝜑 → ((lim sup‘𝐹) < 𝐴 ↔ ∃𝑘 ∈ ℝ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ) < 𝐴)) | ||
| 2-Jan-2022 | df-liminf 39984 | Define the inferior limit of a sequence of extended real numbers. (Contributed by GS, 2-Jan-2022.) |
| ⊢ lim inf = (𝑥 ∈ V ↦ sup(ran (𝑘 ∈ ℝ ↦ inf(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )), ℝ*, < )) | ||
| 2-Jan-2022 | climuz 39976 | Express the predicate: The limit of complex number sequence 𝐹 is 𝐴, or 𝐹 converges to 𝐴. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ Ⅎ𝑘𝐹 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℂ) ⇒ ⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥))) | ||
| 2-Jan-2022 | climuzlem 39975 | Express the predicate: The limit of complex number sequence 𝐹 is 𝐴, or 𝐹 converges to 𝐴. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℂ) ⇒ ⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥))) | ||
| 2-Jan-2022 | 0cnv 39974 | If (/) is a complex number, then it converges to itself. (see 0ncn 9954 and its comment ; see also the comment in climlimsupcex 40001) (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (∅ ∈ ℂ → ∅ ⇝ ∅) | ||
| 2-Jan-2022 | uzubico2 39797 | The upper integers are unbounded above. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) ⇒ ⊢ (𝜑 → ∀𝑥 ∈ ℝ ∃𝑘 ∈ (𝑥[,)+∞)𝑘 ∈ 𝑍) | ||
| 2-Jan-2022 | uzubioo2 39796 | The upper integers are unbounded above. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) ⇒ ⊢ (𝜑 → ∀𝑥 ∈ ℝ ∃𝑘 ∈ (𝑥(,)+∞)𝑘 ∈ 𝑍) | ||
| 2-Jan-2022 | uzubico 39795 | The upper integers are unbounded above. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑋 ∈ ℝ) ⇒ ⊢ (𝜑 → ∃𝑘 ∈ (𝑋[,)+∞)𝑘 ∈ 𝑍) | ||
| 2-Jan-2022 | uzubioo 39794 | The upper integers are unbounded above. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑋 ∈ ℝ) ⇒ ⊢ (𝜑 → ∃𝑘 ∈ (𝑋(,)+∞)𝑘 ∈ 𝑍) | ||
| 2-Jan-2022 | ndmico 39793 | The closed-below, open-above interval function's value is empty outside of its domain. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (¬ (𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴[,)𝐵) = ∅) | ||
| 2-Jan-2022 | dmico 39792 | The domain of the closed-below, open-above interval function. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ dom [,) = (ℝ* × ℝ*) | ||
| 2-Jan-2022 | icossico2 39791 | Condition for a closed-below, open-above interval to be a subset of a closed-below, open-above interval. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐶 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ≤ 𝐴) ⇒ ⊢ (𝜑 → (𝐴[,)𝐶) ⊆ (𝐵[,)𝐶)) | ||
| 2-Jan-2022 | uzinico3 39790 | An upper interval of integers doesn't change when it's intersected with a left-closed, unbounded above interval, with the same lower bound. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) ⇒ ⊢ (𝜑 → 𝑍 = (𝑍 ∩ (𝑀[,)+∞))) | ||
| 2-Jan-2022 | supminfxrrnmpt 39701 | The indexed supremum of a set of reals is the negation of the indexed infimum of that set's image under negation. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*) ⇒ ⊢ (𝜑 → sup(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) = -𝑒inf(ran (𝑥 ∈ 𝐴 ↦ -𝑒𝐵), ℝ*, < )) | ||
| 2-Jan-2022 | xnegred 39700 | An extended real is real if and only if its extended negative is real. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ*) ⇒ ⊢ (𝜑 → (𝐴 ∈ ℝ ↔ -𝑒𝐴 ∈ ℝ)) | ||
| 2-Jan-2022 | supminfxr2 39699 | The extended real suprema of a set of extended reals is the extended real negative of the extended real infima of that set's image under extended real negation. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝜑 → 𝐴 ⊆ ℝ*) ⇒ ⊢ (𝜑 → sup(𝐴, ℝ*, < ) = -𝑒inf({𝑥 ∈ ℝ* ∣ -𝑒𝑥 ∈ 𝐴}, ℝ*, < )) | ||
| 2-Jan-2022 | uzxr 39698 | An upper integer is an extended real. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝐴 ∈ (ℤ≥‘𝑀) → 𝐴 ∈ ℝ*) | ||
| 2-Jan-2022 | xnegrecl2d 39697 | If the extended real negative is real, then the number itself is real. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → -𝑒𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → 𝐴 ∈ ℝ) | ||
| 2-Jan-2022 | xnegre 39696 | An extended real is real if and only if its extended negative is real. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝐴 ∈ ℝ* → (𝐴 ∈ ℝ ↔ -𝑒𝐴 ∈ ℝ)) | ||
| 2-Jan-2022 | infrpgernmpt 39695 | The infimum of a non empty, bounded below, indexed subset of extended reals can be approximated from above by an element of the set. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝐴 ≠ ∅) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝑦 ≤ 𝐵) & ⊢ (𝜑 → 𝐶 ∈ ℝ+) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝐵 ≤ (inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) +𝑒 𝐶)) | ||
| 2-Jan-2022 | supminfxr 39694 | The extended real suprema of a set of reals is the extended real negative of the extended real infima of that set's image under negation. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝜑 → 𝐴 ⊆ ℝ) ⇒ ⊢ (𝜑 → sup(𝐴, ℝ*, < ) = -𝑒inf({𝑥 ∈ ℝ ∣ -𝑥 ∈ 𝐴}, ℝ*, < )) | ||
| 2-Jan-2022 | infxrpnf2 39693 | Removing plus infinity from a set does not affect its infimum. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝐴 ⊆ ℝ* → inf((𝐴 ∖ {+∞}), ℝ*, < ) = inf(𝐴, ℝ*, < )) | ||
| 2-Jan-2022 | uzxrd 39692 | An upper integer is an extended real. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐴 ∈ 𝑍) ⇒ ⊢ (𝜑 → 𝐴 ∈ ℝ*) | ||
| 2-Jan-2022 | nfxneg 39691 | Bound-variable hypothesis builder for the negative of an extended real number. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥-𝑒𝐴 | ||
| 2-Jan-2022 | xnegrecl2 39690 | If the extended real negative is real, then the number itself is real. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ ((𝐴 ∈ ℝ* ∧ -𝑒𝐴 ∈ ℝ) → 𝐴 ∈ ℝ) | ||
| 2-Jan-2022 | uzn0bi 39689 | The upper integers function needs to be applied to an integer, in order to return a nonempty set. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ ((ℤ≥‘𝑀) ≠ ∅ ↔ 𝑀 ∈ ℤ) | ||
| 2-Jan-2022 | max2d 39688 | A number is less than or equal to the maximum of it and another. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → 𝐵 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴)) | ||
| 2-Jan-2022 | leneg3d 39687 | Negative of one side of 'less than or equal to'. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (-𝐴 ≤ 𝐵 ↔ -𝐵 ≤ 𝐴)) | ||
| 2-Jan-2022 | 1xr 39686 | 1 is an extended real number. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ 1 ∈ ℝ* | ||
| 2-Jan-2022 | uzssz2 39685 | An upper set of integers is a subset of all integers. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ 𝑍 = (ℤ≥‘𝑀) ⇒ ⊢ 𝑍 ⊆ ℤ | ||
| 2-Jan-2022 | rphalfltd 39684 | Half of a positive real is less than the original number. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ+) ⇒ ⊢ (𝜑 → (𝐴 / 2) < 𝐴) | ||
| 2-Jan-2022 | infxrgelbrnmpt 39683 | The infimum of an indexed set of extended reals is greater than or equal to a lower bound. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐶 ∈ ℝ*) ⇒ ⊢ (𝜑 → (𝐶 ≤ inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) ↔ ∀𝑥 ∈ 𝐴 𝐶 ≤ 𝐵)) | ||
| 2-Jan-2022 | zxrd 39682 | An integer is an extended real number. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝜑 → 𝐴 ∈ ℤ) ⇒ ⊢ (𝜑 → 𝐴 ∈ ℝ*) | ||
| 2-Jan-2022 | nleltd 39681 | 'Not less than or equal to' implies 'grater than'. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → ¬ 𝐵 ≤ 𝐴) ⇒ ⊢ (𝜑 → 𝐴 < 𝐵) | ||
| 2-Jan-2022 | supxrleubrnmptf 39680 | The supremum of a nonempty bounded indexed set of extended reals is less than or equal to an upper bound. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐶 & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐶 ∈ ℝ*) ⇒ ⊢ (𝜑 → (sup(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) ≤ 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝐶)) | ||
| 2-Jan-2022 | ceilcld 39679 | Closure of the ceiling function. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → (⌈‘𝐴) ∈ ℤ) | ||
| 2-Jan-2022 | max1d 39678 | A number is less than or equal to the maximum of it and another. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → 𝐴 ≤ if(𝐴 ≤ 𝐵, 𝐵, 𝐴)) | ||
| 2-Jan-2022 | supxrltinfxr 39677 | The supremum of the empty set is strictly smaller than the infimum of the empty set. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ sup(∅, ℝ*, < ) < inf(∅, ℝ*, < ) | ||
| 2-Jan-2022 | leneg2d 39676 | Negative of one side of 'less than or equal to'. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝐴 ≤ -𝐵 ↔ 𝐵 ≤ -𝐴)) | ||
| 2-Jan-2022 | infxrrnmptcl 39675 | The infimum of an arbitrary indexed set of extended reals is an extended real. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*) ⇒ ⊢ (𝜑 → inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) ∈ ℝ*) | ||
| 2-Jan-2022 | infxrpnf 39674 | Adding plus infinity to a set does not affect its infimum. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝐴 ⊆ ℝ* → inf((𝐴 ∪ {+∞}), ℝ*, < ) = inf(𝐴, ℝ*, < )) | ||
| 2-Jan-2022 | ceilged 39673 | The ceiling of a real number is greater than or equal to that number. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → 𝐴 ≤ (⌈‘𝐴)) | ||
| 2-Jan-2022 | supminfrnmpt 39672 | The indexed supremum of a bounded-above set of reals is the negation of the indexed infimum of that set's image under negation. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝐴 ≠ ∅) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) & ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦) ⇒ ⊢ (𝜑 → sup(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ, < ) = -inf(ran (𝑥 ∈ 𝐴 ↦ -𝐵), ℝ, < )) | ||
| 2-Jan-2022 | xnegcli 39671 | Closure of extended real negative. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ 𝐴 ∈ ℝ* ⇒ ⊢ -𝑒𝐴 ∈ ℝ* | ||
| 2-Jan-2022 | uzred 39670 | An upper integer is a real number. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐴 ∈ 𝑍) ⇒ ⊢ (𝜑 → 𝐴 ∈ ℝ) | ||
| 2-Jan-2022 | xnegnegd 39669 | Extended real version of negnegd 10383. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ*) ⇒ ⊢ (𝜑 → -𝑒-𝑒𝐴 = 𝐴) | ||
| 2-Jan-2022 | nfxnegd 39668 | Deduction version of nfxneg 39691. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝜑 → Ⅎ𝑥𝐴) ⇒ ⊢ (𝜑 → Ⅎ𝑥-𝑒𝐴) | ||
| 2-Jan-2022 | xnegeqi 39667 | Equality of two extended numbers with -𝑒 in front of them. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ 𝐴 = 𝐵 ⇒ ⊢ -𝑒𝐴 = -𝑒𝐵 | ||
| 2-Jan-2022 | xnegnegi 39666 | Extended real version of negneg 10331. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ 𝐴 ∈ ℝ* ⇒ ⊢ -𝑒-𝑒𝐴 = 𝐴 | ||
| 2-Jan-2022 | xnegrecl 39665 | The extended real negative of a real number is real. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝐴 ∈ ℝ → -𝑒𝐴 ∈ ℝ) | ||
| 2-Jan-2022 | xnegeqd 39664 | Equality of two extended numbers with -𝑒 in front of them. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → -𝑒𝐴 = -𝑒𝐵) | ||
| 2-Jan-2022 | infxrlesupxr 39663 | The supremum of a nonempty set is larger than or equal to the infimum. The second condition is needed, see supxrltinfxr 39677. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝜑 → 𝐴 ⊆ ℝ*) & ⊢ (𝜑 → 𝐴 ≠ ∅) ⇒ ⊢ (𝜑 → inf(𝐴, ℝ*, < ) ≤ sup(𝐴, ℝ*, < )) | ||
| 2-Jan-2022 | uzid3 39662 | Membership of the least member in an upper set of integers. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ 𝑍 = (ℤ≥‘𝑀) ⇒ ⊢ (𝑁 ∈ 𝑍 → 𝑁 ∈ (ℤ≥‘𝑁)) | ||
| 2-Jan-2022 | supxrcli 39661 | The supremum of an arbitrary set of extended reals is an extended real. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ 𝐴 ⊆ ℝ* ⇒ ⊢ sup(𝐴, ℝ*, < ) ∈ ℝ* | ||
| 2-Jan-2022 | supxrmnf2 39660 | Removing minus infinity from a set does not affect its supremum. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝐴 ⊆ ℝ* → sup((𝐴 ∖ {-∞}), ℝ*, < ) = sup(𝐴, ℝ*, < )) | ||
| 2-Jan-2022 | ssrexr 39659 | A subset of the reals is a subset of the extended reals (common case). (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝜑 → 𝐴 ⊆ ℝ) ⇒ ⊢ (𝜑 → 𝐴 ⊆ ℝ*) | ||
| 2-Jan-2022 | imassmpt 39481 | Membership relation for the values of a function whose image is a subclass. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐶)) → 𝐵 ∈ 𝑉) & ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) ⇒ ⊢ (𝜑 → ((𝐹 “ 𝐶) ⊆ 𝐷 ↔ ∀𝑥 ∈ (𝐴 ∩ 𝐶)𝐵 ∈ 𝐷)) | ||
| 2-Jan-2022 | imass2d 39480 | Subset theorem for image. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝜑 → 𝐴 ⊆ 𝐵) ⇒ ⊢ (𝜑 → (𝐶 “ 𝐴) ⊆ (𝐶 “ 𝐵)) | ||
| 2-Jan-2022 | fnfvelrnd 39479 | A function's value belongs to its range. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝜑 → 𝐹 Fn 𝐴) & ⊢ (𝜑 → 𝐵 ∈ 𝐴) ⇒ ⊢ (𝜑 → (𝐹‘𝐵) ∈ ran 𝐹) | ||
| 2-Jan-2022 | fnfvima2 39478 | Given an element of the preimage, its function value is in the image. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝜑 → 𝐹 Fn 𝐴) & ⊢ (𝜑 → 𝐵 ∈ 𝐴) & ⊢ (𝜑 → 𝐵 ∈ 𝐶) ⇒ ⊢ (𝜑 → (𝐹‘𝐵) ∈ (𝐹 “ 𝐶)) | ||
| 2-Jan-2022 | rnmptssbi 39477 | The range of an operation given by the "maps to" notation as a subset. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) ⇒ ⊢ (𝜑 → (ran 𝐹 ⊆ 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶)) | ||
| 2-Jan-2022 | funresd 39476 | A restriction of a function is a function. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝜑 → Fun 𝐹) ⇒ ⊢ (𝜑 → Fun (𝐹 ↾ 𝐴)) | ||
| 2-Jan-2022 | fvelima2 39475 | Function value in an image. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ (𝐹 “ 𝐶)) → ∃𝑥 ∈ (𝐴 ∩ 𝐶)(𝐹‘𝑥) = 𝐵) | ||
| 2-Jan-2022 | ralrnmpt3 39474 | A restricted quantifier over an image set. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) & ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∀𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝜓 ↔ ∀𝑥 ∈ 𝐴 𝜒)) | ||
| 2-Jan-2022 | elmptima 39473 | The image of a function in maps-to notation. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝐶 ∈ 𝑉 → (𝐶 ∈ ((𝑥 ∈ 𝐴 ↦ 𝐵) “ 𝐷) ↔ ∃𝑥 ∈ (𝐴 ∩ 𝐷)𝐶 = 𝐵)) | ||
| 2-Jan-2022 | nel2nelini 39341 | Membership in an intersection implies membership in the second set. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ ¬ 𝐴 ∈ 𝐶 ⇒ ⊢ ¬ 𝐴 ∈ (𝐵 ∩ 𝐶) | ||
| 2-Jan-2022 | nel1nelini 39340 | Membership in an intersection implies membership in the first set. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ ¬ 𝐴 ∈ 𝐵 ⇒ ⊢ ¬ 𝐴 ∈ (𝐵 ∩ 𝐶) | ||
| 2-Jan-2022 | rexlimdva2 39339 | Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝜓) → 𝜒) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → 𝜒)) | ||
| 2-Jan-2022 | nel2nelin 39338 | Membership in an intersection implies membership in the second set. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (¬ 𝐴 ∈ 𝐶 → ¬ 𝐴 ∈ (𝐵 ∩ 𝐶)) | ||
| 2-Jan-2022 | nel1nelin 39337 | Membership in an intersection implies membership in the first set. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (¬ 𝐴 ∈ 𝐵 → ¬ 𝐴 ∈ (𝐵 ∩ 𝐶)) | ||
| 2-Jan-2022 | resabs1i 39336 | Absorption law for restriction. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ 𝐵 ⊆ 𝐶 ⇒ ⊢ ((𝐴 ↾ 𝐶) ↾ 𝐵) = (𝐴 ↾ 𝐵) | ||
| 2-Jan-2022 | rexlimd3 39335 | * Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝜒 & ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝜓) → 𝜒) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → 𝜒)) | ||
| 2-Jan-2022 | rexnegd 39334 | Minus a real number. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → -𝑒𝐴 = -𝐴) | ||
| 2-Jan-2022 | ssrind 39333 | Add right intersection to subclass relation. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ (𝜑 → 𝐴 ⊆ 𝐵) ⇒ ⊢ (𝜑 → (𝐴 ∩ 𝐶) ⊆ (𝐵 ∩ 𝐶)) | ||
| 2-Jan-2022 | rabssd 39332 | Restricted class abstraction in a subclass relationship. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝐵 & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝜒) → 𝑥 ∈ 𝐵) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜒} ⊆ 𝐵) | ||
| 2-Jan-2022 | ssdf2 39331 | A sufficient condition for a subclass relationship. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐵) ⇒ ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | ||
| 2-Jan-2022 | resabs2i 39330 | Absorption law for restriction. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ 𝐵 ⊆ 𝐶 ⇒ ⊢ ((𝐴 ↾ 𝐵) ↾ 𝐶) = (𝐴 ↾ 𝐵) | ||
| 2-Jan-2022 | ifcli 39329 | Membership (closure) of a conditional operator. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| ⊢ 𝐴 ∈ 𝐶 & ⊢ 𝐵 ∈ 𝐶 ⇒ ⊢ if(𝜑, 𝐴, 𝐵) ∈ 𝐶 | ||
| 2-Jan-2022 | frgrwopreglem3 27178 | Lemma 3 for frgrwopreg 27187. The vertices in the sets 𝐴 and 𝐵 have different degrees. (Contributed by Alexander van der Vekens, 30-Dec-2017.) (Revised by AV, 10-May-2021.) (Proof shortened by AV, 2-Jan-2022.) |
| ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐷 = (VtxDeg‘𝐺) & ⊢ 𝐴 = {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾} & ⊢ 𝐵 = (𝑉 ∖ 𝐴) ⇒ ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝐷‘𝑋) ≠ (𝐷‘𝑌)) | ||
| 2-Jan-2022 | frgrwopreglem2 27177 | Lemma 2 for frgrwopreg 27187. If the set 𝐴 of vertices of degree 𝐾 is not empty in a friendship graph with at least two vertices, then 𝐾 must be greater than 1 . This is only an observation, which is not required for the proof the friendship theorem. (Contributed by Alexander van der Vekens, 30-Dec-2017.) (Revised by AV, 2-Jan-2022.) |
| ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐷 = (VtxDeg‘𝐺) & ⊢ 𝐴 = {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾} & ⊢ 𝐵 = (𝑉 ∖ 𝐴) ⇒ ⊢ ((𝐺 ∈ FriendGraph ∧ 1 < (#‘𝑉) ∧ 𝐴 ≠ ∅) → 2 ≤ 𝐾) | ||
| 2-Jan-2022 | eldifeldifsn 4342 | An element of a difference set is an element of the difference with a singleton. (Contributed by AV, 2-Jan-2022.) |
| ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ (𝐵 ∖ 𝐴)) → 𝑌 ∈ (𝐵 ∖ {𝑋})) | ||
| 1-Jan-2022 | hgt750lema 30735 | An upper bound on the contribution of the non-prime terms in the Statement 7.50 of [Helfgott] p. 69. (Contributed by Thierry Arnoux, 1-Jan-2022.) |
| ⊢ 𝑂 = {𝑧 ∈ ℤ ∣ ¬ 2 ∥ 𝑧} & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 2 ≤ 𝑁) & ⊢ 𝐴 = {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ)} & ⊢ 𝐹 = (𝑑 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘𝑎) ∈ (𝑂 ∩ ℙ)} ↦ (𝑑 ∘ if(𝑎 = 0, ( I ↾ (0..^3)), ((pmTrsp‘(0..^3))‘{𝑎, 0})))) ⇒ ⊢ (𝜑 → Σ𝑛 ∈ ((ℕ(repr‘3)𝑁) ∖ ((𝑂 ∩ ℙ)(repr‘3)𝑁))((Λ‘(𝑛‘0)) · ((Λ‘(𝑛‘1)) · (Λ‘(𝑛‘2)))) ≤ (3 · Σ𝑛 ∈ 𝐴 ((Λ‘(𝑛‘0)) · ((Λ‘(𝑛‘1)) · (Λ‘(𝑛‘2)))))) | ||
| 1-Jan-2022 | hgt750lemg 30732 | Lemma for the statement 7.50 of [Helfgott] p. 69. Applying a permutation 𝑇 to the three factors of a product does not change the result. (Contributed by Thierry Arnoux, 1-Jan-2022.) |
| ⊢ 𝐹 = (𝑐 ∈ 𝑅 ↦ (𝑐 ∘ 𝑇)) & ⊢ (𝜑 → 𝑇:(0..^3)–1-1-onto→(0..^3)) & ⊢ (𝜑 → 𝑁:(0..^3)⟶ℕ) & ⊢ (𝜑 → 𝐿:ℕ⟶ℝ) & ⊢ (𝜑 → 𝑁 ∈ 𝑅) ⇒ ⊢ (𝜑 → ((𝐿‘((𝐹‘𝑁)‘0)) · ((𝐿‘((𝐹‘𝑁)‘1)) · (𝐿‘((𝐹‘𝑁)‘2)))) = ((𝐿‘(𝑁‘0)) · ((𝐿‘(𝑁‘1)) · (𝐿‘(𝑁‘2))))) | ||
| 1-Jan-2022 | hgt750lemf 30731 | Lemma for the statement 7.50 of [Helfgott] p. 69. (Contributed by Thierry Arnoux, 1-Jan-2022.) |
| ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝑃 ∈ ℝ) & ⊢ (𝜑 → 𝑄 ∈ ℝ) & ⊢ (𝜑 → 𝐻:ℕ⟶(0[,)+∞)) & ⊢ (𝜑 → 𝐾:ℕ⟶(0[,)+∞)) & ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → (𝑛‘0) ∈ ℕ) & ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → (𝑛‘1) ∈ ℕ) & ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → (𝑛‘2) ∈ ℕ) & ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐾‘𝑚) ≤ 𝑃) & ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐻‘𝑚) ≤ 𝑄) ⇒ ⊢ (𝜑 → Σ𝑛 ∈ 𝐴 (((Λ‘(𝑛‘0)) · (𝐻‘(𝑛‘0))) · (((Λ‘(𝑛‘1)) · (𝐾‘(𝑛‘1))) · ((Λ‘(𝑛‘2)) · (𝐾‘(𝑛‘2))))) ≤ (((𝑃↑2) · 𝑄) · Σ𝑛 ∈ 𝐴 ((Λ‘(𝑛‘0)) · ((Λ‘(𝑛‘1)) · (Λ‘(𝑛‘2)))))) | ||
| 1-Jan-2022 | divsqrtid 30672 | A real number divided by its square root. (Contributed by Thierry Arnoux, 1-Jan-2022.) |
| ⊢ (𝐴 ∈ ℝ+ → (𝐴 / (√‘𝐴)) = (√‘𝐴)) | ||
| 1-Jan-2022 | pmtridf1o 29856 | Transpositions of 𝑋 and 𝑌 (understood to be the identity when 𝑋 = 𝑌), are bijections. (Contributed by Thierry Arnoux, 1-Jan-2022.) |
| ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → 𝑌 ∈ 𝐴) & ⊢ 𝑇 = if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌})) ⇒ ⊢ (𝜑 → 𝑇:𝐴–1-1-onto→𝐴) | ||
| 1-Jan-2022 | prodtp 29573 | A product over a triple is the product of the elements. (Contributed by Thierry Arnoux, 1-Jan-2022.) |
| ⊢ (𝑘 = 𝐴 → 𝐷 = 𝐸) & ⊢ (𝑘 = 𝐵 → 𝐷 = 𝐹) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ (𝜑 → 𝐸 ∈ ℂ) & ⊢ (𝜑 → 𝐹 ∈ ℂ) & ⊢ (𝜑 → 𝐴 ≠ 𝐵) & ⊢ (𝑘 = 𝐶 → 𝐷 = 𝐺) & ⊢ (𝜑 → 𝐶 ∈ 𝑋) & ⊢ (𝜑 → 𝐺 ∈ ℂ) & ⊢ (𝜑 → 𝐴 ≠ 𝐶) & ⊢ (𝜑 → 𝐵 ≠ 𝐶) ⇒ ⊢ (𝜑 → ∏𝑘 ∈ {𝐴, 𝐵, 𝐶}𝐷 = ((𝐸 · 𝐹) · 𝐺)) | ||
| 1-Jan-2022 | prodpr 29572 | A product over a pair is the product of the elements. (Contributed by Thierry Arnoux, 1-Jan-2022.) |
| ⊢ (𝑘 = 𝐴 → 𝐷 = 𝐸) & ⊢ (𝑘 = 𝐵 → 𝐷 = 𝐹) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ (𝜑 → 𝐸 ∈ ℂ) & ⊢ (𝜑 → 𝐹 ∈ ℂ) & ⊢ (𝜑 → 𝐴 ≠ 𝐵) ⇒ ⊢ (𝜑 → ∏𝑘 ∈ {𝐴, 𝐵}𝐷 = (𝐸 · 𝐹)) | ||
| 31-Dec-2021 | dfgcd3 33170 | Alternate definition of the gcd operator. (Contributed by Jim Kingdon, 31-Dec-2021.) |
| ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 gcd 𝑁) = (℩𝑑 ∈ ℕ0 ∀𝑧 ∈ ℤ (𝑧 ∥ 𝑑 ↔ (𝑧 ∥ 𝑀 ∧ 𝑧 ∥ 𝑁)))) | ||
| 31-Dec-2021 | iinssiun 29377 | An indexed intersection is a subset of the corresponding indexed union. (Contributed by Thierry Arnoux, 31-Dec-2021.) |
| ⊢ (𝐴 ≠ ∅ → ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) | ||
| 30-Dec-2021 | frgrncvvdeqlem10 27172 | Lemma 10 for frgrncvvdeq 27173. (Contributed by Alexander van der Vekens, 24-Dec-2017.) (Revised by AV, 10-May-2021.) (Proof shortened by AV, 30-Dec-2021.) |
| ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) & ⊢ 𝐷 = (𝐺 NeighbVtx 𝑋) & ⊢ 𝑁 = (𝐺 NeighbVtx 𝑌) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ (𝜑 → 𝑋 ≠ 𝑌) & ⊢ (𝜑 → 𝑌 ∉ 𝐷) & ⊢ (𝜑 → 𝐺 ∈ FriendGraph ) & ⊢ 𝐴 = (𝑥 ∈ 𝐷 ↦ (℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸)) ⇒ ⊢ (𝜑 → 𝐴:𝐷–1-1-onto→𝑁) | ||
| 30-Dec-2021 | frgrncvvdeqlem9 27171 | Lemma 9 for frgrncvvdeq 27173. This corresponds to statement 3 in [Huneke] p. 1: "By symmetry the map is onto". (Contributed by Alexander van der Vekens, 24-Dec-2017.) (Revised by AV, 10-May-2021.) (Proof shortened by AV, 30-Dec-2021.) |
| ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) & ⊢ 𝐷 = (𝐺 NeighbVtx 𝑋) & ⊢ 𝑁 = (𝐺 NeighbVtx 𝑌) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ (𝜑 → 𝑋 ≠ 𝑌) & ⊢ (𝜑 → 𝑌 ∉ 𝐷) & ⊢ (𝜑 → 𝐺 ∈ FriendGraph ) & ⊢ 𝐴 = (𝑥 ∈ 𝐷 ↦ (℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸)) ⇒ ⊢ (𝜑 → 𝐴:𝐷–onto→𝑁) | ||
| 30-Dec-2021 | frgrncvvdeqlem8 27170 | Lemma 8 for frgrncvvdeq 27173. This corresponds to statement 2 in [Huneke] p. 1: "The map is one-to-one since z in N(x) is uniquely determined as the common neighbor of x and a(x)". (Contributed by Alexander van der Vekens, 23-Dec-2017.) (Revised by AV, 10-May-2021.) (Revised by AV, 30-Dec-2021.) |
| ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) & ⊢ 𝐷 = (𝐺 NeighbVtx 𝑋) & ⊢ 𝑁 = (𝐺 NeighbVtx 𝑌) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ (𝜑 → 𝑋 ≠ 𝑌) & ⊢ (𝜑 → 𝑌 ∉ 𝐷) & ⊢ (𝜑 → 𝐺 ∈ FriendGraph ) & ⊢ 𝐴 = (𝑥 ∈ 𝐷 ↦ (℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸)) ⇒ ⊢ (𝜑 → 𝐴:𝐷–1-1→𝑁) | ||
| 30-Dec-2021 | frgrncvvdeqlem6 27168 | Lemma 6 for frgrncvvdeq 27173. (Contributed by Alexander van der Vekens, 23-Dec-2017.) (Revised by AV, 10-May-2021.) (Proof shortened by AV, 30-Dec-2021.) |
| ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) & ⊢ 𝐷 = (𝐺 NeighbVtx 𝑋) & ⊢ 𝑁 = (𝐺 NeighbVtx 𝑌) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ (𝜑 → 𝑋 ≠ 𝑌) & ⊢ (𝜑 → 𝑌 ∉ 𝐷) & ⊢ (𝜑 → 𝐺 ∈ FriendGraph ) & ⊢ 𝐴 = (𝑥 ∈ 𝐷 ↦ (℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸)) ⇒ ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → {𝑥, (𝐴‘𝑥)} ∈ 𝐸) | ||
| 30-Dec-2021 | frgrncvvdeqlem3 27165 | Lemma 3 for frgrncvvdeq 27173. The unique neighbor of a vertex (expressed by a restricted iota) is the intersection of the corresponding neighborhoods. (Contributed by Alexander van der Vekens, 18-Dec-2017.) (Revised by AV, 10-May-2021.) (Proof shortened by AV, 30-Dec-2021.) |
| ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) & ⊢ 𝐷 = (𝐺 NeighbVtx 𝑋) & ⊢ 𝑁 = (𝐺 NeighbVtx 𝑌) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ (𝜑 → 𝑋 ≠ 𝑌) & ⊢ (𝜑 → 𝑌 ∉ 𝐷) & ⊢ (𝜑 → 𝐺 ∈ FriendGraph ) & ⊢ 𝐴 = (𝑥 ∈ 𝐷 ↦ (℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸)) ⇒ ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → {(℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸)} = ((𝐺 NeighbVtx 𝑥) ∩ 𝑁)) | ||
| 30-Dec-2021 | frgrncvvdeqlem2 27164 | Lemma 2 for frgrncvvdeq 27173. In a friendship graph, for each neighbor of a vertex there is exactly one neighbor of another vertex so that there is an edge between these two neighbors. (Contributed by Alexander van der Vekens, 22-Dec-2017.) (Revised by AV, 10-May-2021.) (Proof shortened by AV, 30-Dec-2021.) |
| ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) & ⊢ 𝐷 = (𝐺 NeighbVtx 𝑋) & ⊢ 𝑁 = (𝐺 NeighbVtx 𝑌) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ (𝜑 → 𝑋 ≠ 𝑌) & ⊢ (𝜑 → 𝑌 ∉ 𝐷) & ⊢ (𝜑 → 𝐺 ∈ FriendGraph ) & ⊢ 𝐴 = (𝑥 ∈ 𝐷 ↦ (℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸)) ⇒ ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ∃!𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸) | ||
| 30-Dec-2021 | frcond4 27134 | The friendship condition, alternatively expressed by neighborhoods: in a friendship graph, the neighborhoods of two different vertices have exactly one vertex in common. (Contributed by Alexander van der Vekens, 19-Dec-2017.) (Revised by AV, 29-Mar-2021.) (Proof shortened by AV, 30-Dec-2021.) |
| ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ (𝐺 ∈ FriendGraph → ∀𝑘 ∈ 𝑉 ∀𝑙 ∈ (𝑉 ∖ {𝑘})∃𝑥 ∈ 𝑉 ((𝐺 NeighbVtx 𝑘) ∩ (𝐺 NeighbVtx 𝑙)) = {𝑥}) | ||
| 30-Dec-2021 | frcond3 27133 | The friendship condition, expressed by neighborhoods: in a friendship graph, the neighborhood of a vertex and the neighborhood of a second, different vertex have exactly one vertex in common. (Contributed by Alexander van der Vekens, 19-Dec-2017.) (Revised by AV, 30-Dec-2021.) |
| ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ (𝐺 ∈ FriendGraph → ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐴 ≠ 𝐶) → ∃𝑥 ∈ 𝑉 ((𝐺 NeighbVtx 𝐴) ∩ (𝐺 NeighbVtx 𝐶)) = {𝑥})) | ||
| 30-Dec-2021 | elinsn 4245 | If the intersection of two classes is a (proper) singleton, then the singleton element is a member of both classes. (Contributed by AV, 30-Dec-2021.) |
| ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐵 ∩ 𝐶) = {𝐴}) → (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶)) | ||
| 29-Dec-2021 | hgt750leme 30736 | An upper bound on the contribution of the non-prime terms in the Statement 7.50 of [Helfgott] p. 69. (Contributed by Thierry Arnoux, 29-Dec-2021.) |
| ⊢ 𝑂 = {𝑧 ∈ ℤ ∣ ¬ 2 ∥ 𝑧} & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → (;10↑;27) ≤ 𝑁) & ⊢ (𝜑 → 𝐻:ℕ⟶(0[,)+∞)) & ⊢ (𝜑 → 𝐾:ℕ⟶(0[,)+∞)) & ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐾‘𝑚) ≤ (1._0_7_9_9_55)) & ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐻‘𝑚) ≤ (1._4_14)) ⇒ ⊢ (𝜑 → Σ𝑛 ∈ ((ℕ(repr‘3)𝑁) ∖ ((𝑂 ∩ ℙ)(repr‘3)𝑁))(((Λ‘(𝑛‘0)) · (𝐻‘(𝑛‘0))) · (((Λ‘(𝑛‘1)) · (𝐾‘(𝑛‘1))) · ((Λ‘(𝑛‘2)) · (𝐾‘(𝑛‘2))))) ≤ (((7._3_48) · ((log‘𝑁) / (√‘𝑁))) · (𝑁↑2))) | ||
| 29-Dec-2021 | hgt750lemd 30726 | An upper bound to the summatory function of the von Mangoldt function on non-primes. (Contributed by Thierry Arnoux, 29-Dec-2021.) |
| ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → (;10↑;27) ≤ 𝑁) ⇒ ⊢ (𝜑 → Σ𝑖 ∈ (((1...𝑁) ∖ ℙ) ∪ {2})(Λ‘𝑖) < ((1._4_2_63) · (√‘𝑁))) | ||
| 29-Dec-2021 | hgt750lemc 30725 | An upper bound to the summatory function of the von Mangoldt function. (Contributed by Thierry Arnoux, 29-Dec-2021.) |
| ⊢ (𝜑 → 𝑁 ∈ ℕ) ⇒ ⊢ (𝜑 → Σ𝑗 ∈ (1...𝑁)(Λ‘𝑗) < ((1._0_3_8_83) · 𝑁)) | ||
| 29-Dec-2021 | dpadd2 29618 | Addition with one decimal, no carry. (Contributed by Thierry Arnoux, 29-Dec-2021.) |
| ⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℝ+ & ⊢ 𝐶 ∈ ℕ0 & ⊢ 𝐷 ∈ ℝ+ & ⊢ 𝐸 ∈ ℕ0 & ⊢ 𝐹 ∈ ℝ+ & ⊢ 𝐺 ∈ ℕ0 & ⊢ 𝐻 ∈ ℕ0 & ⊢ (𝐺 + 𝐻) = 𝐼 & ⊢ ((𝐴.𝐵) + (𝐶.𝐷)) = (𝐸.𝐹) ⇒ ⊢ ((𝐺._𝐴𝐵) + (𝐻._𝐶𝐷)) = (𝐼._𝐸𝐹) | ||
| 29-Dec-2021 | uzssico 29546 | Upper integer sets are a subset of the corresponding closed-below, open-above intervals. (Contributed by Thierry Arnoux, 29-Dec-2021.) |
| ⊢ (𝑀 ∈ ℤ → (ℤ≥‘𝑀) ⊆ (𝑀[,)+∞)) | ||
| 28-Dec-2021 | hgt750lemb 30734 | An upper bound on the contribution of the non-prime terms in the Statement 7.50 of [Helfgott] p. 69. (Contributed by Thierry Arnoux, 28-Dec-2021.) |
| ⊢ 𝑂 = {𝑧 ∈ ℤ ∣ ¬ 2 ∥ 𝑧} & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 2 ≤ 𝑁) & ⊢ 𝐴 = {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ)} ⇒ ⊢ (𝜑 → Σ𝑛 ∈ 𝐴 ((Λ‘(𝑛‘0)) · ((Λ‘(𝑛‘1)) · (Λ‘(𝑛‘2)))) ≤ ((log‘𝑁) · (Σ𝑖 ∈ (((1...𝑁) ∖ ℙ) ∪ {2})(Λ‘𝑖) · Σ𝑗 ∈ (1...𝑁)(Λ‘𝑗)))) | ||
| 28-Dec-2021 | ax-ros336 30724 | Theorem 13. of [RosserSchoenfeld] p. 71. Theorem chpchtlim 25168 states that the ψ and θ function are asymtotic to each other; this axiom postulates an upper bound for their difference. This is stated as an axiom until a formal proof can be provided. (Contributed by Thierry Arnoux, 28-Dec-2021.) |
| ⊢ ∀𝑥 ∈ ℝ+ ((ψ‘𝑥) − (θ‘𝑥)) < ((1._4_2_62) · (√‘𝑥)) | ||
| 28-Dec-2021 | ax-ros335 30723 | Theorem 12. of [RosserSchoenfeld] p. 71. Theorem chpo1ubb 25170 states that the ψ function is bounded by a linear term; this axiom postulates an upper bound for that linear term. This is stated as an axiom until a formal proof can be provided. (Contributed by Thierry Arnoux, 28-Dec-2021.) |
| ⊢ ∀𝑥 ∈ ℝ+ (ψ‘𝑥) < ((1._0_3_8_83) · 𝑥) | ||
| 28-Dec-2021 | chtvalz 30707 | Value of the Chebyshev function for integers. (Contributed by Thierry Arnoux, 28-Dec-2021.) |
| ⊢ (𝑁 ∈ ℤ → (θ‘𝑁) = Σ𝑛 ∈ ((1...𝑁) ∩ ℙ)(log‘𝑛)) | ||
| 28-Dec-2021 | chpvalz 30706 | Value of the second Chebyshev function, or summatory of the von Mangoldt function. (Contributed by Thierry Arnoux, 28-Dec-2021.) |
| ⊢ (𝑁 ∈ ℤ → (ψ‘𝑁) = Σ𝑛 ∈ (1...𝑁)(Λ‘𝑛)) | ||
| 28-Dec-2021 | difininv 29354 | Condition for the intersections of two sets with a given set to be equal. (Contributed by Thierry Arnoux, 28-Dec-2021.) |
| ⊢ ((((𝐴 ∖ 𝐶) ∩ 𝐵) = ∅ ∧ ((𝐶 ∖ 𝐴) ∩ 𝐵) = ∅) → (𝐴 ∩ 𝐵) = (𝐶 ∩ 𝐵)) | ||
| 28-Dec-2021 | frgrncvvdeqlem1 27163 | Lemma 1 for frgrncvvdeq 27173. (Contributed by Alexander van der Vekens, 23-Dec-2017.) (Revised by AV, 8-May-2021.) (Proof shortened by AV, 28-Dec-2021.) |
| ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) & ⊢ 𝐷 = (𝐺 NeighbVtx 𝑋) & ⊢ 𝑁 = (𝐺 NeighbVtx 𝑌) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ (𝜑 → 𝑋 ≠ 𝑌) & ⊢ (𝜑 → 𝑌 ∉ 𝐷) & ⊢ (𝜑 → 𝐺 ∈ FriendGraph ) & ⊢ 𝐴 = (𝑥 ∈ 𝐷 ↦ (℩𝑦 ∈ 𝑁 {𝑥, 𝑦} ∈ 𝐸)) ⇒ ⊢ (𝜑 → 𝑋 ∉ 𝑁) | ||
| 27-Dec-2021 | dfnelbr2 41290 | Alternate definition of the negated membership as binary relation. (Proposed by BJ, 27-Dec-2021.) (Contributed by AV, 27-Dec-2021.) |
| ⊢ _∉ = ((V × V) ∖ E ) | ||
| 27-Dec-2021 | bj-evalidval 33031 | Closed general form of strndxid 15885. Both sides are equal to (𝐹‘𝐴) by bj-evalid 33028 and bj-evalval 33027 respectively, but bj-evalidval 33031 adds something to bj-evalid 33028 and bj-evalval 33027 in that Slot 𝐴 appears on both sides. (Contributed by BJ, 27-Dec-2021.) |
| ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉 ∧ 𝐹 ∈ 𝑈) → (𝐹‘(Slot 𝐴‘( I ↾ 𝑉))) = (Slot 𝐴‘𝐹)) | ||
| 27-Dec-2021 | bj-ndxid 33030 | Proof of ndxid 15883 from ndxarg 15882. (Contributed by BJ, 27-Dec-2021.) (Proof modification is discouraged.) |
| ⊢ 𝐸 = Slot 𝑁 & ⊢ 𝑁 ∈ ℕ ⇒ ⊢ 𝐸 = Slot (𝐸‘ndx) | ||
| 27-Dec-2021 | bj-ndxarg 33029 | Proof of ndxarg 15882 from bj-evalid 33028. (Contributed by BJ, 27-Dec-2021.) (Proof modification is discouraged.) |
| ⊢ 𝐸 = Slot 𝑁 & ⊢ 𝑁 ∈ ℕ ⇒ ⊢ (𝐸‘ndx) = 𝑁 | ||
| 27-Dec-2021 | bj-evalid 33028 | The evaluation at a set of the identity function is that set. (General form of ndxarg 15882.) The restriction to a set 𝑉 is necessary since the argument of the function Slot 𝐴 (like that of any function) has to be a set for the evaluation to be meaningful. (Contributed by BJ, 27-Dec-2021.) |
| ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → (Slot 𝐴‘( I ↾ 𝑉)) = 𝐴) | ||
| 27-Dec-2021 | bj-evalval 33027 | Value of the evaluation at a class. (Closed form of strfvnd 15876 and strfvn 15879). (Contributed by NM, 9-Sep-2011.) (Revised by Mario Carneiro, 15-Nov-2014.) (Revised by BJ, 27-Dec-2021.) |
| ⊢ (𝐹 ∈ 𝑉 → (Slot 𝐴‘𝐹) = (𝐹‘𝐴)) | ||
| 27-Dec-2021 | bj-evalfn 33026 | The evaluation at a class is a function on the universal class. (General form of slotfn 15875). (Contributed by Mario Carneiro, 22-Sep-2015.) (Revised by BJ, 27-Dec-2021.) |
| ⊢ Slot 𝐴 Fn V | ||
| 27-Dec-2021 | bj-evalfun 33025 | The evaluation at a class is a function. (Contributed by BJ, 27-Dec-2021.) |
| ⊢ Fun Slot 𝐴 | ||
| 27-Dec-2021 | bj-evaleq 33024 | Equality theorem for the Slot construction. This is currently a duplicate of sloteq 15862 but may diverge from it if/when a token Eval is introduced for evaluation in order to separate it from Slot and any of its possible modifications. (Contributed by BJ, 27-Dec-2021.) (Proof modification is discouraged.) |
| ⊢ (𝐴 = 𝐵 → Slot 𝐴 = Slot 𝐵) | ||
| 27-Dec-2021 | oddprm2 30733 | Two ways to write the set of odd primes. (Contributed by Thierry Arnoux, 27-Dec-2021.) |
| ⊢ 𝑂 = {𝑧 ∈ ℤ ∣ ¬ 2 ∥ 𝑧} ⇒ ⊢ (ℙ ∖ {2}) = (𝑂 ∩ ℙ) | ||
| 27-Dec-2021 | reprdifc 30705 | Express the representations as a sum of integers in a difference of sets using conditions on each of the indices. (Contributed by Thierry Arnoux, 27-Dec-2021.) |
| ⊢ 𝐶 = {𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∣ ¬ (𝑐‘𝑥) ∈ 𝐵} & ⊢ (𝜑 → 𝐴 ⊆ ℕ) & ⊢ (𝜑 → 𝐵 ⊆ ℕ) & ⊢ (𝜑 → 𝑀 ∈ ℕ0) & ⊢ (𝜑 → 𝑆 ∈ ℕ0) ⇒ ⊢ (𝜑 → ((𝐴(repr‘𝑆)𝑀) ∖ (𝐵(repr‘𝑆)𝑀)) = ∪ 𝑥 ∈ (0..^𝑆)𝐶) | ||
| 27-Dec-2021 | reprpmtf1o 30704 | Transposing 0 and 𝑋 maps representations with a condition on the first index to transpositions with the same condition on the index 𝑋. (Contributed by Thierry Arnoux, 27-Dec-2021.) |
| ⊢ (𝜑 → 𝑆 ∈ ℕ) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐴 ⊆ ℕ) & ⊢ (𝜑 → 𝑋 ∈ (0..^𝑆)) & ⊢ 𝑂 = {𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∣ ¬ (𝑐‘0) ∈ 𝐵} & ⊢ 𝑃 = {𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∣ ¬ (𝑐‘𝑋) ∈ 𝐵} & ⊢ 𝑇 = if(𝑋 = 0, ( I ↾ (0..^𝑆)), ((pmTrsp‘(0..^𝑆))‘{𝑋, 0})) & ⊢ 𝐹 = (𝑐 ∈ 𝑃 ↦ (𝑐 ∘ 𝑇)) ⇒ ⊢ (𝜑 → 𝐹:𝑃–1-1-onto→𝑂) | ||
| 27-Dec-2021 | reprle 30692 | Upper bound to the terms in the representations of 𝑀 as the sum of 𝑆 nonnegative integers from set 𝐴. (Contributed by Thierry Arnoux, 27-Dec-2021.) |
| ⊢ (𝜑 → 𝐴 ⊆ ℕ) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑆 ∈ ℕ0) & ⊢ (𝜑 → 𝐶 ∈ (𝐴(repr‘𝑆)𝑀)) & ⊢ (𝜑 → 𝑋 ∈ (0..^𝑆)) ⇒ ⊢ (𝜑 → (𝐶‘𝑋) ≤ 𝑀) | ||
| 27-Dec-2021 | 1mhdrd 29624 | Example theorem demonstrating decimal expansions. (Contributed by Thierry Arnoux, 27-Dec-2021.) |
| ⊢ ((0._99) + (0._01)) = 1 | ||
| 27-Dec-2021 | threehalves 29623 | Example theorem demonstrating decimal expansions. (Contributed by Thierry Arnoux, 27-Dec-2021.) |
| ⊢ (3 / 2) = (1.5) | ||
| 27-Dec-2021 | dpadd3 29620 | Addition with two decimals. (Contributed by Thierry Arnoux, 27-Dec-2021.) |
| ⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 & ⊢ 𝐶 ∈ ℕ0 & ⊢ 𝐷 ∈ ℕ0 & ⊢ 𝐸 ∈ ℕ0 & ⊢ 𝐺 ∈ ℕ0 & ⊢ 𝐹 ∈ ℕ0 & ⊢ 𝐻 ∈ ℕ0 & ⊢ 𝐼 ∈ ℕ0 & ⊢ (;;𝐴𝐵𝐶 + ;;𝐷𝐸𝐹) = ;;𝐺𝐻𝐼 ⇒ ⊢ ((𝐴._𝐵𝐶) + (𝐷._𝐸𝐹)) = (𝐺._𝐻𝐼) | ||
| 27-Dec-2021 | dpadd 29619 | Addition with one decimal. (Contributed by Thierry Arnoux, 27-Dec-2021.) |
| ⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 & ⊢ 𝐶 ∈ ℕ0 & ⊢ 𝐷 ∈ ℕ0 & ⊢ 𝐸 ∈ ℕ0 & ⊢ 𝐹 ∈ ℕ0 & ⊢ (;𝐴𝐵 + ;𝐶𝐷) = ;𝐸𝐹 ⇒ ⊢ ((𝐴.𝐵) + (𝐶.𝐷)) = (𝐸.𝐹) | ||
| 27-Dec-2021 | fsumub 29574 | An upper bound for a term of a positive finite sum. (Contributed by Thierry Arnoux, 27-Dec-2021.) |
| ⊢ (𝑘 = 𝐾 → 𝐵 = 𝐷) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 = 𝐶) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ+) & ⊢ (𝜑 → 𝐾 ∈ 𝐴) ⇒ ⊢ (𝜑 → 𝐷 ≤ 𝐶) | ||
| 27-Dec-2021 | fusgrvtxdgonume 26450 | The number of vertices of odd degree is even in a finite simple graph. Proposition 1.2.1 in [Diestel] p. 5. See also remark about equation (2) in section I.1 in [Bollobas] p. 4. (Contributed by AV, 27-Dec-2021.) |
| ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ 𝐷 = (VtxDeg‘𝐺) ⇒ ⊢ (𝐺 ∈ FinUSGraph → 2 ∥ (#‘{𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)})) | ||
| 27-Dec-2021 | fusgrfupgrfs 26223 | A finite simple graph is a finite pseudograph of finite size. (Contributed by AV, 27-Dec-2021.) |
| ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ (𝐺 ∈ FinUSGraph → (𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin)) | ||
| 27-Dec-2021 | ndxid 15883 |
A structure component extractor is defined by its own index. This
theorem, together with strfv 15907 below, is useful for avoiding direct
reference to the hard-coded numeric index in component extractor
definitions, such as the 1 in df-base 15863 and the ;10 in
df-ple 15961, making it easier to change should the need
arise.
For example, we can refer to a specific poset with base set 𝐵 and order relation 𝐿 using {⟨(Base‘ndx), 𝐵⟩, ⟨(le‘ndx), 𝐿⟩} rather than {⟨1, 𝐵⟩, ⟨;10, 𝐿⟩}. The latter, while shorter to state, requires revision if we later change ;10 to some other number, and it may also be harder to remember. (Contributed by NM, 19-Oct-2012.) (Revised by Mario Carneiro, 6-Oct-2013.) (Proof shortened by BJ, 27-Dec-2021.) |
| ⊢ 𝐸 = Slot 𝑁 & ⊢ 𝑁 ∈ ℕ ⇒ ⊢ 𝐸 = Slot (𝐸‘ndx) | ||
| 27-Dec-2021 | sloteq 15862 | Equality theorem for the Slot construction. (Contributed by BJ, 27-Dec-2021.) |
| ⊢ (𝐴 = 𝐵 → Slot 𝐴 = Slot 𝐵) | ||
| 26-Dec-2021 | sbgoldbmb 41674 | The strong binary Goldbach conjecture and the modern version of the original formulation of the Goldbach conjecture are equivalent. (Contributed by AV, 26-Dec-2021.) |
| ⊢ (∀𝑛 ∈ Even (4 < 𝑛 → 𝑛 ∈ GoldbachEven ) ↔ ∀𝑛 ∈ (ℤ≥‘6)∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟)) | ||
| 26-Dec-2021 | mogoldbb 41673 | If the modern version of the original formulation of the Goldbach conjecture is valid, the (weak) binary Goldbach conjecture also holds. (Contributed by AV, 26-Dec-2021.) |
| ⊢ (∀𝑛 ∈ (ℤ≥‘6)∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟) → ∀𝑛 ∈ Even (2 < 𝑛 → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))) | ||
| 26-Dec-2021 | mogoldbblem 41629 | Lemma for mogoldbb 41673. (Contributed by AV, 26-Dec-2021.) |
| ⊢ (((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ∧ 𝑅 ∈ ℙ) ∧ 𝑁 ∈ Even ∧ (𝑁 + 2) = ((𝑃 + 𝑄) + 𝑅)) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑁 = (𝑝 + 𝑞)) | ||
| 26-Dec-2021 | odd2prm2 41627 | If an odd number is the sum of two prime numbers, one of the prime numbers must be 2. (Contributed by AV, 26-Dec-2021.) |
| ⊢ ((𝑁 ∈ Odd ∧ (𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) ∧ 𝑁 = (𝑃 + 𝑄)) → (𝑃 = 2 ∨ 𝑄 = 2)) | ||
| 26-Dec-2021 | nelbrnelim 41294 | If a set is related to another set by the negated membership relation, then it is not a member of the other set. (Contributed by AV, 26-Dec-2021.) |
| ⊢ (𝐴 _∉ 𝐵 → 𝐴 ∉ 𝐵) | ||
| 26-Dec-2021 | nelbrnel 41293 | A set is related to another set by the negated membership relation iff it is not a member of the other set. (Contributed by AV, 26-Dec-2021.) |
| ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 _∉ 𝐵 ↔ 𝐴 ∉ 𝐵)) | ||
| 26-Dec-2021 | nelbrim 41292 | If a set is related to another set by the negated membership relation, then it is not a member of the other set. The other direction of the implication is not generally true, because if 𝐴 is a proper class, then ¬ 𝐴 ∈ 𝐵 would be true, but not 𝐴 _∉ 𝐵. (Contributed by AV, 26-Dec-2021.) |
| ⊢ (𝐴 _∉ 𝐵 → ¬ 𝐴 ∈ 𝐵) | ||
| 26-Dec-2021 | nelbr 41291 | The binary relation of a set not being a member of another set. (Contributed by AV, 26-Dec-2021.) |
| ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 _∉ 𝐵 ↔ ¬ 𝐴 ∈ 𝐵)) | ||
| 26-Dec-2021 | df-nelbr 41289 | Define negated membership as binary relation. Analogous to df-eprel 5029 (the epsilon relation). (Contributed by AV, 26-Dec-2021.) |
| ⊢ _∉ = {⟨𝑥, 𝑦⟩ ∣ ¬ 𝑥 ∈ 𝑦} | ||
| 26-Dec-2021 | hgt750lem2 30730 | Decimal multiplication galore! (Contributed by Thierry Arnoux, 26-Dec-2021.) |
| ⊢ (3 · ((((1._0_7_9_9_55)↑2) · (1._4_14)) · ((1._4_2_63) · (1._0_3_8_83)))) < (7._3_48) | ||
| 26-Dec-2021 | dpmul 29621 | Multiplication with one decimal point. (Contributed by Thierry Arnoux, 26-Dec-2021.) |
| ⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 & ⊢ 𝐶 ∈ ℕ0 & ⊢ 𝐷 ∈ ℕ0 & ⊢ 𝐸 ∈ ℕ0 & ⊢ 𝐺 ∈ ℕ0 & ⊢ 𝐽 ∈ ℕ0 & ⊢ 𝐾 ∈ ℕ0 & ⊢ (𝐴 · 𝐶) = 𝐹 & ⊢ (𝐴 · 𝐷) = 𝑀 & ⊢ (𝐵 · 𝐶) = 𝐿 & ⊢ (𝐵 · 𝐷) = ;𝐸𝐾 & ⊢ ((𝐿 + 𝑀) + 𝐸) = ;𝐺𝐽 & ⊢ (𝐹 + 𝐺) = 𝐼 ⇒ ⊢ ((𝐴.𝐵) · (𝐶.𝐷)) = (𝐼._𝐽𝐾) | ||
| 26-Dec-2021 | fusgr1th 26447 | The sum of the degrees of all vertices of a finite simple graph is twice the size of the graph. See equation (1) in section I.1 in [Bollobas] p. 4. Also known as the "First Theorem of Graph Theory" (see https://charlesreid1.com/wiki/First_Theorem_of_Graph_Theory). (Contributed by AV, 26-Dec-2021.) |
| ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ 𝐷 = (VtxDeg‘𝐺) ⇒ ⊢ (𝐺 ∈ FinUSGraph → Σ𝑣 ∈ 𝑉 (𝐷‘𝑣) = (2 · (#‘𝐼))) | ||
| 25-Dec-2021 | sbgoldbo 41675 | If the strong binary Goldbach conjecture is valid, the original formulation of the Goldbach conjecture also holds: Every integer greater than 2 can be expressed as the sum of three "primes" with regarding 1 to be a prime (as Goldbach did). Original text: "Es scheint wenigstens, dass eine jede Zahl, die groesser ist als 2, ein aggregatum trium numerorum primorum sey." (Goldbach, 1742). (Contributed by AV, 25-Dec-2021.) |
| ⊢ 𝑃 = ({1} ∪ ℙ) ⇒ ⊢ (∀𝑛 ∈ Even (4 < 𝑛 → 𝑛 ∈ GoldbachEven ) → ∀𝑛 ∈ (ℤ≥‘3)∃𝑝 ∈ 𝑃 ∃𝑞 ∈ 𝑃 ∃𝑟 ∈ 𝑃 𝑛 = ((𝑝 + 𝑞) + 𝑟)) | ||
| 25-Dec-2021 | even3prm2 41628 | If an even number is the sum of three prime numbers, one of the prime numbers must be 2. (Contributed by AV, 25-Dec-2021.) |
| ⊢ ((𝑁 ∈ Even ∧ (𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ ∧ 𝑅 ∈ ℙ) ∧ 𝑁 = ((𝑃 + 𝑄) + 𝑅)) → (𝑃 = 2 ∨ 𝑄 = 2 ∨ 𝑅 = 2)) | ||
| 25-Dec-2021 | evenltle 41626 | If an even number is greater than another even number, then it is greater than or equal to the other even number plus 2. (Contributed by AV, 25-Dec-2021.) |
| ⊢ ((𝑁 ∈ Even ∧ 𝑀 ∈ Even ∧ 𝑀 < 𝑁) → (𝑀 + 2) ≤ 𝑁) | ||
| 25-Dec-2021 | dpmul4 29622 | An upper bound to multiplication of decimal numbers with 4 digits. (Contributed by Thierry Arnoux, 25-Dec-2021.) |
| ⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 & ⊢ 𝐶 ∈ ℕ0 & ⊢ 𝐷 ∈ ℕ0 & ⊢ 𝐸 ∈ ℕ0 & ⊢ 𝐺 ∈ ℕ0 & ⊢ 𝐽 ∈ ℕ0 & ⊢ 𝐾 ∈ ℕ0 & ⊢ 𝐹 ∈ ℕ0 & ⊢ 𝐻 ∈ ℕ0 & ⊢ 𝐼 ∈ ℕ0 & ⊢ 𝐿 ∈ ℕ0 & ⊢ 𝑀 ∈ ℕ0 & ⊢ 𝑁 ∈ ℕ0 & ⊢ 𝑂 ∈ ℕ0 & ⊢ 𝑃 ∈ ℕ0 & ⊢ 𝑄 ∈ ℕ0 & ⊢ 𝑅 ∈ ℕ0 & ⊢ 𝑆 ∈ ℕ0 & ⊢ 𝑇 ∈ ℕ0 & ⊢ 𝑈 ∈ ℕ0 & ⊢ 𝑊 ∈ ℕ0 & ⊢ 𝑋 ∈ ℕ0 & ⊢ 𝑌 ∈ ℕ0 & ⊢ 𝑍 ∈ ℕ0 & ⊢ 𝑈 < ;10 & ⊢ 𝑃 < ;10 & ⊢ 𝑄 < ;10 & ⊢ (;;𝐿𝑀𝑁 + 𝑂) = ;;;𝑅𝑆𝑇𝑈 & ⊢ ((𝐴.𝐵) · (𝐸.𝐹)) = (𝐼._𝐽𝐾) & ⊢ ((𝐶.𝐷) · (𝐺.𝐻)) = (𝑂._𝑃𝑄) & ⊢ (;;;𝐼𝐽𝐾1 + ;;𝑅𝑆𝑇) = ;;;𝑊𝑋𝑌𝑍 & ⊢ (((𝐴.𝐵) + (𝐶.𝐷)) · ((𝐸.𝐹) + (𝐺.𝐻))) = (((𝐼._𝐽𝐾) + (𝐿._𝑀𝑁)) + (𝑂._𝑃𝑄)) ⇒ ⊢ ((𝐴._𝐵_𝐶𝐷) · (𝐸._𝐹_𝐺𝐻)) < (𝑊._𝑋_𝑌𝑍) | ||
| 25-Dec-2021 | dpmul1000 29607 | Multiply by 1000 a decimal expansion. (Contributed by Thierry Arnoux, 25-Dec-2021.) |
| ⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 & ⊢ 𝐶 ∈ ℕ0 & ⊢ 𝐷 ∈ ℝ ⇒ ⊢ ((𝐴._𝐵_𝐶𝐷) · ;;;1000) = ;;;𝐴𝐵𝐶𝐷 | ||
| 25-Dec-2021 | dp3mul10 29606 | Multiply by 10 a decimal expansion with 3 digits. (Contributed by Thierry Arnoux, 25-Dec-2021.) |
| ⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 & ⊢ 𝐶 ∈ ℝ ⇒ ⊢ ((𝐴._𝐵𝐶) · ;10) = (;𝐴𝐵.𝐶) | ||
| 25-Dec-2021 | dpmul100 29605 | Multiply by 100 a decimal expansion. (Contributed by Thierry Arnoux, 25-Dec-2021.) |
| ⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 & ⊢ 𝐶 ∈ ℝ ⇒ ⊢ ((𝐴._𝐵𝐶) · ;;100) = ;;𝐴𝐵𝐶 | ||
| 25-Dec-2021 | decdiv10 29604 | Divide a decimal number by 10. (Contributed by Thierry Arnoux, 25-Dec-2021.) |
| ⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ (;𝐴𝐵 / ;10) = (𝐴.𝐵) | ||
| 25-Dec-2021 | dpmul10 29603 | Multiply by 10 a decimal expansion. (Contributed by Thierry Arnoux, 25-Dec-2021.) |
| ⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ ((𝐴.𝐵) · ;10) = ;𝐴𝐵 | ||
| 25-Dec-2021 | dp2ltsuc 29593 | Comparing a decimal fraction with the next integer. (Contributed by Thierry Arnoux, 25-Dec-2021.) |
| ⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℝ+ & ⊢ 𝐵 < ;10 & ⊢ (𝐴 + 1) = 𝐶 ⇒ ⊢ _𝐴𝐵 < 𝐶 | ||
| 25-Dec-2021 | rpdp2cl2 29590 | Closure for a decimal fraction with no decimal expansion in the positive real numbers. (Contributed by Thierry Arnoux, 25-Dec-2021.) |
| ⊢ 𝐴 ∈ ℕ ⇒ ⊢ _𝐴0 ∈ ℝ+ | ||
| 25-Dec-2021 | dfdec100 29576 | Split the hundreds from a decimal value. (Contributed by Thierry Arnoux, 25-Dec-2021.) |
| ⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℕ0 & ⊢ 𝐶 ∈ ℝ ⇒ ⊢ ;;𝐴𝐵𝐶 = ((;;100 · 𝐴) + ;𝐵𝐶) | ||
| 24-Dec-2021 | sbgoldbm 41672 | If the strong binary Goldbach conjecture is valid, the modern version of the original formulation of the Goldbach conjecture also holds: Every integer greater than 5 can be expressed as the sum of three primes. (Contributed by AV, 24-Dec-2021.) |
| ⊢ (∀𝑛 ∈ Even (4 < 𝑛 → 𝑛 ∈ GoldbachEven ) → ∀𝑛 ∈ (ℤ≥‘6)∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟)) | ||
| 24-Dec-2021 | sgoldbeven3prm 41671 | If the binary Goldbach conjecture is valid, then an even integer greater than 5 can be expressed as the sum of three primes: Since (𝑁 − 2) is even iff 𝑁 is even, there would be primes 𝑝 and 𝑞 with (𝑁 − 2) = (𝑝 + 𝑞), and therefore 𝑁 = ((𝑝 + 𝑞) + 2). (Contributed by AV, 24-Dec-2021.) |
| ⊢ (∀𝑛 ∈ Even (4 < 𝑛 → 𝑛 ∈ GoldbachEven ) → ((𝑁 ∈ Even ∧ 6 ≤ 𝑁) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑁 = ((𝑝 + 𝑞) + 𝑟))) | ||
| 23-Dec-2021 | sbgoldbb 41670 | If the strong binary Goldbach conjecture is valid, the binary Goldbach conjecture is valid. (Contributed by AV, 23-Dec-2021.) |
| ⊢ (∀𝑛 ∈ Even (4 < 𝑛 → 𝑛 ∈ GoldbachEven ) → ∀𝑛 ∈ Even (2 < 𝑛 → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))) | ||
| 23-Dec-2021 | injresinjlem 12588 | Lemma for injresinj 12589. (Contributed by Alexander van der Vekens, 31-Oct-2017.) (Proof shortened by AV, 14-Feb-2021.) (Revised by Thierry Arnoux, 23-Dec-2021.) |
| ⊢ (¬ 𝑌 ∈ (1..^𝐾) → ((𝐹‘0) ≠ (𝐹‘𝐾) → ((𝐹:(0...𝐾)⟶𝑉 ∧ 𝐾 ∈ ℕ0) → (((𝐹 “ {0, 𝐾}) ∩ (𝐹 “ (1..^𝐾))) = ∅ → ((𝑋 ∈ (0...𝐾) ∧ 𝑌 ∈ (0...𝐾)) → ((𝐹‘𝑋) = (𝐹‘𝑌) → 𝑋 = 𝑌)))))) | ||
| 22-Dec-2021 | ax-tgoldbachgt 41699 | Temporary duplicate of tgoldbachgt 30741, provided as "axiom" as long as this theorem is in the mathbox of Thierry Arnoux: Odd integers greater than (;10↑;27) have at least a representation as a sum of three odd primes. Final statement in section 7.4 of [Helfgott] p. 70 , expressed using the set 𝐺 of odd numbers which can be written as a sum of three odd primes. (Contributed by Thierry Arnoux, 22-Dec-2021.) |
| ⊢ 𝑂 = {𝑧 ∈ ℤ ∣ ¬ 2 ∥ 𝑧} & ⊢ 𝐺 = {𝑧 ∈ 𝑂 ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ ((𝑝 ∈ 𝑂 ∧ 𝑞 ∈ 𝑂 ∧ 𝑟 ∈ 𝑂) ∧ 𝑧 = ((𝑝 + 𝑞) + 𝑟))} ⇒ ⊢ ∃𝑚 ∈ ℕ (𝑚 ≤ (;10↑;27) ∧ ∀𝑛 ∈ 𝑂 (𝑚 < 𝑛 → 𝑛 ∈ 𝐺)) | ||
| 22-Dec-2021 | df-elwise 33033 | Define the elementwise operation associated with a given operation. For instance, + is the addition of complex numbers (axaddf 9966), so if 𝐴 and 𝐵 are sets of complex numbers, then (𝐴(elwise‘ + )𝐵) is the set of numbers of the form (𝑥 + 𝑦) with 𝑥 ∈ 𝐴 and 𝑦 ∈ 𝐵. The set of odd natural numbers is (({2}(elwise‘ · )ℕ0)(elwise‘ + ){1}), or less formally 2ℕ0 + 1. (Contributed by BJ, 22-Dec-2021.) |
| ⊢ elwise = (𝑜 ∈ V ↦ (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑧 ∣ ∃𝑢 ∈ 𝑥 ∃𝑣 ∈ 𝑦 𝑧 = (𝑢𝑜𝑣)})) | ||
| 22-Dec-2021 | tgoldbachgt 30741 | Odd integers greater than (;10↑;27) have at least a representation as a sum of three odd primes. Final statement in section 7.4 of [Helfgott] p. 70 , expressed using the set 𝐺 of odd numbers which can be written as a sum of three odd primes. (Contributed by Thierry Arnoux, 22-Dec-2021.) |
| ⊢ 𝑂 = {𝑧 ∈ ℤ ∣ ¬ 2 ∥ 𝑧} & ⊢ 𝐺 = {𝑧 ∈ 𝑂 ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ ((𝑝 ∈ 𝑂 ∧ 𝑞 ∈ 𝑂 ∧ 𝑟 ∈ 𝑂) ∧ 𝑧 = ((𝑝 + 𝑞) + 𝑟))} ⇒ ⊢ ∃𝑚 ∈ ℕ (𝑚 ≤ (;10↑;27) ∧ ∀𝑛 ∈ 𝑂 (𝑚 < 𝑛 → 𝑛 ∈ 𝐺)) | ||
| 22-Dec-2021 | circlemethhgt 30721 | The circle method, where the Vinogradov sums are weighted using the Von Mangoldt function and smoothed using functions 𝐻 and 𝐾. Statement 7.49 of [Helfgott] p. 69. At this point there is no further constraint on the smoothing functions. (Contributed by Thierry Arnoux, 22-Dec-2021.) |
| ⊢ (𝜑 → 𝐻:ℕ⟶ℝ) & ⊢ (𝜑 → 𝐾:ℕ⟶ℝ) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) ⇒ ⊢ (𝜑 → Σ𝑛 ∈ (ℕ(repr‘3)𝑁)(((Λ‘(𝑛‘0)) · (𝐻‘(𝑛‘0))) · (((Λ‘(𝑛‘1)) · (𝐾‘(𝑛‘1))) · ((Λ‘(𝑛‘2)) · (𝐾‘(𝑛‘2))))) = ∫(0(,)1)(((((Λ ∘𝑓 · 𝐻)vts𝑁)‘𝑥) · ((((Λ ∘𝑓 · 𝐾)vts𝑁)‘𝑥)↑2)) · (exp‘((i · (2 · π)) · (-𝑁 · 𝑥)))) d𝑥) | ||
| 22-Dec-2021 | wrdfd 30616 | A word is a zero-based sequence with a recoverable upper limit, deduction version. (Contributed by Thierry Arnoux, 22-Dec-2021.) |
| ⊢ (𝜑 → 𝑁 = (#‘𝑊)) & ⊢ (𝜑 → 𝑊 ∈ Word 𝑆) ⇒ ⊢ (𝜑 → 𝑊:(0..^𝑁)⟶𝑆) | ||
| 22-Dec-2021 | vtxdgoddnumeven 26449 | The number of vertices of odd degree is even in a finite pseudograph of finite size. Proposition 1.2.1 in [Diestel] p. 5. See also remark about equation (2) in section I.1 in [Bollobas] p. 4. (Contributed by AV, 22-Dec-2021.) |
| ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ 𝐷 = (VtxDeg‘𝐺) ⇒ ⊢ ((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) → 2 ∥ (#‘{𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)})) | ||
| 22-Dec-2021 | finsumvtxdgeven 26448 | The sum of the degrees of all vertices of a finite pseudograph of finite size is even. See equation (2) in section I.1 in [Bollobas] p. 4, where it is also called the handshaking lemma. (Contributed by AV, 22-Dec-2021.) |
| ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ 𝐷 = (VtxDeg‘𝐺) ⇒ ⊢ ((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) → 2 ∥ Σ𝑣 ∈ 𝑉 (𝐷‘𝑣)) | ||
| 22-Dec-2021 | elfznelfzob 12574 | A value in a finite set of sequential integers is a border value if and only if it is not contained in the half-open integer range contained in the finite set of sequential integers. (Contributed by Alexander van der Vekens, 17-Jan-2018.) (Revised by Thierry Arnoux, 22-Dec-2021.) |
| ⊢ (𝑀 ∈ (0...𝐾) → (¬ 𝑀 ∈ (1..^𝐾) ↔ (𝑀 = 0 ∨ 𝑀 = 𝐾))) | ||
| 22-Dec-2021 | elfznelfzo 12573 | A value in a finite set of sequential integers is a border value if it is not contained in the half-open integer range contained in the finite set of sequential integers. (Contributed by Alexander van der Vekens, 31-Oct-2017.) (Revised by Thierry Arnoux, 22-Dec-2021.) |
| ⊢ ((𝑀 ∈ (0...𝐾) ∧ ¬ 𝑀 ∈ (1..^𝐾)) → (𝑀 = 0 ∨ 𝑀 = 𝐾)) | ||
| 20-Dec-2021 | logdivsqrle 30728 | Conditions for ((log x ) / ( sqrt 𝑥)) to be decreasing. (Contributed by Thierry Arnoux, 20-Dec-2021.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ+) & ⊢ (𝜑 → 𝐵 ∈ ℝ+) & ⊢ (𝜑 → (exp‘2) ≤ 𝐴) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) ⇒ ⊢ (𝜑 → ((log‘𝐵) / (√‘𝐵)) ≤ ((log‘𝐴) / (√‘𝐴))) | ||
| 20-Dec-2021 | fdvnegge 30680 | Functions with a non-positive derivative, i.e. decreasing functions, preserve ordering. (Contributed by Thierry Arnoux, 20-Dec-2021.) |
| ⊢ 𝐸 = (𝐶(,)𝐷) & ⊢ (𝜑 → 𝐴 ∈ 𝐸) & ⊢ (𝜑 → 𝐵 ∈ 𝐸) & ⊢ (𝜑 → 𝐹:𝐸⟶ℝ) & ⊢ (𝜑 → (ℝ D 𝐹) ∈ (𝐸–cn→ℝ)) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑥) ≤ 0) ⇒ ⊢ (𝜑 → (𝐹‘𝐵) ≤ (𝐹‘𝐴)) | ||
| 20-Dec-2021 | fdvposle 30679 | Functions with a nonnegative derivative, i.e. monotonously growing functions, preserve ordering. (Contributed by Thierry Arnoux, 20-Dec-2021.) |
| ⊢ 𝐸 = (𝐶(,)𝐷) & ⊢ (𝜑 → 𝐴 ∈ 𝐸) & ⊢ (𝜑 → 𝐵 ∈ 𝐸) & ⊢ (𝜑 → 𝐹:𝐸⟶ℝ) & ⊢ (𝜑 → (ℝ D 𝐹) ∈ (𝐸–cn→ℝ)) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 0 ≤ ((ℝ D 𝐹)‘𝑥)) ⇒ ⊢ (𝜑 → (𝐹‘𝐴) ≤ (𝐹‘𝐵)) | ||
| 20-Dec-2021 | fdvneggt 30678 | Functions with a negative derivative, i.e. monotonously decreasing functions, inverse strict ordering. (Contributed by Thierry Arnoux, 20-Dec-2021.) |
| ⊢ 𝐸 = (𝐶(,)𝐷) & ⊢ (𝜑 → 𝐴 ∈ 𝐸) & ⊢ (𝜑 → 𝐵 ∈ 𝐸) & ⊢ (𝜑 → 𝐹:𝐸⟶ℝ) & ⊢ (𝜑 → (ℝ D 𝐹) ∈ (𝐸–cn→ℝ)) & ⊢ (𝜑 → 𝐴 < 𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑥) < 0) ⇒ ⊢ (𝜑 → (𝐹‘𝐵) < (𝐹‘𝐴)) | ||
| 20-Dec-2021 | fdvposlt 30677 | Functions with a positive derivative, i.e. monotonously growing functions, preserve strict ordering. (Contributed by Thierry Arnoux, 20-Dec-2021.) |
| ⊢ 𝐸 = (𝐶(,)𝐷) & ⊢ (𝜑 → 𝐴 ∈ 𝐸) & ⊢ (𝜑 → 𝐵 ∈ 𝐸) & ⊢ (𝜑 → 𝐹:𝐸⟶ℝ) & ⊢ (𝜑 → (ℝ D 𝐹) ∈ (𝐸–cn→ℝ)) & ⊢ (𝜑 → 𝐴 < 𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 0 < ((ℝ D 𝐹)‘𝑥)) ⇒ ⊢ (𝜑 → (𝐹‘𝐴) < (𝐹‘𝐵)) | ||
| 20-Dec-2021 | fct2relem 30675 | Lemma for ftc2re 30676. (Contributed by Thierry Arnoux, 20-Dec-2021.) |
| ⊢ 𝐸 = (𝐶(,)𝐷) & ⊢ (𝜑 → 𝐴 ∈ 𝐸) & ⊢ (𝜑 → 𝐵 ∈ 𝐸) ⇒ ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ 𝐸) | ||
| 20-Dec-2021 | cxpcncf1 30673 | The power function on complex numbers, for fixed exponent A, is continuous. Similar to cxpcn 24486. (Contributed by Thierry Arnoux, 20-Dec-2021.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐷 ⊆ (ℂ ∖ (-∞(,]0))) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ (𝑥↑𝑐𝐴)) ∈ (𝐷–cn→ℂ)) | ||
| 20-Dec-2021 | rpsqrtcn 30671 | Continuity of the real positive square root function. (Contributed by Thierry Arnoux, 20-Dec-2021.) |
| ⊢ (√ ↾ ℝ+) ∈ (ℝ+–cn→ℝ+) | ||
| 19-Dec-2021 | finsumvtxdg2size 26446 |
The sum of the degrees of all vertices of a finite pseudograph of finite
size is twice the size of the pseudograph. See equation (1) in section
I.1 in [Bollobas] p. 4. Here, the
"proof" is simply the statement
"Since each edge has two endvertices, the sum of the degrees is
exactly
twice the number of edges". The formal proof of this theorem (for
pseudographs) is much more complicated, taking also the used auxiliary
theorems into account. The proof for a (finite) simple graph (see
fusgr1th 26447) would be shorter, but nevertheless still
laborious.
Although this theorem would hold also for infinite pseudographs and
pseudographs of infinite size, the proof of this most general version
(see theorem "sumvtxdg2size" below) would require many more
auxiliary
theorems (e.g., the extension of the sum Σ
over an arbitrary
set).
I dedicate this theorem and its proof to Norman Megill, who deceased too early on December 9, 2021. This proof is an example for the rigor which was the main motivation for Norman Megill to invent and develop Metamath, see section 1.1.6 "Rigor" on page 19 of the Metamath book: "... it is usually assumed in mathematical literature that the person reading the proof is a mathematician familiar with the specialty being described, and that the missing steps are obvious to such a reader or at least the reader is capable of filling them in." I filled in the missing steps of Bollobas' proof as Norm would have liked it... (Contributed by Alexander van der Vekens, 19-Dec-2021.) |
| ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ 𝐷 = (VtxDeg‘𝐺) ⇒ ⊢ ((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) → Σ𝑣 ∈ 𝑉 (𝐷‘𝑣) = (2 · (#‘𝐼))) | ||
| 19-Dec-2021 | finsumvtxdg2sstep 26445 | Induction step of finsumvtxdg2size 26446: In a finite pseudograph of finite size, the sum of the degrees of all vertices of the pseudograph is twice the size of the pseudograph if the sum of the degrees of all vertices of the subgraph of the pseudograph not containing one of the vertices is twice the size of the subgraph. (Contributed by AV, 19-Dec-2021.) |
| ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (iEdg‘𝐺) & ⊢ 𝐾 = (𝑉 ∖ {𝑁}) & ⊢ 𝐼 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑖)} & ⊢ 𝑃 = (𝐸 ↾ 𝐼) & ⊢ 𝑆 = ⟨𝐾, 𝑃⟩ ⇒ ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → ((𝑃 ∈ Fin → Σ𝑣 ∈ 𝐾 ((VtxDeg‘𝑆)‘𝑣) = (2 · (#‘𝑃))) → Σ𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = (2 · (#‘𝐸)))) | ||
| 19-Dec-2021 | finsumvtxdg2ssteplem3 26443 | Lemma for finsumvtxdg2sstep 26445. (Contributed by AV, 19-Dec-2021.) |
| ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (iEdg‘𝐺) & ⊢ 𝐾 = (𝑉 ∖ {𝑁}) & ⊢ 𝐼 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑖)} & ⊢ 𝑃 = (𝐸 ↾ 𝐼) & ⊢ 𝑆 = ⟨𝐾, 𝑃⟩ & ⊢ 𝐽 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑖)} ⇒ ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → (Σ𝑣 ∈ (𝑉 ∖ {𝑁})(#‘{𝑖 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑖)}) + (#‘{𝑖 ∈ dom 𝐸 ∣ (𝐸‘𝑖) = {𝑁}})) = (#‘𝐽)) | ||
| 19-Dec-2021 | usgrres 26200 | A subgraph obtained by removing one vertex and all edges incident with this vertex from a simple graph (see uhgrspan1 26195) is a simple graph. (Contributed by Alexander van der Vekens, 2-Jan-2018.) (Revised by AV, 19-Dec-2021.) |
| ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (iEdg‘𝐺) & ⊢ 𝐹 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑖)} & ⊢ 𝑆 = ⟨(𝑉 ∖ {𝑁}), (𝐸 ↾ 𝐹)⟩ ⇒ ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → 𝑆 ∈ USGraph ) | ||
| 19-Dec-2021 | umgrres 26199 | A subgraph obtained by removing one vertex and all edges incident with this vertex from a multigraph (see uhgrspan1 26195) is a multigraph. (Contributed by AV, 27-Nov-2020.) (Revised by AV, 19-Dec-2021.) |
| ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (iEdg‘𝐺) & ⊢ 𝐹 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑖)} & ⊢ 𝑆 = ⟨(𝑉 ∖ {𝑁}), (𝐸 ↾ 𝐹)⟩ ⇒ ⊢ ((𝐺 ∈ UMGraph ∧ 𝑁 ∈ 𝑉) → 𝑆 ∈ UMGraph ) | ||
| 19-Dec-2021 | upgrres 26198 | A subgraph obtained by removing one vertex and all edges incident with this vertex from a pseudograph (see uhgrspan1 26195) is a pseudograph. (Contributed by AV, 8-Nov-2020.) (Revised by AV, 19-Dec-2021.) |
| ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (iEdg‘𝐺) & ⊢ 𝐹 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑖)} & ⊢ 𝑆 = ⟨(𝑉 ∖ {𝑁}), (𝐸 ↾ 𝐹)⟩ ⇒ ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → 𝑆 ∈ UPGraph ) | ||
| 19-Dec-2021 | umgrreslem 26197 | Lemma for umgrres 26199 and usgrres 26200. (Contributed by AV, 27-Nov-2020.) (Revised by AV, 19-Dec-2021.) |
| ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (iEdg‘𝐺) & ⊢ 𝐹 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑖)} ⇒ ⊢ ((𝐺 ∈ UMGraph ∧ 𝑁 ∈ 𝑉) → ran (𝐸 ↾ 𝐹) ⊆ {𝑝 ∈ 𝒫 (𝑉 ∖ {𝑁}) ∣ (#‘𝑝) = 2}) | ||
| 19-Dec-2021 | upgrreslem 26196 | Lemma for upgrres 26198. (Contributed by AV, 27-Nov-2020.) (Revised by AV, 19-Dec-2021.) |
| ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (iEdg‘𝐺) & ⊢ 𝐹 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑖)} ⇒ ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → ran (𝐸 ↾ 𝐹) ⊆ {𝑝 ∈ (𝒫 (𝑉 ∖ {𝑁}) ∖ {∅}) ∣ (#‘𝑝) ≤ 2}) | ||
| 19-Dec-2021 | numedglnl 26039 | The number of edges incident with a vertex 𝑁 is the number of edges joining 𝑁 with other vertices and the number of loops on 𝑁 in a pseudograph of finite size. (Contributed by AV, 19-Dec-2021.) |
| ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (iEdg‘𝐺) ⇒ ⊢ ((𝐺 ∈ UPGraph ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin) ∧ 𝑁 ∈ 𝑉) → (Σ𝑣 ∈ (𝑉 ∖ {𝑁})(#‘{𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖))}) + (#‘{𝑖 ∈ dom 𝐸 ∣ (𝐸‘𝑖) = {𝑁}})) = (#‘{𝑖 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑖)})) | ||
| 19-Dec-2021 | iedgedg 25943 | An indexed edge is an edge. (Contributed by AV, 19-Dec-2021.) |
| ⊢ 𝐸 = (iEdg‘𝐺) ⇒ ⊢ ((Fun 𝐸 ∧ 𝐼 ∈ dom 𝐸) → (𝐸‘𝐼) ∈ (Edg‘𝐺)) | ||
| 19-Dec-2021 | 3elpr2eq 4435 | If there are three elements in a proper unordered pair, and two of them are different from the third one, the two must be equal. (Contributed by AV, 19-Dec-2021.) |
| ⊢ (((𝑋 ∈ {𝐴, 𝐵} ∧ 𝑌 ∈ {𝐴, 𝐵} ∧ 𝑍 ∈ {𝐴, 𝐵}) ∧ (𝑌 ≠ 𝑋 ∧ 𝑍 ≠ 𝑋)) → 𝑌 = 𝑍) | ||
| 18-Dec-2021 | ideq2 34078 | For sets, the identity binary relation is the same as equality. (Contributed by Peter Mazsa, 24-Jun-2020.) (Revised by Peter Mazsa, 18-Dec-2021.) |
| ⊢ (𝐴 ∈ 𝑉 → (𝐴 I 𝐵 ↔ 𝐴 = 𝐵)) | ||
| 18-Dec-2021 | brid 34077 | Property of the identity binary relation. (Contributed by Peter Mazsa, 18-Dec-2021.) |
| ⊢ (𝐴 I 𝐵 ↔ 𝐵 I 𝐴) | ||
| 18-Dec-2021 | vtxdginducedm1fi 26440 | The degree of a vertex 𝑣 in the induced subgraph 𝑆 of a pseudograph 𝐺 of finite size obtained by removing one vertex 𝑁 plus the number of edges joining the vertex 𝑣 and the vertex 𝑁 is the degree of the vertex 𝑣 in the pseudograph 𝐺. (Contributed by AV, 18-Dec-2021.) |
| ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (iEdg‘𝐺) & ⊢ 𝐾 = (𝑉 ∖ {𝑁}) & ⊢ 𝐼 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑖)} & ⊢ 𝑃 = (𝐸 ↾ 𝐼) & ⊢ 𝑆 = ⟨𝐾, 𝑃⟩ & ⊢ 𝐽 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑖)} ⇒ ⊢ (𝐸 ∈ Fin → ∀𝑣 ∈ (𝑉 ∖ {𝑁})((VtxDeg‘𝐺)‘𝑣) = (((VtxDeg‘𝑆)‘𝑣) + (#‘{𝑙 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑙)}))) | ||
| 18-Dec-2021 | edglnl 26038 | The edges incident with a vertex 𝑁 are the edges joining 𝑁 with other vertices and the loops on 𝑁 in a pseudograph. (Contributed by AV, 18-Dec-2021.) |
| ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (iEdg‘𝐺) ⇒ ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → (∪ 𝑣 ∈ (𝑉 ∖ {𝑁}){𝑖 ∈ dom 𝐸 ∣ (𝑁 ∈ (𝐸‘𝑖) ∧ 𝑣 ∈ (𝐸‘𝑖))} ∪ {𝑖 ∈ dom 𝐸 ∣ (𝐸‘𝑖) = {𝑁}}) = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑖)}) | ||
| 18-Dec-2021 | prproe 4434 | For an element of a proper unordered pair of elements of a class 𝑉, there is another (different) element of the class 𝑉 which is an element of the proper pair. (Contributed by AV, 18-Dec-2021.) |
| ⊢ ((𝐶 ∈ {𝐴, 𝐵} ∧ 𝐴 ≠ 𝐵 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → ∃𝑣 ∈ (𝑉 ∖ {𝐶})𝑣 ∈ {𝐴, 𝐵}) | ||
| 17-Dec-2021 | madeval2 31936 | Alternative characterization of the made by function. (Contributed by Scott Fenton, 17-Dec-2021.) |
| ⊢ (𝐴 ∈ On → ( M ‘𝐴) = {𝑥 ∈ No ∣ ∃𝑎 ∈ 𝒫 ∪ ( M “ 𝐴)∃𝑏 ∈ 𝒫 ∪ ( M “ 𝐴)(𝑎 <<s 𝑏 ∧ (𝑎 |s 𝑏) = 𝑥)}) | ||
| 17-Dec-2021 | madeval 31935 | The value of the made by function. (Contributed by Scott Fenton, 17-Dec-2021.) |
| ⊢ (𝐴 ∈ On → ( M ‘𝐴) = ( |s “ (𝒫 ∪ ( M “ 𝐴) × 𝒫 ∪ ( M “ 𝐴)))) | ||
| 17-Dec-2021 | df-right 31934 | Define the left options of a surreal. This is the set of surreals that are "closest" on the right to the given surreal. (Contributed by Scott Fenton, 17-Dec-2021.) |
| ⊢ R = (𝑥 ∈ No ↦ {𝑦 ∈ ( O ‘( bday ‘𝑥)) ∣ ∀𝑧 ∈ No ((𝑥 <s 𝑧 ∧ 𝑧 <s 𝑦) → ( bday ‘𝑦) ∈ ( bday ‘𝑧))}) | ||
| 17-Dec-2021 | df-left 31933 | Define the left options of a surreal. This is the set of surreals that are "closest" on the left to the given surreal. (Contributed by Scott Fenton, 17-Dec-2021.) |
| ⊢ L = (𝑥 ∈ No ↦ {𝑦 ∈ ( O ‘( bday ‘𝑥)) ∣ ∀𝑧 ∈ No ((𝑦 <s 𝑧 ∧ 𝑧 <s 𝑥) → ( bday ‘𝑦) ∈ ( bday ‘𝑧))}) | ||
| 17-Dec-2021 | df-new 31932 | Define the newer than function. This function carries an ordinal to all surreals made on that day. (Contributed by Scott Fenton, 17-Dec-2021.) |
| ⊢ N = (𝑥 ∈ On ↦ (( O ‘𝑥) ∖ ( M ‘𝑥))) | ||
| 17-Dec-2021 | df-old 31931 | Define the older than function. This function carries an ordinal to all surreals made by a previous ordinal. (Contributed by Scott Fenton, 17-Dec-2021.) |
| ⊢ O = (𝑥 ∈ On ↦ ∪ ( M “ 𝑥)) | ||
| 17-Dec-2021 | df-made 31930 | Define the made by function. This function carries an ordinal to all surreals made by sections of surreals older than it. (Contributed by Scott Fenton, 17-Dec-2021.) |
| ⊢ M = recs((𝑓 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑓 × 𝒫 ∪ ran 𝑓)))) | ||
| 17-Dec-2021 | imaindm 31682 | The image is unaffected by intersection with the domain. (Contributed by Scott Fenton, 17-Dec-2021.) |
| ⊢ (𝑅 “ 𝐴) = (𝑅 “ (𝐴 ∩ dom 𝑅)) | ||
| 17-Dec-2021 | hgt750lem 30729 | Lemma for tgoldbachgtd 30740. (Contributed by Thierry Arnoux, 17-Dec-2021.) |
| ⊢ ((𝑁 ∈ ℕ0 ∧ (;10↑;27) ≤ 𝑁) → ((7._3_48) · ((log‘𝑁) / (√‘𝑁))) < (0._0_0_0_4_2_2_48)) | ||
| 17-Dec-2021 | vtxdginducedm1 26439 | The degree of a vertex 𝑣 in the induced subgraph 𝑆 of a pseudograph 𝐺 obtained by removing one vertex 𝑁 plus the number of edges joining the vertex 𝑣 and the vertex 𝑁 is the degree of the vertex 𝑣 in the pseudograph 𝐺. (Contributed by AV, 17-Dec-2021.) |
| ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (iEdg‘𝐺) & ⊢ 𝐾 = (𝑉 ∖ {𝑁}) & ⊢ 𝐼 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑖)} & ⊢ 𝑃 = (𝐸 ↾ 𝐼) & ⊢ 𝑆 = ⟨𝐾, 𝑃⟩ & ⊢ 𝐽 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑖)} ⇒ ⊢ ∀𝑣 ∈ (𝑉 ∖ {𝑁})((VtxDeg‘𝐺)‘𝑣) = (((VtxDeg‘𝑆)‘𝑣) +𝑒 (#‘{𝑙 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑙)})) | ||
| 17-Dec-2021 | vtxdginducedm1lem4 26438 | Lemma 4 for vtxdginducedm1 26439. (Contributed by AV, 17-Dec-2021.) |
| ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (iEdg‘𝐺) & ⊢ 𝐾 = (𝑉 ∖ {𝑁}) & ⊢ 𝐼 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑖)} & ⊢ 𝑃 = (𝐸 ↾ 𝐼) & ⊢ 𝑆 = ⟨𝐾, 𝑃⟩ & ⊢ 𝐽 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑖)} ⇒ ⊢ (𝑊 ∈ (𝑉 ∖ {𝑁}) → (#‘{𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑊}}) = 0) | ||
| 17-Dec-2021 | fsumsplitsnun 14484 | Separate out a term in a finite sum by splitting the sum into two parts. (Contributed by Alexander van der Vekens, 1-Sep-2018.) (Revised by AV, 17-Dec-2021.) |
| ⊢ ((𝐴 ∈ Fin ∧ (𝑍 ∈ 𝑉 ∧ 𝑍 ∉ 𝐴) ∧ ∀𝑘 ∈ (𝐴 ∪ {𝑍})𝐵 ∈ ℤ) → Σ𝑘 ∈ (𝐴 ∪ {𝑍})𝐵 = (Σ𝑘 ∈ 𝐴 𝐵 + ⦋𝑍 / 𝑘⦌𝐵)) | ||
| 17-Dec-2021 | fsummsnunz 14483 | A finite sum all of whose summands are integers is itself an integer (case where the summation set is the union of a finite set and a singleton). (Contributed by Alexander van der Vekens, 1-Sep-2018.) (Revised by AV, 17-Dec-2021.) |
| ⊢ ((𝐴 ∈ Fin ∧ ∀𝑘 ∈ (𝐴 ∪ {𝑍})𝐵 ∈ ℤ) → Σ𝑘 ∈ (𝐴 ∪ {𝑍})𝐵 ∈ ℤ) | ||
| 17-Dec-2021 | elnelun 3964 | The union of the set of elements 𝑠 determining classes 𝐶 (which may depend on 𝑠) containing a special element and the set of elements 𝑠 determining classes 𝐶 not containing the special element yields the original set. (Contributed by Alexander van der Vekens, 11-Jan-2018.) (Revised by AV, 9-Nov-2020.) (Revised by AV, 17-Dec-2021.) |
| ⊢ 𝐸 = {𝑠 ∈ 𝐴 ∣ 𝐵 ∈ 𝐶} & ⊢ 𝑁 = {𝑠 ∈ 𝐴 ∣ 𝐵 ∉ 𝐶} ⇒ ⊢ (𝐸 ∪ 𝑁) = 𝐴 | ||
| 17-Dec-2021 | elneldisj 3963 | The set of elements 𝑠 determining classes 𝐶 (which may depend on 𝑠) containing a special element and the set of elements 𝑠 determining classes 𝐶 not containing the special element are disjoint. (Contributed by Alexander van der Vekens, 11-Jan-2018.) (Revised by AV, 9-Nov-2020.) (Revised by AV, 17-Dec-2021.) |
| ⊢ 𝐸 = {𝑠 ∈ 𝐴 ∣ 𝐵 ∈ 𝐶} & ⊢ 𝑁 = {𝑠 ∈ 𝐴 ∣ 𝐵 ∉ 𝐶} ⇒ ⊢ (𝐸 ∩ 𝑁) = ∅ | ||
| 16-Dec-2021 | brres2 34035 | Binary relation on a restriction. (Contributed by Peter Mazsa, 2-Jan-2019.) (Revised by Peter Mazsa, 16-Dec-2021.) |
| ⊢ (𝐵(𝑅 ↾ 𝐴)𝐶 ↔ 𝐵(𝑅 ∩ (𝐴 × ran (𝑅 ↾ 𝐴)))𝐶) | ||
| 16-Dec-2021 | 0dp2dp 29617 | Multiply by 10 a decimal expansion which starts with a zero. (Contributed by Thierry Arnoux, 16-Dec-2021.) |
| ⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℝ+ ⇒ ⊢ ((0._𝐴𝐵) · ;10) = (𝐴.𝐵) | ||
| 16-Dec-2021 | dpexpp1 29616 | Add one zero to the mantisse, and a one to the exponent in a scientific notation. (Contributed by Thierry Arnoux, 16-Dec-2021.) |
| ⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℝ+ & ⊢ (𝑃 + 1) = 𝑄 & ⊢ 𝑃 ∈ ℤ & ⊢ 𝑄 ∈ ℤ ⇒ ⊢ ((𝐴.𝐵) · (;10↑𝑃)) = ((0._𝐴𝐵) · (;10↑𝑄)) | ||
| 16-Dec-2021 | dpltc 29615 | Comparing two decimal integers (unequal higher places). (Contributed by Thierry Arnoux, 16-Dec-2021.) |
| ⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℝ+ & ⊢ 𝐶 ∈ ℕ0 & ⊢ 𝐷 ∈ ℝ+ & ⊢ 𝐴 < 𝐶 & ⊢ 𝐵 < ;10 ⇒ ⊢ (𝐴.𝐵) < (𝐶.𝐷) | ||
| 16-Dec-2021 | dpgti 29614 | Comparing a decimal expansions with the next lower integer. (Contributed by Thierry Arnoux, 16-Dec-2021.) |
| ⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℝ+ ⇒ ⊢ 𝐴 < (𝐴.𝐵) | ||
| 16-Dec-2021 | dplti 29613 | Comparing a decimal expansions with the next higher integer. (Contributed by Thierry Arnoux, 16-Dec-2021.) |
| ⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℝ+ & ⊢ 𝐶 ∈ ℕ0 & ⊢ 𝐵 < ;10 & ⊢ (𝐴 + 1) = 𝐶 ⇒ ⊢ (𝐴.𝐵) < 𝐶 | ||
| 16-Dec-2021 | dplt 29612 | Comparing two decimal expansions (equal higher places). (Contributed by Thierry Arnoux, 16-Dec-2021.) |
| ⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℝ+ & ⊢ 𝐶 ∈ ℝ+ & ⊢ 𝐵 < 𝐶 ⇒ ⊢ (𝐴.𝐵) < (𝐴.𝐶) | ||
| 16-Dec-2021 | rpdpcl 29611 | Closure of the decimal point in the positive real numbers. (Contributed by Thierry Arnoux, 16-Dec-2021.) |
| ⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℝ+ ⇒ ⊢ (𝐴.𝐵) ∈ ℝ+ | ||
| 16-Dec-2021 | dp0h 29610 | Remove a zero in the units places. (Contributed by Thierry Arnoux, 16-Dec-2021.) |
| ⊢ 𝐴 ∈ ℝ+ ⇒ ⊢ (0.𝐴) = (𝐴 / ;10) | ||
| 16-Dec-2021 | dp0u 29609 | Add a zero in the tenths place. (Contributed by Thierry Arnoux, 16-Dec-2021.) |
| ⊢ 𝐴 ∈ ℕ0 ⇒ ⊢ (𝐴.0) = 𝐴 | ||
| 16-Dec-2021 | dpval3rp 29608 | Value of the decimal point construct. (Contributed by Thierry Arnoux, 16-Dec-2021.) |
| ⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℝ+ ⇒ ⊢ (𝐴.𝐵) = _𝐴𝐵 | ||
| 16-Dec-2021 | dpval3 29602 | Value of the decimal point construct. (Contributed by Thierry Arnoux, 16-Dec-2021.) |
| ⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ (𝐴.𝐵) = _𝐴𝐵 | ||
| 16-Dec-2021 | dpval2 29601 | Value of the decimal point construct. (Contributed by Thierry Arnoux, 16-Dec-2021.) |
| ⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℝ ⇒ ⊢ (𝐴.𝐵) = (𝐴 + (𝐵 / ;10)) | ||
| 16-Dec-2021 | dp2ltc 29594 | Comparing two decimal expansions (unequal higher places). (Contributed by Thierry Arnoux, 16-Dec-2021.) |
| ⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℝ+ & ⊢ 𝐶 ∈ ℕ0 & ⊢ 𝐷 ∈ ℝ+ & ⊢ 𝐵 < ;10 & ⊢ 𝐴 < 𝐶 ⇒ ⊢ _𝐴𝐵 < _𝐶𝐷 | ||
| 16-Dec-2021 | dp2lt 29592 | Comparing two decimal fractions (equal unit places). (Contributed by Thierry Arnoux, 16-Dec-2021.) |
| ⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℝ+ & ⊢ 𝐶 ∈ ℝ+ & ⊢ 𝐵 < 𝐶 ⇒ ⊢ _𝐴𝐵 < _𝐴𝐶 | ||
| 16-Dec-2021 | dp2lt10 29591 | Decimal fraction builds real numbers less than 10. (Contributed by Thierry Arnoux, 16-Dec-2021.) |
| ⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℝ+ & ⊢ 𝐴 < ;10 & ⊢ 𝐵 < ;10 ⇒ ⊢ _𝐴𝐵 < ;10 | ||
| 16-Dec-2021 | rpdp2cl 29589 | Closure for a decimal fraction in the positive real numbers. (Contributed by Thierry Arnoux, 16-Dec-2021.) |
| ⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℝ+ ⇒ ⊢ _𝐴𝐵 ∈ ℝ+ | ||
| 16-Dec-2021 | dp2clq 29588 | Closure for a decimal fraction. (Contributed by Thierry Arnoux, 16-Dec-2021.) |
| ⊢ 𝐴 ∈ ℕ0 & ⊢ 𝐵 ∈ ℚ ⇒ ⊢ _𝐴𝐵 ∈ ℚ | ||
| 16-Dec-2021 | dp20h 29586 | Add a zero in the unit places. (Contributed by Thierry Arnoux, 16-Dec-2021.) |
| ⊢ 𝐴 ∈ ℝ+ ⇒ ⊢ _0𝐴 = (𝐴 / ;10) | ||
| 16-Dec-2021 | dp20u 29585 | Add a zero in the tenths (lower) place. (Contributed by Thierry Arnoux, 16-Dec-2021.) |
| ⊢ 𝐴 ∈ ℕ0 ⇒ ⊢ _𝐴0 = 𝐴 | ||
| 16-Dec-2021 | vtxdginducedm1lem3 26437 | Lemma 3 for vtxdginducedm1 26439: an edge in the induced subgraph 𝑆 of a pseudograph 𝐺 obtained by removing one vertex 𝑁. (Contributed by AV, 16-Dec-2021.) |
| ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (iEdg‘𝐺) & ⊢ 𝐾 = (𝑉 ∖ {𝑁}) & ⊢ 𝐼 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑖)} & ⊢ 𝑃 = (𝐸 ↾ 𝐼) & ⊢ 𝑆 = ⟨𝐾, 𝑃⟩ ⇒ ⊢ (𝐻 ∈ 𝐼 → ((iEdg‘𝑆)‘𝐻) = (𝐸‘𝐻)) | ||
| 16-Dec-2021 | vtxdginducedm1lem2 26436 | Lemma 2 for vtxdginducedm1 26439: the domain of the edge function in the induced subgraph 𝑆 of a pseudograph 𝐺 obtained by removing one vertex 𝑁. (Contributed by AV, 16-Dec-2021.) |
| ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (iEdg‘𝐺) & ⊢ 𝐾 = (𝑉 ∖ {𝑁}) & ⊢ 𝐼 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑖)} & ⊢ 𝑃 = (𝐸 ↾ 𝐼) & ⊢ 𝑆 = ⟨𝐾, 𝑃⟩ ⇒ ⊢ dom (iEdg‘𝑆) = 𝐼 | ||
| 16-Dec-2021 | vtxdginducedm1lem1 26435 | Lemma 1 for vtxdginducedm1 26439: the edge function in the induced subgraph 𝑆 of a pseudograph 𝐺 obtained by removing one vertex 𝑁. (Contributed by AV, 16-Dec-2021.) |
| ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (iEdg‘𝐺) & ⊢ 𝐾 = (𝑉 ∖ {𝑁}) & ⊢ 𝐼 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑖)} & ⊢ 𝑃 = (𝐸 ↾ 𝐼) & ⊢ 𝑆 = ⟨𝐾, 𝑃⟩ ⇒ ⊢ (iEdg‘𝑆) = 𝑃 | ||
| 15-Dec-2021 | scutf 31919 | Functionhood statement for the surreal cut operator. (Contributed by Scott Fenton, 15-Dec-2021.) |
| ⊢ |s : <<s ⟶ No | ||
| 15-Dec-2021 | tgoldbachgtd 30740 | Odd integers greater than (;10↑;27) have at least a representation as a sum of three odd primes. Final statement in section 7.4 of [Helfgott] p. 70 (Contributed by Thierry Arnoux, 15-Dec-2021.) |
| ⊢ 𝑂 = {𝑧 ∈ ℤ ∣ ¬ 2 ∥ 𝑧} & ⊢ (𝜑 → 𝑁 ∈ 𝑂) & ⊢ (𝜑 → (;10↑;27) ≤ 𝑁) ⇒ ⊢ (𝜑 → 0 < (#‘((𝑂 ∩ ℙ)(repr‘3)𝑁))) | ||
| 15-Dec-2021 | tgoldbachgtda 30739 | Lemma for tgoldbachgtd 30740. (Contributed by Thierry Arnoux, 15-Dec-2021.) |
| ⊢ 𝑂 = {𝑧 ∈ ℤ ∣ ¬ 2 ∥ 𝑧} & ⊢ (𝜑 → 𝑁 ∈ 𝑂) & ⊢ (𝜑 → (;10↑;27) ≤ 𝑁) & ⊢ (𝜑 → 𝐻:ℕ⟶(0[,)+∞)) & ⊢ (𝜑 → 𝐾:ℕ⟶(0[,)+∞)) & ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐾‘𝑚) ≤ (1._0_7_9_9_55)) & ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐻‘𝑚) ≤ (1._4_14)) & ⊢ (𝜑 → ((0._0_0_0_4_2_2_48) · (𝑁↑2)) ≤ ∫(0(,)1)(((((Λ ∘𝑓 · 𝐻)vts𝑁)‘𝑥) · ((((Λ ∘𝑓 · 𝐾)vts𝑁)‘𝑥)↑2)) · (exp‘((i · (2 · π)) · (-𝑁 · 𝑥)))) d𝑥) ⇒ ⊢ (𝜑 → 0 < (#‘((𝑂 ∩ ℙ)(repr‘3)𝑁))) | ||
| 15-Dec-2021 | tgoldbachgtde 30738 | Lemma for tgoldbachgtd 30740. (Contributed by Thierry Arnoux, 15-Dec-2021.) |
| ⊢ 𝑂 = {𝑧 ∈ ℤ ∣ ¬ 2 ∥ 𝑧} & ⊢ (𝜑 → 𝑁 ∈ 𝑂) & ⊢ (𝜑 → (;10↑;27) ≤ 𝑁) & ⊢ (𝜑 → 𝐻:ℕ⟶(0[,)+∞)) & ⊢ (𝜑 → 𝐾:ℕ⟶(0[,)+∞)) & ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐾‘𝑚) ≤ (1._0_7_9_9_55)) & ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐻‘𝑚) ≤ (1._4_14)) & ⊢ (𝜑 → ((0._0_0_0_4_2_2_48) · (𝑁↑2)) ≤ ∫(0(,)1)(((((Λ ∘𝑓 · 𝐻)vts𝑁)‘𝑥) · ((((Λ ∘𝑓 · 𝐾)vts𝑁)‘𝑥)↑2)) · (exp‘((i · (2 · π)) · (-𝑁 · 𝑥)))) d𝑥) ⇒ ⊢ (𝜑 → 0 < Σ𝑛 ∈ ((𝑂 ∩ ℙ)(repr‘3)𝑁)(((Λ‘(𝑛‘0)) · (𝐻‘(𝑛‘0))) · (((Λ‘(𝑛‘1)) · (𝐾‘(𝑛‘1))) · ((Λ‘(𝑛‘2)) · (𝐾‘(𝑛‘2)))))) | ||
| 15-Dec-2021 | tgoldbachgnn 30737 | Lemma for tgoldbachgtd 30740. (Contributed by Thierry Arnoux, 15-Dec-2021.) |
| ⊢ 𝑂 = {𝑧 ∈ ℤ ∣ ¬ 2 ∥ 𝑧} & ⊢ (𝜑 → 𝑁 ∈ 𝑂) & ⊢ (𝜑 → (;10↑;27) ≤ 𝑁) ⇒ ⊢ (𝜑 → 𝑁 ∈ ℕ) | ||
| 15-Dec-2021 | hgt749d 30727 | A deduction version of ax-hgt749 30722. (Contributed by Thierry Arnoux, 15-Dec-2021.) |
| ⊢ 𝑂 = {𝑧 ∈ ℤ ∣ ¬ 2 ∥ 𝑧} & ⊢ (𝜑 → 𝑁 ∈ 𝑂) & ⊢ (𝜑 → (;10↑;27) ≤ 𝑁) ⇒ ⊢ (𝜑 → ∃ℎ ∈ ((0[,)+∞) ↑𝑚 ℕ)∃𝑘 ∈ ((0[,)+∞) ↑𝑚 ℕ)(∀𝑚 ∈ ℕ (𝑘‘𝑚) ≤ (1._0_7_9_9_55) ∧ ∀𝑚 ∈ ℕ (ℎ‘𝑚) ≤ (1._4_14) ∧ ((0._0_0_0_4_2_2_48) · (𝑁↑2)) ≤ ∫(0(,)1)(((((Λ ∘𝑓 · ℎ)vts𝑁)‘𝑥) · ((((Λ ∘𝑓 · 𝑘)vts𝑁)‘𝑥)↑2)) · (exp‘((i · (2 · π)) · (-𝑁 · 𝑥)))) d𝑥)) | ||
| 15-Dec-2021 | ax-hgt749 30722 | Statement 7.49 of [Helfgott] p. 70. For a sufficiently big odd 𝑁, this postulates the existence of smoothing functions ℎ (eta star) and 𝑘 (eta plus) such that the lower bound for the circle integral is big enough. (Contributed by Thierry Arnoux, 15-Dec-2021.) |
| ⊢ ∀𝑛 ∈ {𝑧 ∈ ℤ ∣ ¬ 2 ∥ 𝑧} ((;10↑;27) ≤ 𝑛 → ∃ℎ ∈ ((0[,)+∞) ↑𝑚 ℕ)∃𝑘 ∈ ((0[,)+∞) ↑𝑚 ℕ)(∀𝑚 ∈ ℕ (𝑘‘𝑚) ≤ (1._0_7_9_9_55) ∧ ∀𝑚 ∈ ℕ (ℎ‘𝑚) ≤ (1._4_14) ∧ ((0._0_0_0_4_2_2_48) · (𝑛↑2)) ≤ ∫(0(,)1)(((((Λ ∘𝑓 · ℎ)vts𝑛)‘𝑥) · ((((Λ ∘𝑓 · 𝑘)vts𝑛)‘𝑥)↑2)) · (exp‘((i · (2 · π)) · (-𝑛 · 𝑥)))) d𝑥)) | ||
| 15-Dec-2021 | reprfi2 30701 | Corollary of reprinfz1 30700. (Contributed by Thierry Arnoux, 15-Dec-2021.) |
| ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → 𝑆 ∈ ℕ0) & ⊢ (𝜑 → 𝐴 ⊆ ℕ) ⇒ ⊢ (𝜑 → (𝐴(repr‘𝑆)𝑁) ∈ Fin) | ||
| 15-Dec-2021 | finsumvtxdg2ssteplem1 26441 | Lemma for finsumvtxdg2sstep 26445. (Contributed by AV, 15-Dec-2021.) |
| ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (iEdg‘𝐺) & ⊢ 𝐾 = (𝑉 ∖ {𝑁}) & ⊢ 𝐼 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑖)} & ⊢ 𝑃 = (𝐸 ↾ 𝐼) & ⊢ 𝑆 = ⟨𝐾, 𝑃⟩ & ⊢ 𝐽 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑖)} ⇒ ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → (#‘𝐸) = ((#‘𝑃) + (#‘𝐽))) | ||
| 15-Dec-2021 | hashreshashfun 13226 | The number of elements of a finite function expressed by a restriction. (Contributed by AV, 15-Dec-2021.) |
| ⊢ ((Fun 𝐴 ∧ 𝐴 ∈ Fin ∧ 𝐵 ⊆ dom 𝐴) → (#‘𝐴) = ((#‘(𝐴 ↾ 𝐵)) + (#‘(dom 𝐴 ∖ 𝐵)))) | ||
| 15-Dec-2021 | hashres 13225 | The number of elements of a finite function restricted to a subset of its domain is equal to the number of elements of that subset. (Contributed by AV, 15-Dec-2021.) |
| ⊢ ((Fun 𝐴 ∧ 𝐴 ∈ Fin ∧ 𝐵 ⊆ dom 𝐴) → (#‘(𝐴 ↾ 𝐵)) = (#‘𝐵)) | ||
| 14-Dec-2021 | vtscl 30716 | Closure of the Vinogradov trigonometric sums (Contributed by Thierry Arnoux, 14-Dec-2021.) |
| ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → 𝑋 ∈ ℂ) & ⊢ (𝜑 → 𝐿:ℕ⟶ℂ) ⇒ ⊢ (𝜑 → ((𝐿vts𝑁)‘𝑋) ∈ ℂ) | ||
| 14-Dec-2021 | hashrepr 30703 | Develop the number of representations of an integer 𝑀 as a sum of nonnegative integers in set 𝐴. (Contributed by Thierry Arnoux, 14-Dec-2021.) |
| ⊢ (𝜑 → 𝐴 ⊆ ℕ) & ⊢ (𝜑 → 𝑀 ∈ ℕ0) & ⊢ (𝜑 → 𝑆 ∈ ℕ0) ⇒ ⊢ (𝜑 → (#‘(𝐴(repr‘𝑆)𝑀)) = Σ𝑐 ∈ (ℕ(repr‘𝑆)𝑀)∏𝑎 ∈ (0..^𝑆)(((𝟭‘ℕ)‘𝐴)‘(𝑐‘𝑎))) | ||
| 14-Dec-2021 | reprfz1 30702 | Corollary of reprinfz1 30700. (Contributed by Thierry Arnoux, 14-Dec-2021.) |
| ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → 𝑆 ∈ ℕ0) ⇒ ⊢ (𝜑 → (ℕ(repr‘𝑆)𝑁) = ((1...𝑁)(repr‘𝑆)𝑁)) | ||
| 14-Dec-2021 | prodfzo03 30681 | A product of three factors, indexed starting with zero. (Contributed by Thierry Arnoux, 14-Dec-2021.) |
| ⊢ (𝑘 = 0 → 𝐷 = 𝐴) & ⊢ (𝑘 = 1 → 𝐷 = 𝐵) & ⊢ (𝑘 = 2 → 𝐷 = 𝐶) & ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^3)) → 𝐷 ∈ ℂ) ⇒ ⊢ (𝜑 → ∏𝑘 ∈ (0..^3)𝐷 = (𝐴 · (𝐵 · 𝐶))) | ||
| 13-Dec-2021 | jath 31609 | Closed form of ja 173. Proved using the completeness script. (Proof modification is discouraged.) (Contributed by Scott Fenton, 13-Dec-2021.) |
| ⊢ ((¬ 𝜑 → 𝜒) → ((𝜓 → 𝜒) → ((𝜑 → 𝜓) → 𝜒))) | ||
| 13-Dec-2021 | circlevma 30720 | The Circle Method, where the Vinogradov sums are weighted using the von Mangoldt function, as it appears as proposition 1.1 of [Helfgott] p. 5. (Contributed by Thierry Arnoux, 13-Dec-2021.) |
| ⊢ (𝜑 → 𝑁 ∈ ℕ0) ⇒ ⊢ (𝜑 → Σ𝑛 ∈ (ℕ(repr‘3)𝑁)((Λ‘(𝑛‘0)) · ((Λ‘(𝑛‘1)) · (Λ‘(𝑛‘2)))) = ∫(0(,)1)((((Λvts𝑁)‘𝑥)↑3) · (exp‘((i · (2 · π)) · (-𝑁 · 𝑥)))) d𝑥) | ||
| 13-Dec-2021 | circlemethnat 30719 | The Hardy, Littlewood and Ramanujan Circle Method, Chapter 5.1 of [Nathanson] p. 123. This expresses 𝑅, the number of different ways a nonnegative integer 𝑁 can be represented as the sum of at most 𝑆 integers in the set 𝐴 as an integral of Vinogradov trigonometric sums. (Contributed by Thierry Arnoux, 13-Dec-2021.) |
| ⊢ 𝑅 = (#‘(𝐴(repr‘𝑆)𝑁)) & ⊢ 𝐹 = ((((𝟭‘ℕ)‘𝐴)vts𝑁)‘𝑥) & ⊢ 𝑁 ∈ ℕ0 & ⊢ 𝐴 ⊆ ℕ & ⊢ 𝑆 ∈ ℕ ⇒ ⊢ 𝑅 = ∫(0(,)1)((𝐹↑𝑆) · (exp‘((i · (2 · π)) · (-𝑁 · 𝑥)))) d𝑥 | ||
| 13-Dec-2021 | circlemeth 30718 | The Hardy, Littlewood and Ramanujan Circle Method, in a generic form, with different weighting / smoothing functions. (Contributed by Thierry Arnoux, 13-Dec-2021.) |
| ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → 𝑆 ∈ ℕ) & ⊢ (𝜑 → 𝐿:(0..^𝑆)⟶(ℂ ↑𝑚 ℕ)) ⇒ ⊢ (𝜑 → Σ𝑐 ∈ (ℕ(repr‘𝑆)𝑁)∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) = ∫(0(,)1)(∏𝑎 ∈ (0..^𝑆)(((𝐿‘𝑎)vts𝑁)‘𝑥) · (exp‘((i · (2 · π)) · (-𝑁 · 𝑥)))) d𝑥) | ||
| 13-Dec-2021 | reprinfz1 30700 | For the representation of 𝑁, it is sufficient to consider nonnegative integers up to 𝑁. Remark of [Nathanson] p. 123 (Contributed by Thierry Arnoux, 13-Dec-2021.) |
| ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → 𝑆 ∈ ℕ0) & ⊢ (𝜑 → 𝐴 ⊆ ℕ) ⇒ ⊢ (𝜑 → (𝐴(repr‘𝑆)𝑁) = ((𝐴 ∩ (1...𝑁))(repr‘𝑆)𝑁)) | ||
| 13-Dec-2021 | efmul2picn 30674 | Multiplying by (i · (2 · π)) and taking the exponential preserves continuity. (Contributed by Thierry Arnoux, 13-Dec-2021.) |
| ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (𝐴–cn→ℂ)) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (exp‘((i · (2 · π)) · 𝐵))) ∈ (𝐴–cn→ℂ)) | ||
| 13-Dec-2021 | iblidicc 30670 | The identity function is integrable on any closed interval. (Contributed by Thierry Arnoux, 13-Dec-2021.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (𝑥 ∈ (𝐴[,]𝐵) ↦ 𝑥) ∈ 𝐿1) | ||
| 13-Dec-2021 | nnmulge 29515 | Multiplying by an integer 𝑀 yields greater or equal nonnegative integers. (Contributed by Thierry Arnoux, 13-Dec-2021.) |
| ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ0) → 𝑁 ≤ (𝑀 · 𝑁)) | ||
| 13-Dec-2021 | ifeq3da 29365 | Given an expression 𝐶 containing if(𝜓, 𝐸, 𝐹), substitute (hypotheses .1 and .2) and evaluate (hypotheses .3 and .4) it for both cases at the same time. (Contributed by Thierry Arnoux, 13-Dec-2021.) |
| ⊢ (if(𝜓, 𝐸, 𝐹) = 𝐸 → 𝐶 = 𝐺) & ⊢ (if(𝜓, 𝐸, 𝐹) = 𝐹 → 𝐶 = 𝐻) & ⊢ (𝜑 → 𝐺 = 𝐴) & ⊢ (𝜑 → 𝐻 = 𝐵) ⇒ ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) = 𝐶) | ||
| 12-Dec-2021 | vtsprod 30717 | Express the Vinogradov trigonometric sums to the power of 𝑆 (Contributed by Thierry Arnoux, 12-Dec-2021.) |
| ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → 𝑋 ∈ ℂ) & ⊢ (𝜑 → 𝑆 ∈ ℕ0) & ⊢ (𝜑 → 𝐿:(0..^𝑆)⟶(ℂ ↑𝑚 ℕ)) ⇒ ⊢ (𝜑 → ∏𝑎 ∈ (0..^𝑆)(((𝐿‘𝑎)vts𝑁)‘𝑋) = Σ𝑚 ∈ (0...(𝑆 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (exp‘((i · (2 · π)) · (𝑚 · 𝑋))))) | ||
| 12-Dec-2021 | finsumvtxdg2ssteplem4 26444 | Lemma for finsumvtxdg2sstep 26445. (Contributed by AV, 12-Dec-2021.) |
| ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (iEdg‘𝐺) & ⊢ 𝐾 = (𝑉 ∖ {𝑁}) & ⊢ 𝐼 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑖)} & ⊢ 𝑃 = (𝐸 ↾ 𝐼) & ⊢ 𝑆 = ⟨𝐾, 𝑃⟩ & ⊢ 𝐽 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑖)} ⇒ ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) ∧ Σ𝑣 ∈ 𝐾 ((VtxDeg‘𝑆)‘𝑣) = (2 · (#‘𝑃))) → (Σ𝑣 ∈ (𝑉 ∖ {𝑁})((VtxDeg‘𝐺)‘𝑣) + ((#‘𝐽) + (#‘{𝑖 ∈ dom 𝐸 ∣ (𝐸‘𝑖) = {𝑁}}))) = (2 · ((#‘𝑃) + (#‘𝐽)))) | ||
| 12-Dec-2021 | finsumvtxdg2ssteplem2 26442 | Lemma for finsumvtxdg2sstep 26445. (Contributed by AV, 12-Dec-2021.) |
| ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (iEdg‘𝐺) & ⊢ 𝐾 = (𝑉 ∖ {𝑁}) & ⊢ 𝐼 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑖)} & ⊢ 𝑃 = (𝐸 ↾ 𝐼) & ⊢ 𝑆 = ⟨𝐾, 𝑃⟩ & ⊢ 𝐽 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑖)} ⇒ ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → ((VtxDeg‘𝐺)‘𝑁) = ((#‘𝐽) + (#‘{𝑖 ∈ dom 𝐸 ∣ (𝐸‘𝑖) = {𝑁}}))) | ||
| 12-Dec-2021 | vtxdgop 26366 | The vertex degree expressed as operation. (Contributed by AV, 12-Dec-2021.) |
| ⊢ (𝐺 ∈ 𝑊 → (VtxDeg‘𝐺) = ((Vtx‘𝐺)VtxDeg(iEdg‘𝐺))) | ||
| 12-Dec-2021 | upgrop 25989 | A pseudograph represented by an ordered pair. (Contributed by AV, 12-Dec-2021.) |
| ⊢ (𝐺 ∈ UPGraph → ⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩ ∈ UPGraph ) | ||
| 11-Dec-2021 | sltrec 31924 | A comparison law for surreals considered as cuts of sets of surreals. (Contributed by Scott Fenton, 11-Dec-2021.) |
| ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → (𝑋 <s 𝑌 ↔ (∃𝑐 ∈ 𝐶 𝑋 ≤s 𝑐 ∨ ∃𝑏 ∈ 𝐵 𝑏 ≤s 𝑌))) | ||
| 11-Dec-2021 | slerec 31923 | A comparison law for surreals considered as cuts of sets of surreals. In Conway's treatment, this is the definition of less than or equal. (Contributed by Scott Fenton, 11-Dec-2021.) |
| ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → (𝑋 ≤s 𝑌 ↔ (∀𝑑 ∈ 𝐷 𝑋 <s 𝑑 ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑌))) | ||
| 11-Dec-2021 | scutbdaylt 31922 | If a surreal lies in a gap and is not equal to the cut, its birthday is greater than the cut's. (Contributed by Scott Fenton, 11-Dec-2021.) |
| ⊢ ((𝑋 ∈ No ∧ (𝐴 <<s {𝑋} ∧ {𝑋} <<s 𝐵) ∧ 𝑋 ≠ (𝐴 |s 𝐵)) → ( bday ‘(𝐴 |s 𝐵)) ∈ ( bday ‘𝑋)) | ||
| 11-Dec-2021 | breprexpnat 30712 | Express the 𝑆 th power of the finite series in terms of the number of representations of integers 𝑚 as sums of 𝑆 terms of elements of 𝐴, bounded by 𝑁. Proposition of [Nathanson] p. 123. (Contributed by Thierry Arnoux, 11-Dec-2021.) |
| ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → 𝑆 ∈ ℕ0) & ⊢ (𝜑 → 𝑍 ∈ ℂ) & ⊢ (𝜑 → 𝐴 ⊆ ℕ) & ⊢ 𝑃 = Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍↑𝑏) & ⊢ 𝑅 = (#‘((𝐴 ∩ (1...𝑁))(repr‘𝑆)𝑚)) ⇒ ⊢ (𝜑 → (𝑃↑𝑆) = Σ𝑚 ∈ (0...(𝑆 · 𝑁))(𝑅 · (𝑍↑𝑚))) | ||
| 11-Dec-2021 | hashreprin 30698 | Express a sum of representations over an intersection using a product of the indicator function (Contributed by Thierry Arnoux, 11-Dec-2021.) |
| ⊢ (𝜑 → 𝐴 ⊆ ℕ) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑆 ∈ ℕ0) & ⊢ (𝜑 → 𝐵 ∈ Fin) & ⊢ (𝜑 → 𝐵 ⊆ ℕ) ⇒ ⊢ (𝜑 → (#‘((𝐴 ∩ 𝐵)(repr‘𝑆)𝑀)) = Σ𝑐 ∈ (𝐵(repr‘𝑆)𝑀)∏𝑎 ∈ (0..^𝑆)(((𝟭‘ℕ)‘𝐴)‘(𝑐‘𝑎))) | ||
| 11-Dec-2021 | reprinrn 30696 | Representations with term in an intersection. (Contributed by Thierry Arnoux, 11-Dec-2021.) |
| ⊢ (𝜑 → 𝐴 ⊆ ℕ) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑆 ∈ ℕ0) ⇒ ⊢ (𝜑 → (𝑐 ∈ ((𝐴 ∩ 𝐵)(repr‘𝑆)𝑀) ↔ (𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ran 𝑐 ⊆ 𝐵))) | ||
| 11-Dec-2021 | reprss 30695 | Representations with terms in a subset. (Contributed by Thierry Arnoux, 11-Dec-2021.) |
| ⊢ (𝜑 → 𝐴 ⊆ ℕ) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑆 ∈ ℕ0) & ⊢ (𝜑 → 𝐵 ⊆ 𝐴) ⇒ ⊢ (𝜑 → (𝐵(repr‘𝑆)𝑀) ⊆ (𝐴(repr‘𝑆)𝑀)) | ||
| 11-Dec-2021 | prodindf 30085 | The product of indicators is one if and only if all values are in the set. (Contributed by Thierry Arnoux, 11-Dec-2021.) |
| ⊢ (𝜑 → 𝑂 ∈ 𝑉) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐵 ⊆ 𝑂) & ⊢ (𝜑 → 𝐹:𝐴⟶𝑂) ⇒ ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 (((𝟭‘𝑂)‘𝐵)‘(𝐹‘𝑘)) = if(ran 𝐹 ⊆ 𝐵, 1, 0)) | ||
| 11-Dec-2021 | indsumin 30084 | Finite sum of a product with the indicator function / Cartesian product with the indicator function. (Contributed by Thierry Arnoux, 11-Dec-2021.) |
| ⊢ (𝜑 → 𝑂 ∈ 𝑉) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐴 ⊆ 𝑂) & ⊢ (𝜑 → 𝐵 ⊆ 𝑂) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 ((((𝟭‘𝑂)‘𝐵)‘𝑘) · 𝐶) = Σ𝑘 ∈ (𝐴 ∩ 𝐵)𝐶) | ||
| 11-Dec-2021 | fprodex01 29571 | A product of factors equal to zero or one is zero exactly when one of the factors is zero. (Contributed by Thierry Arnoux, 11-Dec-2021.) |
| ⊢ (𝑘 = 𝑙 → 𝐵 = 𝐶) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ {0, 1}) ⇒ ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 = if(∀𝑙 ∈ 𝐴 𝐶 = 1, 1, 0)) | ||
| 11-Dec-2021 | fprodeq02 29569 | If one of the factors is zero the product is zero. (Contributed by Thierry Arnoux, 11-Dec-2021.) |
| ⊢ (𝑘 = 𝐾 → 𝐵 = 𝐶) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐾 ∈ 𝐴) & ⊢ (𝜑 → 𝐶 = 0) ⇒ ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 = 0) | ||
| 10-Dec-2021 | scutbdaybnd 31921 | An upper bound on the birthday of a surreal cut. (Contributed by Scott Fenton, 10-Dec-2021.) |
| ⊢ (𝐴 <<s 𝐵 → ( bday ‘(𝐴 |s 𝐵)) ⊆ suc ∪ ( bday “ (𝐴 ∪ 𝐵))) | ||
| 10-Dec-2021 | etasslt 31920 | A restatement of noeta 31868 using set less than. (Contributed by Scott Fenton, 10-Dec-2021.) |
| ⊢ (𝐴 <<s 𝐵 → ∃𝑥 ∈ No (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵 ∧ ( bday ‘𝑥) ⊆ suc ∪ ( bday “ (𝐴 ∪ 𝐵)))) | ||
| 9-Dec-2021 | bj-snmoore 33068 | A singleton is a Moore collection. (Contributed by BJ, 9-Dec-2021.) |
| ⊢ (𝐴 ∈ V ↔ {𝐴} ∈ Moore) | ||
| 9-Dec-2021 | bj-0nmoore 33067 | The empty set is not a Moore collection. (Contributed by BJ, 9-Dec-2021.) |
| ⊢ ¬ ∅ ∈ Moore | ||
| 9-Dec-2021 | bj-discrmoore 33066 | The discrete Moore collection on a set. (Contributed by BJ, 9-Dec-2021.) |
| ⊢ (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ Moore) | ||
| 9-Dec-2021 | bj-ismooredr2 33065 | Sufficient condition to be a Moore collection. (Contributed by BJ, 9-Dec-2021.) |
| ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → ∪ 𝐴 ∈ 𝐴) & ⊢ (((𝜑 ∧ 𝑥 ⊆ 𝐴) ∧ 𝑥 ≠ ∅) → ∩ 𝑥 ∈ 𝐴) ⇒ ⊢ (𝜑 → 𝐴 ∈ Moore) | ||
| 9-Dec-2021 | bj-ismooredr 33064 | Sufficient condition to be a Moore collection. (Contributed by BJ, 9-Dec-2021.) |
| ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝐴) → (∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴) ⇒ ⊢ (𝜑 → 𝐴 ∈ Moore) | ||
| 9-Dec-2021 | bj-ismoored2 33063 | Necessary condition to be a Moore collection. (Contributed by BJ, 9-Dec-2021.) |
| ⊢ (𝜑 → 𝐴 ∈ Moore) & ⊢ (𝜑 → 𝐵 ⊆ 𝐴) & ⊢ (𝜑 → 𝐵 ≠ ∅) ⇒ ⊢ (𝜑 → ∩ 𝐵 ∈ 𝐴) | ||
| 9-Dec-2021 | bj-ismoored 33062 | Necessary condition to be a Moore collection. (Contributed by BJ, 9-Dec-2021.) |
| ⊢ (𝜑 → 𝐴 ∈ Moore) & ⊢ (𝜑 → 𝐵 ⊆ 𝐴) ⇒ ⊢ (𝜑 → (∪ 𝐴 ∩ ∩ 𝐵) ∈ 𝐴) | ||
| 9-Dec-2021 | bj-ismoored0 33061 | Necessary condition to be a Moore collection. (Contributed by BJ, 9-Dec-2021.) |
| ⊢ (𝐴 ∈ Moore → ∪ 𝐴 ∈ 𝐴) | ||
| 9-Dec-2021 | bj-ismoorec 33060 | Characterization of Moore collections. (Contributed by BJ, 9-Dec-2021.) |
| ⊢ (𝐴 ∈ Moore ↔ (𝐴 ∈ V ∧ ∀𝑥 ∈ 𝒫 𝐴(∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴)) | ||
| 9-Dec-2021 | bj-ismoore 33059 | Characterization of Moore collections among sets. (Contributed by BJ, 9-Dec-2021.) |
| ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ Moore ↔ ∀𝑥 ∈ 𝒫 𝐴(∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴)) | ||
| 9-Dec-2021 | scutun12 31917 | Union law for surreal cuts. (Contributed by Scott Fenton, 9-Dec-2021.) |
| ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐷) → ((𝐴 ∪ 𝐶) |s (𝐵 ∪ 𝐷)) = (𝐴 |s 𝐵)) | ||
| 9-Dec-2021 | ssltun2 31916 | Union law for surreal set less than. (Contributed by Scott Fenton, 9-Dec-2021.) |
| ⊢ ((𝐴 <<s 𝐵 ∧ 𝐴 <<s 𝐶) → 𝐴 <<s (𝐵 ∪ 𝐶)) | ||
| 9-Dec-2021 | ssltun1 31915 | Union law for surreal set less than. (Contributed by Scott Fenton, 9-Dec-2021.) |
| ⊢ ((𝐴 <<s 𝐶 ∧ 𝐵 <<s 𝐶) → (𝐴 ∪ 𝐵) <<s 𝐶) | ||
| 9-Dec-2021 | sslttr 31914 | Transitive law for surreal set less than. (Contributed by Scott Fenton, 9-Dec-2021.) |
| ⊢ ((𝐴 <<s 𝐵 ∧ 𝐵 <<s 𝐶 ∧ 𝐵 ≠ ∅) → 𝐴 <<s 𝐶) | ||
| 9-Dec-2021 | sssslt2 31907 | Relationship between surreal set less than and subset. (Contributed by Scott Fenton, 9-Dec-2021.) |
| ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐵) → 𝐴 <<s 𝐶) | ||
| 9-Dec-2021 | sssslt1 31906 | Relationship between surreal set less than and subset. (Contributed by Scott Fenton, 9-Dec-2021.) |
| ⊢ ((𝐴 <<s 𝐵 ∧ 𝐶 ⊆ 𝐴) → 𝐶 <<s 𝐵) | ||
| 9-Dec-2021 | elintfv 31662 | Membership in an intersection of function values. (Contributed by Scott Fenton, 9-Dec-2021.) |
| ⊢ 𝑋 ∈ V ⇒ ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝑋 ∈ ∩ (𝐹 “ 𝐵) ↔ ∀𝑦 ∈ 𝐵 𝑋 ∈ (𝐹‘𝑦))) | ||
| 9-Dec-2021 | actfunsnrndisj 30683 | The action 𝐹 of extending function from 𝐵 to 𝐶 with new values at point 𝐼 yields different functions. (Contributed by Thierry Arnoux, 9-Dec-2021.) |
| ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → 𝐴 ⊆ (𝐶 ↑𝑚 𝐵)) & ⊢ (𝜑 → 𝐶 ∈ V) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → ¬ 𝐼 ∈ 𝐵) & ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝑥 ∪ {⟨𝐼, 𝑘⟩})) ⇒ ⊢ (𝜑 → Disj 𝑘 ∈ 𝐶 ran 𝐹) | ||
| 9-Dec-2021 | actfunsnf1o 30682 | The action 𝐹 of extending function from 𝐵 to 𝐶 with new values at point 𝐼 is a bijection. (Contributed by Thierry Arnoux, 9-Dec-2021.) |
| ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → 𝐴 ⊆ (𝐶 ↑𝑚 𝐵)) & ⊢ (𝜑 → 𝐶 ∈ V) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → ¬ 𝐼 ∈ 𝐵) & ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝑥 ∪ {⟨𝐼, 𝑘⟩})) ⇒ ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → 𝐹:𝐴–1-1-onto→ran 𝐹) | ||
| 9-Dec-2021 | fnunres1 29417 | Restriction of a disjoint union to the domain of the first function. (Contributed by Thierry Arnoux, 9-Dec-2021.) |
| ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐹 ∪ 𝐺) ↾ 𝐴) = 𝐹) | ||
| 9-Dec-2021 | funresdm1 29416 | Restriction of a disjoint union to the domain of the first term. (Contributed by Thierry Arnoux, 9-Dec-2021.) |
| ⊢ ((Rel 𝐴 ∧ (dom 𝐴 ∩ dom 𝐵) = ∅) → ((𝐴 ∪ 𝐵) ↾ dom 𝐴) = 𝐴) | ||
| 9-Dec-2021 | edginwlk 26530 | The value of the edge function for an index of an edge within a walk is an edge. (Contributed by AV, 2-Jan-2021.) (Revised by AV, 9-Dec-2021.) |
| ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ ((Fun 𝐼 ∧ 𝐹 ∈ Word dom 𝐼 ∧ 𝐾 ∈ (0..^(#‘𝐹))) → (𝐼‘(𝐹‘𝐾)) ∈ 𝐸) | ||
| 8-Dec-2021 | bj-mooreset 33056 |
A Moore collection is a set. That is, if we define a "Moore
predicate"
by (Moore𝐴 ↔ ∀𝑥 ∈ 𝒫 𝐴(∪ 𝐴 ∩ ∩ 𝑥)
∈ 𝐴), then any
class satisfying that predicate is actually a set. Therefore, the
definition df-bj-moore 33058 is sufficient. Note that the closed sets of
a
topology form a Moore collection, so this remark also applies to
topologies and many other families of sets (namely, as soon as the whole
set is required to be a closed set, as can be seen from the proof, which
relies crucially on uniexr 6972).
Note: if, in the above predicate, we substitute 𝒫 𝑋 for 𝐴, then the last ∈ 𝒫 𝑋 could be weakened to ⊆ 𝒫 𝑋, and then the predicate would be obviously satisfied since ∪ 𝒫 𝑋 = 𝑋, making 𝒫 𝑋 a Moore collection in this weaker sense, even if 𝑋 is a proper class, but the addition of this single case does not add anything interesting. (Contributed by BJ, 8-Dec-2021.) |
| ⊢ (∀𝑥 ∈ 𝒫 𝐴(∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴 → 𝐴 ∈ V) | ||
| 8-Dec-2021 | dmscut 31918 | The domain of the surreal cut operation is all separated surreal sets. (Contributed by Scott Fenton, 8-Dec-2021.) |
| ⊢ dom |s = <<s | ||
| 8-Dec-2021 | scutbday 31913 | The birthday of the surreal cut is equal to the minimum birthday in the gap. (Contributed by Scott Fenton, 8-Dec-2021.) |
| ⊢ (𝐴 <<s 𝐵 → ( bday ‘(𝐴 |s 𝐵)) = ∩ ( bday “ {𝑥 ∈ No ∣ (𝐴 <<s {𝑥} ∧ {𝑥} <<s 𝐵)})) | ||
| 8-Dec-2021 | scutcut 31912 | Cut properties of the surreal cut operation. (Contributed by Scott Fenton, 8-Dec-2021.) |
| ⊢ (𝐴 <<s 𝐵 → ((𝐴 |s 𝐵) ∈ No ∧ 𝐴 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐵)) | ||
| 8-Dec-2021 | scutval 31911 | The value of the surreal cut operation. (Contributed by Scott Fenton, 8-Dec-2021.) |
| ⊢ (𝐴 <<s 𝐵 → (𝐴 |s 𝐵) = (℩𝑥 ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ( bday ‘𝑥) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}))) | ||
| 8-Dec-2021 | conway 31910 | Conway's Simplicity Theorem. Given 𝐴 preceeding 𝐵, there is a unique surreal of minimal length separating them. This is a fundamental property of surreals and will be used (via surreal cuts) to prove many properties later on. (Contributed by Scott Fenton, 8-Dec-2021.) |
| ⊢ (𝐴 <<s 𝐵 → ∃!𝑥 ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ( bday ‘𝑥) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)})) | ||
| 8-Dec-2021 | nulssgt 31909 | The empty set is greater than any set of surreals. (Contributed by Scott Fenton, 8-Dec-2021.) |
| ⊢ (𝐴 ∈ 𝒫 No → 𝐴 <<s ∅) | ||
| 8-Dec-2021 | nulsslt 31908 | The empty set is less than any set of surreals. (Contributed by Scott Fenton, 8-Dec-2021.) |
| ⊢ (𝐴 ∈ 𝒫 No → ∅ <<s 𝐴) | ||
| 8-Dec-2021 | ssltsep 31905 | The separation property of surreal set less than. (Contributed by Scott Fenton, 8-Dec-2021.) |
| ⊢ (𝐴 <<s 𝐵 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦) | ||
| 8-Dec-2021 | ssltss2 31904 | The second argument of surreal set is a set of surreals. (Contributed by Scott Fenton, 8-Dec-2021.) |
| ⊢ (𝐴 <<s 𝐵 → 𝐵 ⊆ No ) | ||
| 8-Dec-2021 | ssltss1 31903 | The first argument of surreal set is a set of surreals. (Contributed by Scott Fenton, 8-Dec-2021.) |
| ⊢ (𝐴 <<s 𝐵 → 𝐴 ⊆ No ) | ||
| 8-Dec-2021 | ssltex2 31902 | The second argument of surreal set less than exists. (Contributed by Scott Fenton, 8-Dec-2021.) |
| ⊢ (𝐴 <<s 𝐵 → 𝐵 ∈ V) | ||
| 8-Dec-2021 | ssltex1 31901 | The first argument of surreal set less than exists. (Contributed by Scott Fenton, 8-Dec-2021.) |
| ⊢ (𝐴 <<s 𝐵 → 𝐴 ∈ V) | ||
| 8-Dec-2021 | brsslt 31900 | Binary relation form of the surreal set less-than relation. (Contributed by Scott Fenton, 8-Dec-2021.) |
| ⊢ (𝐴 <<s 𝐵 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (𝐴 ⊆ No ∧ 𝐵 ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦))) | ||
| 8-Dec-2021 | bdayelon 31892 | The value of the birthday function is always an ordinal. (Contributed by Scott Fenton, 14-Jun-2011.) (Proof shortened by Scott Fenton, 8-Dec-2021.) |
| ⊢ ( bday ‘𝐴) ∈ On | ||
| 8-Dec-2021 | sletrd 31887 | Surreal less than or equal is transitive. (Contributed by Scott Fenton, 8-Dec-2021.) |
| ⊢ (𝜑 → 𝐴 ∈ No ) & ⊢ (𝜑 → 𝐵 ∈ No ) & ⊢ (𝜑 → 𝐶 ∈ No ) & ⊢ (𝜑 → 𝐴 ≤s 𝐵) & ⊢ (𝜑 → 𝐵 ≤s 𝐶) ⇒ ⊢ (𝜑 → 𝐴 ≤s 𝐶) | ||
| 8-Dec-2021 | slelttrd 31886 | Surreal less than is transitive. (Contributed by Scott Fenton, 8-Dec-2021.) |
| ⊢ (𝜑 → 𝐴 ∈ No ) & ⊢ (𝜑 → 𝐵 ∈ No ) & ⊢ (𝜑 → 𝐶 ∈ No ) & ⊢ (𝜑 → 𝐴 ≤s 𝐵) & ⊢ (𝜑 → 𝐵 <s 𝐶) ⇒ ⊢ (𝜑 → 𝐴 <s 𝐶) | ||
| 8-Dec-2021 | sltletrd 31885 | Surreal less than is transitive. (Contributed by Scott Fenton, 8-Dec-2021.) |
| ⊢ (𝜑 → 𝐴 ∈ No ) & ⊢ (𝜑 → 𝐵 ∈ No ) & ⊢ (𝜑 → 𝐶 ∈ No ) & ⊢ (𝜑 → 𝐴 <s 𝐵) & ⊢ (𝜑 → 𝐵 ≤s 𝐶) ⇒ ⊢ (𝜑 → 𝐴 <s 𝐶) | ||
| 8-Dec-2021 | slttrd 31884 | Surreal less than is transitive. (Contributed by Scott Fenton, 8-Dec-2021.) |
| ⊢ (𝜑 → 𝐴 ∈ No ) & ⊢ (𝜑 → 𝐵 ∈ No ) & ⊢ (𝜑 → 𝐶 ∈ No ) & ⊢ (𝜑 → 𝐴 <s 𝐵) & ⊢ (𝜑 → 𝐵 <s 𝐶) ⇒ ⊢ (𝜑 → 𝐴 <s 𝐶) | ||
| 8-Dec-2021 | sletr 31883 | Surreal transitive law. (Contributed by Scott Fenton, 8-Dec-2021.) |
| ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ((𝐴 ≤s 𝐵 ∧ 𝐵 ≤s 𝐶) → 𝐴 ≤s 𝐶)) | ||
| 8-Dec-2021 | slelttr 31882 | Surreal transitive law. (Contributed by Scott Fenton, 8-Dec-2021.) |
| ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ((𝐴 ≤s 𝐵 ∧ 𝐵 <s 𝐶) → 𝐴 <s 𝐶)) | ||
| 8-Dec-2021 | sltletr 31881 | Surreal transitive law. (Contributed by Scott Fenton, 8-Dec-2021.) |
| ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ((𝐴 <s 𝐵 ∧ 𝐵 ≤s 𝐶) → 𝐴 <s 𝐶)) | ||
| 8-Dec-2021 | sletri3 31880 | Trichotomy law for surreal less than or equal. (Contributed by Scott Fenton, 8-Dec-2021.) |
| ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 = 𝐵 ↔ (𝐴 ≤s 𝐵 ∧ 𝐵 ≤s 𝐴))) | ||
| 8-Dec-2021 | sleloe 31879 | Surreal less than or equal in terms of less than. (Contributed by Scott Fenton, 8-Dec-2021.) |
| ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 ≤s 𝐵 ↔ (𝐴 <s 𝐵 ∨ 𝐴 = 𝐵))) | ||
| 8-Dec-2021 | sltnle 31878 | Surreal less than in terms of less than or equal. (Contributed by Scott Fenton, 8-Dec-2021.) |
| ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 <s 𝐵 ↔ ¬ 𝐵 ≤s 𝐴)) | ||
| 8-Dec-2021 | slenlt 31877 | Surreal less than or equal in terms of less than. (Contributed by Scott Fenton, 8-Dec-2021.) |
| ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 ≤s 𝐵 ↔ ¬ 𝐵 <s 𝐴)) | ||
| 8-Dec-2021 | df-sle 31870 | Define the surreal less than or equal predicate. Compare df-le 10080. (Contributed by Scott Fenton, 8-Dec-2021.) |
| ⊢ ≤s = (( No × No ) ∖ ◡ <s ) | ||
| 8-Dec-2021 | sotrine 31658 | Trichotomy law for strict orderings. (Contributed by Scott Fenton, 8-Dec-2021.) |
| ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵 ≠ 𝐶 ↔ (𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵))) | ||
| 8-Dec-2021 | fz2ssnn0 29547 | A finite set of sequential integers that is a subset of ℕ0. (Contributed by Thierry Arnoux, 8-Dec-2021.) |
| ⊢ (𝑀 ∈ ℕ0 → (𝑀...𝑁) ⊆ ℕ0) | ||
| 8-Dec-2021 | uhgriedg0edg0 26022 | A hypergraph has no edges iff its edge function is empty. (Contributed by AV, 21-Oct-2020.) (Proof shortened by AV, 8-Dec-2021.) |
| ⊢ (𝐺 ∈ UHGraph → ((Edg‘𝐺) = ∅ ↔ (iEdg‘𝐺) = ∅)) | ||
| 8-Dec-2021 | edg0iedg0 25949 | There is no edge in a graph iff its edge function is empty. (Contributed by AV, 15-Dec-2020.) (Revised by AV, 8-Dec-2021.) |
| ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ (Fun 𝐼 → (𝐸 = ∅ ↔ 𝐼 = ∅)) | ||
| 8-Dec-2021 | edgiedgb 25947 | A set is an edge iff it is an indexed edge. (Contributed by AV, 17-Oct-2020.) (Revised by AV, 8-Dec-2021.) |
| ⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ (Fun 𝐼 → (𝐸 ∈ (Edg‘𝐺) ↔ ∃𝑥 ∈ dom 𝐼 𝐸 = (𝐼‘𝑥))) | ||
| 8-Dec-2021 | edgval 25941 | The edges of a graph. (Contributed by AV, 1-Jan-2020.) (Revised by AV, 13-Oct-2020.) (Revised by AV, 8-Dec-2021.) |
| ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺) | ||
| 7-Dec-2021 | bj-0int 33055 | If 𝐴 is a collection of subsets of 𝑋, like a topology, two equivalent ways to say that arbitrary intersections of elements of 𝐴 relative to 𝑋 belong to some class 𝐵 (in typical applications, 𝐴 itself). (Contributed by BJ, 7-Dec-2021.) |
| ⊢ (𝐴 ⊆ 𝒫 𝑋 → ((𝑋 ∈ 𝐵 ∧ ∀𝑥 ∈ (𝒫 𝐴 ∖ {∅})∩ 𝑥 ∈ 𝐵) ↔ ∀𝑥 ∈ 𝒫 𝐴(𝑋 ∩ ∩ 𝑥) ∈ 𝐵)) | ||
| 7-Dec-2021 | bj-raldifsn 33054 | All elements in a set satisfy a given property if and only if all but one satisfy that property and that one also does. Typically, this can be used for characterizations that are proved using different methods for a given element and for all others, for instance zero and nonzero numbers, or the empty set and nonempty sets. (Contributed by BJ, 7-Dec-2021.) |
| ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝐵 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 𝜑 ↔ (∀𝑥 ∈ (𝐴 ∖ {𝐵})𝜑 ∧ 𝜓))) | ||
| 7-Dec-2021 | bj-intss 33053 | A nonempty intersection of a family of subsets of a class is included in that class. (Contributed by BJ, 7-Dec-2021.) |
| ⊢ (𝐴 ⊆ 𝒫 𝑋 → (𝐴 ≠ ∅ → ∩ 𝐴 ⊆ 𝑋)) | ||
| 7-Dec-2021 | df-scut 31899 | Define the cut operator on surreal numbers. This operator, which Conway takes as the primitive operator over surreals, picks the surreal lying between two sets of surreals of minimal birthday. (Contributed by Scott Fenton, 7-Dec-2021.) |
| ⊢ |s = (𝑎 ∈ 𝒫 No , 𝑏 ∈ ( <<s “ {𝑎}) ↦ (℩𝑥 ∈ {𝑦 ∈ No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)} ( bday ‘𝑥) = ∩ ( bday “ {𝑦 ∈ No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)}))) | ||
| 7-Dec-2021 | df-sslt 31897 | Define the relationship that holds iff one set of surreals completely precedes another. (Contributed by Scott Fenton, 7-Dec-2021.) |
| ⊢ <<s = {⟨𝑎, 𝑏⟩ ∣ (𝑎 ⊆ No ∧ 𝑏 ⊆ No ∧ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 𝑥 <s 𝑦)} | ||
| 7-Dec-2021 | noeta 31868 | The full-eta axiom for the surreal numbers. This is the single most important property of the surreals. It says that, given two sets of surreals such that one comes completely before the other, there is a surreal lying strictly between the two. Furthermore, there is an upper bound on the birthday of that surreal. Alling's axiom FE. (Contributed by Scott Fenton, 7-Dec-2021.) |
| ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑊) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑥 <s 𝑦) → ∃𝑧 ∈ No (∀𝑥 ∈ 𝐴 𝑥 <s 𝑧 ∧ ∀𝑦 ∈ 𝐵 𝑧 <s 𝑦 ∧ ( bday ‘𝑧) ⊆ suc ∪ ( bday “ (𝐴 ∪ 𝐵)))) | ||
| 7-Dec-2021 | noetalem5 31867 | Lemma for noeta 31868. The full statement of the theorem with hypotheses. (Contributed by Scott Fenton, 7-Dec-2021.) |
| ⊢ 𝑆 = if(∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦, ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2𝑜⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥)))) & ⊢ 𝑍 = (𝑆 ∪ ((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1𝑜})) ⇒ ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ 𝑊) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 𝑎 <s 𝑏) → ∃𝑧 ∈ No (∀𝑎 ∈ 𝐴 𝑎 <s 𝑧 ∧ ∀𝑏 ∈ 𝐵 𝑧 <s 𝑏 ∧ ( bday ‘𝑧) ⊆ suc ∪ ( bday “ (𝐴 ∪ 𝐵)))) | ||
| 7-Dec-2021 | noetalem3 31865 | Lemma for noeta 31868. When 𝐴 and 𝐵 are separated, then 𝑍 is a lower bound for 𝐵. Part of Theorem 5.1 of [Lipparini] p. 7-8. (Contributed by Scott Fenton, 7-Dec-2021.) |
| ⊢ 𝑆 = if(∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦, ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2𝑜⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥)))) & ⊢ 𝑍 = (𝑆 ∪ ((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1𝑜})) ⇒ ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V) ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 𝑎 <s 𝑏) → ∀𝑏 ∈ 𝐵 𝑍 <s 𝑏) | ||
| 7-Dec-2021 | nosep1o 31832 | If the value of a surreal at a separator is 1𝑜 then the surreal is lesser. (Contributed by Scott Fenton, 7-Dec-2021.) |
| ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 ≠ 𝐵) ∧ (𝐴‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) = 1𝑜) → 𝐴 <s 𝐵) | ||
| 7-Dec-2021 | funeldmb 31661 | If ∅ is not part of the range of a function 𝐹, then 𝐴 is in the domain of 𝐹 iff (𝐹‘𝐴) ≠ ∅. (Contributed by Scott Fenton, 7-Dec-2021.) |
| ⊢ ((Fun 𝐹 ∧ ¬ ∅ ∈ ran 𝐹) → (𝐴 ∈ dom 𝐹 ↔ (𝐹‘𝐴) ≠ ∅)) | ||
| 7-Dec-2021 | breprexplemb 30709 | Lemma for breprexp 30711 (closure) (Contributed by Thierry Arnoux, 7-Dec-2021.) |
| ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → 𝑆 ∈ ℕ0) & ⊢ (𝜑 → 𝑍 ∈ ℂ) & ⊢ (𝜑 → 𝐿:(0..^𝑆)⟶(ℂ ↑𝑚 ℕ)) & ⊢ (𝜑 → 𝑋 ∈ (0..^𝑆)) & ⊢ (𝜑 → 𝑌 ∈ ℕ) ⇒ ⊢ (𝜑 → ((𝐿‘𝑋)‘𝑌) ∈ ℂ) | ||
| 7-Dec-2021 | breprexplema 30708 | Lemma for breprexp 30711 (induction step for weighted sums over representations) (Contributed by Thierry Arnoux, 7-Dec-2021.) |
| ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → 𝑆 ∈ ℕ0) & ⊢ (𝜑 → 𝑀 ∈ ℕ0) & ⊢ (𝜑 → 𝑀 ≤ ((𝑆 + 1) · 𝑁)) & ⊢ (((𝜑 ∧ 𝑥 ∈ (0..^(𝑆 + 1))) ∧ 𝑦 ∈ ℕ) → ((𝐿‘𝑥)‘𝑦) ∈ ℂ) ⇒ ⊢ (𝜑 → Σ𝑑 ∈ ((1...𝑁)(repr‘(𝑆 + 1))𝑀)∏𝑎 ∈ (0..^(𝑆 + 1))((𝐿‘𝑎)‘(𝑑‘𝑎)) = Σ𝑏 ∈ (1...𝑁)Σ𝑑 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))(∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑑‘𝑎)) · ((𝐿‘𝑆)‘𝑏))) | ||
| 7-Dec-2021 | reprgt 30699 | There are no representations of more than (𝑆 · 𝑁) with only 𝑆 terms bounded by 𝑁. Remark of [Nathanson] p. 123 (Contributed by Thierry Arnoux, 7-Dec-2021.) |
| ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → 𝐴 ⊆ (1...𝑁)) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑆 ∈ ℕ0) & ⊢ (𝜑 → (𝑆 · 𝑁) < 𝑀) ⇒ ⊢ (𝜑 → (𝐴(repr‘𝑆)𝑀) = ∅) | ||
| 7-Dec-2021 | reprlt 30697 | There are no representations of 𝑀 with more than 𝑀 terms. Remark of [Nathanson] p. 123 (Contributed by Thierry Arnoux, 7-Dec-2021.) |
| ⊢ (𝜑 → 𝐴 ⊆ ℕ) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑆 ∈ ℕ0) & ⊢ (𝜑 → 𝑀 < 𝑆) ⇒ ⊢ (𝜑 → (𝐴(repr‘𝑆)𝑀) = ∅) | ||
| 7-Dec-2021 | reprfi 30694 | Bounded representations are finite sets. (Contributed by Thierry Arnoux, 7-Dec-2021.) |
| ⊢ (𝜑 → 𝐴 ⊆ ℕ) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑆 ∈ ℕ0) & ⊢ (𝜑 → 𝐴 ∈ Fin) ⇒ ⊢ (𝜑 → (𝐴(repr‘𝑆)𝑀) ∈ Fin) | ||
| 7-Dec-2021 | fsum2dsub 30685 | Lemma for breprexp 30711- Re-index a double sum, using difference of the initial indices. (Contributed by Thierry Arnoux, 7-Dec-2021.) |
| ⊢ (𝜑 → 𝑀 ∈ ℕ0) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝑖 = (𝑘 − 𝑗) → 𝐴 = 𝐵) & ⊢ ((𝜑 ∧ 𝑖 ∈ (ℤ≥‘-𝑗) ∧ 𝑗 ∈ (1...𝑁)) → 𝐴 ∈ ℂ) & ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ (((𝑀 + 𝑗) + 1)...(𝑀 + 𝑁))) → 𝐵 = 0) & ⊢ (((𝜑 ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ (0..^𝑗)) → 𝐵 = 0) ⇒ ⊢ (𝜑 → Σ𝑖 ∈ (0...𝑀)Σ𝑗 ∈ (1...𝑁)𝐴 = Σ𝑘 ∈ (0...(𝑀 + 𝑁))Σ𝑗 ∈ (1...𝑁)𝐵) | ||
| 6-Dec-2021 | noetalem4 31866 | Lemma for noeta 31868. Bound the birthday of 𝑍 above. (Contributed by Scott Fenton, 6-Dec-2021.) |
| ⊢ 𝑆 = if(∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦, ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2𝑜⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥)))) & ⊢ 𝑍 = (𝑆 ∪ ((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1𝑜})) ⇒ ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝐵 ⊆ No ∧ 𝐵 ∈ V)) → ( bday ‘𝑍) ⊆ suc ∪ ( bday “ (𝐴 ∪ 𝐵))) | ||
| 6-Dec-2021 | noetalem1 31863 | Lemma for noeta 31868. Establish that our final surreal really is a surreal. (Contributed by Scott Fenton, 6-Dec-2021.) |
| ⊢ 𝑆 = if(∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦, ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2𝑜⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥)))) & ⊢ 𝑍 = (𝑆 ∪ ((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1𝑜})) ⇒ ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝑍 ∈ No ) | ||
| 6-Dec-2021 | nosupbnd2 31862 | Bounding law from above for the surreal supremum. Proposition 4.3 of [Lipparini] p. 6. (Contributed by Scott Fenton, 6-Dec-2021.) |
| ⊢ 𝑆 = if(∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦, ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2𝑜⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥)))) ⇒ ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) → (∀𝑎 ∈ 𝐴 𝑎 <s 𝑍 ↔ ¬ (𝑍 ↾ dom 𝑆) <s 𝑆)) | ||
| 6-Dec-2021 | nosupbnd2lem1 31861 | Bounding law from above when a set of surreals has a maximum. (Contributed by Scott Fenton, 6-Dec-2021.) |
| ⊢ (((𝑈 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑈 <s 𝑦) ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑍 ∈ No ) ∧ ∀𝑎 ∈ 𝐴 𝑎 <s 𝑍) → ¬ (𝑍 ↾ suc dom 𝑈) <s (𝑈 ∪ {⟨dom 𝑈, 2𝑜⟩})) | ||
| 6-Dec-2021 | nosupbnd1 31860 | Bounding law from below for the surreal supremum. Proposition 4.2 of [Lipparini] p. 6. (Contributed by Scott Fenton, 6-Dec-2021.) |
| ⊢ 𝑆 = if(∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦, ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2𝑜⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥)))) ⇒ ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝑈 ∈ 𝐴) → (𝑈 ↾ dom 𝑆) <s 𝑆) | ||
| 6-Dec-2021 | nosupbnd1lem6 31859 | Lemma for nosupbnd1 31860. Establish a hard upper bound when there is no maximum. (Contributed by Scott Fenton, 6-Dec-2021.) |
| ⊢ 𝑆 = if(∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦, ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2𝑜⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥)))) ⇒ ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ 𝑈 ∈ 𝐴) → (𝑈 ↾ dom 𝑆) <s 𝑆) | ||
| 6-Dec-2021 | nosupbnd1lem5 31858 | Lemma for nosupbnd1 31860. If 𝑈 is a prolongment of 𝑆 and in 𝐴, then (𝑈‘dom 𝑆) is not 1𝑜. (Contributed by Scott Fenton, 6-Dec-2021.) |
| ⊢ 𝑆 = if(∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦, ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2𝑜⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥)))) ⇒ ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) → (𝑈‘dom 𝑆) ≠ 1𝑜) | ||
| 6-Dec-2021 | nosupbnd1lem4 31857 | Lemma for nosupbnd1 31860. If 𝑈 is a prolongment of 𝑆 and in 𝐴, then (𝑈‘dom 𝑆) is not undefined. (Contributed by Scott Fenton, 6-Dec-2021.) |
| ⊢ 𝑆 = if(∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦, ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2𝑜⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥)))) ⇒ ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) → (𝑈‘dom 𝑆) ≠ ∅) | ||
| 6-Dec-2021 | nosupbnd1lem3 31856 | Lemma for nosupbnd1 31860. If 𝑈 is a prolongment of 𝑆 and in 𝐴, then (𝑈‘dom 𝑆) is not 2𝑜. (Contributed by Scott Fenton, 6-Dec-2021.) |
| ⊢ 𝑆 = if(∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦, ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2𝑜⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥)))) ⇒ ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆)) → (𝑈‘dom 𝑆) ≠ 2𝑜) | ||
| 6-Dec-2021 | nosupdm 31850 | The domain of the surreal supremum when there is no maximum. The primary point of this theorem is to change bound variable. (Contributed by Scott Fenton, 6-Dec-2021.) |
| ⊢ 𝑆 = if(∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦, ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2𝑜⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥)))) ⇒ ⊢ (¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 → dom 𝑆 = {𝑧 ∣ ∃𝑝 ∈ 𝐴 (𝑧 ∈ dom 𝑝 ∧ ∀𝑞 ∈ 𝐴 (¬ 𝑞 <s 𝑝 → (𝑝 ↾ suc 𝑧) = (𝑞 ↾ suc 𝑧)))}) | ||
| 6-Dec-2021 | nolt02o 31845 | Given 𝐴 less than 𝐵, equal to 𝐵 up to 𝑋, and undefined at 𝑋, then 𝐵(𝑋) = 2𝑜. (Contributed by Scott Fenton, 6-Dec-2021.) |
| ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ 𝐴 <s 𝐵) ∧ (𝐴‘𝑋) = ∅) → (𝐵‘𝑋) = 2𝑜) | ||
| 6-Dec-2021 | nolt02olem 31844 | Lemma for nolt02o 31845. If 𝐴(𝑋) is undefined with 𝐴 surreal and 𝑋 ordinal, then dom 𝐴 ⊆ 𝑋. (Contributed by Scott Fenton, 6-Dec-2021.) |
| ⊢ ((𝐴 ∈ No ∧ 𝑋 ∈ On ∧ (𝐴‘𝑋) = ∅) → dom 𝐴 ⊆ 𝑋) | ||
| 6-Dec-2021 | nosepssdm 31836 | Given two non-equal surreals, their separator is less than or equal to the domain of one of them. Part of Lemma 2.1.1 of [Lipparini] p. 3. (Contributed by Scott Fenton, 6-Dec-2021.) |
| ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 ≠ 𝐵) → ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)} ⊆ dom 𝐴) | ||
| 6-Dec-2021 | nosepeq 31835 | The values of two surreals at a point less than their separators are equal. (Contributed by Scott Fenton, 6-Dec-2021.) |
| ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 ≠ 𝐵) ∧ 𝑋 ∈ ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) → (𝐴‘𝑋) = (𝐵‘𝑋)) | ||
| 6-Dec-2021 | nolesgn2ores 31825 | Given 𝐴 less than or equal to 𝐵, equal to 𝐵 up to 𝑋, and 𝐴(𝑋) = 2𝑜, then (𝐴 ↾ suc 𝑋) = (𝐵 ↾ suc 𝑋). (Contributed by Scott Fenton, 6-Dec-2021.) |
| ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2𝑜) ∧ ¬ 𝐵 <s 𝐴) → (𝐴 ↾ suc 𝑋) = (𝐵 ↾ suc 𝑋)) | ||
| 6-Dec-2021 | nolesgn2o 31824 | Given 𝐴 less than or equal to 𝐵, equal to 𝐵 up to 𝑋, and 𝐴(𝑋) = 2𝑜, then 𝐵(𝑋) = 2𝑜. (Contributed by Scott Fenton, 6-Dec-2021.) |
| ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝑋 ∈ On) ∧ ((𝐴 ↾ 𝑋) = (𝐵 ↾ 𝑋) ∧ (𝐴‘𝑋) = 2𝑜) ∧ ¬ 𝐵 <s 𝐴) → (𝐵‘𝑋) = 2𝑜) | ||
| 6-Dec-2021 | eqfunressuc 31660 | Law for equality of restriction to successors. This is primarily useful when 𝑋 is an ordinal, but it does not require that. (Contributed by Scott Fenton, 6-Dec-2021.) |
| ⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐹 ↾ 𝑋) = (𝐺 ↾ 𝑋) ∧ (𝑋 ∈ dom 𝐹 ∧ 𝑋 ∈ dom 𝐺 ∧ (𝐹‘𝑋) = (𝐺‘𝑋))) → (𝐹 ↾ suc 𝑋) = (𝐺 ↾ suc 𝑋)) | ||
| 6-Dec-2021 | eqfunresadj 31659 | Law for adjoining an element to restrictions of functions. (Contributed by Scott Fenton, 6-Dec-2021.) |
| ⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐹 ↾ 𝑋) = (𝐺 ↾ 𝑋) ∧ (𝑌 ∈ dom 𝐹 ∧ 𝑌 ∈ dom 𝐺 ∧ (𝐹‘𝑌) = (𝐺‘𝑌))) → (𝐹 ↾ (𝑋 ∪ {𝑌})) = (𝐺 ↾ (𝑋 ∪ {𝑌}))) | ||
| 6-Dec-2021 | breprexp 30711 | Express the 𝑆 th power of the finite series in terms of the number of representations of integers 𝑚 as sums of 𝑆 terms. This is a general formulation which allows logarithmic weighting of the sums (see https://mathoverflow.net/questions/253246) and a mix of different smoothing functions taken into account in 𝐿. See breprexpnat 30712 for the simple case presented in the proposition of [Nathanson] p. 123. (Contributed by Thierry Arnoux, 6-Dec-2021.) |
| ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → 𝑆 ∈ ℕ0) & ⊢ (𝜑 → 𝑍 ∈ ℂ) & ⊢ (𝜑 → 𝐿:(0..^𝑆)⟶(ℂ ↑𝑚 ℕ)) ⇒ ⊢ (𝜑 → ∏𝑎 ∈ (0..^𝑆)Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑎)‘𝑏) · (𝑍↑𝑏)) = Σ𝑚 ∈ (0...(𝑆 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑐‘𝑎)) · (𝑍↑𝑚))) | ||
| 6-Dec-2021 | breprexplemc 30710 | Lemma for breprexp 30711 (induction step) (Contributed by Thierry Arnoux, 6-Dec-2021.) |
| ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → 𝑆 ∈ ℕ0) & ⊢ (𝜑 → 𝑍 ∈ ℂ) & ⊢ (𝜑 → 𝐿:(0..^𝑆)⟶(ℂ ↑𝑚 ℕ)) & ⊢ (𝜑 → 𝑇 ∈ ℕ0) & ⊢ (𝜑 → (𝑇 + 1) ≤ 𝑆) & ⊢ (𝜑 → ∏𝑎 ∈ (0..^𝑇)Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑎)‘𝑏) · (𝑍↑𝑏)) = Σ𝑚 ∈ (0...(𝑇 · 𝑁))Σ𝑑 ∈ ((1...𝑁)(repr‘𝑇)𝑚)(∏𝑎 ∈ (0..^𝑇)((𝐿‘𝑎)‘(𝑑‘𝑎)) · (𝑍↑𝑚))) ⇒ ⊢ (𝜑 → ∏𝑎 ∈ (0..^(𝑇 + 1))Σ𝑏 ∈ (1...𝑁)(((𝐿‘𝑎)‘𝑏) · (𝑍↑𝑏)) = Σ𝑚 ∈ (0...((𝑇 + 1) · 𝑁))Σ𝑑 ∈ ((1...𝑁)(repr‘(𝑇 + 1))𝑚)(∏𝑎 ∈ (0..^(𝑇 + 1))((𝐿‘𝑎)‘(𝑑‘𝑎)) · (𝑍↑𝑚))) | ||
| 5-Dec-2021 | nosupbnd1lem2 31855 | Lemma for nosupbnd1 31860. When there is no maximum, if any member of 𝐴 is a prolongment of 𝑆, then so are all elements of 𝐴 above it. (Contributed by Scott Fenton, 5-Dec-2021.) |
| ⊢ 𝑆 = if(∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦, ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2𝑜⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥)))) ⇒ ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ ((𝑈 ∈ 𝐴 ∧ (𝑈 ↾ dom 𝑆) = 𝑆) ∧ (𝑊 ∈ 𝐴 ∧ ¬ 𝑊 <s 𝑈))) → (𝑊 ↾ dom 𝑆) = 𝑆) | ||
| 5-Dec-2021 | nosupbnd1lem1 31854 | Lemma for nosupbnd1 31860. Establish a soft upper bound. (Contributed by Scott Fenton, 5-Dec-2021.) |
| ⊢ 𝑆 = if(∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦, ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2𝑜⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥)))) ⇒ ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ 𝑈 ∈ 𝐴) → ¬ 𝑆 <s (𝑈 ↾ dom 𝑆)) | ||
| 5-Dec-2021 | nosupres 31853 | A restriction law for surreal supremum when there is no maximum. (Contributed by Scott Fenton, 5-Dec-2021.) |
| ⊢ 𝑆 = if(∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦, ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2𝑜⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥)))) ⇒ ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑈 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) → (𝑆 ↾ suc 𝐺) = (𝑈 ↾ suc 𝐺)) | ||
| 5-Dec-2021 | nosupfv 31852 | The value of surreal supremum when there is no maximum. (Contributed by Scott Fenton, 5-Dec-2021.) |
| ⊢ 𝑆 = if(∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦, ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2𝑜⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥)))) ⇒ ⊢ ((¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 ∧ (𝐴 ⊆ No ∧ 𝐴 ∈ V) ∧ (𝑈 ∈ 𝐴 ∧ 𝐺 ∈ dom 𝑈 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑈 → (𝑈 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)))) → (𝑆‘𝐺) = (𝑈‘𝐺)) | ||
| 5-Dec-2021 | nosupbday 31851 | Birthday bounding law for surreal supremum. (Contributed by Scott Fenton, 5-Dec-2021.) |
| ⊢ 𝑆 = if(∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦, ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2𝑜⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥)))) ⇒ ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V) → ( bday ‘𝑆) ⊆ suc ∪ ( bday “ 𝐴)) | ||
| 5-Dec-2021 | nosupno 31849 | The next several theorems deal with a surreal "supremum". This surreal will ultimately be shown to bound 𝐴 below and bound the restriction of any surreal above. We begin by showing that the given expression actually defines a surreal number. (Contributed by Scott Fenton, 5-Dec-2021.) |
| ⊢ 𝑆 = if(∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦, ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2𝑜⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥)))) ⇒ ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ 𝑉) → 𝑆 ∈ No ) | ||
| 5-Dec-2021 | nomaxmo 31847 | A class of surreals has at most one maximum. (Contributed by Scott Fenton, 5-Dec-2021.) |
| ⊢ (𝑆 ⊆ No → ∃*𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ¬ 𝑥 <s 𝑦) | ||
| 5-Dec-2021 | noresle 31846 | Restriction law for surreals. Lemma 2.1.4 of [Lipparini] p. 3. (Contributed by Scott Fenton, 5-Dec-2021.) |
| ⊢ (((𝑈 ∈ No ∧ 𝑆 ∈ No ) ∧ (dom 𝑈 ⊆ 𝐴 ∧ dom 𝑆 ⊆ 𝐴 ∧ ∀𝑔 ∈ 𝐴 ¬ (𝑆 ↾ suc 𝑔) <s (𝑈 ↾ suc 𝑔))) → ¬ 𝑆 <s 𝑈) | ||
| 5-Dec-2021 | reprsuc 30693 | Express the representations recursively. (Contributed by Thierry Arnoux, 5-Dec-2021.) |
| ⊢ (𝜑 → 𝐴 ⊆ ℕ) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑆 ∈ ℕ0) & ⊢ 𝐹 = (𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏)) ↦ (𝑐 ∪ {⟨𝑆, 𝑏⟩})) ⇒ ⊢ (𝜑 → (𝐴(repr‘(𝑆 + 1))𝑀) = ∪ 𝑏 ∈ 𝐴 ran 𝐹) | ||
| 5-Dec-2021 | reprsum 30691 | Sums of values of the members of the representation of 𝑀 equal 𝑀. (Contributed by Thierry Arnoux, 5-Dec-2021.) |
| ⊢ (𝜑 → 𝐴 ⊆ ℕ) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑆 ∈ ℕ0) & ⊢ (𝜑 → 𝐶 ∈ (𝐴(repr‘𝑆)𝑀)) ⇒ ⊢ (𝜑 → Σ𝑎 ∈ (0..^𝑆)(𝐶‘𝑎) = 𝑀) | ||
| 5-Dec-2021 | reprf 30690 | Members of the representation of 𝑀 as the sum of 𝑆 nonnegative integers from set 𝐴 as functions. (Contributed by Thierry Arnoux, 5-Dec-2021.) |
| ⊢ (𝜑 → 𝐴 ⊆ ℕ) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑆 ∈ ℕ0) & ⊢ (𝜑 → 𝐶 ∈ (𝐴(repr‘𝑆)𝑀)) ⇒ ⊢ (𝜑 → 𝐶:(0..^𝑆)⟶𝐴) | ||
| 5-Dec-2021 | fzodif2 29552 | Split the last element of a half-open range of sequential integers. (Contributed by Thierry Arnoux, 5-Dec-2021.) |
| ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → ((𝑀..^(𝑁 + 1)) ∖ {𝑁}) = (𝑀..^𝑁)) | ||
| 5-Dec-2021 | nsnlpligALT 27334 | Alternate version of nsnlplig 27333 using the predicate ∉ instead of ¬ ∈ and whose proof is shorter. (Contributed by AV, 5-Dec-2021.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ (𝐺 ∈ Plig → {𝐴} ∉ 𝐺) | ||
| 5-Dec-2021 | nsnlplig 27333 | There is no "one-point line" in a planar incidence geometry. (Contributed by BJ, 2-Dec-2021.) (Proof shortened by AV, 5-Dec-2021.) |
| ⊢ (𝐺 ∈ Plig → ¬ {𝐴} ∈ 𝐺) | ||
| 5-Dec-2021 | toponcomb 20733 | Biconditional form of toponcom 20732. (Contributed by BJ, 5-Dec-2021.) |
| ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 ∈ (TopOn‘∪ 𝐾) ↔ 𝐾 ∈ (TopOn‘∪ 𝐽))) | ||
| 5-Dec-2021 | snnex 6966 | The class of all singletons is a proper class. See also pwnex 6968. (Contributed by NM, 10-Oct-2008.) (Proof shortened by Eric Schmidt, 7-Dec-2008.) (Proof shortened by BJ, 5-Dec-2021.) |
| ⊢ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∉ V | ||
| 5-Dec-2021 | pm2.61danel 2911 | Deduction eliminating an elementhood in an antecedent. (Contributed by AV, 5-Dec-2021.) |
| ⊢ ((𝜑 ∧ 𝐴 ∈ 𝐵) → 𝜓) & ⊢ ((𝜑 ∧ 𝐴 ∉ 𝐵) → 𝜓) ⇒ ⊢ (𝜑 → 𝜓) | ||
| 4-Dec-2021 | noetalem2 31864 | Lemma for noeta 31868. 𝑍 is an upper bound for 𝐴. Part of Theorem 5.1 of [Lipparini] p. 7-8. (Contributed by Scott Fenton, 4-Dec-2021.) |
| ⊢ 𝑆 = if(∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦, ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {⟨dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2𝑜⟩}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥)))) & ⊢ 𝑍 = (𝑆 ∪ ((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1𝑜})) ⇒ ⊢ (((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝑋 ∈ 𝐴) → 𝑋 <s 𝑍) | ||
| 3-Dec-2021 | rmodislmod 18931 | The right module 𝑅 induces a left module 𝐿 by replacing the scalar multiplication with a reversed multiplication if the scalar ring is commutative. The hypothesis "rmodislmod.r" is a definition of a right module analogous to the definition df-lmod 18865 of a left module, see also islmod 18867. (Contributed by AV, 3-Dec-2021.) |
| ⊢ 𝑉 = (Base‘𝑅) & ⊢ + = (+g‘𝑅) & ⊢ · = ( ·𝑠 ‘𝑅) & ⊢ 𝐹 = (Scalar‘𝑅) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ ⨣ = (+g‘𝐹) & ⊢ × = (.r‘𝐹) & ⊢ 1 = (1r‘𝐹) & ⊢ (𝑅 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑞 ∈ 𝐾 ∀𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 ∀𝑤 ∈ 𝑉 (((𝑤 · 𝑟) ∈ 𝑉 ∧ ((𝑤 + 𝑥) · 𝑟) = ((𝑤 · 𝑟) + (𝑥 · 𝑟)) ∧ (𝑤 · (𝑞 ⨣ 𝑟)) = ((𝑤 · 𝑞) + (𝑤 · 𝑟))) ∧ ((𝑤 · (𝑞 × 𝑟)) = ((𝑤 · 𝑞) · 𝑟) ∧ (𝑤 · 1 ) = 𝑤))) & ⊢ ∗ = (𝑠 ∈ 𝐾, 𝑣 ∈ 𝑉 ↦ (𝑣 · 𝑠)) & ⊢ 𝐿 = (𝑅 sSet ⟨( ·𝑠 ‘ndx), ∗ ⟩) ⇒ ⊢ (𝐹 ∈ CRing → 𝐿 ∈ LMod) | ||
| 3-Dec-2021 | rmodislmodlem 18930 | Lemma for rmodislmod 18931. This is the part of the proof of rmodislmod 18931 which requires the scalar ring to be commutative. (Contributed by AV, 3-Dec-2021.) |
| ⊢ 𝑉 = (Base‘𝑅) & ⊢ + = (+g‘𝑅) & ⊢ · = ( ·𝑠 ‘𝑅) & ⊢ 𝐹 = (Scalar‘𝑅) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ ⨣ = (+g‘𝐹) & ⊢ × = (.r‘𝐹) & ⊢ 1 = (1r‘𝐹) & ⊢ (𝑅 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑞 ∈ 𝐾 ∀𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 ∀𝑤 ∈ 𝑉 (((𝑤 · 𝑟) ∈ 𝑉 ∧ ((𝑤 + 𝑥) · 𝑟) = ((𝑤 · 𝑟) + (𝑥 · 𝑟)) ∧ (𝑤 · (𝑞 ⨣ 𝑟)) = ((𝑤 · 𝑞) + (𝑤 · 𝑟))) ∧ ((𝑤 · (𝑞 × 𝑟)) = ((𝑤 · 𝑞) · 𝑟) ∧ (𝑤 · 1 ) = 𝑤))) & ⊢ ∗ = (𝑠 ∈ 𝐾, 𝑣 ∈ 𝑉 ↦ (𝑣 · 𝑠)) & ⊢ 𝐿 = (𝑅 sSet ⟨( ·𝑠 ‘ndx), ∗ ⟩) ⇒ ⊢ ((𝐹 ∈ CRing ∧ (𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝐾 ∧ 𝑐 ∈ 𝑉)) → ((𝑎 × 𝑏) ∗ 𝑐) = (𝑎 ∗ (𝑏 ∗ 𝑐))) | ||
| 3-Dec-2021 | mapdm0 7872 | The empty set is the only map with empty domain. (Contributed by Glauco Siliprandi, 11-Oct-2020.) (Proof shortened by Thierry Arnoux, 3-Dec-2021.) |
| ⊢ (𝐵 ∈ 𝑉 → (𝐵 ↑𝑚 ∅) = {∅}) | ||
| 3-Dec-2021 | ralrot3 3102 | Rotate three restricted universal quantifiers. (Contributed by AV, 3-Dec-2021.) |
| ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜑 ↔ ∀𝑧 ∈ 𝐶 ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑) | ||
| 3-Dec-2021 | ralcom13 3100 | Swap first and third restricted universal quantifiers. (Contributed by AV, 3-Dec-2021.) |
| ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜑 ↔ ∀𝑧 ∈ 𝐶 ∀𝑦 ∈ 𝐵 ∀𝑥 ∈ 𝐴 𝜑) | ||
| 3-Dec-2021 | 2ralimi 2953 | Inference quantifying both antecedent and consequent two times, with strong hypothesis. (Contributed by AV, 3-Dec-2021.) |
| ⊢ (𝜑 → 𝜓) ⇒ ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓) | ||
| 2-Dec-2021 | repr0 30689 | There is exactly one representation with no elements (an empty sum), only for 𝑀 = 0. (Contributed by Thierry Arnoux, 2-Dec-2021.) |
| ⊢ (𝜑 → 𝐴 ⊆ ℕ) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑆 ∈ ℕ0) ⇒ ⊢ (𝜑 → (𝐴(repr‘0)𝑀) = if(𝑀 = 0, {∅}, ∅)) | ||
| 2-Dec-2021 | itgexpif 30684 | The basis for the circle method in the form of trigonometric sums. Proposition of [Nathanson] p. 123. (Contributed by Thierry Arnoux, 2-Dec-2021.) |
| ⊢ (𝑁 ∈ ℤ → ∫(0(,)1)(exp‘((i · (2 · π)) · (𝑁 · 𝑥))) d𝑥 = if(𝑁 = 0, 1, 0)) | ||
| 2-Dec-2021 | rabeqsnd 29342 | Conditions for a restricted class abstraction to be a singleton, in deduction form. (Contributed by Thierry Arnoux, 2-Dec-2021.) |
| ⊢ (𝑥 = 𝐵 → (𝜓 ↔ 𝜒)) & ⊢ (𝜑 → 𝐵 ∈ 𝐴) & ⊢ (𝜑 → 𝜒) & ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝜓) → 𝑥 = 𝐵) ⇒ ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝐵}) | ||
| 2-Dec-2021 | eulplig 27337 | Through two distinct points of a planar incidence geometry, there is a unique line. (Contributed by BJ, 2-Dec-2021.) |
| ⊢ 𝑃 = ∪ 𝐺 ⇒ ⊢ ((𝐺 ∈ Plig ∧ ((𝐴 ∈ 𝑃 ∧ 𝐵 ∈ 𝑃) ∧ 𝐴 ≠ 𝐵)) → ∃!𝑙 ∈ 𝐺 (𝐴 ∈ 𝑙 ∧ 𝐵 ∈ 𝑙)) | ||
| 2-Dec-2021 | n0lplig 27335 | There is no "empty line" in a planar incidence geometry. (Contributed by AV, 28-Nov-2021.) (Proof shortened by BJ, 2-Dec-2021.) |
| ⊢ (𝐺 ∈ Plig → ¬ ∅ ∈ 𝐺) | ||
| 2-Dec-2021 | ispligb 27329 | The predicate "is a planar incidence geometry". (Contributed by BJ, 2-Dec-2021.) |
| ⊢ 𝑃 = ∪ 𝐺 ⇒ ⊢ (𝐺 ∈ Plig ↔ (𝐺 ∈ V ∧ (∀𝑎 ∈ 𝑃 ∀𝑏 ∈ 𝑃 (𝑎 ≠ 𝑏 → ∃!𝑙 ∈ 𝐺 (𝑎 ∈ 𝑙 ∧ 𝑏 ∈ 𝑙)) ∧ ∀𝑙 ∈ 𝐺 ∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 (𝑎 ≠ 𝑏 ∧ 𝑎 ∈ 𝑙 ∧ 𝑏 ∈ 𝑙) ∧ ∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 ∃𝑐 ∈ 𝑃 ∀𝑙 ∈ 𝐺 ¬ (𝑎 ∈ 𝑙 ∧ 𝑏 ∈ 𝑙 ∧ 𝑐 ∈ 𝑙)))) | ||
| 2-Dec-2021 | abnexg 6964 | Sufficient condition for a class abstraction to be a proper class. The class 𝐹 can be thought of as an expression in 𝑥 and the abstraction appearing in the statement as the class of values 𝐹 as 𝑥 varies through 𝐴. Assuming the antecedents, if that class is a set, then so is the "domain" 𝐴. The converse holds without antecedent, see abrexexg 7140. Note that the second antecedent ∀𝑥 ∈ 𝐴𝑥 ∈ 𝐹 cannot be translated to 𝐴 ⊆ 𝐹 since 𝐹 may depend on 𝑥. In applications, one may take 𝐹 = {𝑥} or 𝐹 = 𝒫 𝑥 (see snnex 6966 and pwnex 6968 respectively, proved from abnex 6965, which is a consequence of abnexg 6964 with 𝐴 = V). (Contributed by BJ, 2-Dec-2021.) |
| ⊢ (∀𝑥 ∈ 𝐴 (𝐹 ∈ 𝑉 ∧ 𝑥 ∈ 𝐹) → ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐹} ∈ 𝑊 → 𝐴 ∈ V)) | ||
| 2-Dec-2021 | rspc2gv 3321 | Restricted specialization with two quantifiers, using implicit substitution. (Contributed by BJ, 2-Dec-2021.) |
| ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑊 𝜑 → 𝜓)) | ||
| 1-Dec-2021 | vtsval 30715 | Value of the Vinogradov trigonometric sums (Contributed by Thierry Arnoux, 1-Dec-2021.) |
| ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → 𝑋 ∈ ℂ) & ⊢ (𝜑 → 𝐿:ℕ⟶ℂ) ⇒ ⊢ (𝜑 → ((𝐿vts𝑁)‘𝑋) = Σ𝑎 ∈ (1...𝑁)((𝐿‘𝑎) · (exp‘((i · (2 · π)) · (𝑎 · 𝑋))))) | ||
| 1-Dec-2021 | df-vts 30714 | Define the Vinogradov trigonometric sums. (Contributed by Thierry Arnoux, 1-Dec-2021.) |
| ⊢ vts = (𝑙 ∈ (ℂ ↑𝑚 ℕ), 𝑛 ∈ ℕ0 ↦ (𝑥 ∈ ℂ ↦ Σ𝑎 ∈ (1...𝑛)((𝑙‘𝑎) · (exp‘((i · (2 · π)) · (𝑎 · 𝑥)))))) | ||
| 1-Dec-2021 | reprval 30688 | Value of the representations of 𝑀 as the sum of 𝑆 nonnegative integers in a given set 𝐴 (Contributed by Thierry Arnoux, 1-Dec-2021.) |
| ⊢ (𝜑 → 𝐴 ⊆ ℕ) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑆 ∈ ℕ0) ⇒ ⊢ (𝜑 → (𝐴(repr‘𝑆)𝑀) = {𝑐 ∈ (𝐴 ↑𝑚 (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀}) | ||
| 1-Dec-2021 | df-repr 30687 | The representations of a nonnegative 𝑚 as the sum of 𝑠 nonnegative integers from a set 𝑏. Cf. Definition of [Nathanson] p. 123. (Contributed by Thierry Arnoux, 1-Dec-2021.) |
| ⊢ repr = (𝑠 ∈ ℕ0 ↦ (𝑏 ∈ 𝒫 ℕ, 𝑚 ∈ ℤ ↦ {𝑐 ∈ (𝑏 ↑𝑚 (0..^𝑠)) ∣ Σ𝑎 ∈ (0..^𝑠)(𝑐‘𝑎) = 𝑚})) | ||
| 1-Dec-2021 | ftc2re 30676 | The Fundamental Theorem of Calculus, part two, for functions continuous on 𝐷. (Contributed by Thierry Arnoux, 1-Dec-2021.) |
| ⊢ 𝐸 = (𝐶(,)𝐷) & ⊢ (𝜑 → 𝐴 ∈ 𝐸) & ⊢ (𝜑 → 𝐵 ∈ 𝐸) & ⊢ (𝜑 → 𝐴 ≤ 𝐵) & ⊢ (𝜑 → 𝐹:𝐸⟶ℂ) & ⊢ (𝜑 → (ℝ D 𝐹) ∈ (𝐸–cn→ℂ)) ⇒ ⊢ (𝜑 → ∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 = ((𝐹‘𝐵) − (𝐹‘𝐴))) | ||
| 1-Dec-2021 | efcld 30669 | Closure law for the exponential function, deduction version. (Contributed by Thierry Arnoux, 1-Dec-2021.) |
| ⊢ (𝜑 → 𝐴 ∈ ℂ) ⇒ ⊢ (𝜑 → (exp‘𝐴) ∈ ℂ) | ||
| 1-Dec-2021 | eqrd 3622 | Deduce equality of classes from equivalence of membership. (Contributed by Thierry Arnoux, 21-Mar-2017.) (Proof shortened by BJ, 1-Dec-2021.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) ⇒ ⊢ (𝜑 → 𝐴 = 𝐵) | ||
| 30-Nov-2021 | noextendseq 31820 | Extend a surreal by a sequence of ordinals. (Contributed by Scott Fenton, 30-Nov-2021.) |
| ⊢ 𝑋 ∈ {1𝑜, 2𝑜} ⇒ ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ On) → (𝐴 ∪ ((𝐵 ∖ dom 𝐴) × {𝑋})) ∈ No ) | ||
| 28-Nov-2021 | pliguhgr 27338 | Any planar incidence geometry can be regarded as a hypergraph with its points as vertices and its lines as edges. See incistruhgr 25974 for a generalization of this case for arbitrary incidence structures (planar incidence geometries are such incidence structures). (Proposed by Gerard Lang, 24-Nov-2021.) (Contributed by AV, 28-Nov-2021.) |
| ⊢ (𝐺 ∈ Plig → ⟨∪ 𝐺, ( I ↾ 𝐺)⟩ ∈ UHGraph ) | ||
| 28-Nov-2021 | n0lpligALT 27336 | Alternate version of n0lplig 27335 using the predicate ∉ instead of ¬ ∈ and whose proof bypasses nsnlplig 27333. (Contributed by AV, 28-Nov-2021.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ (𝐺 ∈ Plig → ∅ ∉ 𝐺) | ||
| 28-Nov-2021 | l2p 27331 | For any line in a planar incidence geometry, there exist two different points on the line. (Contributed by AV, 28-Nov-2021.) |
| ⊢ 𝑃 = ∪ 𝐺 ⇒ ⊢ ((𝐺 ∈ Plig ∧ 𝐿 ∈ 𝐺) → ∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 (𝑎 ≠ 𝑏 ∧ 𝑎 ∈ 𝐿 ∧ 𝑏 ∈ 𝐿)) | ||
| 27-Nov-2021 | uspgrbisymrel 41762 |
There is a bijection between the "simple pseudographs" for a fixed
set
𝑉 of vertices and the class 𝑅 of the
symmetric relations on the
fixed set 𝑉. The simple pseudographs, which are
graphs without
hyper- or multiedges, but which may contain loops, are expressed as
ordered pairs of the vertices and the edges (as proper or improper
unordered pairs of vertices, not as indexed edges!) in this theorem.
That class 𝐺 of such simple pseudographs is a set
(if 𝑉 is a
set, see uspgrex 41758) of equivalence classes of graphs
abstracting from
the index sets of their edge functions.
Solely for this abstraction, there is a bijection between the "simple pseudographs" as members of 𝐺 and the symmetric relations 𝑅 on the fixed set 𝑉 of vertices. This theorem would not hold for 𝐺 = {𝑔 ∈ USPGraph ∣ (Vtx‘𝑔) = 𝑉} and even not for 𝐺 = {⟨𝑣, 𝑒⟩ ∣ (𝑣 = 𝑉 ∧ ⟨𝑣, 𝑒⟩ ∈ USPGraph )}, because these are much bigger classes. (Proposed by Gerard Lang, 16-Nov-2021.) (Contributed by AV, 27-Nov-2021.) |
| ⊢ 𝐺 = {⟨𝑣, 𝑒⟩ ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))} & ⊢ 𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑟𝑦 ↔ 𝑦𝑟𝑥)} ⇒ ⊢ (𝑉 ∈ 𝑊 → ∃𝑓 𝑓:𝐺–1-1-onto→𝑅) | ||
| 27-Nov-2021 | uspgrymrelen 41761 | The set 𝐺 of the "simple pseudographs" with a fixed set of vertices 𝑉 and the class 𝑅 of the symmetric relations on the fixed set 𝑉 are equinumerous. For more details about the class 𝐺 of all "simple pseudographs" see comments on uspgrbisymrel 41762. (Contributed by AV, 27-Nov-2021.) |
| ⊢ 𝐺 = {⟨𝑣, 𝑒⟩ ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))} & ⊢ 𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑟𝑦 ↔ 𝑦𝑟𝑥)} ⇒ ⊢ (𝑉 ∈ 𝑊 → 𝐺 ≈ 𝑅) | ||
| 27-Nov-2021 | uspgrspren 41760 | The set 𝐺 of the "simple pseudographs" with a fixed set of vertices 𝑉 and the class 𝑃 of subsets of the set of pairs over the fixed set 𝑉 are equinumerous. (Contributed by AV, 27-Nov-2021.) |
| ⊢ 𝑃 = 𝒫 (Pairs‘𝑉) & ⊢ 𝐺 = {⟨𝑣, 𝑒⟩ ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))} ⇒ ⊢ (𝑉 ∈ 𝑊 → 𝐺 ≈ 𝑃) | ||
| 27-Nov-2021 | sprsymrelen 41750 | The class 𝑃 of subsets of the set of pairs over a fixed set 𝑉 and the class 𝑅 of symmetric relations on the fixed set 𝑉 are equinumerous. (Contributed by AV, 27-Nov-2021.) |
| ⊢ 𝑃 = 𝒫 (Pairs‘𝑉) & ⊢ 𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑟𝑦 ↔ 𝑦𝑟𝑥)} ⇒ ⊢ (𝑉 ∈ 𝑊 → 𝑃 ≈ 𝑅) | ||
| 27-Nov-2021 | wl-ax8clv2 33381 | Axiom ax-wl-8cl 33377 carries over to our new definition df-wl-clelv2 33380. (Contributed by Wolf Lammen, 27-Nov-2021.) |
| ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 → 𝑦 ∈ 𝐴)) | ||
| 27-Nov-2021 | df-wl-clelv2 33380 | Define the term 𝑥 ∈ 𝐴, 𝑥 in 𝐴 permitted. (Contributed by Wolf Lammen, 27-Nov-2021.) |
| ⊢ (𝑥 ∈ 𝐴 ↔ ∀𝑢(𝑢 = 𝑥 → 𝑢 ∈ 𝐴)) | ||
| 27-Nov-2021 | wl-clelv2-just 33379 | Show that the definition df-wl-clelv2 33380 is conservative. (Contributed by Wolf Lammen, 27-Nov-2021.) |
| ⊢ (𝑥 ∈ 𝐴 ↔ ∀𝑢(𝑢 = 𝑥 → 𝑢 ∈ 𝐴)) | ||
| 27-Nov-2021 | wl-ax8clv1 33378 | Lifting the distinct variable constraint on 𝑥 and 𝑦 in ax-wl-8cl 33377. (Contributed by Wolf Lammen, 27-Nov-2021.) |
| ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 → 𝑦 ∈ 𝐴)) | ||
| 27-Nov-2021 | ax-wl-8cl 33377 |
In ZFC, as presented in this document, classes are meant to be just a
notational convenience, that can be reduced to pure set theory by means
of df-clab 2609 (there stated in the eliminable property).
That is, in an
expression 𝑥 ∈ 𝐴, the class variable 𝐴 is
implicitely assumed
to represent an expression {𝑧 ∣ 𝜑} with some appropriate 𝜑.
Unfortunately, 𝜑 syntactically covers any well-formed
formula
(wff), including 𝑧 ∈ 𝐴. This choice inevitably breaks the
stated
property. And it potentially carries over to any expression containing
class variables. To fix this, a simple rule could exclude class
variables at all in a class defining wff. A more elaborate rule could
detect, and limit exclusion to proper classes (potentially problematic).
In any case, the verification process should enforce any such rule
during replacement, which it currently does not. The result is that we
rely on the awareness of theorem designers to this problem. It seems,
in ZFC proper classes are reduced to a few instances only, so a careful
study may reveal that this limited use does not impose logical loop
holes. It must be said, still, this necessary extra knowledge
contradicts the general philosophy of Metamath, trying to establish
certainty by a machine executable confirmation.
An extension to ZFC allows classes to exist on their own. Classes are then extensions to sets, also seamlessly extending the idea of elementhood. In order to move from the general to the specific, sets presuppose classes, so classes should be introduced first. This is somewhat in opposition to the classic order of introduction of syntactic elements, but has been carried out in the past, for example by the von-Neumann theory of classes. In the context of Metamath, which is a purely text-based syntactical concept, no semantics are imposed at the very beginning on classes. Instead axioms will narrow down bit by bit how elementhood behaves in proofs, of course always with the intuitive understanding of a human in mind. One basic property of elementhood we expect is that the 'element' is replaceable with something equal to it. Or that equality is finer grained than the elementhood relation. This idea is formally expressed in terms coined 'implicit substitution' in this document: ((𝑥 = 𝑦) → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)). This axiom prepares this notation. Note that particular constructions of classes like that in df-clab 2609 in fact allow to prove this axiom. Can we expect to eliminate this axiom then? No, the generalizing term still refers to an unexplained subterm 𝑥 ∈ 𝐴, so this axiom recurs in the general case. On the other hand, our axiom here stays true, even when just the existence of a class is known, as is often the case after applying the axiom of choice, without a chance to actually construct it. We provide a version of this axiom, that requires all variables to be distinct. Step by step these restrictions are lifted, in the end covering the most general term 𝐴 ∈ 𝐵. This axiom is meant as a replacement for ax-8 1992. (Contributed by Wolf Lammen, 27-Nov-2021.) |
| ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 → 𝑦 ∈ 𝐴)) | ||
| 27-Nov-2021 | wel2-wl 33376 | Redefine ∈ in a set context to avoid syntax check errors in mmj2. It is no syntactic error to assign the same variable to 𝑥 and 𝑦. (Contributed by Wolf Lammen, 27-Nov-2021.) |
| wff 𝑥 ∈ 𝑦 | ||
| 27-Nov-2021 | wel-wl 33374 | Redefine ∈ in a set context to avoid syntax check errors in mmj2. 𝑥 and 𝑦 must be distinct. (Contributed by Wolf Lammen, 27-Nov-2021.) |
| wff 𝑥 ∈ 𝑦 | ||
| 26-Nov-2021 | uspgrbisymrelALT 41763 | Alternate proof of uspgrbisymrel 41762 not using the definition of equinumerosity. (Contributed by AV, 26-Nov-2021.) (New usage is discouraged.) (Proof modification is discouraged.) |
| ⊢ 𝐺 = {⟨𝑣, 𝑒⟩ ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))} & ⊢ 𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑟𝑦 ↔ 𝑦𝑟𝑥)} ⇒ ⊢ (𝑉 ∈ 𝑊 → ∃𝑓 𝑓:𝐺–1-1-onto→𝑅) | ||
| 26-Nov-2021 | uspgrbispr 41759 | There is a bijection between the "simple pseudographs" with a fixed set of vertices 𝑉 and the subsets of the set of pairs over the set 𝑉. (Contributed by AV, 26-Nov-2021.) |
| ⊢ 𝑃 = 𝒫 (Pairs‘𝑉) & ⊢ 𝐺 = {⟨𝑣, 𝑒⟩ ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))} ⇒ ⊢ (𝑉 ∈ 𝑊 → ∃𝑓 𝑓:𝐺–1-1-onto→𝑃) | ||
| 26-Nov-2021 | uspgrex 41758 | The class 𝐺 of all "simple pseudographs" with a fixed set of vertices 𝑉 is a set. (Contributed by AV, 26-Nov-2021.) |
| ⊢ 𝑃 = 𝒫 (Pairs‘𝑉) & ⊢ 𝐺 = {⟨𝑣, 𝑒⟩ ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))} ⇒ ⊢ (𝑉 ∈ 𝑊 → 𝐺 ∈ V) | ||
| 26-Nov-2021 | noprefixmo 31848 | In any class of surreals, there is at most one value of the prefix property. (Contributed by Scott Fenton, 26-Nov-2021.) |
| ⊢ (𝐴 ⊆ No → ∃*𝑥∃𝑢 ∈ 𝐴 (𝐺 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝐺) = (𝑣 ↾ suc 𝐺)) ∧ (𝑢‘𝐺) = 𝑥)) | ||
| 26-Nov-2021 | upgredg 26032 | For each edge in a pseudograph, there are two vertices which are connected by this edge. (Contributed by AV, 4-Nov-2020.) (Proof shortened by AV, 26-Nov-2021.) |
| ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ ((𝐺 ∈ UPGraph ∧ 𝐶 ∈ 𝐸) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝐶 = {𝑎, 𝑏}) | ||
| 26-Nov-2021 | 0nelfun 5906 | A function does not contain the empty set. (Contributed by BJ, 26-Nov-2021.) |
| ⊢ (Fun 𝑅 → ∅ ∉ 𝑅) | ||
| 26-Nov-2021 | cnvcnv 5586 | The double converse of a class strips out all elements that are not ordered pairs. (Contributed by NM, 8-Dec-2003.) (Proof shortened by BJ, 26-Nov-2021.) |
| ⊢ ◡◡𝐴 = (𝐴 ∩ (V × V)) | ||
| 26-Nov-2021 | opabssxpd 5338 | An ordered-pair class abstraction is a subset of an Cartesian product. Formerly part of proof for opabex2 7227. (Contributed by AV, 26-Nov-2021.) |
| ⊢ ((𝜑 ∧ 𝜓) → 𝑥 ∈ 𝐴) & ⊢ ((𝜑 ∧ 𝜓) → 𝑦 ∈ 𝐵) ⇒ ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} ⊆ (𝐴 × 𝐵)) | ||
| 25-Nov-2021 | uspgrsprf1o 41757 | The mapping 𝐹 is a bijection between the "simple pseudographs" with a fixed set of vertices 𝑉 and the subsets of the set of pairs over the set 𝑉. See also the comments on uspgrbisymrel 41762. (Contributed by AV, 25-Nov-2021.) |
| ⊢ 𝑃 = 𝒫 (Pairs‘𝑉) & ⊢ 𝐺 = {⟨𝑣, 𝑒⟩ ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))} & ⊢ 𝐹 = (𝑔 ∈ 𝐺 ↦ (2nd ‘𝑔)) ⇒ ⊢ (𝑉 ∈ 𝑊 → 𝐹:𝐺–1-1-onto→𝑃) | ||
| 25-Nov-2021 | uspgrsprfo 41756 | The mapping 𝐹 is a function from the "simple pseudographs" with a fixed set of vertices 𝑉 onto the subsets of the set of pairs over the set 𝑉. (Contributed by AV, 25-Nov-2021.) |
| ⊢ 𝑃 = 𝒫 (Pairs‘𝑉) & ⊢ 𝐺 = {⟨𝑣, 𝑒⟩ ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))} & ⊢ 𝐹 = (𝑔 ∈ 𝐺 ↦ (2nd ‘𝑔)) ⇒ ⊢ (𝑉 ∈ 𝑊 → 𝐹:𝐺–onto→𝑃) | ||
| 25-Nov-2021 | uspgrsprf1 41755 | The mapping 𝐹 is a one-to-one function from the "simple pseudographs" with a fixed set of vertices 𝑉 into the subsets of the set of pairs over the set 𝑉. (Contributed by AV, 25-Nov-2021.) |
| ⊢ 𝑃 = 𝒫 (Pairs‘𝑉) & ⊢ 𝐺 = {⟨𝑣, 𝑒⟩ ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))} & ⊢ 𝐹 = (𝑔 ∈ 𝐺 ↦ (2nd ‘𝑔)) ⇒ ⊢ 𝐹:𝐺–1-1→𝑃 | ||
| 25-Nov-2021 | isuspgrop 26056 | The property of being an undirected simple pseudograph represented as an ordered pair. The representation as an ordered pair is the usual representation of a graph, see section I.1 of [Bollobas] p. 1. (Contributed by AV, 25-Nov-2021.) |
| ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (⟨𝑉, 𝐸⟩ ∈ USPGraph ↔ 𝐸:dom 𝐸–1-1→{𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑝) ≤ 2})) | ||
| 24-Nov-2021 | uspgrsprf 41754 | The mapping 𝐹 is a function from the "simple pseudographs" with a fixed set of vertices 𝑉 into the subsets of the set of pairs over the set 𝑉. (Contributed by AV, 24-Nov-2021.) |
| ⊢ 𝑃 = 𝒫 (Pairs‘𝑉) & ⊢ 𝐺 = {⟨𝑣, 𝑒⟩ ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))} & ⊢ 𝐹 = (𝑔 ∈ 𝐺 ↦ (2nd ‘𝑔)) ⇒ ⊢ 𝐹:𝐺⟶𝑃 | ||
| 24-Nov-2021 | uspgrsprfv 41753 | The value of the function 𝐹 which maps a "simple pseudograph" for a fixed set 𝑉 of vertices to the set of edges (i.e. range of the edge function) of the graph. Solely for 𝐺 as defined here, the function 𝐹 is a bijection between the "simple pseudographs" and the subsets of the set of pairs 𝑃 over the the fixed set 𝑉 of vertices, see uspgrbispr 41759. (Contributed by AV, 24-Nov-2021.) |
| ⊢ 𝑃 = 𝒫 (Pairs‘𝑉) & ⊢ 𝐺 = {⟨𝑣, 𝑒⟩ ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))} & ⊢ 𝐹 = (𝑔 ∈ 𝐺 ↦ (2nd ‘𝑔)) ⇒ ⊢ (𝑋 ∈ 𝐺 → (𝐹‘𝑋) = (2nd ‘𝑋)) | ||
| 24-Nov-2021 | uspgropssxp 41752 | The set 𝐺 of "simple pseudographs" for a fixed set 𝑉 of vertices is a subset of an Cartesian product. For more details about the class 𝐺 of all "simple pseudographs" see comments on uspgrbisymrel 41762. (Contributed by AV, 24-Nov-2021.) |
| ⊢ 𝑃 = 𝒫 (Pairs‘𝑉) & ⊢ 𝐺 = {⟨𝑣, 𝑒⟩ ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))} ⇒ ⊢ (𝑉 ∈ 𝑊 → 𝐺 ⊆ (𝑊 × 𝑃)) | ||
| 24-Nov-2021 | upgredgssspr 41751 | The set of edges of a pseudograph is a subset of the set of unordered pairs of vertices. (Contributed by AV, 24-Nov-2021.) |
| ⊢ (𝐺 ∈ UPGraph → (Edg‘𝐺) ⊆ (Pairs‘(Vtx‘𝐺))) | ||
| 24-Nov-2021 | sprvalpwle2 41739 | The set of all unordered pairs over a given set 𝑉, expressed by a restricted class abstraction. (Contributed by AV, 24-Nov-2021.) |
| ⊢ (𝑉 ∈ 𝑊 → (Pairs‘𝑉) = {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑝) ≤ 2}) | ||
| 24-Nov-2021 | nosepdm 31834 | The first place two surreals differ is an element of the larger of their domains. (Contributed by Scott Fenton, 24-Nov-2021.) |
| ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 ≠ 𝐵) → ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)} ∈ (dom 𝐴 ∪ dom 𝐵)) | ||
| 24-Nov-2021 | nosepdmlem 31833 | Lemma for nosepdm 31834. (Contributed by Scott Fenton, 24-Nov-2021.) |
| ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 <s 𝐵) → ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)} ∈ (dom 𝐴 ∪ dom 𝐵)) | ||
| 24-Nov-2021 | nosepne 31831 | The value of two non-equal surreals at the first place they differ is different. (Contributed by Scott Fenton, 24-Nov-2021.) |
| ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 ≠ 𝐵) → (𝐴‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) ≠ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)})) | ||
| 24-Nov-2021 | nosepnelem 31830 | Lemma for nosepne 31831. (Contributed by Scott Fenton, 24-Nov-2021.) |
| ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐴 <s 𝐵) → (𝐴‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)}) ≠ (𝐵‘∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ (𝐵‘𝑥)})) | ||
| 24-Nov-2021 | soasym 31657 | Asymmetry law for strict orderings. (Contributed by Scott Fenton, 24-Nov-2021.) |
| ⊢ ((𝑅 Or 𝐴 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴)) → (𝑋𝑅𝑌 → ¬ 𝑌𝑅𝑋)) | ||
| 24-Nov-2021 | sotr3 31656 | Transitivity law for strict orderings. (Contributed by Scott Fenton, 24-Nov-2021.) |
| ⊢ ((𝑅 Or 𝐴 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑍 ∈ 𝐴)) → ((𝑋𝑅𝑌 ∧ ¬ 𝑍𝑅𝑌) → 𝑋𝑅𝑍)) | ||
| 24-Nov-2021 | sotrd 31655 | Transitivity law for strict orderings, deduction form. (Contributed by Scott Fenton, 24-Nov-2021.) |
| ⊢ (𝜑 → 𝑅 Or 𝐴) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → 𝑌 ∈ 𝐴) & ⊢ (𝜑 → 𝑍 ∈ 𝐴) & ⊢ (𝜑 → 𝑋𝑅𝑌) & ⊢ (𝜑 → 𝑌𝑅𝑍) ⇒ ⊢ (𝜑 → 𝑋𝑅𝑍) | ||
| 24-Nov-2021 | hashle2prv 13260 | A nonempty subset of a powerset of a class 𝑉 has size less than or equal to two iff it is an unordered pair of elements of 𝑉. (Contributed by AV, 24-Nov-2021.) |
| ⊢ (𝑃 ∈ (𝒫 𝑉 ∖ {∅}) → ((#‘𝑃) ≤ 2 ↔ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑃 = {𝑎, 𝑏})) | ||
| 24-Nov-2021 | hashle2pr 13259 | A nonempty set of size less than or equal to two is an unordered pair of sets. (Contributed by AV, 24-Nov-2021.) |
| ⊢ ((𝑃 ∈ 𝑉 ∧ 𝑃 ≠ ∅) → ((#‘𝑃) ≤ 2 ↔ ∃𝑎∃𝑏 𝑃 = {𝑎, 𝑏})) | ||
| 24-Nov-2021 | xnn0le2is012 12076 | An extended nonnegative integer which is less than or equal to 2 is either 0 or 1 or 2. (Contributed by AV, 24-Nov-2021.) |
| ⊢ ((𝑁 ∈ ℕ0* ∧ 𝑁 ≤ 2) → (𝑁 = 0 ∨ 𝑁 = 1 ∨ 𝑁 = 2)) | ||
| 24-Nov-2021 | xnn0lenn0nn0 12075 | An extended nonnegative integer which is less than or equal to a nonnegative integer is a nonnegative integer. (Contributed by AV, 24-Nov-2021.) |
| ⊢ ((𝑀 ∈ ℕ0* ∧ 𝑁 ∈ ℕ0 ∧ 𝑀 ≤ 𝑁) → 𝑀 ∈ ℕ0) | ||
| 23-Nov-2021 | sprbisymrel 41749 | There is a bijection between the subsets of the set of pairs over a fixed set 𝑉 and the symmetric relations 𝑅 on the fixed set 𝑉. (Contributed by AV, 23-Nov-2021.) |
| ⊢ 𝑃 = 𝒫 (Pairs‘𝑉) & ⊢ 𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑟𝑦 ↔ 𝑦𝑟𝑥)} ⇒ ⊢ (𝑉 ∈ 𝑊 → ∃𝑓 𝑓:𝑃–1-1-onto→𝑅) | ||
| 23-Nov-2021 | sprsymrelf1o 41748 | The mapping 𝐹 is a bijection between the subsets of the set of pairs over a fixed set 𝑉 into the symmetric relations 𝑅 on the fixed set 𝑉. (Contributed by AV, 23-Nov-2021.) |
| ⊢ 𝑃 = 𝒫 (Pairs‘𝑉) & ⊢ 𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑟𝑦 ↔ 𝑦𝑟𝑥)} & ⊢ 𝐹 = (𝑝 ∈ 𝑃 ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}) ⇒ ⊢ (𝑉 ∈ 𝑊 → 𝐹:𝑃–1-1-onto→𝑅) | ||
| 23-Nov-2021 | sprsymrelfo 41747 | The mapping 𝐹 is a function from the subsets of the set of pairs over a fixed set 𝑉 onto the symmetric relations 𝑅 on the fixed set 𝑉. (Contributed by AV, 23-Nov-2021.) |
| ⊢ 𝑃 = 𝒫 (Pairs‘𝑉) & ⊢ 𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑟𝑦 ↔ 𝑦𝑟𝑥)} & ⊢ 𝐹 = (𝑝 ∈ 𝑃 ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}) ⇒ ⊢ (𝑉 ∈ 𝑊 → 𝐹:𝑃–onto→𝑅) | ||
| 23-Nov-2021 | sprsymrelfolem2 41743 | Lemma 2 for sprsymrelfo 41747. (Contributed by AV, 23-Nov-2021.) |
| ⊢ 𝑄 = {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑅𝑏)} ⇒ ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥)) → (𝑥𝑅𝑦 ↔ ∃𝑐 ∈ 𝑄 𝑐 = {𝑥, 𝑦})) | ||
| 23-Nov-2021 | slttrine 31876 | Trichotomy law for surreals. (Contributed by Scott Fenton, 23-Nov-2021.) |
| ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 ≠ 𝐵 ↔ (𝐴 <s 𝐵 ∨ 𝐵 <s 𝐴))) | ||
| 22-Nov-2021 | sprsymrelfolem1 41742 | Lemma 1 for sprsymrelfo 41747. (Contributed by AV, 22-Nov-2021.) |
| ⊢ 𝑄 = {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑅𝑏)} ⇒ ⊢ 𝑄 ∈ 𝒫 (Pairs‘𝑉) | ||
| 22-Nov-2021 | sprsymrelf1lem 41741 | Lemma for sprsymrelf1 41746. (Contributed by AV, 22-Nov-2021.) |
| ⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑏 ⊆ (Pairs‘𝑉)) → ({⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑎 𝑐 = {𝑥, 𝑦}} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑏 𝑐 = {𝑥, 𝑦}} → 𝑎 ⊆ 𝑏)) | ||
| 22-Nov-2021 | prsprel 41737 | The elements of a pair from the set of all unordered pairs over a given set 𝑉 are elements of the set 𝑉. (Contributed by AV, 22-Nov-2021.) |
| ⊢ (({𝑋, 𝑌} ∈ (Pairs‘𝑉) ∧ (𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑊)) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) | ||
| 22-Nov-2021 | prssspr 41735 | An element of a subset of the set of all unordered pairs over a given set 𝑉, is a pair of elements of the set 𝑉. (Contributed by AV, 22-Nov-2021.) |
| ⊢ ((𝑃 ⊆ (Pairs‘𝑉) ∧ 𝑋 ∈ 𝑃) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑋 = {𝑎, 𝑏}) | ||
| 22-Nov-2021 | sprel 41734 | An element of the set of all unordered pairs over a given set 𝑉 is a pair of elements of the set 𝑉. (Contributed by AV, 22-Nov-2021.) |
| ⊢ (𝑋 ∈ (Pairs‘𝑉) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑋 = {𝑎, 𝑏}) | ||
| 22-Nov-2021 | noextendgt 31823 | Extending a surreal with a positive sign results in a bigger surreal. (Contributed by Scott Fenton, 22-Nov-2021.) |
| ⊢ (𝐴 ∈ No → 𝐴 <s (𝐴 ∪ {⟨dom 𝐴, 2𝑜⟩})) | ||
| 22-Nov-2021 | noextendlt 31822 | Extending a surreal with a negative sign results in a smaller surreal. (Contributed by Scott Fenton, 22-Nov-2021.) |
| ⊢ (𝐴 ∈ No → (𝐴 ∪ {⟨dom 𝐴, 1𝑜⟩}) <s 𝐴) | ||
| 22-Nov-2021 | noextenddif 31821 | Calculate the place where a surreal and its extension differ. (Contributed by Scott Fenton, 22-Nov-2021.) |
| ⊢ 𝑋 ∈ {1𝑜, 2𝑜} ⇒ ⊢ (𝐴 ∈ No → ∩ {𝑥 ∈ On ∣ (𝐴‘𝑥) ≠ ((𝐴 ∪ {⟨dom 𝐴, 𝑋⟩})‘𝑥)} = dom 𝐴) | ||
| 22-Nov-2021 | noextend 31819 | Extending a surreal by one sign value results in a new surreal. (Contributed by Scott Fenton, 22-Nov-2021.) |
| ⊢ 𝑋 ∈ {1𝑜, 2𝑜} ⇒ ⊢ (𝐴 ∈ No → (𝐴 ∪ {⟨dom 𝐴, 𝑋⟩}) ∈ No ) | ||
| 21-Nov-2021 | prsssprel 41738 | The elements of a pair from a subset of the set of all unordered pairs over a given set 𝑉 are elements of the set 𝑉. (Contributed by AV, 21-Nov-2021.) |
| ⊢ ((𝑃 ⊆ (Pairs‘𝑉) ∧ {𝑋, 𝑌} ∈ 𝑃 ∧ (𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑊)) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) | ||
| 21-Nov-2021 | prelspr 41736 | An unordered pair of elements of a fixed set 𝑉 belongs to the set of all unordered pairs over the set 𝑉. (Contributed by AV, 21-Nov-2021.) |
| ⊢ ((𝑉 ∈ 𝑊 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → {𝑋, 𝑌} ∈ (Pairs‘𝑉)) | ||
| 21-Nov-2021 | sprvalpwn0 41733 | The set of all unordered pairs over a given set 𝑉, expressed by a restricted class abstraction. (Contributed by AV, 21-Nov-2021.) |
| ⊢ (𝑉 ∈ 𝑊 → (Pairs‘𝑉) = {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏}}) | ||
| 21-Nov-2021 | spr0el 41732 | The empty set is not an unordered pair over any set 𝑉. (Contributed by AV, 21-Nov-2021.) |
| ⊢ ∅ ∉ (Pairs‘𝑉) | ||
| 21-Nov-2021 | sprssspr 41731 | The set of all unordered pairs over a given set 𝑉 is a subset of the set of all unordered pairs. (Contributed by AV, 21-Nov-2021.) |
| ⊢ (Pairs‘𝑉) ⊆ {𝑝 ∣ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}} | ||
| 21-Nov-2021 | sprvalpw 41730 | The set of all unordered pairs over a given set 𝑉, expressed by a restricted class abstraction. (Contributed by AV, 21-Nov-2021.) |
| ⊢ (𝑉 ∈ 𝑊 → (Pairs‘𝑉) = {𝑝 ∈ 𝒫 𝑉 ∣ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏}}) | ||
| 21-Nov-2021 | sprval 41729 | The set of all unordered pairs over a given set 𝑉. (Contributed by AV, 21-Nov-2021.) |
| ⊢ (𝑉 ∈ 𝑊 → (Pairs‘𝑉) = {𝑝 ∣ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏}}) | ||
| 21-Nov-2021 | df-spr 41728 | Define the function which maps a set 𝑣 to the set of pairs consisting of elements of the set 𝑣. (Contributed by AV, 21-Nov-2021.) |
| ⊢ Pairs = (𝑣 ∈ V ↦ {𝑝 ∣ ∃𝑎 ∈ 𝑣 ∃𝑏 ∈ 𝑣 𝑝 = {𝑎, 𝑏}}) | ||
| 21-Nov-2021 | spr0nelg 41726 | The empty set is not an element of all unordered pairs. (Contributed by AV, 21-Nov-2021.) |
| ⊢ ∅ ∉ {𝑝 ∣ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}} | ||
| 21-Nov-2021 | elsprel 41725 | An unordered pair is an element of all unordered pairs. At least one of the two elements of the unordered pair must be a set. Otherwise, the unordered pair would be the empty set, see prprc 4302, which is not an element of all unordered pairs, see spr0nelg 41726. (Contributed by AV, 21-Nov-2021.) |
| ⊢ ((𝐴 ∈ 𝑉 ∨ 𝐵 ∈ 𝑊) → {𝐴, 𝐵} ∈ {𝑝 ∣ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}}) | ||
| 21-Nov-2021 | sprid 41724 | Two identical representations of the class of all unordered pairs. (Contributed by AV, 21-Nov-2021.) |
| ⊢ {𝑝 ∣ ∃𝑎 ∈ V ∃𝑏 ∈ V 𝑝 = {𝑎, 𝑏}} = {𝑝 ∣ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}} | ||
| 21-Nov-2021 | dford5 31608 | A class is ordinal iff it is a subclass of On and transitive. (Contributed by Scott Fenton, 21-Nov-2021.) |
| ⊢ (Ord 𝐴 ↔ (𝐴 ⊆ On ∧ Tr 𝐴)) | ||
| 19-Nov-2021 | sprsymrelf1 41746 | The mapping 𝐹 is a one-to-one function from the subsets of the set of pairs over a fixed set 𝑉 into the symmetric relations 𝑅 on the fixed set 𝑉. (Contributed by AV, 19-Nov-2021.) |
| ⊢ 𝑃 = 𝒫 (Pairs‘𝑉) & ⊢ 𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑟𝑦 ↔ 𝑦𝑟𝑥)} & ⊢ 𝐹 = (𝑝 ∈ 𝑃 ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}) ⇒ ⊢ 𝐹:𝑃–1-1→𝑅 | ||
| 19-Nov-2021 | sprsymrelf 41745 | The mapping 𝐹 is a function from the subsets of the set of pairs over a fixed set 𝑉 into the symmetric relations 𝑅 on the fixed set 𝑉. (Contributed by AV, 19-Nov-2021.) |
| ⊢ 𝑃 = 𝒫 (Pairs‘𝑉) & ⊢ 𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑟𝑦 ↔ 𝑦𝑟𝑥)} & ⊢ 𝐹 = (𝑝 ∈ 𝑃 ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}) ⇒ ⊢ 𝐹:𝑃⟶𝑅 | ||
| 19-Nov-2021 | sprsymrelfv 41744 | The value of the function 𝐹 which maps a subset of the set of pairs over a fixed set 𝑉 to the relation relating two elements of the set 𝑉 iff they are in a pair of the subset. (Contributed by AV, 19-Nov-2021.) |
| ⊢ 𝑃 = 𝒫 (Pairs‘𝑉) & ⊢ 𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑟𝑦 ↔ 𝑦𝑟𝑥)} & ⊢ 𝐹 = (𝑝 ∈ 𝑃 ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}) ⇒ ⊢ (𝑋 ∈ 𝑃 → (𝐹‘𝑋) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑋 𝑐 = {𝑥, 𝑦}}) | ||
| 19-Nov-2021 | sprsymrelfvlem 41740 | Lemma for sprsymrelf 41745 and sprsymrelfv 41744. (Contributed by AV, 19-Nov-2021.) |
| ⊢ (𝑃 ⊆ (Pairs‘𝑉) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦}} ∈ 𝒫 (𝑉 × 𝑉)) | ||
| 18-Nov-2021 | upgrpredgv 26034 | An edge of a pseudograph always connects two vertices if the edge contains two sets. The two vertices/sets need not necessarily be different (loops are allowed). (Contributed by AV, 18-Nov-2021.) |
| ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ ((𝐺 ∈ UPGraph ∧ (𝑀 ∈ 𝑈 ∧ 𝑁 ∈ 𝑊) ∧ {𝑀, 𝑁} ∈ 𝐸) → (𝑀 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉)) | ||
| 18-Nov-2021 | cnfldfunALT 19759 | Alternate proof of cnfldfun 19758 (much shorter proof, using cnfldstr 19748 and structn0fun 15869: in addition, it must be shown that ∅ ∉ ℂfld). (Contributed by AV, 18-Nov-2021.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ Fun ℂfld | ||
| 17-Nov-2021 | cffldtocusgr 26343 | The field of complex numbers can be made a complete simple graph with the set of pairs of complex numbers regarded as edges. This theorem demonstrates the capabilities of the current definitions for graphs applied to extensible structures. (Contributed by AV, 14-Nov-2021.) (Proof shortened by AV, 17-Nov-2021.) |
| ⊢ 𝑃 = {𝑥 ∈ 𝒫 ℂ ∣ (#‘𝑥) = 2} & ⊢ 𝐺 = (ℂfld sSet ⟨(.ef‘ndx), ( I ↾ 𝑃)⟩) ⇒ ⊢ 𝐺 ∈ ComplUSGraph | ||
| 17-Nov-2021 | structtocusgr 26342 | Any (extensible) structure with a base set can be made a complete simple graph with the set of pairs of elements of the base set regarded as edges. (Contributed by AV, 10-Nov-2021.) (Revised by AV, 17-Nov-2021.) |
| ⊢ 𝑃 = {𝑥 ∈ 𝒫 (Base‘𝑆) ∣ (#‘𝑥) = 2} & ⊢ (𝜑 → 𝑆 Struct 𝑋) & ⊢ 𝐺 = (𝑆 sSet ⟨(.ef‘ndx), ( I ↾ 𝑃)⟩) & ⊢ (𝜑 → (Base‘ndx) ∈ dom 𝑆) ⇒ ⊢ (𝜑 → 𝐺 ∈ ComplUSGraph) | ||
| 17-Nov-2021 | structtousgr 26341 | Any (extensible) structure with a base set can be made a simple graph with the set of pairs of elements of the base set regarded as edges. (Contributed by AV, 10-Nov-2021.) (Revised by AV, 17-Nov-2021.) |
| ⊢ 𝑃 = {𝑥 ∈ 𝒫 (Base‘𝑆) ∣ (#‘𝑥) = 2} & ⊢ (𝜑 → 𝑆 Struct 𝑋) & ⊢ 𝐺 = (𝑆 sSet ⟨(.ef‘ndx), ( I ↾ 𝑃)⟩) & ⊢ (𝜑 → (Base‘ndx) ∈ dom 𝑆) ⇒ ⊢ (𝜑 → 𝐺 ∈ USGraph ) | ||
| 16-Nov-2021 | usgrstrrepe 26127 | Replacing (or adding) the edges (between elements of the base set) of an extensible structure results in a simple graph. Instead of requiring (𝜑 → 𝐺 Struct 𝑋), it would be sufficient to require (𝜑 → Fun (𝐺 ∖ {∅})) and (𝜑 → 𝐺 ∈ V). (Contributed by AV, 13-Nov-2021.) (Proof shortened by AV, 16-Nov-2021.) |
| ⊢ 𝑉 = (Base‘𝐺) & ⊢ 𝐼 = (.ef‘ndx) & ⊢ (𝜑 → 𝐺 Struct 𝑋) & ⊢ (𝜑 → (Base‘ndx) ∈ dom 𝐺) & ⊢ (𝜑 → 𝐸 ∈ 𝑊) & ⊢ (𝜑 → 𝐸:dom 𝐸–1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2}) ⇒ ⊢ (𝜑 → (𝐺 sSet ⟨𝐼, 𝐸⟩) ∈ USGraph ) | ||
| 16-Nov-2021 | uhgrstrrepe 25973 | Replacing (or adding) the edges (between elements of the base set) of an extensible structure results in a hypergraph. Instead of requiring (𝜑 → 𝐺 Struct 𝑋), it would be sufficient to require (𝜑 → Fun (𝐺 ∖ {∅})) and (𝜑 → 𝐺 ∈ V). (Contributed by AV, 18-Jan-2020.) (Revised by AV, 7-Jun-2021.) (Revised by AV, 16-Nov-2021.) |
| ⊢ 𝑉 = (Base‘𝐺) & ⊢ 𝐼 = (.ef‘ndx) & ⊢ (𝜑 → 𝐺 Struct 𝑋) & ⊢ (𝜑 → (Base‘ndx) ∈ dom 𝐺) & ⊢ (𝜑 → 𝐸 ∈ 𝑊) & ⊢ (𝜑 → 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})) ⇒ ⊢ (𝜑 → (𝐺 sSet ⟨𝐼, 𝐸⟩) ∈ UHGraph ) | ||
| 16-Nov-2021 | setsiedg 25928 | The (indexed) edges of a structure with a base set and an inserted resp. replaced slot for the edge function. (Contributed by AV, 7-Jun-2021.) (Revised by AV, 16-Nov-2021.) |
| ⊢ 𝐼 = (.ef‘ndx) & ⊢ (𝜑 → 𝐺 Struct 𝑋) & ⊢ (𝜑 → (Base‘ndx) ∈ dom 𝐺) & ⊢ (𝜑 → 𝐸 ∈ 𝑊) ⇒ ⊢ (𝜑 → (iEdg‘(𝐺 sSet ⟨𝐼, 𝐸⟩)) = 𝐸) | ||
| 16-Nov-2021 | setsvtx 25927 | The vertices of a structure with a base set and an inserted resp. replaced slot for the edge function. (Contributed by AV, 18-Jan-2020.) (Revised by AV, 16-Nov-2021.) |
| ⊢ 𝐼 = (.ef‘ndx) & ⊢ (𝜑 → 𝐺 Struct 𝑋) & ⊢ (𝜑 → (Base‘ndx) ∈ dom 𝐺) & ⊢ (𝜑 → 𝐸 ∈ 𝑊) ⇒ ⊢ (𝜑 → (Vtx‘(𝐺 sSet ⟨𝐼, 𝐸⟩)) = (Base‘𝐺)) | ||
| 16-Nov-2021 | edgfid 25869 | Utility theorem: index-independent form of df-edgf 25868. (Contributed by AV, 16-Nov-2021.) |
| ⊢ .ef = Slot (.ef‘ndx) | ||
| 16-Nov-2021 | basprssdmsets 15925 | The pair of the base index and another index is a subset of the domain of the structure obtained by replacing/adding a slot at the other index in a structure having a base slot. (Contributed by AV, 7-Jun-2021.) (Revised by AV, 16-Nov-2021.) |
| ⊢ (𝜑 → 𝑆 Struct 𝑋) & ⊢ (𝜑 → 𝐼 ∈ 𝑈) & ⊢ (𝜑 → 𝐸 ∈ 𝑊) & ⊢ (𝜑 → (Base‘ndx) ∈ dom 𝑆) ⇒ ⊢ (𝜑 → {(Base‘ndx), 𝐼} ⊆ dom (𝑆 sSet ⟨𝐼, 𝐸⟩)) | ||
| 16-Nov-2021 | setsn0fun 15895 | The value of the structure replacement function (without the empty set) is a function if the structure (without the empty set)is a function. (Contributed by AV, 7-Jun-2021.) (Revised by AV, 16-Nov-2021.) |
| ⊢ (𝜑 → 𝑆 Struct 𝑋) & ⊢ (𝜑 → 𝐼 ∈ 𝑈) & ⊢ (𝜑 → 𝐸 ∈ 𝑊) ⇒ ⊢ (𝜑 → Fun ((𝑆 sSet ⟨𝐼, 𝐸⟩) ∖ {∅})) | ||
| 15-Nov-2021 | fundmge2nop0 13274 | A function with a domain containing (at least) two different elements is not an ordered pair. This stronger version of fundmge2nop 13275 (with the less restrictive requirement that (𝐺 ∖ {∅}) needs to be a function instead of 𝐺) is useful for proofs for extensible structures, see structn0fun 15869. (Contributed by AV, 12-Oct-2020.) (Revised by AV, 7-Jun-2021.) (Proof shortened by AV, 15-Nov-2021.) |
| ⊢ ((Fun (𝐺 ∖ {∅}) ∧ 2 ≤ (#‘dom 𝐺)) → ¬ 𝐺 ∈ (V × V)) | ||
| 15-Nov-2021 | funopdmsn 6415 | The domain of a function which is an ordered pair is a singleton. (Contributed by AV, 15-Nov-2021.) |
| ⊢ 𝐺 = ⟨𝑋, 𝑌⟩ & ⊢ 𝑋 ∈ 𝑉 & ⊢ 𝑌 ∈ 𝑊 ⇒ ⊢ ((Fun 𝐺 ∧ 𝐴 ∈ dom 𝐺 ∧ 𝐵 ∈ dom 𝐺) → 𝐴 = 𝐵) | ||
| 15-Nov-2021 | 0nelrel 5162 | A binary relation does not contain the empty set. (Contributed by AV, 15-Nov-2021.) |
| ⊢ (Rel 𝑅 → ∅ ∉ 𝑅) | ||
| 15-Nov-2021 | opwo0id 4961 | An ordered pair is equal to the ordered pair without the empty set. This is because no ordered pair contains the empty set. (Contributed by AV, 15-Nov-2021.) |
| ⊢ ⟨𝑋, 𝑌⟩ = (⟨𝑋, 𝑌⟩ ∖ {∅}) | ||
| 14-Nov-2021 | cnfldfun 19758 | The field of complex numbers is a function. (Contributed by AV, 14-Nov-2021.) |
| ⊢ Fun ℂfld | ||
| 14-Nov-2021 | basendxnplusgndx 15989 | The slot for the base set is not the slot for the group operation in an extensible structure. (Contributed by AV, 14-Nov-2021.) |
| ⊢ (Base‘ndx) ≠ (+g‘ndx) | ||
| 14-Nov-2021 | setsstruct 15898 | An extensible structure with a replaced slot is an extensible structure. (Contributed by AV, 9-Jun-2021.) (Revised by AV, 14-Nov-2021.) |
| ⊢ ((𝐸 ∈ 𝑉 ∧ 𝐼 ∈ (ℤ≥‘𝑀) ∧ 𝐺 Struct ⟨𝑀, 𝑁⟩) → (𝐺 sSet ⟨𝐼, 𝐸⟩) Struct ⟨𝑀, if(𝐼 ≤ 𝑁, 𝑁, 𝐼)⟩) | ||
| 14-Nov-2021 | setsexstruct2 15897 | An extensible structure with a replaced slot is an extensible structure. (Contributed by AV, 14-Nov-2021.) |
| ⊢ ((𝐺 Struct 𝑋 ∧ 𝐸 ∈ 𝑉 ∧ 𝐼 ∈ ℕ) → ∃𝑦(𝐺 sSet ⟨𝐼, 𝐸⟩) Struct 𝑦) | ||
| 14-Nov-2021 | setsstruct2 15896 | An extensible structure with a replaced slot is an extensible structure. (Contributed by AV, 14-Nov-2021.) |
| ⊢ (((𝐺 Struct 𝑋 ∧ 𝐸 ∈ 𝑉 ∧ 𝐼 ∈ ℕ) ∧ 𝑌 = ⟨if(𝐼 ≤ (1st ‘𝑋), 𝐼, (1st ‘𝑋)), if(𝐼 ≤ (2nd ‘𝑋), (2nd ‘𝑋), 𝐼)⟩) → (𝐺 sSet ⟨𝐼, 𝐸⟩) Struct 𝑌) | ||
| 14-Nov-2021 | disjtp2 4252 | Two completely distinct unordered triples are disjoint. (Contributed by AV, 14-Nov-2021.) |
| ⊢ (((𝐴 ≠ 𝐷 ∧ 𝐵 ≠ 𝐷 ∧ 𝐶 ≠ 𝐷) ∧ (𝐴 ≠ 𝐸 ∧ 𝐵 ≠ 𝐸 ∧ 𝐶 ≠ 𝐸) ∧ (𝐴 ≠ 𝐹 ∧ 𝐵 ≠ 𝐹 ∧ 𝐶 ≠ 𝐹)) → ({𝐴, 𝐵, 𝐶} ∩ {𝐷, 𝐸, 𝐹}) = ∅) | ||
| 14-Nov-2021 | disjtpsn 4251 | The disjoint intersection of an unordered triple and a singleton. (Contributed by AV, 14-Nov-2021.) |
| ⊢ ((𝐴 ≠ 𝐷 ∧ 𝐵 ≠ 𝐷 ∧ 𝐶 ≠ 𝐷) → ({𝐴, 𝐵, 𝐶} ∩ {𝐷}) = ∅) | ||
| 13-Nov-2021 | structn0fun 15869 | A structure witout the empty set is a function. (Contributed by AV, 13-Nov-2021.) |
| ⊢ (𝐹 Struct 𝑋 → Fun (𝐹 ∖ {∅})) | ||
| 13-Nov-2021 | ssfzunsn 12387 | A subset of a finite sequence of integers extended by an integer is a subset of a (possibly extended) finite sequence of integers. (Contributed by AV, 8-Jun-2021.) (Proof shortened by AV, 13-Nov-2021.) |
| ⊢ ((𝑆 ⊆ (𝑀...𝑁) ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ (ℤ≥‘𝑀)) → (𝑆 ∪ {𝐼}) ⊆ (𝑀...if(𝐼 ≤ 𝑁, 𝑁, 𝐼))) | ||
| 13-Nov-2021 | ssfzunsnext 12386 | A subset of a finite sequence of integers extended by an integer is a subset of a (possibly extended) finite sequence of integers. (Contributed by AV, 13-Nov-2021.) |
| ⊢ ((𝑆 ⊆ (𝑀...𝑁) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐼 ∈ ℤ)) → (𝑆 ∪ {𝐼}) ⊆ (if(𝐼 ≤ 𝑀, 𝐼, 𝑀)...if(𝐼 ≤ 𝑁, 𝑁, 𝐼))) | ||
| 12-Nov-2021 | iedgvalsnop 25934 | Degenerated case 2 for edges: The set of indexed edges of a singleton containing an ordered pair with equal components is the singleton containing the component. (Contributed by AV, 24-Sep-2020.) (Proof shortened by AV, 12-Nov-2021.) |
| ⊢ 𝐵 ∈ V & ⊢ 𝐺 = {⟨𝐵, 𝐵⟩} ⇒ ⊢ (iEdg‘𝐺) = {𝐵} | ||
| 12-Nov-2021 | vtxvalsnop 25933 | Degenerated case 2 for vertices: The set of vertices of a singleton containing an ordered pair with equal components is the singleton containing the component. (Contributed by AV, 24-Sep-2020.) (Proof shortened by AV, 12-Nov-2021.) |
| ⊢ 𝐵 ∈ V & ⊢ 𝐺 = {⟨𝐵, 𝐵⟩} ⇒ ⊢ (Vtx‘𝐺) = {𝐵} | ||
| 12-Nov-2021 | struct2griedg 25920 | The set of indexed edges of a graph represented as an extensible structure with vertices as base set and indexed edges. (Contributed by AV, 23-Sep-2020.) (Proof shortened by AV, 12-Nov-2021.) |
| ⊢ 𝐺 = {⟨(Base‘ndx), 𝑉⟩, ⟨(.ef‘ndx), 𝐸⟩} ⇒ ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (iEdg‘𝐺) = 𝐸) | ||
| 12-Nov-2021 | structgrssiedg 25914 | The set of indexed edges of a graph represented as an extensible structure with vertices as base set and indexed edges. (Contributed by AV, 14-Oct-2020.) (Proof shortened by AV, 12-Nov-2021.) |
| ⊢ (𝜑 → 𝐺 Struct 𝑋) & ⊢ (𝜑 → 𝑉 ∈ 𝑌) & ⊢ (𝜑 → 𝐸 ∈ 𝑍) & ⊢ (𝜑 → {⟨(Base‘ndx), 𝑉⟩, ⟨(.ef‘ndx), 𝐸⟩} ⊆ 𝐺) ⇒ ⊢ (𝜑 → (iEdg‘𝐺) = 𝐸) | ||
| 12-Nov-2021 | structgrssvtx 25913 | The set of vertices of a graph represented as an extensible structure with vertices as base set and indexed edges. (Contributed by AV, 14-Oct-2020.) (Proof shortened by AV, 12-Nov-2021.) |
| ⊢ (𝜑 → 𝐺 Struct 𝑋) & ⊢ (𝜑 → 𝑉 ∈ 𝑌) & ⊢ (𝜑 → 𝐸 ∈ 𝑍) & ⊢ (𝜑 → {⟨(Base‘ndx), 𝑉⟩, ⟨(.ef‘ndx), 𝐸⟩} ⊆ 𝐺) ⇒ ⊢ (𝜑 → (Vtx‘𝐺) = 𝑉) | ||
| 12-Nov-2021 | structgrssvtxlem 25912 | Lemma for structgrssvtx 25913 and structgrssiedg 25914. (Contributed by AV, 14-Oct-2020.) (Proof shortened by AV, 12-Nov-2021.) |
| ⊢ (𝜑 → 𝐺 Struct 𝑋) & ⊢ (𝜑 → 𝑉 ∈ 𝑌) & ⊢ (𝜑 → 𝐸 ∈ 𝑍) & ⊢ (𝜑 → {⟨(Base‘ndx), 𝑉⟩, ⟨(.ef‘ndx), 𝐸⟩} ⊆ 𝐺) ⇒ ⊢ (𝜑 → 2 ≤ (#‘dom 𝐺)) | ||
| 12-Nov-2021 | structiedg0val 25911 | The set of indexed edges of an extensible structure with a base set and another slot not being the slot for edge functions is empty. (Contributed by AV, 23-Sep-2020.) (Proof shortened by AV, 12-Nov-2021.) |
| ⊢ 𝑆 ∈ ℕ & ⊢ (Base‘ndx) < 𝑆 & ⊢ 𝐺 = {⟨(Base‘ndx), 𝑉⟩, ⟨𝑆, 𝐸⟩} ⇒ ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑆 ≠ (.ef‘ndx)) → (iEdg‘𝐺) = ∅) | ||
| 12-Nov-2021 | structvtxval 25910 | The set of vertices of an extensible structure with a base set and another slot. (Contributed by AV, 23-Sep-2020.) (Proof shortened by AV, 12-Nov-2021.) |
| ⊢ 𝑆 ∈ ℕ & ⊢ (Base‘ndx) < 𝑆 & ⊢ 𝐺 = {⟨(Base‘ndx), 𝑉⟩, ⟨𝑆, 𝐸⟩} ⇒ ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (Vtx‘𝐺) = 𝑉) | ||
| 12-Nov-2021 | structvtxvallem 25909 | Lemma for structvtxval 25910 and structiedg0val 25911. (Contributed by AV, 23-Sep-2020.) (Revised by AV, 12-Nov-2021.) |
| ⊢ 𝑆 ∈ ℕ & ⊢ (Base‘ndx) < 𝑆 & ⊢ 𝐺 = {⟨(Base‘ndx), 𝑉⟩, ⟨𝑆, 𝐸⟩} ⇒ ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → 2 ≤ (#‘dom 𝐺)) | ||
| 12-Nov-2021 | funiedgval 25906 | The set of indexed edges of a graph represented as an extensible structure with vertices as base set and indexed edges. (Contributed by AV, 21-Sep-2020.) (Revised by AV, 7-Jun-2021.) (Revised by AV, 12-Nov-2021.) |
| ⊢ ((Fun (𝐺 ∖ {∅}) ∧ {(Base‘ndx), (.ef‘ndx)} ⊆ dom 𝐺) → (iEdg‘𝐺) = (.ef‘𝐺)) | ||
| 12-Nov-2021 | funvtxval 25905 | The set of vertices of a graph represented as an extensible structure with vertices as base set and indexed edges. (Contributed by AV, 22-Sep-2020.) (Revised by AV, 7-Jun-2021.) (Revised by AV, 12-Nov-2021.) |
| ⊢ ((Fun (𝐺 ∖ {∅}) ∧ {(Base‘ndx), (.ef‘ndx)} ⊆ dom 𝐺) → (Vtx‘𝐺) = (Base‘𝐺)) | ||
| 12-Nov-2021 | edgfiedgval 25902 | The set of indexed edges of a graph represented as an extensible structure with the indexed edges in the slot for edge functions. (Contributed by AV, 14-Oct-2020.) (Revised by AV, 12-Nov-2021.) |
| ⊢ (𝜑 → 𝐺 Struct 𝑋) & ⊢ (𝜑 → 2 ≤ (#‘dom 𝐺)) & ⊢ (𝜑 → 𝐸 ∈ 𝑌) & ⊢ (𝜑 → ⟨(.ef‘ndx), 𝐸⟩ ∈ 𝐺) ⇒ ⊢ (𝜑 → (iEdg‘𝐺) = 𝐸) | ||
| 12-Nov-2021 | basvtxval 25901 | The set of vertices of a graph represented as an extensible structure with the set of vertices as base set. (Contributed by AV, 14-Oct-2020.) (Revised by AV, 12-Nov-2021.) |
| ⊢ (𝜑 → 𝐺 Struct 𝑋) & ⊢ (𝜑 → 2 ≤ (#‘dom 𝐺)) & ⊢ (𝜑 → 𝑉 ∈ 𝑌) & ⊢ (𝜑 → ⟨(Base‘ndx), 𝑉⟩ ∈ 𝐺) ⇒ ⊢ (𝜑 → (Vtx‘𝐺) = 𝑉) | ||
| 12-Nov-2021 | funvtxval0 25897 | The set of vertices of an extensible structure with a base set and (at least) another slot. (Contributed by AV, 22-Sep-2020.) (Revised by AV, 7-Jun-2021.) (Revised by AV, 12-Nov-2021.) |
| ⊢ 𝑆 ∈ V ⇒ ⊢ ((Fun (𝐺 ∖ {∅}) ∧ 𝑆 ≠ (Base‘ndx) ∧ {(Base‘ndx), 𝑆} ⊆ dom 𝐺) → (Vtx‘𝐺) = (Base‘𝐺)) | ||
| 12-Nov-2021 | funiedgdm2val 25894 | The set of indexed edges of an extensible structure with (at least) two slots. (Contributed by AV, 22-Sep-2020.) (Revised by AV, 7-Jun-2021.) (Revised by AV, 12-Nov-2021.) |
| ⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ ((Fun (𝐺 ∖ {∅}) ∧ 𝐴 ≠ 𝐵 ∧ {𝐴, 𝐵} ⊆ dom 𝐺) → (iEdg‘𝐺) = (.ef‘𝐺)) | ||
| 12-Nov-2021 | funvtxdm2val 25893 | The set of vertices of an extensible structure with (at least) two slots. (Contributed by AV, 22-Sep-2020.) (Revised by AV, 7-Jun-2021.) (Revised by AV, 12-Nov-2021.) |
| ⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ ((Fun (𝐺 ∖ {∅}) ∧ 𝐴 ≠ 𝐵 ∧ {𝐴, 𝐵} ⊆ dom 𝐺) → (Vtx‘𝐺) = (Base‘𝐺)) | ||
| 12-Nov-2021 | funiedgdmge2val 25892 | The set of indexed edges of an extensible structure with (at least) two slots. (Contributed by AV, 12-Oct-2020.) (Revised by AV, 7-Jun-2021.) (Revised by AV, 12-Nov-2021.) |
| ⊢ ((Fun (𝐺 ∖ {∅}) ∧ 2 ≤ (#‘dom 𝐺)) → (iEdg‘𝐺) = (.ef‘𝐺)) | ||
| 12-Nov-2021 | funvtxdmge2val 25891 | The set of vertices of an extensible structure with (at least) two slots. (Contributed by AV, 12-Oct-2020.) (Revised by AV, 7-Jun-2021.) (Revised by AV, 12-Nov-2021.) |
| ⊢ ((Fun (𝐺 ∖ {∅}) ∧ 2 ≤ (#‘dom 𝐺)) → (Vtx‘𝐺) = (Base‘𝐺)) | ||
| 12-Nov-2021 | structfun 15873 | Convert between two kinds of structure closure. (Contributed by Mario Carneiro, 29-Aug-2015.) (Proof shortened by AV, 12-Nov-2021.) |
| ⊢ 𝐹 Struct 𝑋 ⇒ ⊢ Fun ◡◡𝐹 | ||
| 12-Nov-2021 | structfung 15872 | The converse of the converse of a structure is a function. Closed form of structfun 15873. (Contributed by AV, 12-Nov-2021.) |
| ⊢ (𝐹 Struct 𝑋 → Fun ◡◡𝐹) | ||
| 12-Nov-2021 | hashdmpropge2 13265 | The size of the domain of a class which contains two ordered pairs with different first componens is greater than or mequal to 2. (Contributed by AV, 12-Nov-2021.) |
| ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ (𝜑 → 𝐶 ∈ 𝑋) & ⊢ (𝜑 → 𝐷 ∈ 𝑌) & ⊢ (𝜑 → 𝐹 ∈ 𝑍) & ⊢ (𝜑 → 𝐴 ≠ 𝐵) & ⊢ (𝜑 → {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} ⊆ 𝐹) ⇒ ⊢ (𝜑 → 2 ≤ (#‘dom 𝐹)) | ||
| 11-Nov-2021 | compne 38643 | The complement of 𝐴 is not equal to 𝐴. (Contributed by Andrew Salmon, 15-Jul-2011.) (Proof shortened by BJ, 11-Nov-2021.) |
| ⊢ (V ∖ 𝐴) ≠ 𝐴 | ||
| 11-Nov-2021 | bj-sscon 33014 | Contraposition law for relative subsets. Relative and generalized version of ssconb 3743, which it can shorten, as well as conss2 38647. (Contributed by BJ, 11-Nov-2021.) |
| ⊢ ((𝐴 ∩ 𝑉) ⊆ (𝑉 ∖ 𝐵) ↔ (𝐵 ∩ 𝑉) ⊆ (𝑉 ∖ 𝐴)) | ||
| 11-Nov-2021 | bj-disj2r 33013 | Relative version of ssdifin0 4050, allowing a biconditional, and of disj2 4024. This proof does not rely, even indirectly, on ssdifin0 4050 nor disj2 4024. (Contributed by BJ, 11-Nov-2021.) |
| ⊢ ((𝐴 ∩ 𝑉) ⊆ (𝑉 ∖ 𝐵) ↔ ((𝐴 ∩ 𝐵) ∩ 𝑉) = ∅) | ||
| 10-Nov-2021 | setsv 41348 | The value of the structure replacement function is a set. (Contributed by AV, 10-Nov-2021.) |
| ⊢ ((𝑆 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑆 sSet ⟨𝐴, 𝐵⟩) ∈ V) | ||
| 10-Nov-2021 | setsnidel 41347 | The injected slot is an element of the structure with replacement. (Contributed by AV, 10-Nov-2021.) |
| ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ 𝑅 = (𝑆 sSet ⟨𝐴, 𝐵⟩) & ⊢ (𝜑 → 𝐶 ∈ 𝑋) & ⊢ (𝜑 → 𝐷 ∈ 𝑌) & ⊢ (𝜑 → ⟨𝐶, 𝐷⟩ ∈ 𝑆) & ⊢ (𝜑 → 𝐴 ≠ 𝐶) ⇒ ⊢ (𝜑 → ⟨𝐶, 𝐷⟩ ∈ 𝑅) | ||
| 10-Nov-2021 | setsidel 41346 | The injected slot is an element of the structure with replacement. (Contributed by AV, 10-Nov-2021.) |
| ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ 𝑅 = (𝑆 sSet ⟨𝐴, 𝐵⟩) ⇒ ⊢ (𝜑 → ⟨𝐴, 𝐵⟩ ∈ 𝑅) | ||
| 10-Nov-2021 | cusgrexi 26339 | An arbitrary set regarded as vertices together with the set of pairs of elements of this set regarded as edges is a complete simple graph. (Contributed by Alexander van der Vekens, 12-Jan-2018.) (Revised by AV, 5-Nov-2020.) (Proof shortened by AV, 10-Nov-2021.) |
| ⊢ 𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2} ⇒ ⊢ (𝑉 ∈ 𝑊 → ⟨𝑉, ( I ↾ 𝑃)⟩ ∈ ComplUSGraph) | ||
| 10-Nov-2021 | cusgrexilem2 26338 | Lemma 2 for cusgrexi 26339. (Contributed by AV, 12-Jan-2018.) (Revised by AV, 10-Nov-2021.) |
| ⊢ 𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2} ⇒ ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑣 ∈ 𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → ∃𝑒 ∈ ran ( I ↾ 𝑃){𝑣, 𝑛} ⊆ 𝑒) | ||
| 10-Nov-2021 | usgrexi 26337 | An arbitrary set regarded as vertices together with the set of pairs of elements of this set regarded as edges is a simple graph. (Contributed by Alexander van der Vekens, 12-Jan-2018.) (Revised by AV, 5-Nov-2020.) (Proof shortened by AV, 10-Nov-2021.) |
| ⊢ 𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2} ⇒ ⊢ (𝑉 ∈ 𝑊 → ⟨𝑉, ( I ↾ 𝑃)⟩ ∈ USGraph ) | ||
| 10-Nov-2021 | usgrexilem 26336 | Lemma for usgrexi 26337. (Contributed by AV, 12-Jan-2018.) (Revised by AV, 10-Nov-2021.) |
| ⊢ 𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2} ⇒ ⊢ (𝑉 ∈ 𝑊 → ( I ↾ 𝑃):dom ( I ↾ 𝑃)–1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2}) | ||
| 10-Nov-2021 | opelstrbas 15978 | The base set of a structure with a base set. (Contributed by AV, 10-Nov-2021.) |
| ⊢ (𝜑 → 𝑆 Struct 𝑋) & ⊢ (𝜑 → 𝑉 ∈ 𝑌) & ⊢ (𝜑 → ⟨(Base‘ndx), 𝑉⟩ ∈ 𝑆) ⇒ ⊢ (𝜑 → 𝑉 = (Base‘𝑆)) | ||
| 10-Nov-2021 | structex 15868 | A structure is a set. (Contributed by AV, 10-Nov-2021.) |
| ⊢ (𝐺 Struct 𝑋 → 𝐺 ∈ V) | ||
| 9-Nov-2021 | bj-alequexv 32655 | Version of bj-alequex 32708 with DV(x,y), requiring fewer axioms. (Contributed by BJ, 9-Nov-2021.) |
| ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∃𝑥𝜑) | ||
| 9-Nov-2021 | bj-ssbjust 32618 | Justification theorem for df-ssb 32620 from Tarski's FOL. (Contributed by BJ, 9-Nov-2021.) |
| ⊢ (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) ↔ ∀𝑧(𝑧 = 𝑡 → ∀𝑥(𝑥 = 𝑧 → 𝜑))) | ||
| 8-Nov-2021 | bj-cleljustab 32847 | An instance of df-clel 2618 where the LHS (the definiendum) has the form "setvar ∈ class abstraction". The straightforward yet important fact that this statement can be proved from FOL= and df-clab 2609 (hence without df-clel 2618 or df-cleq 2615) was stressed by Mario Carneiro. The instance of df-clel 2618 where the LHS has the form "setvar ∈ setvar" is proved as cleljust 1998, from FOL= and ax-8 1992. Note: when df-ssb 32620 is the official definition for substitution, one can use bj-ssbequ instead of sbequ 2376 to prove bj-cleljustab 32847 from Tarski's FOL= with df-clab 2609. (Contributed by BJ, 8-Nov-2021.) (Proof modification is discouraged.) |
| ⊢ (𝑥 ∈ {𝑦 ∣ 𝜑} ↔ ∃𝑧(𝑧 = 𝑥 ∧ 𝑧 ∈ {𝑦 ∣ 𝜑})) | ||
| 8-Nov-2021 | bj-spimvwt 32656 | Closed form of spimvw 1927. See also spimt 2253. (Contributed by BJ, 8-Nov-2021.) |
| ⊢ (∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓)) → (∀𝑥𝜑 → 𝜓)) | ||
| 8-Nov-2021 | bj-exalim 32611 | Distributing quantifiers over a double implication. (Contributed by BJ, 8-Nov-2021.) |
| ⊢ (∀𝑥(𝜑 → (𝜓 → 𝜒)) → (∃𝑥𝜑 → (∀𝑥𝜓 → ∃𝑥𝜒))) | ||
| 8-Nov-2021 | bj-alexim 32605 | Closed form of aleximi 1759 (with a shorter proof, so aleximi 1759 could be deduced from it -- exim 1761 would have to be proved first, but its proof is shorter (currently almost a subproof of aleximi 1759)). (Contributed by BJ, 8-Nov-2021.) |
| ⊢ (∀𝑥(𝜑 → (𝜓 → 𝜒)) → (∀𝑥𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))) | ||
| 5-Nov-2021 | isupwlkg 41718 | Generalisation of isupwlk 41717: Conditions for two classes to represent a simple walk. (Contributed by AV, 5-Nov-2021.) |
| ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ (𝐺 ∈ 𝑊 → (𝐹(UPWalks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))) | ||
| 5-Nov-2021 | ifpsnprss 26518 | Lemma for wlkvtxeledg 26519: Two adjacent (not necessarily different) vertices 𝐴 and 𝐵 in a walk are incident with an edge 𝐸. (Contributed by AV, 4-Apr-2021.) (Revised by AV, 5-Nov-2021.) |
| ⊢ (if-(𝐴 = 𝐵, 𝐸 = {𝐴}, {𝐴, 𝐵} ⊆ 𝐸) → {𝐴, 𝐵} ⊆ 𝐸) | ||
| 5-Nov-2021 | wlkprop 26507 | Properties of a walk. (Contributed by AV, 5-Nov-2021.) |
| ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ (𝐹(Walks‘𝐺)𝑃 → (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))))) | ||
| 3-Nov-2021 | bj-spcimdvv 32885 | Remove from spcimdv 3290 dependency on ax-7 1935, ax-8 1992, ax-10 2019, ax-11 2034, ax-12 2047 ax-13 2246, ax-ext 2602, df-cleq 2615, df-clab 2609 (and df-nfc 2753, df-v 3202, df-or 385, df-tru 1486, df-nf 1710) at the price of adding a DV condition on 𝑥, 𝐵 (but in usages, 𝑥 is typically a dummy, hence fresh, variable). For the version without this DV condition, see bj-spcimdv 32884. (Contributed by BJ, 3-Nov-2021.) (Proof modification is discouraged.) |
| ⊢ (𝜑 → 𝐴 ∈ 𝐵) & ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → (∀𝑥𝜓 → 𝜒)) | ||
| 3-Nov-2021 | 19.21t 2073 | Closed form of Theorem 19.21 of [Margaris] p. 90, see 19.21 2075. (Contributed by NM, 27-May-1997.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 3-Jan-2018.) df-nf 1710 changed. (Revised by Wolf Lammen, 11-Sep-2021.) (Proof shortened by BJ, 3-Nov-2021.) |
| ⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓))) | ||
| 3-Nov-2021 | 19.38b 1768 | Under a non-freeness hypothesis, the implication 19.38 1766 can be strengthened to an equivalence. See also 19.38a 1767. (Contributed by BJ, 3-Nov-2021.) |
| ⊢ (Ⅎ𝑥𝜓 → ((∃𝑥𝜑 → ∀𝑥𝜓) ↔ ∀𝑥(𝜑 → 𝜓))) | ||
| 3-Nov-2021 | 19.38a 1767 | Under a non-freeness hypothesis, the implication 19.38 1766 can be strengthened to an equivalence. See also 19.38b 1768. (Contributed by BJ, 3-Nov-2021.) |
| ⊢ (Ⅎ𝑥𝜑 → ((∃𝑥𝜑 → ∀𝑥𝜓) ↔ ∀𝑥(𝜑 → 𝜓))) | ||
| 31-Oct-2021 | crctcsh 26716 | Cyclically shifting the indices of a circuit ⟨𝐹, 𝑃⟩ results in a circuit ⟨𝐻, 𝑄⟩. (Contributed by AV, 10-Mar-2021.) (Proof shortened by AV, 31-Oct-2021.) |
| ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ (𝜑 → 𝐹(Circuits‘𝐺)𝑃) & ⊢ 𝑁 = (#‘𝐹) & ⊢ (𝜑 → 𝑆 ∈ (0..^𝑁)) & ⊢ 𝐻 = (𝐹 cyclShift 𝑆) & ⊢ 𝑄 = (𝑥 ∈ (0...𝑁) ↦ if(𝑥 ≤ (𝑁 − 𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁)))) ⇒ ⊢ (𝜑 → 𝐻(Circuits‘𝐺)𝑄) | ||
| 31-Oct-2021 | uspgrn2crct 26700 | In a simple pseudograph there are no circuits with length 2 (consisting of two edges). (Contributed by Alexander van der Vekens, 9-Nov-2017.) (Revised by AV, 3-Feb-2021.) (Proof shortened by AV, 31-Oct-2021.) |
| ⊢ ((𝐺 ∈ USPGraph ∧ 𝐹(Circuits‘𝐺)𝑃) → (#‘𝐹) ≠ 2) | ||
| 31-Oct-2021 | usgr2pth 26660 | In a simple graph, there is a path of length 2 iff there are three distinct vertices so that one of them is connected to each of the two others by an edge. (Contributed by Alexander van der Vekens, 27-Jan-2018.) (Revised by AV, 5-Jun-2021.) (Proof shortened by AV, 31-Oct-2021.) |
| ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ (𝐺 ∈ USGraph → ((𝐹(Paths‘𝐺)𝑃 ∧ (#‘𝐹) = 2) ↔ (𝐹:(0..^2)–1-1→dom 𝐼 ∧ 𝑃:(0...2)–1-1→𝑉 ∧ ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ (𝑉 ∖ {𝑥})∃𝑧 ∈ (𝑉 ∖ {𝑥, 𝑦})(((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑦} ∧ (𝐼‘(𝐹‘1)) = {𝑦, 𝑧}))))) | ||
| 31-Oct-2021 | usgr2trlncl 26656 | In a simple graph, any trail of length 2 does not start and end at the same vertex. (Contributed by AV, 5-Jun-2021.) (Proof shortened by AV, 31-Oct-2021.) |
| ⊢ ((𝐺 ∈ USGraph ∧ (#‘𝐹) = 2) → (𝐹(Trails‘𝐺)𝑃 → (𝑃‘0) ≠ (𝑃‘2))) | ||
| 31-Oct-2021 | usgr2wlkspth 26655 | In a simple graph, any walk of length 2 between two different vertices is a simple path. (Contributed by Alexander van der Vekens, 2-Mar-2018.) (Revised by AV, 27-Jan-2021.) (Proof shortened by AV, 31-Oct-2021.) |
| ⊢ ((𝐺 ∈ USGraph ∧ (#‘𝐹) = 2 ∧ 𝐴 ≠ 𝐵) → (𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃 ↔ 𝐹(𝐴(SPathsOn‘𝐺)𝐵)𝑃)) | ||
| 31-Oct-2021 | uhgrwkspth 26651 | Any walk of length 1 between two different vertices is a simple path. (Contributed by AV, 25-Jan-2021.) (Proof shortened by AV, 31-Oct-2021.) |
| ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ ((𝐺 ∈ 𝑊 ∧ (#‘𝐹) = 1 ∧ 𝐴 ≠ 𝐵) → (𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃 ↔ 𝐹(𝐴(SPathsOn‘𝐺)𝐵)𝑃)) | ||
| 31-Oct-2021 | spthonepeq 26648 | The endpoints of a simple path between two vertices are equal iff the path is of length 0. (Contributed by Alexander van der Vekens, 1-Mar-2018.) (Revised by AV, 18-Jan-2021.) (Proof shortened by AV, 31-Oct-2021.) |
| ⊢ (𝐹(𝐴(SPathsOn‘𝐺)𝐵)𝑃 → (𝐴 = 𝐵 ↔ (#‘𝐹) = 0)) | ||
| 31-Oct-2021 | pthdivtx 26625 | The inner vertices of a path are distinct from all other vertices. (Contributed by AV, 5-Feb-2021.) (Proof shortened by AV, 31-Oct-2021.) |
| ⊢ ((𝐹(Paths‘𝐺)𝑃 ∧ (𝐼 ∈ (1..^(#‘𝐹)) ∧ 𝐽 ∈ (0...(#‘𝐹)) ∧ 𝐼 ≠ 𝐽)) → (𝑃‘𝐼) ≠ (𝑃‘𝐽)) | ||
| 30-Oct-2021 | eucrctshift 27103 | Cyclically shifting the indices of an Eulerian circuit ⟨𝐹, 𝑃⟩ results in an Eulerian circuit ⟨𝐻, 𝑄⟩. (Contributed by AV, 15-Mar-2021.) (Proof shortened by AV, 30-Oct-2021.) |
| ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ (𝜑 → 𝐹(Circuits‘𝐺)𝑃) & ⊢ 𝑁 = (#‘𝐹) & ⊢ (𝜑 → 𝑆 ∈ (0..^𝑁)) & ⊢ 𝐻 = (𝐹 cyclShift 𝑆) & ⊢ 𝑄 = (𝑥 ∈ (0...𝑁) ↦ if(𝑥 ≤ (𝑁 − 𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁)))) & ⊢ (𝜑 → 𝐹(EulerPaths‘𝐺)𝑃) ⇒ ⊢ (𝜑 → (𝐻(EulerPaths‘𝐺)𝑄 ∧ 𝐻(Circuits‘𝐺)𝑄)) | ||
| 30-Oct-2021 | eupth2eucrct 27077 | Append one path segment to an Eulerian path ⟨𝐹, 𝑃⟩ which may not be an (Eulerian) circuit to become an Eulerian circuit ⟨𝐻, 𝑄⟩ of the supergraph 𝑆 obtained by adding the new edge to the graph 𝐺. (Contributed by AV, 11-Mar-2021.) (Proof shortened by AV, 30-Oct-2021.) |
| ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ (𝜑 → Fun 𝐼) & ⊢ (𝜑 → 𝐼 ∈ Fin) & ⊢ (𝜑 → 𝐵 ∈ V) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) & ⊢ (𝜑 → ¬ 𝐵 ∈ dom 𝐼) & ⊢ (𝜑 → 𝐹(EulerPaths‘𝐺)𝑃) & ⊢ 𝑁 = (#‘𝐹) & ⊢ (𝜑 → 𝐸 ∈ (Edg‘𝐺)) & ⊢ (𝜑 → {(𝑃‘𝑁), 𝐶} ⊆ 𝐸) & ⊢ (iEdg‘𝑆) = (𝐼 ∪ {⟨𝐵, 𝐸⟩}) & ⊢ 𝐻 = (𝐹 ∪ {⟨𝑁, 𝐵⟩}) & ⊢ 𝑄 = (𝑃 ∪ {⟨(𝑁 + 1), 𝐶⟩}) & ⊢ (Vtx‘𝑆) = 𝑉 & ⊢ ((𝜑 ∧ 𝐶 = (𝑃‘𝑁)) → 𝐸 = {𝐶}) & ⊢ (𝜑 → 𝐶 = (𝑃‘0)) ⇒ ⊢ (𝜑 → (𝐻(EulerPaths‘𝑆)𝑄 ∧ 𝐻(Circuits‘𝑆)𝑄)) | ||
| 30-Oct-2021 | eupthp1 27076 | Append one path segment to an Eulerian path ⟨𝐹, 𝑃⟩ to become an Eulerian path ⟨𝐻, 𝑄⟩ of the supergraph 𝑆 obtained by adding the new edge to the graph 𝐺. (Contributed by Mario Carneiro, 7-Apr-2015.) (Revised by AV, 7-Mar-2021.) (Proof shortened by AV, 30-Oct-2021.) |
| ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ (𝜑 → Fun 𝐼) & ⊢ (𝜑 → 𝐼 ∈ Fin) & ⊢ (𝜑 → 𝐵 ∈ V) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) & ⊢ (𝜑 → ¬ 𝐵 ∈ dom 𝐼) & ⊢ (𝜑 → 𝐹(EulerPaths‘𝐺)𝑃) & ⊢ 𝑁 = (#‘𝐹) & ⊢ (𝜑 → 𝐸 ∈ (Edg‘𝐺)) & ⊢ (𝜑 → {(𝑃‘𝑁), 𝐶} ⊆ 𝐸) & ⊢ (iEdg‘𝑆) = (𝐼 ∪ {⟨𝐵, 𝐸⟩}) & ⊢ 𝐻 = (𝐹 ∪ {⟨𝑁, 𝐵⟩}) & ⊢ 𝑄 = (𝑃 ∪ {⟨(𝑁 + 1), 𝐶⟩}) & ⊢ (Vtx‘𝑆) = 𝑉 & ⊢ ((𝜑 ∧ 𝐶 = (𝑃‘𝑁)) → 𝐸 = {𝐶}) ⇒ ⊢ (𝜑 → 𝐻(EulerPaths‘𝑆)𝑄) | ||
| 30-Oct-2021 | eupthres 27075 | The restriction ⟨𝐻, 𝑄⟩ of an Eulerian path ⟨𝐹, 𝑃⟩ to an initial segment of the path (of length 𝑁) forms an Eulerian path on the subgraph 𝑆 consisting of the edges in the initial segment. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 3-May-2015.) (Revised by AV, 6-Mar-2021.) (Proof shortened by AV, 30-Oct-2021.) |
| ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ (𝜑 → 𝐹(EulerPaths‘𝐺)𝑃) & ⊢ (𝜑 → 𝑁 ∈ (0..^(#‘𝐹))) & ⊢ (𝜑 → (iEdg‘𝑆) = (𝐼 ↾ (𝐹 “ (0..^𝑁)))) & ⊢ 𝐻 = (𝐹 ↾ (0..^𝑁)) & ⊢ 𝑄 = (𝑃 ↾ (0...𝑁)) & ⊢ (Vtx‘𝑆) = 𝑉 ⇒ ⊢ (𝜑 → 𝐻(EulerPaths‘𝑆)𝑄) | ||
| 30-Oct-2021 | eupth0 27074 | There is an Eulerian path on an empty graph, i.e. a graph with at least one vertex, but without an edge. (Contributed by Mario Carneiro, 7-Apr-2015.) (Revised by AV, 5-Mar-2021.) (Proof shortened by AV, 30-Oct-2021.) |
| ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐼 = ∅) → ∅(EulerPaths‘𝐺){⟨0, 𝐴⟩}) | ||
| 30-Oct-2021 | upgreupthi 27068 | Properties of an Eulerian path in a pseudograph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 18-Feb-2021.) (Proof shortened by AV, 30-Oct-2021.) |
| ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹(EulerPaths‘𝐺)𝑃) → (𝐹:(0..^(#‘𝐹))–1-1-onto→dom 𝐼 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})) | ||
| 30-Oct-2021 | upgriseupth 27067 | The property "⟨𝐹, 𝑃⟩ is an Eulerian path on the pseudograph 𝐺". (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 3-May-2015.) (Revised by AV, 18-Feb-2021.) (Revised by AV, 30-Oct-2021.) |
| ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ (𝐺 ∈ UPGraph → (𝐹(EulerPaths‘𝐺)𝑃 ↔ (𝐹:(0..^(#‘𝐹))–1-1-onto→dom 𝐼 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))) | ||
| 30-Oct-2021 | eupthi 27063 | Properties of an Eulerian path. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 18-Feb-2021.) (Proof shortened by AV, 30-Oct-2021.) |
| ⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ (𝐹(EulerPaths‘𝐺)𝑃 → (𝐹(Walks‘𝐺)𝑃 ∧ 𝐹:(0..^(#‘𝐹))–1-1-onto→dom 𝐼)) | ||
| 30-Oct-2021 | iseupthf1o 27062 | The property "⟨𝐹, 𝑃⟩ is an Eulerian path on the graph 𝐺". An Eulerian path is defined as bijection 𝐹 from the edges to a set 0...(𝑁 − 1) and a function 𝑃:(0...𝑁)⟶𝑉 into the vertices such that for each 0 ≤ 𝑘 < 𝑁, 𝐹(𝑘) is an edge from 𝑃(𝑘) to 𝑃(𝑘 + 1). (Since the edges are undirected and there are possibly many edges between any two given vertices, we need to list both the edges and the vertices of the path separately.) (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 3-May-2015.) (Revised by AV, 18-Feb-2021.) (Revised by AV, 30-Oct-2021.) |
| ⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ (𝐹(EulerPaths‘𝐺)𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ 𝐹:(0..^(#‘𝐹))–1-1-onto→dom 𝐼)) | ||
| 30-Oct-2021 | iseupth 27061 | The property "⟨𝐹, 𝑃⟩ is an Eulerian path on the graph 𝐺". An Eulerian path is defined as bijection 𝐹 from the edges to a set 0...(𝑁 − 1) and a function 𝑃:(0...𝑁)⟶𝑉 into the vertices such that for each 0 ≤ 𝑘 < 𝑁, 𝐹(𝑘) is an edge from 𝑃(𝑘) to 𝑃(𝑘 + 1). (Since the edges are undirected and there are possibly many edges between any two given vertices, we need to list both the edges and the vertices of the path separately.) (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 3-May-2015.) (Revised by AV, 18-Feb-2021.) (Revised by AV, 30-Oct-2021.) |
| ⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ (𝐹(EulerPaths‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃 ∧ 𝐹:(0..^(#‘𝐹))–onto→dom 𝐼)) | ||
| 30-Oct-2021 | 3cycld 27038 | Construction of a 3-cycle from three given edges in a graph. (Contributed by Alexander van der Vekens, 13-Nov-2017.) (Revised by AV, 10-Feb-2021.) (Revised by AV, 24-Mar-2021.) (Proof shortened by AV, 30-Oct-2021.) |
| ⊢ 𝑃 = ⟨“𝐴𝐵𝐶𝐷”⟩ & ⊢ 𝐹 = ⟨“𝐽𝐾𝐿”⟩ & ⊢ (𝜑 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉))) & ⊢ (𝜑 → ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷) ∧ 𝐶 ≠ 𝐷)) & ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾) ∧ {𝐶, 𝐷} ⊆ (𝐼‘𝐿))) & ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ (𝜑 → (𝐽 ≠ 𝐾 ∧ 𝐽 ≠ 𝐿 ∧ 𝐾 ≠ 𝐿)) & ⊢ (𝜑 → 𝐴 = 𝐷) ⇒ ⊢ (𝜑 → 𝐹(Cycles‘𝐺)𝑃) | ||
| 30-Oct-2021 | 3spthd 27036 | A simple path of length 3 from one vertex to another, different vertex via a third vertex. (Contributed by AV, 10-Feb-2021.) (Revised by AV, 24-Mar-2021.) (Proof shortened by AV, 30-Oct-2021.) |
| ⊢ 𝑃 = ⟨“𝐴𝐵𝐶𝐷”⟩ & ⊢ 𝐹 = ⟨“𝐽𝐾𝐿”⟩ & ⊢ (𝜑 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉))) & ⊢ (𝜑 → ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷) ∧ 𝐶 ≠ 𝐷)) & ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾) ∧ {𝐶, 𝐷} ⊆ (𝐼‘𝐿))) & ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ (𝜑 → (𝐽 ≠ 𝐾 ∧ 𝐽 ≠ 𝐿 ∧ 𝐾 ≠ 𝐿)) & ⊢ (𝜑 → 𝐴 ≠ 𝐷) ⇒ ⊢ (𝜑 → 𝐹(SPaths‘𝐺)𝑃) | ||
| 30-Oct-2021 | 3trld 27032 | Construction of a trail from two given edges in a graph. (Contributed by Alexander van der Vekens, 13-Nov-2017.) (Revised by AV, 8-Feb-2021.) (Revised by AV, 24-Mar-2021.) (Proof shortened by AV, 30-Oct-2021.) |
| ⊢ 𝑃 = ⟨“𝐴𝐵𝐶𝐷”⟩ & ⊢ 𝐹 = ⟨“𝐽𝐾𝐿”⟩ & ⊢ (𝜑 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉))) & ⊢ (𝜑 → ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ (𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷) ∧ 𝐶 ≠ 𝐷)) & ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾) ∧ {𝐶, 𝐷} ⊆ (𝐼‘𝐿))) & ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ (𝜑 → (𝐽 ≠ 𝐾 ∧ 𝐽 ≠ 𝐿 ∧ 𝐾 ≠ 𝐿)) ⇒ ⊢ (𝜑 → 𝐹(Trails‘𝐺)𝑃) | ||
| 30-Oct-2021 | ntrl2v2e 27018 | A walk which is not a trail: In a graph with two vertices and one edge connecting these two vertices, to go from one vertex to the other and back to the first vertex via the same/only edge is a walk, see wlk2v2e 27017, but not a trail. Notice that 𝐺 is a simple graph (without loops) only if 𝑋 ≠ 𝑌. (Contributed by Alexander van der Vekens, 22-Oct-2017.) (Revised by AV, 8-Jan-2021.) (Proof shortened by AV, 30-Oct-2021.) |
| ⊢ 𝐼 = ⟨“{𝑋, 𝑌}”⟩ & ⊢ 𝐹 = ⟨“00”⟩ & ⊢ 𝑋 ∈ V & ⊢ 𝑌 ∈ V & ⊢ 𝑃 = ⟨“𝑋𝑌𝑋”⟩ & ⊢ 𝐺 = ⟨{𝑋, 𝑌}, 𝐼⟩ ⇒ ⊢ ¬ 𝐹(Trails‘𝐺)𝑃 | ||
| 30-Oct-2021 | lp1cycl 27012 | A loop (which is an edge at index 𝐽) induces a cycle of length 1 in a hypergraph. (Contributed by AV, 2-Feb-2021.) (Proof shortened by AV, 30-Oct-2021.) |
| ⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼‘𝐽) = {𝐴}) → ⟨“𝐽”⟩(Cycles‘𝐺)⟨“𝐴𝐴”⟩) | ||
| 30-Oct-2021 | 1pthd 27003 | In a graph with two vertices and an edge connecting these two vertices, to go from one vertex to the other vertex via this edge is a path. The two vertices need not be distinct (in the case of a loop) - in this case, however, the path is not a simple path. (Contributed by Alexander van der Vekens, 3-Dec-2017.) (Revised by AV, 22-Jan-2021.) (Revised by AV, 23-Mar-2021.) (Proof shortened by AV, 30-Oct-2021.) |
| ⊢ 𝑃 = ⟨“𝑋𝑌”⟩ & ⊢ 𝐹 = ⟨“𝐽”⟩ & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → (𝐼‘𝐽) = {𝑋}) & ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → {𝑋, 𝑌} ⊆ (𝐼‘𝐽)) & ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ (𝜑 → 𝐹(Paths‘𝐺)𝑃) | ||
| 30-Oct-2021 | 1trld 27002 | In a graph with two vertices and an edge connecting these two vertices, to go from one vertex to the other vertex via this edge is a trail. The two vertices need not be distinct (in the case of a loop). (Contributed by Alexander van der Vekens, 3-Dec-2017.) (Revised by AV, 22-Jan-2021.) (Revised by AV, 23-Mar-2021.) (Proof shortened by AV, 30-Oct-2021.) |
| ⊢ 𝑃 = ⟨“𝑋𝑌”⟩ & ⊢ 𝐹 = ⟨“𝐽”⟩ & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → (𝐼‘𝐽) = {𝑋}) & ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → {𝑋, 𝑌} ⊆ (𝐼‘𝐽)) & ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ (𝜑 → 𝐹(Trails‘𝐺)𝑃) | ||
| 30-Oct-2021 | 0cycl 26994 | A pair of an empty set (of edges) and a second set (of vertices) is a cycle if and only if the second set contains exactly one vertex (in an undirected graph). (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 31-Jan-2021.) (Revised by AV, 30-Oct-2021.) |
| ⊢ (𝐺 ∈ 𝑊 → (∅(Cycles‘𝐺)𝑃 ↔ 𝑃:(0...0)⟶(Vtx‘𝐺))) | ||
| 30-Oct-2021 | 0crct 26993 | A pair of an empty set (of edges) and a second set (of vertices) is a circuit if and only if the second set contains exactly one vertex (in an undirected graph). (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 31-Jan-2021.) (Revised by AV, 30-Oct-2021.) |
| ⊢ (𝐺 ∈ 𝑊 → (∅(Circuits‘𝐺)𝑃 ↔ 𝑃:(0...0)⟶(Vtx‘𝐺))) | ||
| 30-Oct-2021 | 0clwlk 26991 | A pair of an empty set (of edges) and a second set (of vertices) is a closed walk if and only if the second set contains exactly one vertex (in an undirected graph). (Contributed by Alexander van der Vekens, 15-Mar-2018.) (Revised by AV, 17-Feb-2021.) (Revised by AV, 30-Oct-2021.) |
| ⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ (𝐺 ∈ 𝑋 → (∅(ClWalks‘𝐺)𝑃 ↔ 𝑃:(0...0)⟶𝑉)) | ||
| 30-Oct-2021 | 0pthon 26988 | A path of length 0 from a vertex to itself. (Contributed by Alexander van der Vekens, 3-Dec-2017.) (Revised by AV, 20-Jan-2021.) (Revised by AV, 30-Oct-2021.) |
| ⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ ((𝑃:(0...0)⟶𝑉 ∧ (𝑃‘0) = 𝑁) → ∅(𝑁(PathsOn‘𝐺)𝑁)𝑃) | ||
| 30-Oct-2021 | 0spth 26987 | A pair of an empty set (of edges) and a second set (of vertices) is a simple path iff the second set contains exactly one vertex. (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 18-Jan-2021.) (Revised by AV, 30-Oct-2021.) |
| ⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ (𝐺 ∈ 𝑊 → (∅(SPaths‘𝐺)𝑃 ↔ 𝑃:(0...0)⟶𝑉)) | ||
| 30-Oct-2021 | 0pth 26986 | A pair of an empty set (of edges) and a second set (of vertices) is a path iff the second set contains exactly one vertex. (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 19-Jan-2021.) (Revised by AV, 30-Oct-2021.) |
| ⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ (𝐺 ∈ 𝑊 → (∅(Paths‘𝐺)𝑃 ↔ 𝑃:(0...0)⟶𝑉)) | ||
| 30-Oct-2021 | 0trlon 26985 | A trail of length 0 from a vertex to itself. (Contributed by Alexander van der Vekens, 2-Dec-2017.) (Revised by AV, 8-Jan-2021.) (Revised by AV, 23-Mar-2021.) (Proof shortened by AV, 30-Oct-2021.) |
| ⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ ((𝑃:(0...0)⟶𝑉 ∧ (𝑃‘0) = 𝑁) → ∅(𝑁(TrailsOn‘𝐺)𝑁)𝑃) | ||
| 30-Oct-2021 | is0trl 26984 | A pair of an empty set (of edges) and a sequence of one vertex is a trail (of length 0). (Contributed by AV, 7-Jan-2021.) (Revised by AV, 23-Mar-2021.) (Proof shortened by AV, 30-Oct-2021.) |
| ⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ ((𝑃 = {⟨0, 𝑁⟩} ∧ 𝑁 ∈ 𝑉) → ∅(Trails‘𝐺)𝑃) | ||
| 30-Oct-2021 | 0trl 26983 | A pair of an empty set (of edges) and a second set (of vertices) is a trail iff the second set contains exactly one vertex. (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 7-Jan-2021.) (Revised by AV, 30-Oct-2021.) |
| ⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ (𝐺 ∈ 𝑈 → (∅(Trails‘𝐺)𝑃 ↔ 𝑃:(0...0)⟶𝑉)) | ||
| 30-Oct-2021 | 0wlkon 26981 | A walk of length 0 from a vertex to itself. (Contributed by Alexander van der Vekens, 2-Dec-2017.) (Revised by AV, 3-Jan-2021.) (Revised by AV, 23-Mar-2021.) (Proof shortened by AV, 30-Oct-2021.) |
| ⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ ((𝑃:(0...0)⟶𝑉 ∧ (𝑃‘0) = 𝑁) → ∅(𝑁(WalksOn‘𝐺)𝑁)𝑃) | ||
| 30-Oct-2021 | is0wlk 26978 | A pair of an empty set (of edges) and a sequence of one vertex is a walk (of length 0). (Contributed by AV, 3-Jan-2021.) (Revised by AV, 23-Mar-2021.) (Proof shortened by AV, 30-Oct-2021.) |
| ⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ ((𝑃 = {⟨0, 𝑁⟩} ∧ 𝑁 ∈ 𝑉) → ∅(Walks‘𝐺)𝑃) | ||
| 30-Oct-2021 | 0wlk 26977 | A pair of an empty set (of edges) and a second set (of vertices) is a walk iff the second set contains exactly one vertex. (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 3-Jan-2021.) (Revised by AV, 30-Oct-2021.) |
| ⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ (𝐺 ∈ 𝑈 → (∅(Walks‘𝐺)𝑃 ↔ 𝑃:(0...0)⟶𝑉)) | ||
| 30-Oct-2021 | 2spthd 26837 | A simple path of length 2 from one vertex to another, different vertex via a third vertex. (Contributed by Alexander van der Vekens, 1-Feb-2018.) (Revised by AV, 24-Jan-2021.) (Revised by AV, 24-Mar-2021.) (Proof shortened by AV, 30-Oct-2021.) |
| ⊢ 𝑃 = ⟨“𝐴𝐵𝐶”⟩ & ⊢ 𝐹 = ⟨“𝐽𝐾”⟩ & ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) & ⊢ (𝜑 → (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶)) & ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾))) & ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ (𝜑 → 𝐽 ≠ 𝐾) & ⊢ (𝜑 → 𝐴 ≠ 𝐶) ⇒ ⊢ (𝜑 → 𝐹(SPaths‘𝐺)𝑃) | ||
| 30-Oct-2021 | 2trld 26834 | Construction of a trail from two given edges in a graph. (Contributed by Alexander van der Vekens, 4-Dec-2017.) (Revised by AV, 24-Jan-2021.) (Revised by AV, 24-Mar-2021.) (Proof shortened by AV, 30-Oct-2021.) |
| ⊢ 𝑃 = ⟨“𝐴𝐵𝐶”⟩ & ⊢ 𝐹 = ⟨“𝐽𝐾”⟩ & ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) & ⊢ (𝜑 → (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶)) & ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾))) & ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ (𝜑 → 𝐽 ≠ 𝐾) ⇒ ⊢ (𝜑 → 𝐹(Trails‘𝐺)𝑃) | ||
| 30-Oct-2021 | crctcshtrl 26715 | Cyclically shifting the indices of a circuit ⟨𝐹, 𝑃⟩ results in a trail ⟨𝐻, 𝑄⟩. (Contributed by AV, 14-Mar-2021.) (Proof shortened by AV, 30-Oct-2021.) |
| ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ (𝜑 → 𝐹(Circuits‘𝐺)𝑃) & ⊢ 𝑁 = (#‘𝐹) & ⊢ (𝜑 → 𝑆 ∈ (0..^𝑁)) & ⊢ 𝐻 = (𝐹 cyclShift 𝑆) & ⊢ 𝑄 = (𝑥 ∈ (0...𝑁) ↦ if(𝑥 ≤ (𝑁 − 𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁)))) ⇒ ⊢ (𝜑 → 𝐻(Trails‘𝐺)𝑄) | ||
| 30-Oct-2021 | cyclispthon 26696 | A cycle is a path starting and ending at its first vertex. (Contributed by Alexander van der Vekens, 8-Nov-2017.) (Revised by AV, 31-Jan-2021.) (Proof shortened by AV, 30-Oct-2021.) |
| ⊢ (𝐹(Cycles‘𝐺)𝑃 → 𝐹((𝑃‘0)(PathsOn‘𝐺)(𝑃‘0))𝑃) | ||
| 30-Oct-2021 | cyclnspth 26695 | A (non trivial) cycle is not a simple path. (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 31-Jan-2021.) (Proof shortened by AV, 30-Oct-2021.) |
| ⊢ (𝐹 ≠ ∅ → (𝐹(Cycles‘𝐺)𝑃 → ¬ 𝐹(SPaths‘𝐺)𝑃)) | ||
| 30-Oct-2021 | cycliscrct 26694 | A cycle is a circuit. (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 31-Jan-2021.) (Proof shortened by AV, 30-Oct-2021.) |
| ⊢ (𝐹(Cycles‘𝐺)𝑃 → 𝐹(Circuits‘𝐺)𝑃) | ||
| 30-Oct-2021 | crctisclwlk 26689 | A circuit is a closed walk. (Contributed by AV, 17-Feb-2021.) (Proof shortened by AV, 30-Oct-2021.) |
| ⊢ (𝐹(Circuits‘𝐺)𝑃 → 𝐹(ClWalks‘𝐺)𝑃) | ||
| 30-Oct-2021 | cyclprop 26688 | The properties of a cycle: A cycle is a closed path. (Contributed by AV, 31-Jan-2021.) (Proof shortened by AV, 30-Oct-2021.) |
| ⊢ (𝐹(Cycles‘𝐺)𝑃 → (𝐹(Paths‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(#‘𝐹)))) | ||
| 30-Oct-2021 | crctprop 26687 | The properties of a circuit: A circuit is a closed trail. (Contributed by AV, 31-Jan-2021.) (Proof shortened by AV, 30-Oct-2021.) |
| ⊢ (𝐹(Circuits‘𝐺)𝑃 → (𝐹(Trails‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(#‘𝐹)))) | ||
| 30-Oct-2021 | iscycl 26686 | Sufficient and necessary conditions for a pair of functions to be a cycle (in an undirected graph): A pair of function "is" (represents) a cycle iff it is a closed path. (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 31-Jan-2021.) (Revised by AV, 30-Oct-2021.) |
| ⊢ (𝐹(Cycles‘𝐺)𝑃 ↔ (𝐹(Paths‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(#‘𝐹)))) | ||
| 30-Oct-2021 | iscrct 26685 | Sufficient and necessary conditions for a pair of functions to be a circuit (in an undirected graph): A pair of function "is" (represents) a circuit iff it is a closed trail. (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 31-Jan-2021.) (Revised by AV, 30-Oct-2021.) |
| ⊢ (𝐹(Circuits‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(#‘𝐹)))) | ||
| 30-Oct-2021 | clwlkwlk 26671 | Closed walks are walks (in an undirected graph). (Contributed by Alexander van der Vekens, 23-Jun-2018.) (Revised by AV, 16-Feb-2021.) (Proof shortened by AV, 30-Oct-2021.) |
| ⊢ (𝑊 ∈ (ClWalks‘𝐺) → 𝑊 ∈ (Walks‘𝐺)) | ||
| 30-Oct-2021 | clwlkiswlk 26670 | A closed walk is a walk (in an undirected graph). (Contributed by Alexander van der Vekens, 15-Mar-2018.) (Revised by AV, 16-Feb-2021.) (Proof shortened by AV, 30-Oct-2021.) |
| ⊢ (𝐹(ClWalks‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃) | ||
| 30-Oct-2021 | isclwlk 26669 | A pair of functions represents a closed walk iff it represents a walk in which the first vertex is equal to the last vertex. (Contributed by Alexander van der Vekens, 24-Jun-2018.) (Revised by AV, 16-Feb-2021.) (Proof shortened by AV, 30-Oct-2021.) |
| ⊢ (𝐹(ClWalks‘𝐺)𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(#‘𝐹)))) | ||
| 30-Oct-2021 | pthd 26665 | Two words representing a trail which also represent a path in a graph. (Contributed by AV, 10-Feb-2021.) (Proof shortened by AV, 30-Oct-2021.) |
| ⊢ (𝜑 → 𝑃 ∈ Word V) & ⊢ 𝑅 = ((#‘𝑃) − 1) & ⊢ (𝜑 → ∀𝑖 ∈ (0..^(#‘𝑃))∀𝑗 ∈ (1..^𝑅)(𝑖 ≠ 𝑗 → (𝑃‘𝑖) ≠ (𝑃‘𝑗))) & ⊢ (#‘𝐹) = 𝑅 & ⊢ (𝜑 → 𝐹(Trails‘𝐺)𝑃) ⇒ ⊢ (𝜑 → 𝐹(Paths‘𝐺)𝑃) | ||
| 30-Oct-2021 | upgrspthswlk 26634 | The set of simple paths in a pseudograph, expressed as walk. Notice that this theorem would not hold for arbitrary hypergraphs, since a walk with distinct vertices does not need to be a trail: let E = { p0, p1, p2 } be a hyperedge, then ( p0, e, p1, e, p2 ) is walk with distinct vertices, but not with distinct edges. Therefore, E is not a trail and, by definition, also no path. (Contributed by AV, 11-Jan-2021.) (Proof shortened by AV, 17-Jan-2021.) (Proof shortened by AV, 30-Oct-2021.) |
| ⊢ (𝐺 ∈ UPGraph → (SPaths‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝐺)𝑝 ∧ Fun ◡𝑝)}) | ||
| 30-Oct-2021 | pthdepisspth 26631 | A path with different start and end points is a simple path (in an undirected graph). (Contributed by Alexander van der Vekens, 31-Oct-2017.) (Revised by AV, 12-Jan-2021.) (Proof shortened by AV, 30-Oct-2021.) |
| ⊢ ((𝐹(Paths‘𝐺)𝑃 ∧ (𝑃‘0) ≠ (𝑃‘(#‘𝐹))) → 𝐹(SPaths‘𝐺)𝑃) | ||
| 30-Oct-2021 | spthdep 26630 | A simple path (at least of length 1) has different start and end points (in an undirected graph). (Contributed by AV, 31-Jan-2021.) (Proof shortened by AV, 30-Oct-2021.) |
| ⊢ ((𝐹(SPaths‘𝐺)𝑃 ∧ (#‘𝐹) ≠ 0) → (𝑃‘0) ≠ (𝑃‘(#‘𝐹))) | ||
| 30-Oct-2021 | spthdifv 26629 | The vertices of a simple path are distinct, so the vertex function is one-to-one. (Contributed by Alexander van der Vekens, 26-Jan-2018.) (Revised by AV, 5-Jun-2021.) (Proof shortened by AV, 30-Oct-2021.) |
| ⊢ (𝐹(SPaths‘𝐺)𝑃 → 𝑃:(0...(#‘𝐹))–1-1→(Vtx‘𝐺)) | ||
| 30-Oct-2021 | spthispth 26622 | A simple path is a path (in an undirected graph). (Contributed by Alexander van der Vekens, 21-Oct-2017.) (Revised by AV, 9-Jan-2021.) (Proof shortened by AV, 30-Oct-2021.) |
| ⊢ (𝐹(SPaths‘𝐺)𝑃 → 𝐹(Paths‘𝐺)𝑃) | ||
| 30-Oct-2021 | pthistrl 26621 | A path is a trail (in an undirected graph). (Contributed by Alexander van der Vekens, 21-Oct-2017.) (Revised by AV, 9-Jan-2021.) (Proof shortened by AV, 30-Oct-2021.) |
| ⊢ (𝐹(Paths‘𝐺)𝑃 → 𝐹(Trails‘𝐺)𝑃) | ||
| 30-Oct-2021 | relpths 26616 | The set (Paths‘𝐺) of all paths on 𝐺 is a set of pairs by our definition of a path, and so is a relation. (Contributed by AV, 30-Oct-2021.) |
| ⊢ Rel (Paths‘𝐺) | ||
| 29-Oct-2021 | eupths 27060 | The Eulerian paths on the graph 𝐺. (Contributed by AV, 18-Feb-2021.) (Revised by AV, 29-Oct-2021.) |
| ⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ (EulerPaths‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝐺)𝑝 ∧ 𝑓:(0..^(#‘𝑓))–onto→dom 𝐼)} | ||
| 29-Oct-2021 | clwlks 26668 | The set of closed walks (in an undirected graph). (Contributed by Alexander van der Vekens, 15-Mar-2018.) (Revised by AV, 16-Feb-2021.) (Revised by AV, 29-Oct-2021.) |
| ⊢ (ClWalks‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝐺)𝑝 ∧ (𝑝‘0) = (𝑝‘(#‘𝑓)))} | ||
| 29-Oct-2021 | isspth 26620 | Conditions for a pair of classes/functions to be a simple path (in an undirected graph). (Contributed by Alexander van der Vekens, 21-Oct-2017.) (Revised by AV, 9-Jan-2021.) (Revised by AV, 29-Oct-2021.) |
| ⊢ (𝐹(SPaths‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃 ∧ Fun ◡𝑃)) | ||
| 29-Oct-2021 | ispth 26619 | Conditions for a pair of classes/functions to be a path (in an undirected graph). (Contributed by Alexander van der Vekens, 21-Oct-2017.) (Revised by AV, 9-Jan-2021.) (Revised by AV, 29-Oct-2021.) |
| ⊢ (𝐹(Paths‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃 ∧ Fun ◡(𝑃 ↾ (1..^(#‘𝐹))) ∧ ((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅)) | ||
| 29-Oct-2021 | spthsfval 26618 | The set of simple paths (in an undirected graph). (Contributed by Alexander van der Vekens, 21-Oct-2017.) (Revised by AV, 9-Jan-2021.) (Revised by AV, 29-Oct-2021.) |
| ⊢ (SPaths‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝐺)𝑝 ∧ Fun ◡𝑝)} | ||
| 29-Oct-2021 | pthsfval 26617 | The set of paths (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Oct-2017.) (Revised by AV, 9-Jan-2021.) (Revised by AV, 29-Oct-2021.) |
| ⊢ (Paths‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝐺)𝑝 ∧ Fun ◡(𝑝 ↾ (1..^(#‘𝑓))) ∧ ((𝑝 “ {0, (#‘𝑓)}) ∩ (𝑝 “ (1..^(#‘𝑓)))) = ∅)} | ||
| 29-Oct-2021 | upgrf1istrl 26600 | Properties of a pair of a one-to-one function into the set of indices of edges and a function into the set of vertices to be a trail in a pseudograph. (Contributed by Alexander van der Vekens, 20-Oct-2017.) (Revised by AV, 7-Jan-2021.) (Revised by AV, 29-Oct-2021.) |
| ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ (𝐺 ∈ UPGraph → (𝐹(Trails‘𝐺)𝑃 ↔ (𝐹:(0..^(#‘𝐹))–1-1→dom 𝐼 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))) | ||
| 29-Oct-2021 | upgristrl 26599 | Properties of a pair of functions to be a trail in a pseudograph, definition of walks expanded. (Contributed by Alexander van der Vekens, 20-Oct-2017.) (Revised by AV, 7-Jan-2021.) (Revised by AV, 29-Oct-2021.) |
| ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ (𝐺 ∈ UPGraph → (𝐹(Trails‘𝐺)𝑃 ↔ ((𝐹 ∈ Word dom 𝐼 ∧ Fun ◡𝐹) ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))) | ||
| 29-Oct-2021 | trlres 26597 | The restriction ⟨𝐻, 𝑄⟩ of a trail ⟨𝐹, 𝑃⟩ to an initial segment of the trail (of length 𝑁) forms a trail on the subgraph 𝑆 consisting of the edges in the initial segment. (Contributed by AV, 6-Mar-2021.) (Proof shortened by AV, 29-Oct-2021.) |
| ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ (𝜑 → 𝐹(Trails‘𝐺)𝑃) & ⊢ (𝜑 → 𝑁 ∈ (0..^(#‘𝐹))) & ⊢ 𝐻 = (𝐹 ↾ (0..^𝑁)) & ⊢ (𝜑 → (Vtx‘𝑆) = 𝑉) & ⊢ (𝜑 → (iEdg‘𝑆) = (𝐼 ↾ (𝐹 “ (0..^𝑁)))) & ⊢ 𝑄 = (𝑃 ↾ (0...𝑁)) ⇒ ⊢ (𝜑 → 𝐻(Trails‘𝑆)𝑄) | ||
| 29-Oct-2021 | trlf1 26595 | The enumeration 𝐹 of a trail ⟨𝐹, 𝑃⟩ is injective. (Contributed by AV, 20-Feb-2021.) (Proof shortened by AV, 29-Oct-2021.) |
| ⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ (𝐹(Trails‘𝐺)𝑃 → 𝐹:(0..^(#‘𝐹))–1-1→dom 𝐼) | ||
| 29-Oct-2021 | trliswlk 26594 | A trail is a walk. (Contributed by Alexander van der Vekens, 20-Oct-2017.) (Revised by AV, 7-Jan-2021.) (Proof shortened by AV, 29-Oct-2021.) |
| ⊢ (𝐹(Trails‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃) | ||
| 29-Oct-2021 | istrl 26593 | Conditions for a pair of classes/functions to be a trail (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Oct-2017.) (Revised by AV, 28-Dec-2020.) (Revised by AV, 29-Oct-2021.) |
| ⊢ (𝐹(Trails‘𝐺)𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ Fun ◡𝐹)) | ||
| 29-Oct-2021 | trlsfval 26592 | The set of trails (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Oct-2017.) (Revised by AV, 28-Dec-2020.) (Revised by AV, 29-Oct-2021.) |
| ⊢ (Trails‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝐺)𝑝 ∧ Fun ◡𝑓)} | ||
| 29-Oct-2021 | reltrls 26591 | The set (Trails‘𝐺) of all trails on 𝐺 is a set of pairs by our definition of a trail, and so is a relation. (Contributed by AV, 29-Oct-2021.) |
| ⊢ Rel (Trails‘𝐺) | ||
| 29-Oct-2021 | fvmptopab 6697 | The function value of a mapping 𝑀 to a restricted binary relation expressed as an ordered-pair class abstraction: The restricted binary relation is a binary relation given as value of a function 𝐹 restricted by the condition 𝜓. (Contributed by AV, 31-Jan-2021.) (Revised by AV, 29-Oct-2021.) |
| ⊢ ((𝜑 ∧ 𝑧 = 𝑍) → (𝜒 ↔ 𝜓)) & ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝑥(𝐹‘𝑍)𝑦} ∈ V) & ⊢ 𝑀 = (𝑧 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹‘𝑧)𝑦 ∧ 𝜒)}) ⇒ ⊢ (𝜑 → (𝑀‘𝑍) = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓)}) | ||
| 29-Oct-2021 | brfvopabrbr 6279 | The binary relation of a function value which is an ordered-pair class abstraction of a restricted binary relation is the restricted binary relation. The first hypothesis can often be obtained by using fvmptopab 6697. (Contributed by AV, 29-Oct-2021.) |
| ⊢ (𝐴‘𝑍) = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐵‘𝑍)𝑦 ∧ 𝜑)} & ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝜑 ↔ 𝜓)) & ⊢ Rel (𝐵‘𝑍) ⇒ ⊢ (𝑋(𝐴‘𝑍)𝑌 ↔ (𝑋(𝐵‘𝑍)𝑌 ∧ 𝜓)) | ||
| 29-Oct-2021 | nprrel12 5160 | Proper classes are not related via any relation. (Contributed by AV, 29-Oct-2021.) |
| ⊢ Rel 𝑅 ⇒ ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ¬ 𝐴𝑅𝐵) | ||
| 29-Oct-2021 | opab0 5007 | Empty ordered pair class abstraction. (Contributed by AV, 29-Oct-2021.) |
| ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} = ∅ ↔ ∀𝑥∀𝑦 ¬ 𝜑) | ||
| 29-Oct-2021 | 4exmid 997 | The disjunction of the four possible combinations of two wffs and their negations is always true. A four-way excluded middle (see exmid 431). (Contributed by David Abernethy, 28-Jan-2014.) (Proof shortened by NM, 29-Oct-2021.) |
| ⊢ (((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)) ∨ ((𝜑 ∧ ¬ 𝜓) ∨ (𝜓 ∧ ¬ 𝜑))) | ||
| 29-Oct-2021 | dfbi3 994 | An alternate definition of the biconditional. Theorem *5.23 of [WhiteheadRussell] p. 124. (Contributed by NM, 27-Jun-2002.) (Proof shortened by Wolf Lammen, 3-Nov-2013.) (Proof shortened by NM, 29-Oct-2021.) |
| ⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓))) | ||
| 28-Oct-2021 | isclwlkupgr 26674 | Properties of a pair of functions to be a closed walk (in a pseudograph). (Contributed by Alexander van der Vekens, 24-Jun-2018.) (Revised by AV, 11-Apr-2021.) (Revised by AV, 28-Oct-2021.) |
| ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ (𝐺 ∈ UPGraph → (𝐹(ClWalks‘𝐺)𝑃 ↔ ((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉) ∧ (∀𝑘 ∈ (0..^(#‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ∧ (𝑃‘0) = (𝑃‘(#‘𝐹)))))) | ||
| 28-Oct-2021 | upgr2wlk 26564 | Properties of a pair of functions to be a walk of length 2 in a pseudograph. Note that the vertices need not to be distinct and the edges can be loops or multiedges. (Contributed by Alexander van der Vekens, 16-Feb-2018.) (Revised by AV, 3-Jan-2021.) (Revised by AV, 28-Oct-2021.) |
| ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ (𝐺 ∈ UPGraph → ((𝐹(Walks‘𝐺)𝑃 ∧ (#‘𝐹) = 2) ↔ (𝐹:(0..^2)⟶dom 𝐼 ∧ 𝑃:(0...2)⟶𝑉 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})))) | ||
| 28-Oct-2021 | upgriswlk 26537 | Properties of a pair of functions to be a walk in a pseudograph. (Contributed by AV, 2-Jan-2021.) (Revised by AV, 28-Oct-2021.) |
| ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ (𝐺 ∈ UPGraph → (𝐹(Walks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))) | ||
| 28-Oct-2021 | dfiota4 5879 | The ℩ operation using the if operator. (Contributed by Scott Fenton, 6-Oct-2017.) (Proof shortened by JJ, 28-Oct-2021.) |
| ⊢ (℩𝑥𝜑) = if(∃!𝑥𝜑, ∪ {𝑥 ∣ 𝜑}, ∅) | ||
| 28-Oct-2021 | dfcleq 2616 | The same as df-cleq 2615 with the hypothesis removed using the Axiom of Extensionality ax-ext 2602. (Contributed by NM, 15-Sep-1993.) Revised to make use of axext3 2604 instead of ax-ext 2602, so that ax-9 1999 will appear in lists of axioms used by a proof, since df-cleq 2615 implies ax-9 1999 by theorem bj-ax9 32890. We may revisit this in the future. (Revised by NM, 28-Oct-2021.) (Proof modification is discouraged.) |
| ⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) | ||
| 25-Oct-2021 | cnv0 5535 | The converse of the empty set. (Contributed by NM, 6-Apr-1998.) Remove dependency on ax-sep 4781, ax-nul 4789, ax-pr 4906. (Revised by KP, 25-Oct-2021.) |
| ⊢ ◡∅ = ∅ | ||
| 25-Oct-2021 | relopabi 5245 | A class of ordered pairs is a relation. (Contributed by Mario Carneiro, 21-Dec-2013.) Remove dependency on ax-sep 4781, ax-nul 4789, ax-pr 4906. (Revised by KP, 25-Oct-2021.) |
| ⊢ 𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝜑} ⇒ ⊢ Rel 𝐴 | ||
| 25-Oct-2021 | ssrel 5207 | A subclass relationship depends only on a relation's ordered pairs. Theorem 3.2(i) of [Monk1] p. 33. (Contributed by NM, 2-Aug-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) Remove dependency on ax-sep 4781, ax-nul 4789, ax-pr 4906. (Revised by KP, 25-Oct-2021.) |
| ⊢ (Rel 𝐴 → (𝐴 ⊆ 𝐵 ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵))) | ||
| 25-Oct-2021 | syldc 48 | Syllogism deduction. Commuted form of syld 47. (Contributed by BJ, 25-Oct-2021.) |
| ⊢ (𝜑 → (𝜓 → 𝜒)) & ⊢ (𝜑 → (𝜒 → 𝜃)) ⇒ ⊢ (𝜓 → (𝜑 → 𝜃)) | ||
| 25-Oct-2021 | syl11 33 | A syllogism inference. Commuted form of an instance of syl 17. (Contributed by BJ, 25-Oct-2021.) |
| ⊢ (𝜑 → (𝜓 → 𝜒)) & ⊢ (𝜃 → 𝜑) ⇒ ⊢ (𝜓 → (𝜃 → 𝜒)) | ||
| 24-Oct-2021 | uspgrunop 26081 | The union of two simple pseudographs (with the same vertex set): If ⟨𝑉, 𝐸⟩ and ⟨𝑉, 𝐹⟩ are simple pseudographs, then ⟨𝑉, 𝐸 ∪ 𝐹⟩ is a pseudograph (the vertex set stays the same, but the edges from both graphs are kept, maybe resulting incident two edges between two vertices). (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 16-Oct-2020.) (Revised by AV, 24-Oct-2021.) |
| ⊢ (𝜑 → 𝐺 ∈ USPGraph ) & ⊢ (𝜑 → 𝐻 ∈ USPGraph ) & ⊢ 𝐸 = (iEdg‘𝐺) & ⊢ 𝐹 = (iEdg‘𝐻) & ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ (𝜑 → (Vtx‘𝐻) = 𝑉) & ⊢ (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅) ⇒ ⊢ (𝜑 → ⟨𝑉, (𝐸 ∪ 𝐹)⟩ ∈ UPGraph ) | ||
| 24-Oct-2021 | upgrunop 26014 | The union of two pseudographs (with the same vertex set): If ⟨𝑉, 𝐸⟩ and ⟨𝑉, 𝐹⟩ are pseudographs, then ⟨𝑉, 𝐸 ∪ 𝐹⟩ is a pseudograph (the vertex set stays the same, but the edges from both graphs are kept). (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 12-Oct-2020.) (Revised by AV, 24-Oct-2021.) |
| ⊢ (𝜑 → 𝐺 ∈ UPGraph ) & ⊢ (𝜑 → 𝐻 ∈ UPGraph ) & ⊢ 𝐸 = (iEdg‘𝐺) & ⊢ 𝐹 = (iEdg‘𝐻) & ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ (𝜑 → (Vtx‘𝐻) = 𝑉) & ⊢ (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅) ⇒ ⊢ (𝜑 → ⟨𝑉, (𝐸 ∪ 𝐹)⟩ ∈ UPGraph ) | ||
| 24-Oct-2021 | upgrun 26013 | The union 𝑈 of two pseudographs 𝐺 and 𝐻 with the same vertex set 𝑉 is a pseudograph with the vertex 𝑉 and the union (𝐸 ∪ 𝐹) of the (indexed) edges. (Contributed by AV, 12-Oct-2020.) (Revised by AV, 24-Oct-2021.) |
| ⊢ (𝜑 → 𝐺 ∈ UPGraph ) & ⊢ (𝜑 → 𝐻 ∈ UPGraph ) & ⊢ 𝐸 = (iEdg‘𝐺) & ⊢ 𝐹 = (iEdg‘𝐻) & ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ (𝜑 → (Vtx‘𝐻) = 𝑉) & ⊢ (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅) & ⊢ (𝜑 → 𝑈 ∈ 𝑊) & ⊢ (𝜑 → (Vtx‘𝑈) = 𝑉) & ⊢ (𝜑 → (iEdg‘𝑈) = (𝐸 ∪ 𝐹)) ⇒ ⊢ (𝜑 → 𝑈 ∈ UPGraph ) | ||
| 24-Oct-2021 | ushgrunop 25972 | The union of two (undirected) simple hypergraphs (with the same vertex set) represented as ordered pair: If ⟨𝑉, 𝐸⟩ and ⟨𝑉, 𝐹⟩ are simple hypergraphs, then ⟨𝑉, 𝐸 ∪ 𝐹⟩ is a (not necessarily simple) hypergraph - the vertex set stays the same, but the edges from both graphs are kept, possibly resulting in two edges between two vertices. (Contributed by AV, 29-Nov-2020.) (Revised by AV, 24-Oct-2021.) |
| ⊢ (𝜑 → 𝐺 ∈ USHGraph ) & ⊢ (𝜑 → 𝐻 ∈ USHGraph ) & ⊢ 𝐸 = (iEdg‘𝐺) & ⊢ 𝐹 = (iEdg‘𝐻) & ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ (𝜑 → (Vtx‘𝐻) = 𝑉) & ⊢ (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅) ⇒ ⊢ (𝜑 → ⟨𝑉, (𝐸 ∪ 𝐹)⟩ ∈ UHGraph ) | ||
| 24-Oct-2021 | ushgrun 25971 | The union 𝑈 of two (undirected) simple hypergraphs 𝐺 and 𝐻 with the same vertex set 𝑉 is a (not necessarily simple) hypergraph with the vertex 𝑉 and the union (𝐸 ∪ 𝐹) of the (indexed) edges. (Contributed by AV, 29-Nov-2020.) (Revised by AV, 24-Oct-2021.) |
| ⊢ (𝜑 → 𝐺 ∈ USHGraph ) & ⊢ (𝜑 → 𝐻 ∈ USHGraph ) & ⊢ 𝐸 = (iEdg‘𝐺) & ⊢ 𝐹 = (iEdg‘𝐻) & ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ (𝜑 → (Vtx‘𝐻) = 𝑉) & ⊢ (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅) & ⊢ (𝜑 → 𝑈 ∈ 𝑊) & ⊢ (𝜑 → (Vtx‘𝑈) = 𝑉) & ⊢ (𝜑 → (iEdg‘𝑈) = (𝐸 ∪ 𝐹)) ⇒ ⊢ (𝜑 → 𝑈 ∈ UHGraph ) | ||
| 24-Oct-2021 | uhgrunop 25970 | The union of two (undirected) hypergraphs (with the same vertex set) represented as ordered pair: If ⟨𝑉, 𝐸⟩ and ⟨𝑉, 𝐹⟩ are hypergraphs, then ⟨𝑉, 𝐸 ∪ 𝐹⟩ is a hypergraph (the vertex set stays the same, but the edges from both graphs are kept, possibly resulting in two edges between two vertices). (Contributed by Alexander van der Vekens, 27-Dec-2017.) (Revised by AV, 11-Oct-2020.) (Revised by AV, 24-Oct-2021.) |
| ⊢ (𝜑 → 𝐺 ∈ UHGraph ) & ⊢ (𝜑 → 𝐻 ∈ UHGraph ) & ⊢ 𝐸 = (iEdg‘𝐺) & ⊢ 𝐹 = (iEdg‘𝐻) & ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ (𝜑 → (Vtx‘𝐻) = 𝑉) & ⊢ (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅) ⇒ ⊢ (𝜑 → ⟨𝑉, (𝐸 ∪ 𝐹)⟩ ∈ UHGraph ) | ||
| 24-Oct-2021 | uhgrun 25969 | The union 𝑈 of two (undirected) hypergraphs 𝐺 and 𝐻 with the same vertex set 𝑉 is a hypergraph with the vertex 𝑉 and the union (𝐸 ∪ 𝐹) of the (indexed) edges. (Contributed by AV, 11-Oct-2020.) (Revised by AV, 24-Oct-2021.) |
| ⊢ (𝜑 → 𝐺 ∈ UHGraph ) & ⊢ (𝜑 → 𝐻 ∈ UHGraph ) & ⊢ 𝐸 = (iEdg‘𝐺) & ⊢ 𝐹 = (iEdg‘𝐻) & ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ (𝜑 → (Vtx‘𝐻) = 𝑉) & ⊢ (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅) & ⊢ (𝜑 → 𝑈 ∈ 𝑊) & ⊢ (𝜑 → (Vtx‘𝑈) = 𝑉) & ⊢ (𝜑 → (iEdg‘𝑈) = (𝐸 ∪ 𝐹)) ⇒ ⊢ (𝜑 → 𝑈 ∈ UHGraph ) | ||
| 24-Oct-2021 | hashle00 13188 | If the size of a set is less than or equal to zero, the set must be empty. (Contributed by Alexander van der Vekens, 6-Jan-2018.) (Proof shortened by AV, 24-Oct-2021.) |
| ⊢ (𝑉 ∈ 𝑊 → ((#‘𝑉) ≤ 0 ↔ 𝑉 = ∅)) | ||
| 24-Oct-2021 | hashf 13125 | The size function maps all finite sets to their cardinality, as members of ℕ0, and infinite sets to +∞. TODO-AV: mark as OBSOLETE and replace it by hashfxnn0 13124? (Contributed by Mario Carneiro, 13-Sep-2013.) (Revised by Mario Carneiro, 13-Jul-2014.) (Proof shortened by AV, 24-Oct-2021.) |
| ⊢ #:V⟶(ℕ0 ∪ {+∞}) | ||
| 24-Oct-2021 | fun2d 6068 | The union of functions with disjoint domains is a function, deduction version of fun2 6067. (Contributed by AV, 11-Oct-2020.) (Revised by AV, 24-Oct-2021.) |
| ⊢ (𝜑 → 𝐹:𝐴⟶𝐶) & ⊢ (𝜑 → 𝐺:𝐵⟶𝐶) & ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅) ⇒ ⊢ (𝜑 → (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐵)⟶𝐶) | ||
| 23-Oct-2021 | smflimsupmpt 41035 | The superior limit of a sequence of sigma-measurable functions is sigma-measurable. Proposition 121F (d) of [Fremlin1] p. 39 . 𝐴 can contain 𝑚 as a free variable, in other words it can be thought of as an indexed collection 𝐴(𝑚). 𝐵 can be thought of as a collection with two indexes 𝐵(𝑚, 𝑥). (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑚𝜑 & ⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑛𝜑 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑊) & ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆)) & ⊢ 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐴 ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ 𝐵)) ∈ ℝ} & ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ (lim sup‘(𝑚 ∈ 𝑍 ↦ 𝐵))) ⇒ ⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆)) | ||
| 23-Oct-2021 | smflimsup 41034 | The superior limit of a sequence of sigma-measurable functions is sigma-measurable. Proposition 121F (d) of [Fremlin1] p. 39 . (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑚𝐹 & ⊢ Ⅎ𝑥𝐹 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) & ⊢ 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ} & ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)))) ⇒ ⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆)) | ||
| 23-Oct-2021 | smflimsuplem8 41033 | The superior limit of a sequence of sigma-measurable functions is sigma-measurable. Proposition 121F (d) of [Fremlin1] p. 39 . (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) & ⊢ 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ} & ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)))) & ⊢ 𝐸 = (𝑘 ∈ 𝑍 ↦ {𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑘)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑘) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ}) & ⊢ 𝐻 = (𝑘 ∈ 𝑍 ↦ (𝑥 ∈ (𝐸‘𝑘) ↦ sup(ran (𝑚 ∈ (ℤ≥‘𝑘) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ))) ⇒ ⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆)) | ||
| 23-Oct-2021 | smflimsuplem7 41032 | The superior limit of a sequence of sigma-measurable functions is sigma-measurable. Proposition 121F (d) of [Fremlin1] p. 39 . (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) & ⊢ 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ} & ⊢ 𝐸 = (𝑘 ∈ 𝑍 ↦ {𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑘)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑘) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ}) & ⊢ 𝐻 = (𝑘 ∈ 𝑍 ↦ (𝑥 ∈ (𝐸‘𝑘) ↦ sup(ran (𝑚 ∈ (ℤ≥‘𝑘) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ))) ⇒ ⊢ (𝜑 → 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑘 ∈ (ℤ≥‘𝑛)dom (𝐻‘𝑘) ∣ (𝑘 ∈ 𝑍 ↦ ((𝐻‘𝑘)‘𝑥)) ∈ dom ⇝ }) | ||
| 23-Oct-2021 | smflimsuplem6 41031 | The superior limit of a sequence of sigma-measurable functions is sigma-measurable. Proposition 121F (d) of [Fremlin1] p. 39 . (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑛𝜑 & ⊢ Ⅎ𝑚𝜑 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) & ⊢ 𝐸 = (𝑛 ∈ 𝑍 ↦ {𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ}) & ⊢ 𝐻 = (𝑛 ∈ 𝑍 ↦ (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ))) & ⊢ (𝜑 → (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋))) ∈ ℝ) & ⊢ (𝜑 → 𝑁 ∈ 𝑍) & ⊢ (𝜑 → 𝑋 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑁)dom (𝐹‘𝑚)) ⇒ ⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ ((𝐻‘𝑛)‘𝑋)) ∈ dom ⇝ ) | ||
| 23-Oct-2021 | smflimsuplem5 41030 | 𝐻 converges to the superior limit of 𝐹. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑛𝜑 & ⊢ Ⅎ𝑚𝜑 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) & ⊢ 𝐸 = (𝑛 ∈ 𝑍 ↦ {𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ}) & ⊢ 𝐻 = (𝑛 ∈ 𝑍 ↦ (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ))) & ⊢ (𝜑 → (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋))) ∈ ℝ) & ⊢ (𝜑 → 𝑁 ∈ 𝑍) & ⊢ (𝜑 → 𝑋 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑁)dom (𝐹‘𝑚)) ⇒ ⊢ (𝜑 → (𝑛 ∈ (ℤ≥‘𝑁) ↦ ((𝐻‘𝑛)‘𝑋)) ⇝ (lim sup‘(𝑚 ∈ (ℤ≥‘𝑁) ↦ ((𝐹‘𝑚)‘𝑋)))) | ||
| 23-Oct-2021 | smflimsuplem4 41029 | If 𝐻 converges, the lim sup of 𝐹 is real. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑛𝜑 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) & ⊢ 𝐸 = (𝑛 ∈ 𝑍 ↦ {𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ}) & ⊢ 𝐻 = (𝑛 ∈ 𝑍 ↦ (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ))) & ⊢ (𝜑 → 𝑁 ∈ 𝑍) & ⊢ (𝜑 → 𝑥 ∈ ∩ 𝑛 ∈ (ℤ≥‘𝑁)dom (𝐻‘𝑛)) & ⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ ((𝐻‘𝑛)‘𝑥)) ∈ dom ⇝ ) ⇒ ⊢ (𝜑 → (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ) | ||
| 23-Oct-2021 | smflimsuplem3 41028 | The limit of the (𝐻‘𝑛) functions is sigma-measurable. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) & ⊢ 𝐸 = (𝑛 ∈ 𝑍 ↦ {𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ}) & ⊢ 𝐻 = (𝑛 ∈ 𝑍 ↦ (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ))) ⇒ ⊢ (𝜑 → (𝑥 ∈ {𝑥 ∈ ∪ 𝑘 ∈ 𝑍 ∩ 𝑛 ∈ (ℤ≥‘𝑘)dom (𝐻‘𝑛) ∣ (𝑛 ∈ 𝑍 ↦ ((𝐻‘𝑛)‘𝑥)) ∈ dom ⇝ } ↦ ( ⇝ ‘(𝑛 ∈ 𝑍 ↦ ((𝐻‘𝑛)‘𝑥)))) ∈ (SMblFn‘𝑆)) | ||
| 23-Oct-2021 | smflimsuplem2 41027 | The superior limit of a sequence of sigma-measurable functions is sigma-measurable. Proposition 121F (d) of [Fremlin1] p. 39 . (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑚𝜑 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) & ⊢ 𝐸 = (𝑛 ∈ 𝑍 ↦ {𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ}) & ⊢ 𝐻 = (𝑛 ∈ 𝑍 ↦ (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ))) & ⊢ (𝜑 → 𝑛 ∈ 𝑍) & ⊢ (𝜑 → (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋))) ∈ ℝ) & ⊢ (𝜑 → 𝑋 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚)) ⇒ ⊢ (𝜑 → 𝑋 ∈ dom (𝐻‘𝑛)) | ||
| 23-Oct-2021 | smflimsuplem1 41026 | If 𝐻 converges, the lim sup of 𝐹 is real. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ 𝐸 = (𝑛 ∈ 𝑍 ↦ {𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ}) & ⊢ 𝐻 = (𝑛 ∈ 𝑍 ↦ (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ))) & ⊢ (𝜑 → 𝐾 ∈ 𝑍) ⇒ ⊢ (𝜑 → dom (𝐻‘𝐾) ⊆ dom (𝐹‘𝐾)) | ||
| 23-Oct-2021 | smfinfmpt 41025 | The infimum of a countable set of sigma-measurable functions is sigma-measurable. Proposition 121F (c) of [Fremlin1] p. 38 . (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑛𝜑 & ⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆)) & ⊢ 𝐷 = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 𝐴 ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑦 ≤ 𝐵} & ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ inf(ran (𝑛 ∈ 𝑍 ↦ 𝐵), ℝ, < )) ⇒ ⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆)) | ||
| 23-Oct-2021 | smfinf 41024 | The infimum of a countable set of sigma-measurable functions is sigma-measurable. Proposition 121F (c) of [Fremlin1] p. 38 . (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑛𝐹 & ⊢ Ⅎ𝑥𝐹 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) & ⊢ 𝐷 = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥)} & ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ inf(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < )) ⇒ ⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆)) | ||
| 23-Oct-2021 | smfinflem 41023 | The infimum of a countable set of sigma-measurable functions is sigma-measurable. Proposition 121F (c) of [Fremlin1] p. 38 . (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) & ⊢ 𝐷 = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥)} & ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ inf(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < )) ⇒ ⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆)) | ||
| 23-Oct-2021 | smfsupxr 41022 | The supremum of a countable set of sigma-measurable functions is sigma-measurable. Proposition 121F (b) of [Fremlin1] p. 38 . (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑛𝐹 & ⊢ Ⅎ𝑥𝐹 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) & ⊢ 𝐷 = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ*, < ) ∈ ℝ} & ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ*, < )) ⇒ ⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆)) | ||
| 23-Oct-2021 | smfsupmpt 41021 | The supremum of a countable set of sigma-measurable functions is sigma-measurable. Proposition 121F (b) of [Fremlin1] p. 38 . (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑛𝜑 & ⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆)) & ⊢ 𝐷 = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 𝐴 ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝐵 ≤ 𝑦} & ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ sup(ran (𝑛 ∈ 𝑍 ↦ 𝐵), ℝ, < )) ⇒ ⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆)) | ||
| 23-Oct-2021 | smfsup 41020 | The supremum of a countable set of sigma-measurable functions is sigma-measurable. Proposition 121F (b) of [Fremlin1] p. 38 . (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑛𝐹 & ⊢ Ⅎ𝑥𝐹 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) & ⊢ 𝐷 = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦} & ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < )) ⇒ ⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆)) | ||
| 23-Oct-2021 | smfsuplem3 41019 | The supremum of a countable set of sigma-measurable functions is sigma-measurable. Proposition 121F (b) of [Fremlin1] p. 38 . (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) & ⊢ 𝐷 = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦} & ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < )) ⇒ ⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆)) | ||
| 23-Oct-2021 | smfsuplem2 41018 | The supremum of a countable set of sigma-measurable functions is sigma-measurable. Proposition 121F (b) of [Fremlin1] p. 38 . (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) & ⊢ 𝐷 = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦} & ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < )) & ⊢ (𝜑 → 𝐴 ∈ ℝ) ⇒ ⊢ (𝜑 → (◡𝐺 “ (-∞(,]𝐴)) ∈ (𝑆 ↾t 𝐷)) | ||
| 23-Oct-2021 | smfsuplem1 41017 | The supremum of a countable set of sigma-measurable functions is sigma-measurable. Proposition 121F (b) of [Fremlin1] p. 38 . (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) & ⊢ 𝐷 = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦} & ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < )) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐻:𝑍⟶𝑆) & ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (◡(𝐹‘𝑛) “ (-∞(,]𝐴)) = ((𝐻‘𝑛) ∩ dom (𝐹‘𝑛))) ⇒ ⊢ (𝜑 → (◡𝐺 “ (-∞(,]𝐴)) ∈ (𝑆 ↾t 𝐷)) | ||
| 23-Oct-2021 | smflimmpt 41016 | The limit of a sequence of sigma-measurable functions is sigma-measurable. Proposition 121F (a) of [Fremlin1] p. 38 . Notice that every function in the sequence can have a different (partial) domain, and the domain of convergence can be decidedly irregular (Remark 121G of [Fremlin1] p. 39 ). 𝐴 can contain 𝑚 as a free variable, in other words it can be thought as an indexed collection 𝐴(𝑚). 𝐵 can be thought as a collection with two indexes 𝐵(𝑚, 𝑥). (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑚𝜑 & ⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑛𝜑 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → 𝐴 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑊) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆)) & ⊢ 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐴 ∣ (𝑚 ∈ 𝑍 ↦ 𝐵) ∈ dom ⇝ } & ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ 𝐵))) ⇒ ⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆)) | ||
| 23-Oct-2021 | issmfle2d 41015 | A sufficient condition for "𝐹 being a measurable function w.r.t. to the sigma-algebra 𝑆". (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑎𝜑 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐷 ⊆ ∪ 𝑆) & ⊢ (𝜑 → 𝐹:𝐷⟶ℝ) & ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → (◡𝐹 “ (-∞(,]𝑎)) ∈ (𝑆 ↾t 𝐷)) ⇒ ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) | ||
| 23-Oct-2021 | smfpimcc 41014 | Given a countable set of sigma-measurable functions, and a Borel set 𝐴 there exists a choice function ℎ that, for each measurable function, chooses a measurable set that, when intersected with the function's domain, gives the preimage of 𝐴. This is a generalization of the observation at the beginning of the proof of Proposition 121F of [Fremlin1] p. 39 . The statement would also be provable for uncountable sets, but in most cases it will suffice to consider the countable case, and only the axiom of countable choice will be needed. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑛𝐹 & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) & ⊢ 𝐽 = (topGen‘ran (,)) & ⊢ 𝐵 = (SalGen‘𝐽) & ⊢ (𝜑 → 𝐴 ∈ 𝐵) ⇒ ⊢ (𝜑 → ∃ℎ(ℎ:𝑍⟶𝑆 ∧ ∀𝑛 ∈ 𝑍 (◡(𝐹‘𝑛) “ 𝐴) = ((ℎ‘𝑛) ∩ dom (𝐹‘𝑛)))) | ||
| 23-Oct-2021 | smfpimcclem 41013 | Lemma for smfpimcc 41014 given the choice function 𝐶. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑛𝜑 & ⊢ 𝑍 ∈ 𝑉 & ⊢ (𝜑 → 𝑆 ∈ 𝑊) & ⊢ ((𝜑 ∧ 𝑦 ∈ ran (𝑛 ∈ 𝑍 ↦ {𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑛))})) → (𝐶‘𝑦) ∈ 𝑦) & ⊢ 𝐻 = (𝑛 ∈ 𝑍 ↦ (𝐶‘{𝑠 ∈ 𝑆 ∣ (◡(𝐹‘𝑛) “ 𝐴) = (𝑠 ∩ dom (𝐹‘𝑛))})) ⇒ ⊢ (𝜑 → ∃ℎ(ℎ:𝑍⟶𝑆 ∧ ∀𝑛 ∈ 𝑍 (◡(𝐹‘𝑛) “ 𝐴) = ((ℎ‘𝑛) ∩ dom (𝐹‘𝑛)))) | ||
| 23-Oct-2021 | smflim2 41012 | The limit of a sequence of sigma-measurable functions is sigma-measurable. Proposition 121F (a) of [Fremlin1] p. 38 . Notice that every function in the sequence can have a different (partial) domain, and the domain of convergence can be decidedly irregular (Remark 121G of [Fremlin1] p. 39 ). TODO this has less distinct variable restrictions than smflim and should replace it. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑚𝐹 & ⊢ Ⅎ𝑥𝐹 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) & ⊢ 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ } & ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)))) ⇒ ⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆)) | ||
| 23-Oct-2021 | smffmpt 41011 | A function measurable w.r.t. to a sigma-algebra, is actually a function. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆)) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℝ) | ||
| 23-Oct-2021 | smfneg 41010 | The negative of a sigma-measurable function is measurable. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝑆 ∈ SAlg) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆)) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ -𝐵) ∈ (SMblFn‘𝑆)) | ||
| 23-Oct-2021 | supcnvlimsupmpt 39973 | If a function on a set of upper integers has a real superior limit, the supremum of the rightmost parts of the function, converges to that superior limit. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑗𝜑 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝐵 ∈ ℝ) & ⊢ (𝜑 → (lim sup‘(𝑗 ∈ 𝑍 ↦ 𝐵)) ∈ ℝ) ⇒ ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ sup(ran (𝑗 ∈ (ℤ≥‘𝑘) ↦ 𝐵), ℝ*, < )) ⇝ (lim sup‘(𝑗 ∈ 𝑍 ↦ 𝐵))) | ||
| 23-Oct-2021 | supcnvlimsup 39972 | If a function on a set of upper integers has a real superior limit, the supremum of the rightmost parts of the function, converges to that superior limit. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ) & ⊢ (𝜑 → (lim sup‘𝐹) ∈ ℝ) ⇒ ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑘)), ℝ*, < )) ⇝ (lim sup‘𝐹)) | ||
| 23-Oct-2021 | limsupreuzmpt 39971 | Given a function on the reals, defined on a set of upper integers, its supremum limit is real if and only if two condition holds: 1. there is a real number that is smaller or equal than the function, infinitely often; 2. there is a real number that is larger or equal than the function. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑗𝜑 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → ((lim sup‘(𝑗 ∈ 𝑍 ↦ 𝐵)) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)𝑥 ≤ 𝐵 ∧ ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑥))) | ||
| 23-Oct-2021 | limsupvaluz2 39970 | The superior limit, when the domain of a real-valued function is a set of upper integers, and the superior limit is real. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ) & ⊢ (𝜑 → (lim sup‘𝐹) ∈ ℝ) ⇒ ⊢ (𝜑 → (lim sup‘𝐹) = inf(ran (𝑘 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑘)), ℝ*, < )), ℝ, < )) | ||
| 23-Oct-2021 | limsupreuz 39969 | Given a function on the reals, its supremum limit is real if and only if two condition holds: 1. there is a real number that is smaller or equal than the function, infinitely often; 2. there is a real number that is larger or equal than the function. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑗𝐹 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ) ⇒ ⊢ (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑗) ∧ ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 (𝐹‘𝑗) ≤ 𝑥))) | ||
| 23-Oct-2021 | limsupre3uz 39968 | Given a function on the extended reals, its supremum limit is real if and only if two condition holds: 1. there is a real number that is smaller or equal than the function, infinitely often; 2. there is a real number that is eventually larger or equal than the function. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑗𝐹 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) ⇒ ⊢ (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑗) ∧ ∃𝑥 ∈ ℝ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥))) | ||
| 23-Oct-2021 | limsupre3uzlem 39967 | Given a function on the extended reals, its supremum limit is real if and only if two condition holds: 1. there is a real number that is smaller or equal than the function, infinitely often; 2. there is a real number that is eventually larger or equal than the function. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑗𝐹 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) ⇒ ⊢ (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑗) ∧ ∃𝑥 ∈ ℝ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥))) | ||
| 23-Oct-2021 | limsupre3mpt 39966 | Given a function on the extended reals, its supremum limit is real if and only if two condition holds: 1. there is a real number that is smaller or equal than the function, at some point, in any upper part of the reals; 2. there is a real number that is eventually larger or equal than the function. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*) ⇒ ⊢ (𝜑 → ((lim sup‘(𝑥 ∈ 𝐴 ↦ 𝐵)) ∈ ℝ ↔ (∃𝑦 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑥 ∈ 𝐴 (𝑘 ≤ 𝑥 ∧ 𝑦 ≤ 𝐵) ∧ ∃𝑦 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑘 ≤ 𝑥 → 𝐵 ≤ 𝑦)))) | ||
| 23-Oct-2021 | limsupre3 39965 | Given a function on the extended reals, its supremum limit is real if and only if two condition holds: 1. there is a real number that is smaller or equal than the function, at some point, in any upper part of the reals; 2. there is a real number that is eventually larger or equal than the function. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑗𝐹 & ⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*) ⇒ ⊢ (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) ∧ ∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)))) | ||
| 23-Oct-2021 | limsupre3lem 39964 | Given a function on the extended reals, its supremum limit is real if and only if two condition holds: 1. there is a real number that is smaller or equal than the function, at some point, in any upper part of the reals; 2. there is a real number that is eventually larger or equal than the function. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑗𝐹 & ⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*) ⇒ ⊢ (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) ∧ ∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)))) | ||
| 23-Oct-2021 | limsupequzmptf 39963 | Two functions that are eventually equal to one another have the same superior limit. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑗𝜑 & ⊢ Ⅎ𝑗𝐴 & ⊢ Ⅎ𝑗𝐵 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ 𝐴 = (ℤ≥‘𝑀) & ⊢ 𝐵 = (ℤ≥‘𝑁) & ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐶 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐵) → 𝐶 ∈ 𝑊) ⇒ ⊢ (𝜑 → (lim sup‘(𝑗 ∈ 𝐴 ↦ 𝐶)) = (lim sup‘(𝑗 ∈ 𝐵 ↦ 𝐶))) | ||
| 23-Oct-2021 | limsupre2mpt 39962 | Given a function on the extended reals, its supremum limit is real if and only if two condition holds: 1. there is a real number that is smaller than the function, at some point, in any upper part of the reals; 2. there is a real number that is eventually larger than the function. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*) ⇒ ⊢ (𝜑 → ((lim sup‘(𝑥 ∈ 𝐴 ↦ 𝐵)) ∈ ℝ ↔ (∃𝑦 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑥 ∈ 𝐴 (𝑘 ≤ 𝑥 ∧ 𝑦 < 𝐵) ∧ ∃𝑦 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑘 ≤ 𝑥 → 𝐵 < 𝑦)))) | ||
| 23-Oct-2021 | limsupequzmpt 39961 | Two functions that are eventually equal to one another have the same superior limit. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑗𝜑 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ 𝐴 = (ℤ≥‘𝑀) & ⊢ 𝐵 = (ℤ≥‘𝑁) & ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐶 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐵) → 𝐶 ∈ 𝑊) ⇒ ⊢ (𝜑 → (lim sup‘(𝑗 ∈ 𝐴 ↦ 𝐶)) = (lim sup‘(𝑗 ∈ 𝐵 ↦ 𝐶))) | ||
| 23-Oct-2021 | limsupequzmptlem 39960 | Two functions that are eventually equal to one another have the same superior limit. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑗𝜑 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ 𝐴 = (ℤ≥‘𝑀) & ⊢ 𝐵 = (ℤ≥‘𝑁) & ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐶 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐵) → 𝐶 ∈ 𝑊) & ⊢ 𝐾 = if(𝑀 ≤ 𝑁, 𝑁, 𝑀) ⇒ ⊢ (𝜑 → (lim sup‘(𝑗 ∈ 𝐴 ↦ 𝐶)) = (lim sup‘(𝑗 ∈ 𝐵 ↦ 𝐶))) | ||
| 23-Oct-2021 | limsupmnfuz 39959 | The superior limit of a function is -∞ if and only if every real number is the upper bound of the restriction of the function to a set of upper integers. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑗𝐹 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) ⇒ ⊢ (𝜑 → ((lim sup‘𝐹) = -∞ ↔ ∀𝑥 ∈ ℝ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥)) | ||
| 23-Oct-2021 | limsupmnfuzlem 39958 | The superior limit of a function is -∞ if and only if every real number is the upper bound of the restriction of the function to a set of upper integers. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) ⇒ ⊢ (𝜑 → ((lim sup‘𝐹) = -∞ ↔ ∀𝑥 ∈ ℝ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥)) | ||
| 23-Oct-2021 | limsupre2 39957 | Given a function on the extended reals, its supremum limit is real if and only if two condition holds: 1. there is a real number that is smaller than the function, at some point, in any upper part of the reals; 2. there is a real number that is eventually larger than the function. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑗𝐹 & ⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*) ⇒ ⊢ (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 < (𝐹‘𝑗)) ∧ ∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < 𝑥)))) | ||
| 23-Oct-2021 | limsupre2lem 39956 | Given a function on the extended reals, its supremum limit is real if and only if two condition holds: 1. there is a real number that is smaller than the function, at some point, in any upper part of the reals; 2. there is a real number that is eventually larger than the function. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑗𝐹 & ⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*) ⇒ ⊢ (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 < (𝐹‘𝑗)) ∧ ∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < 𝑥)))) | ||
| 23-Oct-2021 | limsupequz 39955 | Two functions that are eventually equal to one another have the same superior limit. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑘𝜑 & ⊢ Ⅎ𝑘𝐹 & ⊢ Ⅎ𝑘𝐺 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐹 Fn (ℤ≥‘𝑀)) & ⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ (𝜑 → 𝐺 Fn (ℤ≥‘𝑁)) & ⊢ (𝜑 → 𝐾 ∈ ℤ) & ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝐾)) → (𝐹‘𝑘) = (𝐺‘𝑘)) ⇒ ⊢ (𝜑 → (lim sup‘𝐹) = (lim sup‘𝐺)) | ||
| 23-Oct-2021 | limsupequzlem 39954 | Two functions that are eventually equal to one another have the same superior limit. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑘𝜑 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐹 Fn (ℤ≥‘𝑀)) & ⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ (𝜑 → 𝐺 Fn (ℤ≥‘𝑁)) & ⊢ (𝜑 → 𝐾 ∈ ℤ) & ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝐾)) → (𝐹‘𝑘) = (𝐺‘𝑘)) ⇒ ⊢ (𝜑 → (lim sup‘𝐹) = (lim sup‘𝐺)) | ||
| 23-Oct-2021 | limsupmnf 39953 | The superior limit of a function is -∞ if and only if every real number is the upper bound of the restriction of the function to an upper interval of real numbers. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑗𝐹 & ⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*) ⇒ ⊢ (𝜑 → ((lim sup‘𝐹) = -∞ ↔ ∀𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥))) | ||
| 23-Oct-2021 | limsupmnflem 39952 | The superior limit of a function is -∞ if and only if every real number is the upper bound of the restriction of the function to an upper interval of real numbers. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*) & ⊢ 𝐺 = (𝑘 ∈ ℝ ↦ sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < )) ⇒ ⊢ (𝜑 → ((lim sup‘𝐹) = -∞ ↔ ∀𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥))) | ||
| 23-Oct-2021 | limsupubuzmpt 39951 | If the limsup is not +∞, then the function is eventually bounded. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑗𝜑 & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝐵 ∈ ℝ) & ⊢ (𝜑 → (lim sup‘(𝑗 ∈ 𝑍 ↦ 𝐵)) ≠ +∞) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑥) | ||
| 23-Oct-2021 | limsupequzmpt2 39950 | Two functions that are eventually equal to one another have the same superior limit. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑗𝜑 & ⊢ Ⅎ𝑗𝐴 & ⊢ Ⅎ𝑗𝐵 & ⊢ 𝐴 = (ℤ≥‘𝑀) & ⊢ 𝐵 = (ℤ≥‘𝑁) & ⊢ (𝜑 → 𝐾 ∈ 𝐴) & ⊢ (𝜑 → 𝐾 ∈ 𝐵) & ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝐾)) → 𝐶 ∈ 𝑉) ⇒ ⊢ (𝜑 → (lim sup‘(𝑗 ∈ 𝐴 ↦ 𝐶)) = (lim sup‘(𝑗 ∈ 𝐵 ↦ 𝐶))) | ||
| 23-Oct-2021 | limsupvaluzmpt 39949 | The superior limit, when the domain of the function is a set of upper integers (the first condition is needed, otherwise the l.h.s. would be -∞ and the r.h.s. would be +∞). (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑗𝜑 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝐵 ∈ ℝ*) ⇒ ⊢ (𝜑 → (lim sup‘(𝑗 ∈ 𝑍 ↦ 𝐵)) = inf(ran (𝑘 ∈ 𝑍 ↦ sup(ran (𝑗 ∈ (ℤ≥‘𝑘) ↦ 𝐵), ℝ*, < )), ℝ*, < )) | ||
| 23-Oct-2021 | climinf3 39948 | A convergent, non-increasing sequence, converges to the infimum of its range. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑘𝜑 & ⊢ Ⅎ𝑘𝐹 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘)) & ⊢ (𝜑 → 𝐹 ∈ dom ⇝ ) ⇒ ⊢ (𝜑 → 𝐹 ⇝ inf(ran 𝐹, ℝ*, < )) | ||
| 23-Oct-2021 | climinfmpt 39947 | A bounded below, monotonic non increasing sequence converges to the infimum of its range. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑘𝜑 & ⊢ Ⅎ𝑗𝜑 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℝ) & ⊢ (𝑘 = 𝑗 → 𝐵 = 𝐶) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍 ∧ 𝑗 = (𝑘 + 1)) → 𝐶 ≤ 𝐵) & ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 𝑥 ≤ 𝐵) ⇒ ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ 𝐵) ⇝ inf(ran (𝑘 ∈ 𝑍 ↦ 𝐵), ℝ*, < )) | ||
| 23-Oct-2021 | climinf2mpt 39946 | A bounded below, monotonic non increasing sequence converges to the infimum of its range. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑘𝜑 & ⊢ Ⅎ𝑗𝜑 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℝ) & ⊢ (𝑘 = 𝑗 → 𝐵 = 𝐶) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍 ∧ 𝑗 = (𝑘 + 1)) → 𝐶 ≤ 𝐵) & ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ 𝐵) ∈ dom ⇝ ) ⇒ ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ 𝐵) ⇝ inf(ran (𝑘 ∈ 𝑍 ↦ 𝐵), ℝ*, < )) | ||
| 23-Oct-2021 | limsupubuz 39945 | For a real-valued function on a set of upper integers, if the superior limit is not +∞, then the function is bounded above. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑗𝐹 & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ) & ⊢ (𝜑 → (lim sup‘𝐹) ≠ +∞) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 (𝐹‘𝑗) ≤ 𝑥) | ||
| 23-Oct-2021 | limsupubuzlem 39944 | If the limsup is not +∞, then the function is bounded. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑗𝜑 & ⊢ Ⅎ𝑗𝑋 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ) & ⊢ (𝜑 → 𝑌 ∈ ℝ) & ⊢ (𝜑 → 𝐾 ∈ ℝ) & ⊢ (𝜑 → ∀𝑗 ∈ 𝑍 (𝐾 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑌)) & ⊢ 𝑁 = if((⌈‘𝐾) ≤ 𝑀, 𝑀, (⌈‘𝐾)) & ⊢ 𝑊 = sup(ran (𝑗 ∈ (𝑀...𝑁) ↦ (𝐹‘𝑗)), ℝ, < ) & ⊢ 𝑋 = if(𝑊 ≤ 𝑌, 𝑌, 𝑊) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 (𝐹‘𝑗) ≤ 𝑥) | ||
| 23-Oct-2021 | limsuppnf 39943 | If the restriction of a function to every upper interval is unbounded above, its lim sup is +∞. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑗𝐹 & ⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*) ⇒ ⊢ (𝜑 → ((lim sup‘𝐹) = +∞ ↔ ∀𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)))) | ||
| 23-Oct-2021 | limsuppnflem 39942 | If the restriction of a function to every upper interval is unbounded above, its lim sup is +∞. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑗𝐹 & ⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*) ⇒ ⊢ (𝜑 → ((lim sup‘𝐹) = +∞ ↔ ∀𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)))) | ||
| 23-Oct-2021 | limsupresuz2 39941 | If the domain of a function is a subset of the integers, the superior limit doesn't change when the function is restricted to an upper set of integers. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹 ∈ 𝑉) & ⊢ (𝜑 → dom 𝐹 ⊆ ℤ) ⇒ ⊢ (𝜑 → (lim sup‘(𝐹 ↾ 𝑍)) = (lim sup‘𝐹)) | ||
| 23-Oct-2021 | limsupvaluz 39940 | The superior limit, when the domain of the function is a set of upper integers (the first condition is needed, otherwise the l.h.s. would be -∞ and the r.h.s. would be +∞). (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) ⇒ ⊢ (𝜑 → (lim sup‘𝐹) = inf(ran (𝑘 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑘)), ℝ*, < )), ℝ*, < )) | ||
| 23-Oct-2021 | climinf2 39939 | A convergent, non-increasing sequence, converges to the infimum of its range. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑘𝜑 & ⊢ Ⅎ𝑘𝐹 & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘)) & ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑘)) ⇒ ⊢ (𝜑 → 𝐹 ⇝ inf(ran 𝐹, ℝ*, < )) | ||
| 23-Oct-2021 | climinf2lem 39938 | A convergent, non-increasing sequence, converges to the infimum of its range. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐹:𝑍⟶ℝ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘)) & ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑘)) ⇒ ⊢ (𝜑 → 𝐹 ⇝ inf(ran 𝐹, ℝ*, < )) | ||
| 23-Oct-2021 | limsupres 39937 | The superior limit of a restriction is less than or equal to the original superior limit. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ (𝜑 → 𝐹 ∈ 𝑉) ⇒ ⊢ (𝜑 → (lim sup‘(𝐹 ↾ 𝐶)) ≤ (lim sup‘𝐹)) | ||
| 23-Oct-2021 | limsupub 39936 | If the limsup is not +∞, then the function is eventually bounded. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑗𝜑 & ⊢ Ⅎ𝑗𝐹 & ⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*) & ⊢ (𝜑 → (lim sup‘𝐹) ≠ +∞) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) | ||
| 23-Oct-2021 | limsupresuz 39935 | If the real part of the domain of a function is a subset of the integers, the superior limit doesn't change when the function is restricted to an upper set of integers. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹 ∈ 𝑉) & ⊢ (𝜑 → dom (𝐹 ↾ ℝ) ⊆ ℤ) ⇒ ⊢ (𝜑 → (lim sup‘(𝐹 ↾ 𝑍)) = (lim sup‘𝐹)) | ||
| 23-Oct-2021 | limsuppnfd 39934 | If the restriction of a function to every upper interval is unbounded above, its lim sup is +∞. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑗𝐹 & ⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*) & ⊢ (𝜑 → ∀𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))) ⇒ ⊢ (𝜑 → (lim sup‘𝐹) = +∞) | ||
| 23-Oct-2021 | limsuppnfdlem 39933 | If the restriction of a function to every upper interval is unbounded above, its lim sup is +∞. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*) & ⊢ (𝜑 → ∀𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))) & ⊢ 𝐺 = (𝑘 ∈ ℝ ↦ sup(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) ⇒ ⊢ (𝜑 → (lim sup‘𝐹) = +∞) | ||
| 23-Oct-2021 | limsupresico 39932 | The superior limit doesn't change when a function is restricted to the upper part of the reals. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ (𝜑 → 𝑀 ∈ ℝ) & ⊢ 𝑍 = (𝑀[,)+∞) & ⊢ (𝜑 → 𝐹 ∈ 𝑉) ⇒ ⊢ (𝜑 → (lim sup‘(𝐹 ↾ 𝑍)) = (lim sup‘𝐹)) | ||
| 23-Oct-2021 | limsuplesup 39931 | An upper bound for the superior limit. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ (𝜑 → 𝐹 ∈ 𝑉) & ⊢ (𝜑 → 𝐾 ∈ ℝ) ⇒ ⊢ (𝜑 → (lim sup‘𝐹) ≤ sup(((𝐹 “ (𝐾[,)+∞)) ∩ ℝ*), ℝ*, < )) | ||
| 23-Oct-2021 | climfvd 39930 | The limit of a convergent sequence, expressed as the function value of the convergence relation. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ (𝜑 → 𝐹 ⇝ 𝐴) ⇒ ⊢ (𝜑 → 𝐴 = ( ⇝ ‘𝐹)) | ||
| 23-Oct-2021 | climeqmpt 39929 | Two functions that are eventually equal to one another have the same limit. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑍 ⊆ 𝐴) & ⊢ (𝜑 → 𝑍 ⊆ 𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝐶 ∈ 𝑈) ⇒ ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐶) ⇝ 𝐷 ↔ (𝑥 ∈ 𝐵 ↦ 𝐶) ⇝ 𝐷)) | ||
| 23-Oct-2021 | limsupresre 39928 | The supremum limit of a function only depends on the real part of its domain. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ (𝜑 → 𝐹 ∈ 𝑉) ⇒ ⊢ (𝜑 → (lim sup‘(𝐹 ↾ ℝ)) = (lim sup‘𝐹)) | ||
| 23-Oct-2021 | climeldmeqmpt2 39927 | Two functions that are eventually equal, either both are convergent or both are divergent. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑘𝜑 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐴 ∈ 𝑊) & ⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → 𝑍 ⊆ 𝐴) & ⊢ (𝜑 → 𝑍 ⊆ 𝐵) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐶 ∈ 𝑈) ⇒ ⊢ (𝜑 → ((𝑘 ∈ 𝐴 ↦ 𝐶) ∈ dom ⇝ ↔ (𝑘 ∈ 𝐵 ↦ 𝐶) ∈ dom ⇝ )) | ||
| 23-Oct-2021 | limsup0 39926 | The superior limit of the empty set (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ (lim sup‘∅) = -∞ | ||
| 23-Oct-2021 | climfveqmpt2 39925 | Two functions that are eventually equal to one another have the same limit. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑘𝜑 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ (𝜑 → 𝑍 ⊆ 𝐴) & ⊢ (𝜑 → 𝑍 ⊆ 𝐵) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐶 ∈ 𝑈) ⇒ ⊢ (𝜑 → ( ⇝ ‘(𝑘 ∈ 𝐴 ↦ 𝐶)) = ( ⇝ ‘(𝑘 ∈ 𝐵 ↦ 𝐶))) | ||
| 23-Oct-2021 | limsupval3 39924 | The superior limit of an infinite sequence 𝐹 of extended real numbers. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑘𝜑 & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*) & ⊢ 𝐺 = (𝑘 ∈ ℝ ↦ sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < )) ⇒ ⊢ (𝜑 → (lim sup‘𝐹) = inf(ran 𝐺, ℝ*, < )) | ||
| 23-Oct-2021 | climfv 39923 | The limit of a convergent sequence, expressed as the function value of the convergence relation. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ (𝐹 ⇝ 𝐴 → 𝐴 = ( ⇝ ‘𝐹)) | ||
| 23-Oct-2021 | limsupcld 39922 | Closure of the superior limit. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ (𝜑 → 𝐹 ∈ 𝑉) ⇒ ⊢ (𝜑 → (lim sup‘𝐹) ∈ ℝ*) | ||
| 23-Oct-2021 | climeldmeqmpt3 39921 | Two functions that are eventually equal, either both are convergent or both are divergent. TODO: this is more general than climeldmeqmpt 39900 and should replace it. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑘𝜑 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐶 ∈ 𝑊) & ⊢ (𝜑 → 𝑍 ⊆ 𝐴) & ⊢ (𝜑 → 𝑍 ⊆ 𝐶) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ 𝑈) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 = 𝐷) ⇒ ⊢ (𝜑 → ((𝑘 ∈ 𝐴 ↦ 𝐵) ∈ dom ⇝ ↔ (𝑘 ∈ 𝐶 ↦ 𝐷) ∈ dom ⇝ )) | ||
| 23-Oct-2021 | climeqf 39920 | Two functions that are eventually equal to one another have the same limit. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑘𝜑 & ⊢ Ⅎ𝑘𝐹 & ⊢ Ⅎ𝑘𝐺 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹 ∈ 𝑉) & ⊢ (𝜑 → 𝐺 ∈ 𝑊) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = (𝐺‘𝑘)) ⇒ ⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ 𝐺 ⇝ 𝐴)) | ||
| 23-Oct-2021 | climbddf 39919 | A converging sequence of complex numbers is bounded. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑘𝐹 & ⊢ 𝑍 = (ℤ≥‘𝑀) ⇒ ⊢ ((𝑀 ∈ ℤ ∧ 𝐹 ∈ dom ⇝ ∧ ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ) → ∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 (abs‘(𝐹‘𝑘)) ≤ 𝑥) | ||
| 23-Oct-2021 | limsupbnd1f 39918 | If a sequence is eventually at most 𝐴, then the limsup is also at most 𝐴. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑗𝐹 & ⊢ (𝜑 → 𝐵 ⊆ ℝ) & ⊢ (𝜑 → 𝐹:𝐵⟶ℝ*) & ⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝐴)) ⇒ ⊢ (𝜑 → (lim sup‘𝐹) ≤ 𝐴) | ||
| 23-Oct-2021 | limsupref 39917 | If a sequence is bounded, then the limsup is real. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑗𝐹 & ⊢ (𝜑 → 𝐴 ⊆ ℝ) & ⊢ (𝜑 → sup(𝐴, ℝ*, < ) = +∞) & ⊢ (𝜑 → 𝐹:𝐴⟶ℝ) & ⊢ (𝜑 → ∃𝑏 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (abs‘(𝐹‘𝑗)) ≤ 𝑏)) ⇒ ⊢ (𝜑 → (lim sup‘𝐹) ∈ ℝ) | ||
| 23-Oct-2021 | climreclmpt 39916 | The limit of B convergent real sequence is real. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑘𝜑 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℝ) & ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ 𝐴) ⇝ 𝐵) ⇒ ⊢ (𝜑 → 𝐵 ∈ ℝ) | ||
| 23-Oct-2021 | climeldmeqf 39915 | Two functions that are eventually equal, either both are convergent or both are divergent. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑘𝜑 & ⊢ Ⅎ𝑘𝐹 & ⊢ Ⅎ𝑘𝐺 & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹 ∈ 𝑉) & ⊢ (𝜑 → 𝐺 ∈ 𝑊) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = (𝐺‘𝑘)) ⇒ ⊢ (𝜑 → (𝐹 ∈ dom ⇝ ↔ 𝐺 ∈ dom ⇝ )) | ||
| 23-Oct-2021 | climfveqmpt3 39914 | Two functions that are eventually equal to one another have the same limit. TODO: this is more general than climfveqmpt 39903 and should replace it. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑘𝜑 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐶 ∈ 𝑊) & ⊢ (𝜑 → 𝑍 ⊆ 𝐴) & ⊢ (𝜑 → 𝑍 ⊆ 𝐶) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ 𝑈) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 = 𝐷) ⇒ ⊢ (𝜑 → ( ⇝ ‘(𝑘 ∈ 𝐴 ↦ 𝐵)) = ( ⇝ ‘(𝑘 ∈ 𝐶 ↦ 𝐷))) | ||
| 23-Oct-2021 | climmptf 39913 | Exhibit a function 𝐺 with the same convergence properties as the not-quite-function 𝐹. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑘𝐹 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐹 ∈ 𝑉) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ 𝐺 = (𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘)) ⇒ ⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ 𝐺 ⇝ 𝐴)) | ||
| 23-Oct-2021 | climfveqf 39912 | Two functions that are eventually equal to one another have the same limit. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑘𝜑 & ⊢ Ⅎ𝑘𝐹 & ⊢ Ⅎ𝑘𝐺 & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐹 ∈ 𝑉) & ⊢ (𝜑 → 𝐺 ∈ 𝑊) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = (𝐺‘𝑘)) ⇒ ⊢ (𝜑 → ( ⇝ ‘𝐹) = ( ⇝ ‘𝐺)) | ||
| 23-Oct-2021 | uzinico2 39789 | An upper interval of integers is the intersection of a larger upper interval of integers with an upper part of the reals. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) ⇒ ⊢ (𝜑 → (ℤ≥‘𝑁) = ((ℤ≥‘𝑀) ∩ (𝑁[,)+∞))) | ||
| 23-Oct-2021 | preimaiocmnf 39788 | Preimage of a right-closed interval, unbounded below. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ (𝜑 → 𝐹:𝐴⟶ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) ⇒ ⊢ (𝜑 → (◡𝐹 “ (-∞(,]𝐵)) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≤ 𝐵}) | ||
| 23-Oct-2021 | uzinico 39787 | An upper interval of integers is the intersection of the integers with an upper part of the reals. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) ⇒ ⊢ (𝜑 → 𝑍 = (ℤ ∩ (𝑀[,)+∞))) | ||
| 23-Oct-2021 | iocleubd 39786 | An element of a left open right closed interval is smaller or equal to its upper bound. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐶 ∈ (𝐴(,]𝐵)) ⇒ ⊢ (𝜑 → 𝐶 ≤ 𝐵) | ||
| 23-Oct-2021 | icogelbd 39785 | An element of a left closed right open interval is larger or equal to its lower bound. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐶 ∈ (𝐴[,)𝐵)) ⇒ ⊢ (𝜑 → 𝐴 ≤ 𝐶) | ||
| 23-Oct-2021 | uzub 39658 | A set of reals, indexed by upper integers, is bound if and only if any upper part is bound. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑗𝜑 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ ℝ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)𝐵 ≤ 𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑥)) | ||
| 23-Oct-2021 | uzublem 39657 | A set of reals, indexed by upper integers, is bound if and only if any upper part is bound. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑗𝜑 & ⊢ Ⅎ𝑗𝑋 & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑌 ∈ ℝ) & ⊢ 𝑊 = sup(ran (𝑗 ∈ (𝑀...𝐾) ↦ 𝐵), ℝ, < ) & ⊢ 𝑋 = if(𝑊 ≤ 𝑌, 𝑌, 𝑊) & ⊢ (𝜑 → 𝐾 ∈ 𝑍) & ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝐵 ∈ ℝ) & ⊢ (𝜑 → ∀𝑗 ∈ (ℤ≥‘𝐾)𝐵 ≤ 𝑌) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑥) | ||
| 23-Oct-2021 | supxrre3rnmpt 39656 | The indexed supremum of a nonempty set of reals, is real if and only if it is bounded-above . (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝐴 ≠ ∅) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (sup(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) ∈ ℝ ↔ ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦)) | ||
| 23-Oct-2021 | infxrunb3rnmpt 39655 | The infimum of an unbounded-below set of extended reals is minus infinity. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑦𝜑 & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*) ⇒ ⊢ (𝜑 → (∀𝑦 ∈ ℝ ∃𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ↔ inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) = -∞)) | ||
| 23-Oct-2021 | rexabsle2 39654 | An indexed set of absolute values of real numbers is bounded if and only if the original values are bounded above and below. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 (abs‘𝐵) ≤ 𝑦 ↔ (∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ∧ ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝑦 ≤ 𝐵))) | ||
| 23-Oct-2021 | uzssd3 39653 | Subset relationship for two sets of upper integers. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ 𝑍 = (ℤ≥‘𝑀) ⇒ ⊢ (𝑁 ∈ 𝑍 → (ℤ≥‘𝑁) ⊆ 𝑍) | ||
| 23-Oct-2021 | uzn0d 39652 | The upper integers are all nonempty. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) ⇒ ⊢ (𝜑 → 𝑍 ≠ ∅) | ||
| 23-Oct-2021 | infxrunb3 39651 | The infimum of an unbounded-below set of extended reals is minus infinity. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ (𝐴 ⊆ ℝ* → (∀𝑥 ∈ ℝ ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ↔ inf(𝐴, ℝ*, < ) = -∞)) | ||
| 23-Oct-2021 | infrnmptle 39650 | An indexed infimum of extended reals is smaller than another indexed infimum of extended reals, when every indexed element is smaller than the corresponding one. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℝ*) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ≤ 𝐶) ⇒ ⊢ (𝜑 → inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) ≤ inf(ran (𝑥 ∈ 𝐴 ↦ 𝐶), ℝ*, < )) | ||
| 23-Oct-2021 | suprleubrnmpt 39649 | The supremum of a nonempty bounded indexed set of reals is less than or equal to an upper bound. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝐴 ≠ ∅) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) & ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦) & ⊢ (𝜑 → 𝐶 ∈ ℝ) ⇒ ⊢ (𝜑 → (sup(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ, < ) ≤ 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝐶)) | ||
| 23-Oct-2021 | supxrrernmpt 39648 | The real and extended real indexed suprema match when the indexed real supremum exists. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝐴 ≠ ∅) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) & ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦) ⇒ ⊢ (𝜑 → sup(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) = sup(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ, < )) | ||
| 23-Oct-2021 | allbutfiinf 39647 | Given a "for all but finitely many" condition, the condition holds from 𝑁 on. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ 𝐴 = ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)𝐵 & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ 𝑁 = inf({𝑛 ∈ 𝑍 ∣ ∀𝑚 ∈ (ℤ≥‘𝑛)𝑋 ∈ 𝐵}, ℝ, < ) ⇒ ⊢ (𝜑 → (𝑁 ∈ 𝑍 ∧ ∀𝑚 ∈ (ℤ≥‘𝑁)𝑋 ∈ 𝐵)) | ||
| 23-Oct-2021 | rexabsle 39646 | An indexed set of absolute values of real numbers is bounded if and only if the original values are bounded above and below. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 (abs‘𝐵) ≤ 𝑦 ↔ (∃𝑤 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑤 ∧ ∃𝑧 ∈ ℝ ∀𝑥 ∈ 𝐴 𝑧 ≤ 𝐵))) | ||
| 23-Oct-2021 | rexabslelem 39645 | An indexed set of absolute values of real numbers is bounded if and only if the original values are bounded above and below. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → (∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 (abs‘𝐵) ≤ 𝑦 ↔ (∃𝑤 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑤 ∧ ∃𝑧 ∈ ℝ ∀𝑥 ∈ 𝐴 𝑧 ≤ 𝐵))) | ||
| 23-Oct-2021 | uzssd2 39644 | Subset relationship for two sets of upper integers. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑁 ∈ 𝑍) ⇒ ⊢ (𝜑 → (ℤ≥‘𝑁) ⊆ 𝑍) | ||
| 23-Oct-2021 | uzidd2 39643 | Membership of the least member in an upper set of integers. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ 𝑍 = (ℤ≥‘𝑀) ⇒ ⊢ (𝜑 → 𝑀 ∈ 𝑍) | ||
| 23-Oct-2021 | unb2ltle 39642 | "Unbounded below" expressed with < and with ≤. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ (𝐴 ⊆ ℝ* → (∀𝑤 ∈ ℝ ∃𝑦 ∈ 𝐴 𝑦 < 𝑤 ↔ ∀𝑥 ∈ ℝ ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)) | ||
| 23-Oct-2021 | infleinf2 39641 | If any element in 𝐵 is larger or equal to an element in 𝐴, then the infimum of 𝐴 is smaller or equal to the infimum of 𝐵. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → 𝐴 ⊆ ℝ*) & ⊢ (𝜑 → 𝐵 ⊆ ℝ*) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) ⇒ ⊢ (𝜑 → inf(𝐴, ℝ*, < ) ≤ inf(𝐵, ℝ*, < )) | ||
| 23-Oct-2021 | eluzelz2d 39640 | A member of an upper set of integers is an integer. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑁 ∈ 𝑍) ⇒ ⊢ (𝜑 → 𝑁 ∈ ℤ) | ||
| 23-Oct-2021 | uz0 39639 | The upper integers function applied to a non integer, is the empty set. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ (¬ 𝑀 ∈ ℤ → (ℤ≥‘𝑀) = ∅) | ||
| 23-Oct-2021 | xrre4 39638 | An extended real is real iff it is not an infinty. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ (𝐴 ∈ ℝ* → (𝐴 ∈ ℝ ↔ (𝐴 ≠ -∞ ∧ 𝐴 ≠ +∞))) | ||
| 23-Oct-2021 | infxrlbrnmpt2 39637 | A member of a nonempty indexed set of reals is greater than or equal to the set's lower bound. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐶 ∈ 𝐴) & ⊢ (𝜑 → 𝐷 ∈ ℝ*) & ⊢ (𝑥 = 𝐶 → 𝐵 = 𝐷) ⇒ ⊢ (𝜑 → inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) ≤ 𝐷) | ||
| 23-Oct-2021 | elfzd 39636 | Membership in a finite set of sequential integers. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ (𝜑 → 𝐾 ∈ ℤ) & ⊢ (𝜑 → 𝑀 ≤ 𝐾) & ⊢ (𝜑 → 𝐾 ≤ 𝑁) ⇒ ⊢ (𝜑 → 𝐾 ∈ (𝑀...𝑁)) | ||
| 23-Oct-2021 | eluzd 39635 | Membership in an upper set of integers. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ (𝜑 → 𝑀 ≤ 𝑁) ⇒ ⊢ (𝜑 → 𝑁 ∈ 𝑍) | ||
| 23-Oct-2021 | uzssd 39634 | Subset relationship for two sets of upper integers. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) ⇒ ⊢ (𝜑 → (ℤ≥‘𝑁) ⊆ (ℤ≥‘𝑀)) | ||
| 23-Oct-2021 | uzssre2 39633 | An upper set of integers is a subset of the Reals. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ 𝑍 = (ℤ≥‘𝑀) ⇒ ⊢ 𝑍 ⊆ ℝ | ||
| 23-Oct-2021 | supxrleubrnmpt 39632 | The supremum of a nonempty bounded indexed set of extended reals is less than or equal to an upper bound. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐶 ∈ ℝ*) ⇒ ⊢ (𝜑 → (sup(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) ≤ 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝐶)) | ||
| 23-Oct-2021 | uzidd 39631 | Membership of the least member in an upper set of integers. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ (𝜑 → 𝑀 ∈ ℤ) ⇒ ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘𝑀)) | ||
| 23-Oct-2021 | uzid2 39630 | Membership of the least member in an upper set of integers. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ (𝑀 ∈ (ℤ≥‘𝑁) → 𝑀 ∈ (ℤ≥‘𝑀)) | ||
| 23-Oct-2021 | resabs2d 39629 | Absorption law for restriction. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ (𝜑 → 𝐵 ⊆ 𝐶) ⇒ ⊢ (𝜑 → ((𝐴 ↾ 𝐵) ↾ 𝐶) = (𝐴 ↾ 𝐵)) | ||
| 23-Oct-2021 | pnfnre2 39628 | Plus infinity is not a real number. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ ¬ +∞ ∈ ℝ | ||
| 23-Oct-2021 | eluzelz2 39627 | A member of an upper set of integers is an integer. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ 𝑍 = (ℤ≥‘𝑀) ⇒ ⊢ (𝑁 ∈ 𝑍 → 𝑁 ∈ ℤ) | ||
| 23-Oct-2021 | ren0 39626 | The set of reals is nonempty. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ ℝ ≠ ∅ | ||
| 23-Oct-2021 | fimaxre4 39625 | A nonempty finite set of real numbers is bounded (image set version). (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) ⇒ ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦) | ||
| 23-Oct-2021 | elfzod 39624 | Membership in a half-open integer interval. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ (𝜑 → 𝐾 ∈ (ℤ≥‘𝑀)) & ⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ (𝜑 → 𝐾 < 𝑁) ⇒ ⊢ (𝜑 → 𝐾 ∈ (𝑀..^𝑁)) | ||
| 23-Oct-2021 | supxrunb3 39623 | The supremum of an unbounded-above set of extended reals is plus infinity. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ (𝐴 ⊆ ℝ* → (∀𝑥 ∈ ℝ ∃𝑦 ∈ 𝐴 𝑥 ≤ 𝑦 ↔ sup(𝐴, ℝ*, < ) = +∞)) | ||
| 23-Oct-2021 | fisupclrnmpt 39622 | A nonempty finite indexed set contains its supremum. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝑅 Or 𝐴) & ⊢ (𝜑 → 𝐵 ∈ Fin) & ⊢ (𝜑 → 𝐵 ≠ ∅) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ 𝐴) ⇒ ⊢ (𝜑 → sup(ran (𝑥 ∈ 𝐵 ↦ 𝐶), 𝐴, 𝑅) ∈ 𝐴) | ||
| 23-Oct-2021 | zssxr 39621 | The integers are a subset of the extended reals. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ ℤ ⊆ ℝ* | ||
| 23-Oct-2021 | uzssre 39620 | An upper set of integers is a subset of the Reals. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ (ℤ≥‘𝑀) ⊆ ℝ | ||
| 23-Oct-2021 | mnfnre2 39619 | Minus infinity is not a real number. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ ¬ -∞ ∈ ℝ | ||
| 23-Oct-2021 | xreqnltd 39618 | A consequence of trichotomy. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → ¬ 𝐴 < 𝐵) | ||
| 23-Oct-2021 | rnmptss2 39472 | The range of an operation given by the "maps to" notation as a subset. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝐴 ⊆ 𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑉) ⇒ ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ 𝐶) ⊆ ran (𝑥 ∈ 𝐵 ↦ 𝐶)) | ||
| 23-Oct-2021 | rnmptbd 39471 | Boundness above of the range of a function in map-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) ⇒ ⊢ (𝜑 → (∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ↔ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦)) | ||
| 23-Oct-2021 | rnmptbdlem 39470 | Boundness above of the range of a function in map-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑦𝜑 & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) ⇒ ⊢ (𝜑 → (∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ↔ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦)) | ||
| 23-Oct-2021 | rnmptssdf 39469 | The range of an operation given by the "maps to" notation as a subset. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝐶 & ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶) ⇒ ⊢ (𝜑 → ran 𝐹 ⊆ 𝐶) | ||
| 23-Oct-2021 | suprubrnmpt 39468 | A member of a nonempty indexed set of reals is less than or equal to the set's upper bound. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) & ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦) ⇒ ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ≤ sup(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ, < )) | ||
| 23-Oct-2021 | suprubrnmpt2 39467 | A member of a nonempty indexed set of reals is less than or equal to the set's upper bound. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) & ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦) & ⊢ (𝜑 → 𝐶 ∈ 𝐴) & ⊢ (𝜑 → 𝐷 ∈ ℝ) & ⊢ (𝑥 = 𝐶 → 𝐵 = 𝐷) ⇒ ⊢ (𝜑 → 𝐷 ≤ sup(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ, < )) | ||
| 23-Oct-2021 | suprclrnmpt 39466 | Closure of the indexed supremum of a nonempty bounded set of reals. Range of a function in map-to notation can be used, to express an indexed supremum. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝐴 ≠ ∅) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) & ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦) ⇒ ⊢ (𝜑 → sup(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ, < ) ∈ ℝ) | ||
| 23-Oct-2021 | infnsuprnmpt 39465 | The indexed infimum of real numbers is the negative of the indexed supremum of the negative values. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝐴 ≠ ∅) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) & ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝑦 ≤ 𝐵) ⇒ ⊢ (𝜑 → inf(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ, < ) = -sup(ran (𝑥 ∈ 𝐴 ↦ -𝐵), ℝ, < )) | ||
| 23-Oct-2021 | rnmptbd2 39464 | Boundness below of the range of a function in map-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) ⇒ ⊢ (𝜑 → (∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝑦 ≤ 𝐵 ↔ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑦 ≤ 𝑧)) | ||
| 23-Oct-2021 | rnmptbd2lem 39463 | Boundness below of the range of a function in map-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) ⇒ ⊢ (𝜑 → (∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝑦 ≤ 𝐵 ↔ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑦 ≤ 𝑧)) | ||
| 23-Oct-2021 | rnmptssf 39462 | The range of an operation given by the "maps to" notation as a subset. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑥𝐶 & ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) ⇒ ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ran 𝐹 ⊆ 𝐶) | ||
| 23-Oct-2021 | funimaeq 39461 | Membership relation for the values of a function whose image is a subclass. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → Fun 𝐹) & ⊢ (𝜑 → Fun 𝐺) & ⊢ (𝜑 → 𝐴 ⊆ dom 𝐹) & ⊢ (𝜑 → 𝐴 ⊆ dom 𝐺) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐺‘𝑥)) ⇒ ⊢ (𝜑 → (𝐹 “ 𝐴) = (𝐺 “ 𝐴)) | ||
| 23-Oct-2021 | fmptd2 39460 | Domain and codomain of the mapping operation; deduction form. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝐶) | ||
| 23-Oct-2021 | fnfvimad 39459 | A function's value belongs to the image. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ (𝜑 → 𝐹 Fn 𝐴) & ⊢ (𝜑 → 𝐵 ∈ 𝐴) & ⊢ (𝜑 → 𝐵 ∈ 𝐶) ⇒ ⊢ (𝜑 → (𝐹‘𝐵) ∈ (𝐹 “ 𝐶)) | ||
| 23-Oct-2021 | fvelimad 39458 | Function value in an image. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑥𝐹 & ⊢ (𝜑 → 𝐹 Fn 𝐴) & ⊢ (𝜑 → 𝐶 ∈ (𝐹 “ 𝐵)) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ (𝐴 ∩ 𝐵)(𝐹‘𝑥) = 𝐶) | ||
| 23-Oct-2021 | mptima2 39457 | Image of a function in map-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ (𝜑 → 𝐶 ⊆ 𝐴) ⇒ ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) “ 𝐶) = ran (𝑥 ∈ 𝐶 ↦ 𝐵)) | ||
| 23-Oct-2021 | rnmptbdd 39456 | Boundness of the range of a function in map-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦) ⇒ ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦) | ||
| 23-Oct-2021 | rnmptbddlem 39455 | Boundness of the range of a function in map-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦) ⇒ ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦) | ||
| 23-Oct-2021 | elpreimad 39454 | Membership in the preimage of a set under a function. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ (𝜑 → 𝐹 Fn 𝐴) & ⊢ (𝜑 → 𝐵 ∈ 𝐴) & ⊢ (𝜑 → (𝐹‘𝐵) ∈ 𝐶) ⇒ ⊢ (𝜑 → 𝐵 ∈ (◡𝐹 “ 𝐶)) | ||
| 23-Oct-2021 | rnmptlb 39453 | Boundness below of the range of a function in map-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝑦 ≤ 𝐵) ⇒ ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑦 ≤ 𝑧) | ||
| 23-Oct-2021 | mpteq12da 39452 | An equality inference for the maps to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑥𝜑 & ⊢ (𝜑 → 𝐴 = 𝐶) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐷) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷)) | ||
| 23-Oct-2021 | mptssid 39450 | The mapping operation expressed with its actual domain. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑥𝐴 & ⊢ 𝐶 = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} ⇒ ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐵) | ||
| 23-Oct-2021 | resimass 39449 | The image of a restriction is a subset of the original image. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ ((𝐴 ↾ 𝐵) “ 𝐶) ⊆ (𝐴 “ 𝐶) | ||
| 23-Oct-2021 | fmptf 39448 | Functionality of the mapping operation. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ Ⅎ𝑥𝐵 & ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶) ⇒ ⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ↔ 𝐹:𝐴⟶𝐵) | ||
| 23-Oct-2021 | fvmptd3 39447 | Deduction version of fvmpt 6282. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) & ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) & ⊢ (𝜑 → 𝐴 ∈ 𝐷) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝐹‘𝐴) = 𝐶) | ||
| 23-Oct-2021 | fvmpt4 39446 | Value of a function given by the "maps to" notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝐶) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵) | ||
| 23-Oct-2021 | fvmptd2 39445 | Deduction version of fvmpt 6282. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
| ⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) & ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 = 𝐶) & ⊢ (𝜑 → 𝐴 ∈ 𝐷) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝐹‘𝐴) = 𝐶) | ||
(29-Jul-2020) Mario Carneiro presented MM0 at the CICM conference. See this Google Group post which includes a YouTube link.
(20-Jul-2020) Rohan Ridenour found 5 shorter D-proofs in our Shortest known proofs... file. In particular, he reduced *4.39 from 901 to 609 steps. A note on the Metamath Solitaire page mentions a tool that he worked with.
(19-Jul-2020) David A. Wheeler posted a video (https://youtu.be/3R27Qx69jHc) on how to (re)prove Schwabhäuser 4.6 for the Metamath Proof Explorer. See also his older videos.
(19-Jul-2020) In version 0.184 of the metamath program, "verify markup" now checks that mathboxes are independent i.e. do not cross-reference each other. To turn off this check, use "/mathbox_skip"
(30-Jun-2020) In version 0.183 of the metamath program, (1) "verify markup" now has checking for (i) underscores in labels, (ii) that *ALT and *OLD theorems have both discouragement tags, and (iii) that lines don't have trailing spaces. (2) "save proof.../rewrap" no longer left-aligns $p/$a comments that contain the string "<HTML>"; see this note.
(5-Apr-2020) Glauco Siliprandi added a new proof to the 100 theorem list, e is Transcendental etransc, bringing the Metamath total to 74.
(12-Feb-2020) A bug in the 'minimize' command of metamath.exe versions 0.179 (29-Nov-2019) and 0.180 (10-Dec-2019) may incorrectly bring in the use of new axioms. Version 0.181 fixes it.
(20-Jan-2020) David A. Wheeler created a video called Walkthrough of the tutorial in mmj2. See the Google Group announcement for more details. (All of his videos are listed on the Other Metamath-Related Topics page.)
(18-Jan-2020) The FOMM 2020 talks are on youtube now. Mario Carneiro's talk is Metamath Zero, or: How to Verify a Verifier. Since they are washed out in the video, the PDF slides are available separately.
(14-Dec-2019) Glauco Siliprandi added a new proof to the 100 theorem list, Fourier series convergence fourier, bringing the Metamath total to 73.
(25-Nov-2019) Alexander van der Vekens added a new proof to the 100 theorem list, The Cayley-Hamilton Theorem cayleyhamilton, bringing the Metamath total to 72.
(25-Oct-2019) Mario Carneiro's paper "Metamath Zero: The Cartesian Theorem Prover" (submitted to CPP 2020) is now available on arXiv: https://arxiv.org/abs/1910.10703. There is a related discussion on Hacker News.
(30-Sep-2019) Mario Carneiro's talk about MM0 at ITP 2019 is available on YouTube: x86 verification from scratch (24 minutes). Google Group discussion: Metamath Zero.
(29-Sep-2019) David Wheeler created a fascinating Gource video that animates the construction of set.mm, available on YouTube: Metamath set.mm contributions viewed with Gource through 2019-09-26 (4 minutes). Google Group discussion: Gource video of set.mm contributions.
(24-Sep-2019) nLab added a page for Metamath. It mentions Stefan O'Rear's Busy Beaver work using the set.mm axiomatization (and fails to mention Mario's definitional soundness checker)
(1-Sep-2019) Xuanji Li published a Visual Studio Code extension to support metamath syntax highlighting.
(10-Aug-2019) (revised 21-Sep-2019) Version 0.178 of the metamath program has the following changes: (1) "minimize_with" will now prevent dependence on new $a statements unless the new qualifier "/allow_new_axioms" is specified. For routine usage, it is suggested that you use "minimize_with * /allow_new_axioms * /no_new_axioms_from ax-*" instead of just "minimize_with *". See "help minimize_with" and this Google Group post. Also note that the qualifier "/allow_growth" has been renamed to "/may_grow". (2) "/no_versioning" was added to "write theorem_list".
(8-Jul-2019) Jon Pennant announced the creation of a Metamath search engine. Try it and feel free to comment on it at https://groups.google.com/d/msg/metamath/cTeU5AzUksI/5GesBfDaCwAJ.
(16-May-2019) Set.mm now has a major new section on elementary geometry. This begins with definitions that implement Tarski's axioms of geometry (including concepts such as congruence and betweenness). This uses set.mm's extensible structures, making them easier to use for many circumstances. The section then connects Tarski geometry with geometry in Euclidean places. Most of the work in this section is due to Thierry Arnoux, with earlier work by Mario Carneiro and Scott Fenton. [Reported by DAW.]
(9-May-2019) We are sad to report that long-time contributor Alan Sare passed away on Mar. 23. There is some more information at the top of his mathbox (click on "Mathbox for Alan Sare") and his obituary. We extend our condolences to his family.
(10-Mar-2019) Jon Pennant and Mario Carneiro added a new proof to the 100 theorem list, Heron's formula heron, bringing the Metamath total to 71.
(22-Feb-2019) Alexander van der Vekens added a new proof to the 100 theorem list, Cramer's rule cramer, bringing the Metamath total to 70.
(6-Feb-2019) David A. Wheeler has made significant improvements and updates to the Metamath book. Any comments, errors found, or suggestions are welcome and should be turned into an issue or pull request at https://github.com/metamath/metamath-book (or sent to me if you prefer).
(26-Dec-2018) I added Appendix 8 to the MPE Home Page that cross-references new and old axiom numbers.
(20-Dec-2018) The axioms have been renumbered according to this Google Groups post.
(24-Nov-2018) Thierry Arnoux created a new page on topological structures. The page along with its SVG files are maintained on GitHub.
(11-Oct-2018) Alexander van der Vekens added a new proof to the 100 theorem list, the Friendship Theorem friendship, bringing the Metamath total to 69.
(1-Oct-2018) Naip Moro has written gramm, a Metamath proof verifier written in Antlr4/Java.
(16-Sep-2018) The definition df-riota has been simplified so that it evaluates to the empty set instead of an Undef value. This change affects a significant part of set.mm.
(2-Sep-2018) Thierry Arnoux added a new proof to the 100 theorem list, Euler's partition theorem eulerpart, bringing the Metamath total to 68.
(1-Sep-2018) The Kate editor now has Metamath syntax highlighting built in. (Communicated by Wolf Lammen.)
(15-Aug-2018) The Intuitionistic Logic Explorer now has a Most Recent Proofs page.
(4-Aug-2018) Version 0.163 of the metamath program now indicates (with an asterisk) which Table of Contents headers have associated comments.
(10-May-2018) George Szpiro, journalist and author of several books on popular mathematics such as Poincare's Prize and Numbers Rule, used a genetic algorithm to find shorter D-proofs of "*3.37" and "meredith" in our Shortest known proofs... file.
(19-Apr-2018) The EMetamath Eclipse plugin has undergone many improvements since its initial release as the change log indicates. Thierry uses it as his main proof assistant and writes, "I added support for mmj2's auto-transformations, which allows it to infer several steps when building proofs. This added a lot of comfort for writing proofs.... I can now switch back and forth between the proof assistant and editing the Metamath file.... I think no other proof assistant has this feature."
(11-Apr-2018) Benoît Jubin solved an open problem about the "Axiom of Twoness," showing that it is necessary for completeness. See item 14 on the "Open problems and miscellany" page.
(25-Mar-2018) Giovanni Mascellani has announced mmpp, a new proof editing environment for the Metamath language.
(27-Feb-2018) Bill Hale has released an app for the Apple iPad and desktop computer that allows you to browse Metamath theorems and their proofs.
(17-Jan-2018) Dylan Houlihan has kindly provided a new mirror site. He has also provided an rsync server; type "rsync uk.metamath.org::" in a bash shell to check its status (it should return "metamath metamath").
(15-Jan-2018) The metamath program, version 0.157, has been updated to implement the file inclusion conventions described in the 21-Dec-2017 entry of mmnotes.txt.
(11-Dec-2017) I added a paragraph, suggested by Gérard Lang, to the distinct variable description here.
(10-Dec-2017) Per FL's request, his mathbox will be removed from set.mm. If you wish to export any of his theorems, today's version (master commit 1024a3a) is the last one that will contain it.
(11-Nov-2017) Alan Sare updated his completeusersproof program.
(3-Oct-2017) Sean B. Palmer created a web page that runs the metamath program under emulated Linux in JavaScript. He also wrote some programs to work with our shortest known proofs of the PM propositional calculus theorems.
(28-Sep-2017) Ivan Kuckir wrote a tutorial blog entry, Introduction to Metamath, that summarizes the language syntax. (It may have been written some time ago, but I was not aware of it before.)
(26-Sep-2017) The default directory for the Metamath Proof Explorer (MPE) has been changed from the GIF version (mpegif) to the Unicode version (mpeuni) throughout the site. Please let me know if you find broken links or other issues.
(24-Sep-2017) Saveliy Skresanov added a new proof to the 100 theorem list, Ceva's Theorem cevath, bringing the Metamath total to 67.
(3-Sep-2017) Brendan Leahy added a new proof to the 100 theorem list, Area of a Circle areacirc, bringing the Metamath total to 66.
(7-Aug-2017) Mario Carneiro added a new proof to the 100 theorem list, Principle of Inclusion/Exclusion incexc, bringing the Metamath total to 65.
(1-Jul-2017) Glauco Siliprandi added a new proof to the 100 theorem list, Stirling's Formula stirling, bringing the Metamath total to 64. Related theorems include 2 versions of Wallis' formula for π (wallispi and wallispi2).
(7-May-2017) Thierry Arnoux added a new proof to the 100 theorem list, Betrand's Ballot Problem ballotth, bringing the Metamath total to 63.
(20-Apr-2017) Glauco Siliprandi added a new proof in the supplementary list on the 100 theorem list, Stone-Weierstrass Theorem stowei.
(28-Feb-2017) David Moews added a new proof to the 100 theorem list, Product of Segments of Chords chordthm, bringing the Metamath total to 62.
(1-Jan-2017) Saveliy Skresanov added a new proof to the 100 theorem list, Isosceles triangle theorem isosctr, bringing the Metamath total to 61.
(1-Jan-2017) Mario Carneiro added 2 new proofs to the 100 theorem list, L'Hôpital's Rule lhop and Taylor's Theorem taylth, bringing the Metamath total to 60.
(28-Dec-2016) David A. Wheeler is putting together a page on Metamath (specifically set.mm) conventions. Comments are welcome on the Google Group thread.
(24-Dec-2016) Mario Carneiro introduced the abbreviation "F/ x ph" (symbols: turned F, x, phi) in df-nf to represent the "effectively not free" idiom "A. x ( ph -> A. x ph )". Theorem nf2 shows a version without nested quantifiers.
(22-Dec-2016) Naip Moro has developed a Metamath database for G. Spencer-Brown's Laws of Form. You can follow the Google Group discussion here.
(20-Dec-2016) In metamath program version 0.137, 'verify markup *' now checks that ax-XXX $a matches axXXX $p when the latter exists, per the discussion at https://groups.google.com/d/msg/metamath/Vtz3CKGmXnI/Fxq3j1I_EQAJ.
(24-Nov-2016) Mingl Yuan has kindly provided a mirror site in Beijing, China. He has also provided an rsync server; type "rsync cn.metamath.org::" in a bash shell to check its status (it should return "metamath metamath").
(14-Aug-2016) All HTML pages on this site should now be mobile-friendly and pass the Mobile-Friendly Test. If you find one that does not, let me know.
(14-Aug-2016) Daniel Whalen wrote a paper describing the use of using deep learning to prove 14% of test theorems taken from set.mm: Holophrasm: a neural Automated Theorem Prover for higher-order logic. The associated program is called Holophrasm.
(14-Aug-2016) David A. Wheeler created a video called Metamath Proof Explorer: A Modern Principia Mathematica
(12-Aug-2016) A Gitter chat room has been created for Metamath.
(9-Aug-2016) Mario Carneiro wrote a Metamath proof verifier in the Scala language as part of the ongoing Metamath -> MMT import project
(9-Aug-2016) David A. Wheeler created a GitHub project called metamath-test (last execution run) to check that different verifiers both pass good databases and detect errors in defective ones.
(4-Aug-2016) Mario gave two presentations at CICM 2016.
(17-Jul-2016) Thierry Arnoux has written EMetamath, a Metamath plugin for the Eclipse IDE.
(16-Jul-2016) Mario recovered Chris Capel's collapsible proof demo.
(13-Jul-2016) FL sent me an updated version of PDF (LaTeX source) developed with Lamport's pf2 package. See the 23-Apr-2012 entry below.
(12-Jul-2016) David A. Wheeler produced a new video for mmj2 called "Creating functions in Metamath". It shows a more efficient approach than his previous recent video "Creating functions in Metamath" (old) but it can be of interest to see both approaches.
(10-Jul-2016) Metamath program version 0.132 changes the command 'show restricted' to 'show discouraged' and adds a new command, 'set discouragement'. See the mmnotes.txt entry of 11-May-2016 (updated 10-Jul-2016).
(12-Jun-2016) Dan Getz has written Metamath.jl, a Metamath proof verifier written in the Julia language.
(10-Jun-2016) If you are using metamath program versions 0.128, 0.129, or 0.130, please update to version 0.131. (In the bad versions, 'minimize_with' ignores distinct variable violations.)
(1-Jun-2016) Mario Carneiro added new proofs to the 100 theorem list, the Prime Number Theorem pnt and the Perfect Number Theorem perfect, bringing the Metamath total to 58.
(12-May-2016) Mario Carneiro added a new proof to the 100 theorem list, Dirichlet's theorem dirith, bringing the Metamath total to 56. (Added 17-May-2016) An informal exposition of the proof can be found at http://metamath-blog.blogspot.com/2016/05/dirichlets-theorem.html
(10-Mar-2016) Metamath program version 0.125 adds a new qualifier, /fast, to 'save proof'. See the mmnotes.txt entry of 10-Mar-2016.
(6-Mar-2016) The most recent set.mm has a large update converting variables from letters to symbols. See this Google Groups post.
(16-Feb-2016) Mario Carneiro's new paper "Models for Metamath" can be found here and on arxiv.org.
(6-Feb-2016) There are now 22 math symbols that can be used as variable names. See mmascii.html near the 50th table row, starting with "./\".
(29-Jan-2016) Metamath program version 0.123 adds /packed and /explicit qualifiers to 'save proof' and 'show proof'. See this Google Groups post.
(13-Jan-2016) The Unicode math symbols now provide for external CSS and use the XITS web font. Thanks to David A. Wheeler, Mario Carneiro, Cris Perdue, Jason Orendorff, and Frédéric Liné for discussions on this topic. Two commands, htmlcss and htmlfont, were added to the $t comment in set.mm and are recognized by Metamath program version 0.122.
(21-Dec-2015) Axiom ax-12, now renamed ax-12o, was replaced by a new shorter equivalent, ax-12. The equivalence is provided by theorems ax12o and ax12.
(13-Dec-2015) A new section on the theory of classes was added to the MPE Home Page. Thanks to Gérard Lang for suggesting this section and improvements to it.
(17-Nov-2015) Metamath program version 0.121: 'verify markup' was added to check comment markup consistency; see 'help verify markup'. You are encouraged to make sure 'verify markup */f' has no warnings prior to mathbox submissions. The date consistency rules are given in this Google Groups post.
(23-Sep-2015) Drahflow wrote, "I am currently working on yet another proof assistant, main reason being: I understand stuff best if I code it. If anyone is interested: https://github.com/Drahflow/Igor (but in my own programming language, so expect a complicated build process :P)"
(23-Aug-2015) Ivan Kuckir created MM Tool, a Metamath proof verifier and editor written in JavaScript that runs in a browser.
(25-Jul-2015) Axiom ax-10 is shown to be redundant by theorem ax10 , so it was removed from the predicate calculus axiom list.
(19-Jul-2015) Mario Carneiro gave two talks related to Metamath at CICM 2015, which are linked to at Other Metamath-Related Topics.
(18-Jul-2015) The metamath program has been updated to version 0.118. 'show trace_back' now has a '/to' qualifier to show the path back to a specific axiom such as ax-ac. See 'help show trace_back'.
(12-Jul-2015) I added the HOL Explorer for Mario Carneiro's hol.mm database. Although the home page needs to be filled out, the proofs can be accessed.
(11-Jul-2015) I started a new page, Other Metamath-Related Topics, that will hold miscellaneous material that doesn't fit well elsewhere (or is hard to find on this site). Suggestions welcome.
(23-Jun-2015) Metamath's mascot, Penny the cat (2007 photo), passed away today. She was 18 years old.
(21-Jun-2015) Mario Carneiro added 3 new proofs to the 100 theorem list: All Primes (1 mod 4) Equal the Sum of Two Squares 2sq, The Law of Quadratic Reciprocity lgsquad and the AM-GM theorem amgm, bringing the Metamath total to 55.
(13-Jun-2015) Stefan O'Rear's smm, written in JavaScript, can now be used as a standalone proof verifier. This brings the total number of independent Metamath verifiers to 8, written in just as many languages (C, Java. JavaScript, Python, Haskell, Lua, C#, C++).
(12-Jun-2015) David A. Wheeler added 2 new proofs to the 100 theorem list: The Law of Cosines lawcos and Ptolemy's Theorem ptolemy, bringing the Metamath total to 52.
(30-May-2015) The metamath program has been updated to version 0.117. (1) David A. Wheeler provided an enhancement to speed up the 'improve' command by 28%; see README.TXT for more information. (2) In web pages with proofs, local hyperlinks on step hypotheses no longer clip the Expression cell at the top of the page.
(9-May-2015) Stefan O'Rear has created an archive of older set.mm releases back to 1998: https://github.com/sorear/set.mm-history/.
(7-May-2015) The set.mm dated 7-May-2015 is a major revision, updated by Mario, that incorporates the new ordered pair definition df-op that was agreed upon. There were 700 changes, listed at the top of set.mm. Mathbox users are advised to update their local mathboxes. As usual, if any mathbox user has trouble incorporating these changes into their mathbox in progress, Mario or I will be glad to do them for you.
(7-May-2015) Mario has added 4 new theorems to the 100 theorem list: Ramsey's Theorem ramsey, The Solution of a Cubic cubic, The Solution of the General Quartic Equation quart, and The Birthday Problem birthday. In the Supplementary List, Stefan O'Rear added the Hilbert Basis Theorem hbt.
(28-Apr-2015) A while ago, Mario Carneiro wrote up a proof of the unambiguity of set.mm's grammar, which has now been added to this site: grammar-ambiguity.txt.
(22-Apr-2015) The metamath program has been updated to version 0.114. In MM-PA, 'show new_proof/unknown' now shows the relative offset (-1, -2,...) used for 'assign' arguments, suggested by Stefan O'Rear.
(20-Apr-2015) I retrieved an old version of the missing "Metamath 100" page from archive.org and updated it to what I think is the current state: mm_100.html. Anyone who wants to edit it can email updates to this page to me.
(19-Apr-2015) The metamath program has been updated to version 0.113, mostly with patches provided by Stefan O'Rear. (1) 'show statement %' (or any command allowing label wildcards) will select statements whose proofs were changed in current session. ('help search' will show all wildcard matching rules.) (2) 'show statement =' will select the statement being proved in MM-PA. (3) The proof date stamp is now created only if the proof is complete.
(18-Apr-2015) There is now a section for Scott Fenton's NF database: New Foundations Explorer.
(16-Apr-2015) Mario describes his recent additions to set.mm at https://groups.google.com/forum/#!topic/metamath/VAGNmzFkHCs. It include 2 new additions to the Formalizing 100 Theorems list, Leibniz' series for pi (leibpi) and the Konigsberg Bridge problem (konigsberg)
(10-Mar-2015) Mario Carneiro has written a paper, "Arithmetic in Metamath, Case Study: Bertrand's Postulate," for CICM 2015. A preprint is available at arXiv:1503.02349.
(23-Feb-2015) Scott Fenton has created a Metamath formalization of NF set theory: https://github.com/sctfn/metamath-nf/. For more information, see the Metamath Google Group posting.
(28-Jan-2015) Mario Carneiro added Wilson's Theorem (wilth), Ascending or Descending Sequences (erdsze, erdsze2), and Derangements Formula (derangfmla, subfaclim), bringing the Metamath total for Formalizing 100 Theorems to 44.
(19-Jan-2015) Mario Carneiro added Sylow's Theorem (sylow1, sylow2, sylow2b, sylow3), bringing the Metamath total for Formalizing 100 Theorems to 41.
(9-Jan-2015) The hypothesis order of mpbi*an* was changed. See the Notes entry of 9-Jan-2015.
(1-Jan-2015) Mario Carneiro has written a paper, "Conversion of HOL Light proofs into Metamath," that has been submitted to the Journal of Formalized Reasoning. A preprint is available on arxiv.org.
(22-Nov-2014) Stefan O'Rear added the Solutions to Pell's Equation (rmxycomplete) and Liouville's Theorem and the Construction of Transcendental Numbers (aaliou), bringing the Metamath total for Formalizing 100 Theorems to 40.
(22-Nov-2014) The metamath program has been updated with version 0.111. (1) Label wildcards now have a label range indicator "~" so that e.g. you can show or search all of the statements in a mathbox. See 'help search'. (Stefan O'Rear added this to the program.) (2) A qualifier was added to 'minimize_with' to prevent the use of any axioms not already used in the proof e.g. 'minimize_with * /no_new_axioms_from ax-*' will prevent the use of ax-ac if the proof doesn't already use it. See 'help minimize_with'.
(10-Oct-2014) Mario Carneiro has encoded the axiomatic basis for the HOL theorem prover into a Metamath source file, hol.mm.
(24-Sep-2014) Mario Carneiro added the Sum of the Angles of a Triangle (ang180), bringing the Metamath total for Formalizing 100 Theorems to 38.
(15-Sep-2014) Mario Carneiro added the Fundamental Theorem of Algebra (fta), bringing the Metamath total for Formalizing 100 Theorems to 37.
(3-Sep-2014) Mario Carneiro added the Fundamental Theorem of Integral Calculus (ftc1, ftc2). This brings the Metamath total for Formalizing 100 Theorems to 35. (added 14-Sep-2014) Along the way, he added the Mean Value Theorem (mvth), bringing the total to 36.
(16-Aug-2014) Mario Carneiro started a Metamath blog at http://metamath-blog.blogspot.com/.
(10-Aug-2014) Mario Carneiro added Erdős's proof of the divergence of the inverse prime series (prmrec). This brings the Metamath total for Formalizing 100 Theorems to 34.
(31-Jul-2014) Mario Carneiro added proofs for Euler's Summation of 1 + (1/2)^2 + (1/3)^2 + .... (basel) and The Factor and Remainder Theorems (facth, plyrem). This brings the Metamath total for Formalizing 100 Theorems to 33.
(16-Jul-2014) Mario Carneiro added proofs for Four Squares Theorem (4sq), Formula for the Number of Combinations (hashbc), and Divisibility by 3 Rule (3dvds). This brings the Metamath total for Formalizing 100 Theorems to 31.
(11-Jul-2014) Mario Carneiro added proofs for Divergence of the Harmonic Series (harmonic), Order of a Subgroup (lagsubg), and Lebesgue Measure and Integration (itgcl). This brings the Metamath total for Formalizing 100 Theorems to 28.
(7-Jul-2014) Mario Carneiro presented a talk, "Natural Deduction in the Metamath Proof Language," at the 6PCM conference. Slides Audio
(25-Jun-2014) In version 0.108 of the metamath program, the 'minimize_with' command is now more automated. It now considers compressed proof length; it scans the statements in forward and reverse order and chooses the best; and it avoids $d conflicts. The '/no_distinct', '/brief', and '/reverse' qualifiers are obsolete, and '/verbose' no longer lists all statements scanned but gives more details about decision criteria.
(12-Jun-2014) To improve naming uniformity, theorems about operation values now use the abbreviation "ov". For example, df-opr, opreq1, oprabval5, and oprvres are now called df-ov, oveq1, ov5, and ovres respectively.
(11-Jun-2014) Mario Carneiro finished a major revision of set.mm. His notes are under the 11-Jun-2014 entry in the Notes
(4-Jun-2014) Mario Carneiro provided instructions and screenshots for syntax highlighting for the jEdit editor for use with Metamath and mmj2 source files.
(19-May-2014) Mario Carneiro added a feature to mmj2, in the build at
https://github.com/digama0/mmj2/raw/dev-build/mmj2jar/mmj2.jar, which
tests all but 5 definitions in set.mm for soundness. You can turn on
the test by adding
SetMMDefinitionsCheckWithExclusions,ax-*,df-bi,df-clab,df-cleq,df-clel,df-sbc
to your RunParms.txt file.
(17-May-2014) A number of labels were changed in set.mm, listed at the top of set.mm as usual. Note in particular that the heavily-used visset, elisseti, syl11anc, syl111anc were changed respectively to vex, elexi, syl2anc, syl3anc.
(16-May-2014) Scott Fenton formalized a proof for "Sum of kth powers": fsumkthpow. This brings the Metamath total for Formalizing 100 Theorems to 25.
(9-May-2014) I (Norm Megill) presented an overview of Metamath at the "Formalization of mathematics in proof assistants" workshop at the Institut Henri Poincaré in Paris. The slides for this talk are here.
(22-Jun-2014) Version 0.107 of the metamath program adds a "PART" indention level to the Statement List table of contents, adds 'show proof ... /size' to show source file bytes used, and adds 'show elapsed_time'. The last one is helpful for measuring the run time of long commands. See 'help write theorem_list', 'help show proof', and 'help show elapsed_time' for more information.
(2-May-2014) Scott Fenton formalized a proof of Sum of the Reciprocals of the Triangular Numbers: trirecip. This brings the Metamath total for Formalizing 100 Theorems to 24.
(19-Apr-2014) Scott Fenton formalized a proof of the Formula for Pythagorean Triples: pythagtrip. This brings the Metamath total for Formalizing 100 Theorems to 23.
(11-Apr-2014) David A. Wheeler produced a much-needed and well-done video for mmj2, called "Introduction to Metamath & mmj2". Thanks, David!
(15-Mar-2014) Mario Carneiro formalized a proof of Bertrand's postulate: bpos. This brings the Metamath total for Formalizing 100 Theorems to 22.
(18-Feb-2014) Mario Carneiro proved that complex number axiom ax-cnex is redundant (theorem cnex). See also Real and Complex Numbers.
(11-Feb-2014) David A. Wheeler has created a theorem compilation that tracks those theorems in Freek Wiedijk's Formalizing 100 Theorems list that have been proved in set.mm. If you find a error or omission in this list, let me know so it can be corrected. (Update 1-Mar-2014: Mario has added eulerth and bezout to the list.)
(4-Feb-2014) Mario Carneiro writes:
The latest commit on the mmj2 development branch introduced an exciting new feature, namely syntax highlighting for mmp files in the main window. (You can pick up the latest mmj2.jar at https://github.com/digama0/mmj2/blob/develop/mmj2jar/mmj2.jar .) The reason I am asking for your help at this stage is to help with design for the syntax tokenizer, which is responsible for breaking down the input into various tokens with names like "comment", "set", and "stephypref", which are then colored according to the user's preference. As users of mmj2 and metamath, what types of highlighting would be useful to you?One limitation of the tokenizer is that since (for performance reasons) it can be started at any line in the file, highly contextual coloring, like highlighting step references that don't exist previously in the file, is difficult to do. Similarly, true parsing of the formulas using the grammar is possible but likely to be unmanageably slow. But things like checking theorem labels against the database is quite simple to do under the current setup.
That said, how can this new feature be optimized to help you when writing proofs?
(13-Jan-2014) Mathbox users: the *19.21a*, *19.23a* series of theorems have been renamed to *alrim*, *exlim*. You can update your mathbox with a global replacement of string '19.21a' with 'alrim' and '19.23a' with 'exlim'.
(5-Jan-2014) If you downloaded mmj2 in the past 3 days, please update it with the current version, which fixes a bug introduced by the recent changes that made it unable to read in most of the proofs in the textarea properly.
(4-Jan-2014) I added a list of "Allowed substitutions" under the "Distinct variable groups" list on the theorem web pages, for example axsep. This is an experimental feature and comments are welcome.
(3-Jan-2014) Version 0.102 of the metamath program produces more space-efficient compressed proofs (still compatible with the specification in Appendix B of the Metamath book) using an algorithm suggested by Mario Carneiro. See 'help save proof' in the program. Also, mmj2 now generates proofs in the new format. The new mmj2 also has a mandatory update that fixes a bug related to the new format; you must update your mmj2 copy to use it with the latest set.mm.
(23-Dec-2013) Mario Carneiro has updated many older definitions to use the maps-to notation. If you have difficulty updating your local mathbox, contact him or me for assistance.
(1-Nov-2013) 'undo' and 'redo' commands were added to the Proof Assistant in metamath program version 0.07.99. See 'help undo' in the program.
(8-Oct-2013) Today's Notes entry describes some proof repair techniques.
(5-Oct-2013) Today's Notes entry explains some recent extensible structure improvements.
(8-Sep-2013) Mario Carneiro has revised the square root and sequence generator definitions. See today's Notes entry.
(3-Aug-2013) Mario Carneiro writes: "I finally found enough time to create a GitHub repository for development at https://github.com/digama0/mmj2. A permalink to the latest version plus source (akin to mmj2.zip) is https://github.com/digama0/mmj2/zipball/, and the jar file on its own (mmj2.jar) is at https://github.com/digama0/mmj2/blob/master/mmj2jar/mmj2.jar?raw=true. Unfortunately there is no easy way to automatically generate mmj2jar.zip, but this is available as part of the zip distribution for mmj2.zip. History tracking will be handled by the repository now. Do you have old versions of the mmj2 directory? I could add them as historical commits if you do."
(18-Jun-2013) Mario Carneiro has done a major revision and cleanup of the construction of real and complex numbers. In particular, rather than using equivalence classes as is customary for the construction of the temporary rationals, he used only "reduced fractions", so that the use of the axiom of infinity is avoided until it becomes necessary for the construction of the temporary reals.
(18-May-2013) Mario Carneiro has added the ability to produce compressed proofs to mmj2. This is not an official release but can be downloaded here if you want to try it: mmj2.jar. If you have any feedback, send it to me (NM), and I will forward it to Mario. (Disclaimer: this release has not been endorsed by Mel O'Cat. If anyone has been in contact with him, please let me know.)
(29-Mar-2013) Charles Greathouse reduced the size of our PNG symbol images using the pngout program.
(8-Mar-2013) Wolf Lammen has reorganized the theorems in the "Logical negation" section of set.mm into a more orderly, less scattered arrangement.
(27-Feb-2013) Scott Fenton has done a large cleanup of set.mm, eliminating *OLD references in 144 proofs. See the Notes entry for 27-Feb-2013.
(21-Feb-2013) *ATTENTION MATHBOX USERS* The order of hypotheses of many syl* theorems were changed, per a suggestion of Mario Carneiro. You need to update your local mathbox copy for compatibility with the new set.mm, or I can do it for you if you wish. See the Notes entry for 21-Feb-2013.
(16-Feb-2013) Scott Fenton shortened the direct-from-axiom proofs of *3.1, *3.43, *4.4, *4.41, *4.5, *4.76, *4.83, *5.33, *5.35, *5.36, and meredith in the "Shortest known proofs of the propositional calculus theorems from Principia Mathematica" (pmproofs.txt).
(27-Jan-2013) Scott Fenton writes, "I've updated Ralph Levien's mmverify.py. It's now a Python 3 program, and supports compressed proofs and file inclusion statements. This adds about fifty lines to the original program. Enjoy!"
(10-Jan-2013) A new mathbox was added for Mario Carneiro, who has contributed a number of cardinality theorems without invoking the Axiom of Choice. This is nice work, and I will be using some of these (those suffixed with "NEW") to replace the existing ones in the main part of set.mm that currently invoke AC unnecessarily.
(4-Jan-2013) As mentioned in the 19-Jun-2012 item below, Eric Schmidt discovered that the complex number axioms axaddcom (now addcom) and ax0id (now addid1) are redundant (schmidt-cnaxioms.pdf, .tex). In addition, ax1id (now mulid1) can be weakened to ax1rid. Scott Fenton has now formalized this work, so that now there are 23 instead of 25 axioms for real and complex numbers in set.mm. The Axioms for Complex Numbers page has been updated with these results. An interesting part of the proof, showing how commutativity of addition follows from other laws, is in addcomi.
(27-Nov-2012) The frequently-used theorems "an1s", "an1rs", "ancom13s", "ancom31s" were renamed to "an12s", "an32s", "an13s", "an31s" to conform to the convention for an12 etc.
(4-Nov-2012) The changes proposed in the Notes, renaming Grp to GrpOp etc., have been incorporated into set.mm. See the list of changes at the top of set.mm. If you want me to update your mathbox with these changes, send it to me along with the version of set.mm that it works with.
(20-Sep-2012) Mel O'Cat updated http://us2.metamath.org:88/ocat/mmj2/TESTmmj2jar.zip. See the README.TXT for a description of the new features.
(21-Aug-2012) Mel O'Cat has uploaded SearchOptionsMockup9.zip, a mockup for the new search screen in mmj2. See the README.txt file for instructions. He will welcome feedback via x178g243 at yahoo.com.
(19-Jun-2012) Eric Schmidt has discovered that in our axioms for complex numbers, axaddcom and ax0id are redundant. (At some point these need to be formalized for set.mm.) He has written up these and some other nice results, including some independence results for the axioms, in schmidt-cnaxioms.pdf (schmidt-cnaxioms.tex).
(23-Apr-2012) Frédéric Liné sent me a PDF (LaTeX source) developed with Lamport's pf2 package. He wrote: "I think it works well with Metamath since the proofs are in a tree form. I use it to have a sketch of a proof. I get this way a better understanding of the proof and I can cut down its size. For instance, inpreima5 was reduced by 50% when I wrote the corresponding proof with pf2."
(5-Mar-2012) I added links to Wikiproofs and its recent changes in the "Wikis" list at the top of this page.
(12-Jan-2012) Thanks to William Hoza who sent me a ZFC T-shirt, and thanks to the ZFC models (courtesy of the Inaccessible Cardinals agency).
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(24-Nov-2011) In metamath program version 0.07.71, the 'minimize_with' command by default now scans from bottom to top instead of top to bottom, since empirically this often (although not always) results in a shorter proof. A top to bottom scan can be specified with a new qualifier '/reverse'. You can try both methods (starting from the same original proof, of course) and pick the shorter proof.
(15-Oct-2011) From Mel O'Cat:
I just uploaded mmj2.zip containing the 1-Nov-2011 (20111101)
release:
http://us2.metamath.org:88/ocat/mmj2/mmj2.zip
http://us2.metamath.org:88/ocat/mmj2/mmj2.md5
A few last minute tweaks:
1. I now bless double-click starting of mmj2.bat (MacMMJ2.command in Mac OS-X)!
See mmj2\QuickStart.html
2. Much improved support of Mac OS-X systems.
See mmj2\QuickStart.html
3. I tweaked the Command Line Argument Options report to
a) print every time;
b) print as much as possible even if
there are errors in the command line arguments -- and the
last line printed corresponds to the argument in error;
c) removed Y/N argument on the command line to enable/disable
the report. this simplifies things.
4) Documentation revised, including the PATutorial.
See CHGLOG.TXT for list of all changes.
Good luck. And thanks for all of your help!
(15-Sep-2011) MATHBOX USERS: I made a large number of label name changes to set.mm to improve naming consistency. There is a script at the top of the current set.mm that you can use to update your mathbox or older set.mm. Or if you wish, I can do the update on your next mathbox submission - in that case, please include a .zip of the set.mm version you used.
(30-Aug-2011) Scott Fenton shortened the direct-from-axiom proofs of *3.33, *3.45, *4.36, and meredith in the "Shortest known proofs of the propositional calculus theorems from Principia Mathematica" (pmproofs.txt).
(21-Aug-2011) A post on reddit generated 60,000 hits (and a TOS violation notice from my provider...),
(18-Aug-2011) The Metamath Google Group has a discussion of my canonical conjunctions proposal. Any feedback directly to me (Norm Megill) is also welcome.
(4-Jul-2011) John Baker has provided (metamath_kindle.zip) "a modified version of [the] metamath.tex [Metamath] book source that is formatted for the Kindle. If you compile the document the resulting PDF can be loaded into into a Kindle and easily read." (Update: the PDF file is now included also.)
(3-Jul-2011) Nested 'submit' calls are now allowed, in metamath program version 0.07.68. Thus you can create or modify a command file (script) from within a command file then 'submit' it. While 'submit' cannot pass arguments (nor are there plans to add this feature), you can 'substitute' strings in the 'submit' target file before calling it in order to emulate this.
(28-Jun-2011)The metamath program version 0.07.64 adds the '/include_mathboxes' qualifier to 'minimize_with'; by default, 'minimize_with *' will now skip checking user mathboxes. Since mathboxes should be independent from each other, this will help prevent accidental cross-"contamination". Also, '/rewrap' was added to 'write source' to automatically wrap $a and $p comments so as to conform to the current formatting conventions used in set.mm. This means you no longer have to be concerned about line length < 80 etc.
(19-Jun-2011) ATTENTION MATHBOX USERS: The wff variables et, ze, si, and rh are now global. This change was made primarily to resolve some conflicts between mathboxes, but it will also let you avoid having to constantly redeclare these locally in the future. Unfortunately, this change can affect the $f hypothesis order, which can cause proofs referencing theorems that use these variables to fail. All mathbox proofs currently in set.mm have been corrected for this, and you should refresh your local copy for further development of your mathbox. You can correct your proofs that are not in set.mm as follows. Only the proofs that fail under the current set.mm (using version 0.07.62 or later of the metamath program) need to be modified.
To fix a proof that references earlier theorems using et, ze, si, and rh, do the following (using a hypothetical theorem 'abc' as an example): 'prove abc' (ignore error messages), 'delete floating', 'initialize all', 'unify all/interactive', 'improve all', 'save new_proof/compressed'. If your proof uses dummy variables, these must be reassigned manually.
To fix a proof that uses et, ze, si, and rh as local variables, make sure the proof is saved in 'compressed' format. Then delete the local declarations ($v and $f statements) and follow the same steps above to correct the proof.
I apologize for the inconvenience. If you have trouble fixing your proofs, you can contact me for assistance.
Note: Versions of the metamath program before 0.07.62 did not flag an error when global variables were redeclared locally, as it should have according to the spec. This caused these spec violations to go unnoticed in some older set.mm versions. The new error messages are in fact just informational and can be ignored when working with older set.mm versions.
(7-Jun-2011) The metamath program version 0.07.60 fixes a bug with the 'minimize_with' command found by Andrew Salmon.
(12-May-2010) Andrew Salmon shortened many proofs, shown above. For comparison, I have temporarily kept the old version, which is suffixed with OLD, such as oridmOLD for oridm.
(9-Dec-2010) Eric Schmidt has written a Metamath proof verifier in C++, called checkmm.cpp.
(3-Oct-2010) The following changes were made to the tokens in set.mm. The subset and proper subset symbol changes to C_ and C. were made to prevent defeating the parenthesis matching in Emacs. Other changes were made so that all letters a-z and A-Z are now available for variable names. One-letter constants such as _V, _e, and _i are now shown on the web pages with Roman instead of italic font, to disambiguate italic variable names. The new convention is that a prefix of _ indicates Roman font and a prefix of ~ indicates a script (curly) font. Thanks to Stefan Allan and Frédéric Liné for discussions leading to this change.
| Old | New | Description |
|---|---|---|
| C. | _C | binomial coefficient |
| E | _E | epsilon relation |
| e | _e | Euler's constant |
| I | _I | identity relation |
| i | _i | imaginary unit |
| V | _V | universal class |
| (_ | C_ | subset |
| (. | C. | proper subset |
| P~ | ~P | power class |
| H~ | ~H | Hilbert space |
(25-Sep-2010) The metamath program (version 0.07.54) now implements the current Metamath spec, so footnote 2 on p. 92 of the Metamath book can be ignored.
(24-Sep-2010) The metamath program (version 0.07.53) fixes bug 2106, reported by Michal Burger.
(14-Sep-2010) The metamath program (version 0.07.52) has a revamped LaTeX output with 'show statement xxx /tex', which produces the combined statement, description, and proof similar to the web page generation. Also, 'show proof xxx /lemmon/renumber' now matches the web page step numbers. ('show proof xxx/renumber' still has the indented form conforming to the actual RPN proof, with slightly different numbering.)
(9-Sep-2010) The metamath program (version 0.07.51) was updated with a modification by Stefan Allan that adds hyperlinks the the Ref column of proofs.
(12-Jun-2010) Scott Fenton contributed a D-proof (directly from axioms) of Meredith's single axiom (see the end of pmproofs.txt). A description of Meredith's axiom can be found in theorem meredith.
(11-Jun-2010) A new Metamath mirror was added in Austria, courtesy of Kinder-Enduro.
(28-Feb-2010) Raph Levien's Ghilbert project now has a new Ghilbert site and a Google Group.
(26-Jan-2010) Dmitri Vlasov writes, "I admire the simplicity and power of the metamath language, but still I see its great disadvantage - the proofs in metamath are completely non-manageable by humans without proof assistants. Therefore I decided to develop another language, which would be a higher-level superstructure language towards metamath, and which will support human-readable/writable proofs directly, without proof assistants. I call this language mdl (acronym for 'mathematics development language')." The latest version of Dmitri's translators from metamath to mdl and back can be downloaded from http://mathdevlanguage.sourceforge.net/. Currently only Linux is supported, but Dmitri says is should not be difficult to port it to other platforms that have a g++ compiler.
(11-Sep-2009) The metamath program (version 0.07.48) has been updated to enforce the whitespace requirement of the current spec.
(10-Sep-2009) Matthew Leitch has written an nice article, "How to write mathematics clearly", that briefly mentions Metamath. Overall it makes some excellent points. (I have written to him about a few things I disagree with.)
(28-May-2009) AsteroidMeta is back on-line. Note the URL change.
(12-May-2009) Charles Greathouse wrote a Greasemonkey script to reformat the axiom list on Metamath web site proof pages. This is a beta version; he will appreciate feedback.
(11-May-2009) Stefan Allan modified the metamath program to add the command "show statement xxx /mnemonics", which produces the output file Mnemosyne.txt for use with the Mnemosyne project. The current Metamath program download incorporates this command. Instructions: Create the file mnemosyne.txt with e.g. "show statement ax-* /mnemonics". In the Mnemosyne program, load the file by choosing File->Import then file format "Q and A on separate lines". Notes: (1) Don't try to load all of set.mm, it will crash the program due to a bug in Mnemosyne. (2) On my computer, the arrows in ax-1 don't display. Stefan reports that they do on his computer. (Both are Windows XP.)
(3-May-2009) Steven Baldasty wrote a Metamath syntax highlighting file for the gedit editor. Screenshot.
(1-May-2009) Users on a gaming forum discuss our 2+2=4 proof. Notable comments include "Ew math!" and "Whoever wrote this has absolutely no life."
(12-Mar-2009) Chris Capel has created a Javascript theorem viewer demo that (1) shows substitutions and (2) allows expanding and collapsing proof steps. You are invited to take a look and give him feedback at his Metablog.
(28-Feb-2009) Chris Capel has written a Metamath proof verifier in C#, available at http://pdf23ds.net/bzr/MathEditor/Verifier/Verifier.cs and weighing in at 550 lines. Also, that same URL without the file on it is a Bazaar repository.
(2-Dec-2008) A new section was added to the Deduction Theorem page, called Logic, Metalogic, Metametalogic, and Metametametalogic.
(24-Aug-2008) (From ocat): The 1-Aug-2008 version of mmj2 is ready (mmj2.zip), size = 1,534,041 bytes. This version contains the Theorem Loader enhancement which provides a "sandboxing" capability for user theorems and dynamic update of new theorems to the Metamath database already loaded in memory by mmj2. Also, the new "mmj2 Service" feature enables calling mmj2 as a subroutine, or having mmj2 call your program, and provides access to the mmj2 data structures and objects loaded in memory (i.e. get started writing those Jython programs!) See also mmj2 on AsteroidMeta.
(23-May-2008) Gérard Lang pointed me to Bob Solovay's note on AC and strongly inaccessible cardinals. One of the eventual goals for set.mm is to prove the Axiom of Choice from Grothendieck's axiom, like Mizar does, and this note may be helpful for anyone wanting to attempt that. Separately, I also came across a history of the size reduction of grothprim (viewable in Firefox and some versions of Internet Explorer).
(14-Apr-2008) A "/join" qualifier was added to the "search" command in the metamath program (version 0.07.37). This qualifier will join the $e hypotheses to the $a or $p for searching, so that math tokens in the $e's can be matched as well. For example, "search *com* +v" produces no results, but "search *com* +v /join" yields commutative laws involving vector addition. Thanks to Stefan Allan for suggesting this idea.
(8-Apr-2008) The 8,000th theorem, hlrel, was added to the Metamath Proof Explorer part of the database.
(2-Mar-2008) I added a small section to the end of the Deduction Theorem page.
(17-Feb-2008) ocat has uploaded the "1-Mar-2008" mmj2: mmj2.zip. See the description.
(16-Jan-2008) O'Cat has written mmj2 Proof Assistant Quick Tips.
(30-Dec-2007) "How to build a library of formalized mathematics".
(22-Dec-2007) The Metamath Proof Explorer was included in the top 30 science resources for 2007 by the University at Albany Science Library.
(17-Dec-2007) Metamath's Wikipedia entry says, "This article may require cleanup to meet Wikipedia's quality standards" (see its discussion page). Volunteers are welcome. :) (In the interest of objectivity, I don't edit this entry.)
(20-Nov-2007) Jeff Hoffman created nicod.mm and posted it to the Google Metamath Group.
(19-Nov-2007) Reinder Verlinde suggested adding tooltips to the hyperlinks on the proof pages, which I did for proof step hyperlinks. Discussion.
(5-Nov-2007) A Usenet challenge. :)
(4-Aug-2007) I added a "Request for comments on proposed 'maps to' notation" at the bottom of the AsteroidMeta set.mm discussion page.
(21-Jun-2007) A preprint (PDF file) describing Kurt Maes' axiom of choice with 5 quantifiers, proved in set.mm as ackm.
(20-Jun-2007) The 7,000th theorem, ifpr, was added to the Metamath Proof Explorer part of the database.
(29-Apr-2007) Blog mentions of Metamath: here and here.
(21-Mar-2007) Paul Chapman is working on a new proof browser, which has highlighting that allows you to see the referenced theorem before and after the substitution was made. Here is a screenshot of theorem 0nn0 and a screenshot of theorem 2p2e4.
(15-Mar-2007) A picture of Penny the cat guarding the us2.metamath.org:8888 server and making the rounds.
(16-Feb-2007) For convenience, the program "drule.c" (pronounced "D-rule", not "drool") mentioned in pmproofs.txt can now be downloaded (drule.c) without having to ask me for it. The same disclaimer applies: even though this program works and has no known bugs, it was not intended for general release. Read the comments at the top of the program for instructions.
(28-Jan-2007) Jason Orendorff set up a new mailing list for Metamath: http://groups.google.com/group/metamath.
(20-Jan-2007) Bob Solovay provided a revised version of his Metamath database for Peano arithmetic, peano.mm.
(2-Jan-2007) Raph Levien has set up a wiki called Barghest for the Ghilbert language and software.
(26-Dec-2006) I posted an explanation of theorem ecoprass on Usenet.
(2-Dec-2006) Berislav Žarnić translated the Metamath Solitaire applet to Croatian.
(26-Nov-2006) Dan Getz has created an RSS feed for new theorems as they appear on this page.
(6-Nov-2006) The first 3 paragraphs in Appendix 2: Note on the Axioms were rewritten to clarify the connection between Tarski's axiom system and Metamath.
(31-Oct-2006) ocat asked for a do-over due to a bug in mmj2 -- if you downloaded the mmj2.zip version dated 10/28/2006, then download the new version dated 10/30.
(29-Oct-2006) ocat has announced that the
long-awaited 1-Nov-2006 release of mmj2 is available now.
The new "Unify+Get Hints" is quite
useful, and any proof can be generated as follows. With "?" in the Hyp
field and Ref field blank, select "Unify+Get Hints". Select a hint from
the list and put it in the Ref field. Edit any $n dummy variables to
become the desired wffs. Rinse and repeat for the new proof steps
generated, until the proof is done.
The new tutorial, mmj2PATutorial.bat,
explains this in detail. One way to reduce or avoid dummy $n's is to
fill in the Hyp field with a comma-separated list of any known
hypothesis matches to earlier proof steps, keeping a "?" in the list to
indicate that the remaining hypotheses are unknown. Then "Unify+Get
Hints" can be applied. The tutorial page
\mmj2\data\mmp\PATutorial\Page405.mmp has an example.
Don't forget that the eimm
export/import program lets you go back and forth between the mmj2 and
the metamath program proof assistants, without exiting from either one,
to exploit the best features of each as required.
(21-Oct-2006) Martin Kiselkov has written a Metamath proof verifier in the Lua scripting language, called verify.lua. While it is not practical as an everyday verifier - he writes that it takes about 40 minutes to verify set.mm on a a Pentium 4 - it could be useful to someone learning Lua or Metamath, and importantly it provides another independent way of verifying the correctness of Metamath proofs. His code looks like it is nicely structured and very readable. He is currently working on a faster version in C++.
(19-Oct-2006) New AsteroidMeta page by Raph, Distinctors_vs_binders.
(13-Oct-2006) I put a simple Metamath browser on my PDA (Palm Tungsten E) so that I don't have to lug around my laptop. Here is a screenshot. It isn't polished, but I'll provide the file + instructions if anyone wants it.
(3-Oct-2006) A blog entry, Principia for Reverse Mathematics.
(28-Sep-2006) A blog entry, Metamath responds.
(26-Sep-2006) A blog entry, Metamath isn't hygienic.
(11-Aug-2006) A blog entry, Metamath and the Peano Induction Axiom.
(26-Jul-2006) A new open problem in predicate calculus was added.
(18-Jun-2006) The 6,000th theorem, mt4d, was added to the Metamath Proof Explorer part of the database.
(9-May-2006) Luca Ciciriello has upgraded the t2mf program, which is a C
program used to create the MIDI files on the
Metamath Music Page, so
that it works on MacOS X. This is a nice accomplishment, since the
original program was written before C was standardized by ANSI and will
not compile on modern compilers.
Unfortunately, the original program source states no copyright terms.
The main author, Tim Thompson, has kindly agreed to release his code to
public domain, but two other authors have also contributed to the code,
and so far I have been unable to contact them for copyright clearance.
Therefore I cannot offer the MacOS X version for public download on this
site until this is resolved. Update 10-May-2006: Another author,
M. Czeiszperger, has released his contribution to public domain.
If you are interested in Luca's modified source code,
please contact me directly.
(18-Apr-2006) Incomplete proofs in progress can now be interchanged between the Metamath program's CLI Proof Assistant and mmj2's GUI Proof Assistant, using a new export-import program called eimm. This can be done without exiting either proof assistant, so that the strengths of each approach can be exploited during proof development. See "Use Case 5a" and "Use Case 5b" at mmj2ProofAssistantFeedback.
(28-Mar-2006) Scott Fenton updated his second version of Metamath Solitaire (the one that uses external axioms). He writes: "I've switched to making it a standalone program, as it seems silly to have an applet that can't be run in a web browser. Check the README file for further info." The download is mmsol-0.5.tar.gz.
(27-Mar-2006) Scott Fenton has updated the Metamath Solitaire Java
applet to Java 1.5: (1) QSort has been stripped out: its functionality
is in the Collections class that Sun ships; (2) all Vectors have been
replaced by ArrayLists; (3) generic types have been tossed in wherever
they fit: this cuts back drastically on casting; and (4) any warnings
Eclipse spouted out have been dealt with. I haven't yet updated it
officially, because I don't know if it will work with Microsoft's JVM in
older versions of Internet Explorer. The current official version is
compiled with Java 1.3, because it won't work with Microsoft's JVM if it
is compiled with Java 1.4. (As distasteful as that seems,
I will get complaints from users if it
doesn't work with Microsoft's JVM.) If anyone can verify that Scott's new
version runs on Microsoft's JVM, I would be grateful. Scott's new
version is mm.java-1.5.gz; after
uncompressing it, rename it to mm.java,
use it to replace the existing mm.java file in the
Metamath Solitaire download, and recompile according to instructions
in the mm.java comments.
Scott has also created a second version, mmsol-0.2.tar.gz, that reads
the axioms from ASCII files, instead of having the axioms hard-coded in
the program. This can be very useful if you want to play with custom
axioms, and you can also add a collection of starting theorems as
"axioms" to work from. However, it must be run from the local directory
with appletviewer, since the default Java security model doesn't allow
reading files from a browser. It works with the JDK 5 Update 6
Java download.
To compile (from Windows Command Prompt): C:\Program
Files\Java\jdk1.5.0_06\bin\javac.exe mm.java
To run (from Windows Command Prompt): C:\Program
Files\Java\jdk1.5.0_06\bin\appletviewer.exe mms.html
(21-Jan-2006) Juha Arpiainen proved the independence of axiom ax-11 from the others. This was published as an open problem in my 1995 paper (Remark 9.5 on PDF page 17). See Item 9a on the Workshop Miscellany for his seven-line proof. See also the Asteroid Meta metamathMathQuestions page under the heading "Axiom of variable substitution: ax-11". Congratulations, Juha!
(20-Oct-2005) Juha Arpiainen is working on a proof verifier in Common Lisp called Bourbaki. Its proof language has its roots in Metamath, with the goal of providing a more powerful syntax and definitional soundness checking. See its documentation and related discussion.
(17-Oct-2005) Marnix Klooster has written a Metamath proof verifier in Haskell, called Hmm. Also see his Announcement. The complete program (Hmm.hs, HmmImpl.hs, and HmmVerify.hs) has only 444 lines of code, excluding comments and blank lines. It verifies compressed as well as regular proofs; moreover, it transparently verifies both per-spec compressed proofs and the flawed format he uncovered (see comment below of 16-Oct-05).
(16-Oct-2005) Marnix Klooster noticed that for large proofs, the compressed proof format did not match the spec in the book. His algorithm to correct the problem has been put into the Metamath program (version 0.07.6). The program still verifies older proofs with the incorrect format, but the user will be nagged to update them with 'save proof *'. In set.mm, 285 out of 6376 proofs are affected. (The incorrect format did not affect proof correctness or verification, since the compression and decompression algorithms matched each other.)
(13-Sep-2005) Scott Fenton found an interesting axiom, ax46, which could be used to replace both ax-4 and ax-6.
(29-Jul-2005) Metamath was selected as site of the week by American Scientist Online.
(8-Jul-2005) Roy Longton has contributed 53 new theorems to the Quantum Logic Explorer. You can see them in the Theorem List starting at lem3.3.3lem1. He writes, "If you want, you can post an open challenge to see if anyone can find shorter proofs of the theorems I submitted."
(10-May-2005) A Usenet post I posted about the infinite prime proof; another one about indexed unions.
(3-May-2005) The theorem divexpt is the 5,000th theorem added to the Metamath Proof Explorer database.
(12-Apr-2005) Raph Levien solved the open problem in item 16 on the Workshop Miscellany page and as a corollary proved that axiom ax-9 is independent from the other axioms of predicate calculus and equality. This is the first such independence proof so far; a goal is to prove all of them independent (or to derive any redundant ones from the others).
(8-Mar-2005) I added a paragraph above our complex number axioms table, summarizing the construction and indicating where Dedekind cuts are defined. Thanks to Andrew Buhr for comments on this.
(16-Feb-2005) The Metamath Music Page is mentioned as a reference or resource for a university course called Math, Mind, and Music. .
(28-Jan-2005) Steven Cullinane parodied the Metamath Music Page in his blog.
(18-Jan-2005) Waldek Hebisch upgraded the Metamath program to run on the AMD64 64-bit processor.
(17-Jan-2005) A symbol list summary was added to the beginning of the Hilbert Space Explorer Home Page. Thanks to Mladen Pavicic for suggesting this.
(6-Jan-2005) Someone assembled an amazon.com list of some of the books in the Metamath Proof Explorer Bibliography.
(4-Jan-2005) The definition of ordinal exponentiation was decided on after this Usenet discussion.
(19-Dec-2004) A bit of trivia: my Erdös number is 2, as you can see from this list.
(20-Oct-2004) I started this Usenet discussion about the "reals are uncountable" proof (127 comments; last one on Nov. 12).
(12-Oct-2004) gch-kn shows the equivalence of the Generalized Continuum Hypothesis and Prof. Nambiar's Axiom of Combinatorial Sets. This proof answers his Open Problem 2 (PDF file).
(5-Aug-2004) I gave a talk on "Hilbert Lattice Equations" at the Argonne workshop.
(25-Jul-2004) The theorem nthruz is the 4,000th theorem added to the Metamath Proof Explorer database.
(27-May-2004) Josiah Burroughs contributed the proofs u1lemn1b, u1lem3var1, oi3oa3lem1, and oi3oa3 to the Quantum Logic Explorer database ql.mm.
(23-May-2004) Some minor typos found by Josh Purinton were corrected in the Metamath book. In addition, Josh simplified the definition of the closure of a pre-statement of a formal system in Appendix C.
(5-May-2004) Gregory Bush has found shorter proofs for 67 of the 193 propositional calculus theorems listed in Principia Mathematica, thus establishing 67 new records. (This was challenge #4 on the open problems page.)
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