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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | dfso3 31601* | Expansion of the definition of a strict order. (Contributed by Scott Fenton, 6-Jun-2016.) |
⊢ (𝑅 Or 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧) ∧ (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥))) | ||
Theorem | brtpid1 31602 | A binary relation involving unordered triplets. (Contributed by Scott Fenton, 7-Jun-2016.) |
⊢ 𝐴{〈𝐴, 𝐵〉, 𝐶, 𝐷}𝐵 | ||
Theorem | brtpid2 31603 | A binary relation involving unordered triplets. (Contributed by Scott Fenton, 7-Jun-2016.) |
⊢ 𝐴{𝐶, 〈𝐴, 𝐵〉, 𝐷}𝐵 | ||
Theorem | brtpid3 31604 | A binary relation involving unordered triplets. (Contributed by Scott Fenton, 7-Jun-2016.) |
⊢ 𝐴{𝐶, 𝐷, 〈𝐴, 𝐵〉}𝐵 | ||
Theorem | ceqsrexv2 31605* | Alternate elimitation of a restricted existential quantifier, using implicit substitution. (Contributed by Scott Fenton, 5-Sep-2017.) |
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃𝑥 ∈ 𝐵 (𝑥 = 𝐴 ∧ 𝜑) ↔ (𝐴 ∈ 𝐵 ∧ 𝜓)) | ||
Theorem | iota5f 31606* | A method for computing iota. (Contributed by Scott Fenton, 13-Dec-2017.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝐴 & ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → (𝜓 ↔ 𝑥 = 𝐴)) ⇒ ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → (℩𝑥𝜓) = 𝐴) | ||
Theorem | ceqsralv2 31607* | Alternate elimination of a restricted universal quantifier, using implicit substitution. (Contributed by Scott Fenton, 7-Dec-2020.) |
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥 ∈ 𝐵 (𝑥 = 𝐴 → 𝜑) ↔ (𝐴 ∈ 𝐵 → 𝜓)) | ||
Theorem | 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)) | ||
Theorem | 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))) | ||
Theorem | sqdivzi 31610 | Distribution of square over division. (Contributed by Scott Fenton, 7-Jun-2013.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ ⇒ ⊢ (𝐵 ≠ 0 → ((𝐴 / 𝐵)↑2) = ((𝐴↑2) / (𝐵↑2))) | ||
Theorem | subdivcomb1 31611 | Bring a term in a subtraction into the numerator. (Contributed by Scott Fenton, 3-Jul-2013.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → (((𝐶 · 𝐴) − 𝐵) / 𝐶) = (𝐴 − (𝐵 / 𝐶))) | ||
Theorem | subdivcomb2 31612 | Bring a term in a subtraction into the numerator. (Contributed by Scott Fenton, 3-Jul-2013.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → ((𝐴 − (𝐶 · 𝐵)) / 𝐶) = ((𝐴 / 𝐶) − 𝐵)) | ||
Theorem | supfz 31613 | The supremum of a finite sequence of integers. (Contributed by Scott Fenton, 8-Aug-2013.) |
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → sup((𝑀...𝑁), ℤ, < ) = 𝑁) | ||
Theorem | inffz 31614 | The infimum of a finite sequence of integers. (Contributed by Scott Fenton, 8-Aug-2013.) (Revised by AV, 10-Oct-2021.) |
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → inf((𝑀...𝑁), ℤ, < ) = 𝑀) | ||
Theorem | inffzOLD 31615 | The infimum of a finite sequence of integers. (Contributed by Scott Fenton, 8-Aug-2013.) Obsolete version of inffz 31614 as of 10-Oct-2021. (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → sup((𝑀...𝑁), ℤ, ◡ < ) = 𝑀) | ||
Theorem | fz0n 31616 | The sequence (0...(𝑁 − 1)) is empty iff 𝑁 is zero. (Contributed by Scott Fenton, 16-May-2014.) |
⊢ (𝑁 ∈ ℕ0 → ((0...(𝑁 − 1)) = ∅ ↔ 𝑁 = 0)) | ||
Theorem | shftvalg 31617 | Value of a sequence shifted by 𝐴. (Contributed by Scott Fenton, 16-Dec-2017.) |
⊢ ((𝐹 ∈ 𝑉 ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐹 shift 𝐴)‘𝐵) = (𝐹‘(𝐵 − 𝐴))) | ||
Theorem | divcnvlin 31618* | Limit of the ratio of two linear functions. (Contributed by Scott Fenton, 17-Dec-2017.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℤ) & ⊢ (𝜑 → 𝐹 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = ((𝑘 + 𝐴) / (𝑘 + 𝐵))) ⇒ ⊢ (𝜑 → 𝐹 ⇝ 1) | ||
Theorem | climlec3 31619* | Comparison of a constant to the limit of a sequence. (Contributed by Scott Fenton, 5-Jan-2018.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐹 ⇝ 𝐴) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ≤ 𝐵) ⇒ ⊢ (𝜑 → 𝐴 ≤ 𝐵) | ||
Theorem | logi 31620 | Calculate the logarithm of i. (Contributed by Scott Fenton, 13-Apr-2020.) |
⊢ (log‘i) = (i · (π / 2)) | ||
Theorem | iexpire 31621 | i raised to itself is real. (Contributed by Scott Fenton, 13-Apr-2020.) |
⊢ (i↑𝑐i) ∈ ℝ | ||
Theorem | bcneg1 31622 | The binomial coefficent over negative one is zero. (Contributed by Scott Fenton, 29-May-2020.) |
⊢ (𝑁 ∈ ℕ0 → (𝑁C-1) = 0) | ||
Theorem | bcm1nt 31623 | The proportion of one bionmial coefficient to another with 𝑁 decreased by 1. (Contributed by Scott Fenton, 23-Jun-2020.) |
⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...(𝑁 − 1))) → (𝑁C𝐾) = (((𝑁 − 1)C𝐾) · (𝑁 / (𝑁 − 𝐾)))) | ||
Theorem | bcprod 31624* | A product identity for binomial coefficents. (Contributed by Scott Fenton, 23-Jun-2020.) |
⊢ (𝑁 ∈ ℕ → ∏𝑘 ∈ (1...(𝑁 − 1))((𝑁 − 1)C𝑘) = ∏𝑘 ∈ (1...(𝑁 − 1))(𝑘↑((2 · 𝑘) − 𝑁))) | ||
Theorem | bccolsum 31625* | A column-sum rule for binomial coefficents. (Contributed by Scott Fenton, 24-Jun-2020.) |
⊢ ((𝑁 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) → Σ𝑘 ∈ (0...𝑁)(𝑘C𝐶) = ((𝑁 + 1)C(𝐶 + 1))) | ||
Theorem | iprodefisumlem 31626 | Lemma for iprodefisum 31627. (Contributed by Scott Fenton, 11-Feb-2018.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐹:𝑍⟶ℂ) ⇒ ⊢ (𝜑 → seq𝑀( · , (exp ∘ 𝐹)) = (exp ∘ seq𝑀( + , 𝐹))) | ||
Theorem | iprodefisum 31627* | Applying the exponential function to an infinite sum yields an infinite product. (Contributed by Scott Fenton, 11-Feb-2018.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐵) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℂ) & ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ ) ⇒ ⊢ (𝜑 → ∏𝑘 ∈ 𝑍 (exp‘𝐵) = (exp‘Σ𝑘 ∈ 𝑍 𝐵)) | ||
Theorem | iprodgam 31628* | An infinite product version of Euler's gamma function. (Contributed by Scott Fenton, 12-Feb-2018.) |
⊢ (𝜑 → 𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ))) ⇒ ⊢ (𝜑 → (Γ‘𝐴) = (∏𝑘 ∈ ℕ (((1 + (1 / 𝑘))↑𝑐𝐴) / (1 + (𝐴 / 𝑘))) / 𝐴)) | ||
Theorem | faclimlem1 31629* | Lemma for faclim 31632. Closed form for a particular sequence. (Contributed by Scott Fenton, 15-Dec-2017.) |
⊢ (𝑀 ∈ ℕ0 → seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (𝑀 / 𝑛)) · (1 + (1 / 𝑛))) / (1 + ((𝑀 + 1) / 𝑛))))) = (𝑥 ∈ ℕ ↦ ((𝑀 + 1) · ((𝑥 + 1) / (𝑥 + (𝑀 + 1)))))) | ||
Theorem | faclimlem2 31630* | Lemma for faclim 31632. Show a limit for the inductive step. (Contributed by Scott Fenton, 15-Dec-2017.) |
⊢ (𝑀 ∈ ℕ0 → seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (𝑀 / 𝑛)) · (1 + (1 / 𝑛))) / (1 + ((𝑀 + 1) / 𝑛))))) ⇝ (𝑀 + 1)) | ||
Theorem | faclimlem3 31631 | Lemma for faclim 31632. Algebraic manipulation for the final induction. (Contributed by Scott Fenton, 15-Dec-2017.) |
⊢ ((𝑀 ∈ ℕ0 ∧ 𝐵 ∈ ℕ) → (((1 + (1 / 𝐵))↑(𝑀 + 1)) / (1 + ((𝑀 + 1) / 𝐵))) = ((((1 + (1 / 𝐵))↑𝑀) / (1 + (𝑀 / 𝐵))) · (((1 + (𝑀 / 𝐵)) · (1 + (1 / 𝐵))) / (1 + ((𝑀 + 1) / 𝐵))))) | ||
Theorem | faclim 31632* | An infinite product expression relating to factorials. Originally due to Euler. (Contributed by Scott Fenton, 22-Nov-2017.) |
⊢ 𝐹 = (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝐴) / (1 + (𝐴 / 𝑛)))) ⇒ ⊢ (𝐴 ∈ ℕ0 → seq1( · , 𝐹) ⇝ (!‘𝐴)) | ||
Theorem | iprodfac 31633* | An infinite product expression for factorial. (Contributed by Scott Fenton, 15-Dec-2017.) |
⊢ (𝐴 ∈ ℕ0 → (!‘𝐴) = ∏𝑘 ∈ ℕ (((1 + (1 / 𝑘))↑𝐴) / (1 + (𝐴 / 𝑘)))) | ||
Theorem | faclim2 31634* | Another factorial limit due to Euler. (Contributed by Scott Fenton, 17-Dec-2017.) |
⊢ 𝐹 = (𝑛 ∈ ℕ ↦ (((!‘𝑛) · ((𝑛 + 1)↑𝑀)) / (!‘(𝑛 + 𝑀)))) ⇒ ⊢ (𝑀 ∈ ℕ0 → 𝐹 ⇝ 1) | ||
Theorem | pdivsq 31635 | Condition for a prime dividing a square. (Contributed by Scott Fenton, 8-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
⊢ ((𝑃 ∈ ℙ ∧ 𝑀 ∈ ℤ) → (𝑃 ∥ 𝑀 ↔ 𝑃 ∥ (𝑀↑2))) | ||
Theorem | dvdspw 31636 | Exponentiation law for divisibility. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝐾 ∥ 𝑀 → 𝐾 ∥ (𝑀↑𝑁))) | ||
Theorem | gcd32 31637 | Swap the second and third arguments of a gcd. (Contributed by Scott Fenton, 8-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → ((𝐴 gcd 𝐵) gcd 𝐶) = ((𝐴 gcd 𝐶) gcd 𝐵)) | ||
Theorem | gcdabsorb 31638 | Absorption law for gcd. (Contributed by Scott Fenton, 8-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴 gcd 𝐵) gcd 𝐵) = (𝐴 gcd 𝐵)) | ||
Theorem | brtp 31639 | A condition for a binary relation over an unordered triple. (Contributed by Scott Fenton, 8-Jun-2011.) |
⊢ 𝑋 ∈ V & ⊢ 𝑌 ∈ V ⇒ ⊢ (𝑋{〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉, 〈𝐸, 𝐹〉}𝑌 ↔ ((𝑋 = 𝐴 ∧ 𝑌 = 𝐵) ∨ (𝑋 = 𝐶 ∧ 𝑌 = 𝐷) ∨ (𝑋 = 𝐸 ∧ 𝑌 = 𝐹))) | ||
Theorem | dftr6 31640 | A potential definition of transitivity for sets. (Contributed by Scott Fenton, 18-Mar-2012.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (Tr 𝐴 ↔ 𝐴 ∈ (V ∖ ran (( E ∘ E ) ∖ E ))) | ||
Theorem | coep 31641* | Composition with epsilon. (Contributed by Scott Fenton, 18-Feb-2013.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴( E ∘ 𝑅)𝐵 ↔ ∃𝑥 ∈ 𝐵 𝐴𝑅𝑥) | ||
Theorem | coepr 31642* | Composition with the converse of epsilon. (Contributed by Scott Fenton, 18-Feb-2013.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴(𝑅 ∘ ◡ E )𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑥𝑅𝐵) | ||
Theorem | dffr5 31643 | A quantifier free definition of a well-founded relationship. (Contributed by Scott Fenton, 11-Apr-2011.) |
⊢ (𝑅 Fr 𝐴 ↔ (𝒫 𝐴 ∖ {∅}) ⊆ ran ( E ∖ ( E ∘ ◡𝑅))) | ||
Theorem | dfso2 31644 | Quantifier free definition of a strict order. (Contributed by Scott Fenton, 22-Feb-2013.) |
⊢ (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ (𝐴 × 𝐴) ⊆ (𝑅 ∪ ( I ∪ ◡𝑅)))) | ||
Theorem | dfpo2 31645 | Quantifier free definition of a partial ordering. (Contributed by Scott Fenton, 22-Feb-2013.) |
⊢ (𝑅 Po 𝐴 ↔ ((𝑅 ∩ ( I ↾ 𝐴)) = ∅ ∧ ((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅)) | ||
Theorem | br8 31646* | Substitution for an eight-place predicate. (Contributed by Scott Fenton, 26-Sep-2013.) (Revised by Mario Carneiro, 3-May-2015.) |
⊢ (𝑎 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ (𝑏 = 𝐵 → (𝜓 ↔ 𝜒)) & ⊢ (𝑐 = 𝐶 → (𝜒 ↔ 𝜃)) & ⊢ (𝑑 = 𝐷 → (𝜃 ↔ 𝜏)) & ⊢ (𝑒 = 𝐸 → (𝜏 ↔ 𝜂)) & ⊢ (𝑓 = 𝐹 → (𝜂 ↔ 𝜁)) & ⊢ (𝑔 = 𝐺 → (𝜁 ↔ 𝜎)) & ⊢ (ℎ = 𝐻 → (𝜎 ↔ 𝜌)) & ⊢ (𝑥 = 𝑋 → 𝑃 = 𝑄) & ⊢ 𝑅 = {〈𝑝, 𝑞〉 ∣ ∃𝑥 ∈ 𝑆 ∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 ∃𝑐 ∈ 𝑃 ∃𝑑 ∈ 𝑃 ∃𝑒 ∈ 𝑃 ∃𝑓 ∈ 𝑃 ∃𝑔 ∈ 𝑃 ∃ℎ ∈ 𝑃 (𝑝 = 〈〈𝑎, 𝑏〉, 〈𝑐, 𝑑〉〉 ∧ 𝑞 = 〈〈𝑒, 𝑓〉, 〈𝑔, ℎ〉〉 ∧ 𝜑)} ⇒ ⊢ (((𝑋 ∈ 𝑆 ∧ 𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄) ∧ (𝐶 ∈ 𝑄 ∧ 𝐷 ∈ 𝑄 ∧ 𝐸 ∈ 𝑄) ∧ (𝐹 ∈ 𝑄 ∧ 𝐺 ∈ 𝑄 ∧ 𝐻 ∈ 𝑄)) → (〈〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉〉𝑅〈〈𝐸, 𝐹〉, 〈𝐺, 𝐻〉〉 ↔ 𝜌)) | ||
Theorem | br6 31647* | Substitution for a six-place predicate. (Contributed by Scott Fenton, 4-Oct-2013.) (Revised by Mario Carneiro, 3-May-2015.) |
⊢ (𝑎 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ (𝑏 = 𝐵 → (𝜓 ↔ 𝜒)) & ⊢ (𝑐 = 𝐶 → (𝜒 ↔ 𝜃)) & ⊢ (𝑑 = 𝐷 → (𝜃 ↔ 𝜏)) & ⊢ (𝑒 = 𝐸 → (𝜏 ↔ 𝜂)) & ⊢ (𝑓 = 𝐹 → (𝜂 ↔ 𝜁)) & ⊢ (𝑥 = 𝑋 → 𝑃 = 𝑄) & ⊢ 𝑅 = {〈𝑝, 𝑞〉 ∣ ∃𝑥 ∈ 𝑆 ∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 ∃𝑐 ∈ 𝑃 ∃𝑑 ∈ 𝑃 ∃𝑒 ∈ 𝑃 ∃𝑓 ∈ 𝑃 (𝑝 = 〈𝑎, 〈𝑏, 𝑐〉〉 ∧ 𝑞 = 〈𝑑, 〈𝑒, 𝑓〉〉 ∧ 𝜑)} ⇒ ⊢ ((𝑋 ∈ 𝑆 ∧ (𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄 ∧ 𝐶 ∈ 𝑄) ∧ (𝐷 ∈ 𝑄 ∧ 𝐸 ∈ 𝑄 ∧ 𝐹 ∈ 𝑄)) → (〈𝐴, 〈𝐵, 𝐶〉〉𝑅〈𝐷, 〈𝐸, 𝐹〉〉 ↔ 𝜁)) | ||
Theorem | br4 31648* | Substitution for a four-place predicate. (Contributed by Scott Fenton, 9-Oct-2013.) (Revised by Mario Carneiro, 14-Oct-2013.) |
⊢ (𝑎 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ (𝑏 = 𝐵 → (𝜓 ↔ 𝜒)) & ⊢ (𝑐 = 𝐶 → (𝜒 ↔ 𝜃)) & ⊢ (𝑑 = 𝐷 → (𝜃 ↔ 𝜏)) & ⊢ (𝑥 = 𝑋 → 𝑃 = 𝑄) & ⊢ 𝑅 = {〈𝑝, 𝑞〉 ∣ ∃𝑥 ∈ 𝑆 ∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 ∃𝑐 ∈ 𝑃 ∃𝑑 ∈ 𝑃 (𝑝 = 〈𝑎, 𝑏〉 ∧ 𝑞 = 〈𝑐, 𝑑〉 ∧ 𝜑)} ⇒ ⊢ ((𝑋 ∈ 𝑆 ∧ (𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄) ∧ (𝐶 ∈ 𝑄 ∧ 𝐷 ∈ 𝑄)) → (〈𝐴, 𝐵〉𝑅〈𝐶, 𝐷〉 ↔ 𝜏)) | ||
Theorem | cnvco1 31649 | Another distributive law of converse over class composition. (Contributed by Scott Fenton, 3-May-2014.) |
⊢ ◡(◡𝐴 ∘ 𝐵) = (◡𝐵 ∘ 𝐴) | ||
Theorem | cnvco2 31650 | Another distributive law of converse over class composition. (Contributed by Scott Fenton, 3-May-2014.) |
⊢ ◡(𝐴 ∘ ◡𝐵) = (𝐵 ∘ ◡𝐴) | ||
Theorem | eldm3 31651 | Quantifier-free definition of membership in a domain. (Contributed by Scott Fenton, 21-Jan-2017.) |
⊢ (𝐴 ∈ dom 𝐵 ↔ (𝐵 ↾ {𝐴}) ≠ ∅) | ||
Theorem | elrn3 31652 | Quantifier-free definition of membership in a range. (Contributed by Scott Fenton, 21-Jan-2017.) |
⊢ (𝐴 ∈ ran 𝐵 ↔ (𝐵 ∩ (V × {𝐴})) ≠ ∅) | ||
Theorem | pocnv 31653 | The converse of a partial ordering is still a partial ordering. (Contributed by Scott Fenton, 13-Jun-2018.) |
⊢ (𝑅 Po 𝐴 → ◡𝑅 Po 𝐴) | ||
Theorem | socnv 31654 | The converse of a strict ordering is still a strict ordering. (Contributed by Scott Fenton, 13-Jun-2018.) |
⊢ (𝑅 Or 𝐴 → ◡𝑅 Or 𝐴) | ||
Theorem | 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) | ||
Theorem | 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)) | ||
Theorem | soasym 31657 | Asymmetry law for strict orderings. (Contributed by Scott Fenton, 24-Nov-2021.) |
⊢ ((𝑅 Or 𝐴 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴)) → (𝑋𝑅𝑌 → ¬ 𝑌𝑅𝑋)) | ||
Theorem | 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))) | ||
Theorem | 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}))) | ||
Theorem | 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)) | ||
Theorem | 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) ≠ ∅)) | ||
Theorem | 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))) | ||
Theorem | funpsstri 31663 | A condition for subset trichotomy for functions. (Contributed by Scott Fenton, 19-Apr-2011.) |
⊢ ((Fun 𝐻 ∧ (𝐹 ⊆ 𝐻 ∧ 𝐺 ⊆ 𝐻) ∧ (dom 𝐹 ⊆ dom 𝐺 ∨ dom 𝐺 ⊆ dom 𝐹)) → (𝐹 ⊊ 𝐺 ∨ 𝐹 = 𝐺 ∨ 𝐺 ⊊ 𝐹)) | ||
Theorem | fundmpss 31664 | If a class 𝐹 is a proper subset of a function 𝐺, then dom 𝐹 ⊊ dom 𝐺. (Contributed by Scott Fenton, 20-Apr-2011.) |
⊢ (Fun 𝐺 → (𝐹 ⊊ 𝐺 → dom 𝐹 ⊊ dom 𝐺)) | ||
Theorem | fvresval 31665 | The value of a function at a restriction is either null or the same as the function itself. (Contributed by Scott Fenton, 4-Sep-2011.) |
⊢ (((𝐹 ↾ 𝐵)‘𝐴) = (𝐹‘𝐴) ∨ ((𝐹 ↾ 𝐵)‘𝐴) = ∅) | ||
Theorem | funsseq 31666 | Given two functions with equal domains, equality only requires one direction of the subset relationship. (Contributed by Scott Fenton, 24-Apr-2012.) (Proof shortened by Mario Carneiro, 3-May-2015.) |
⊢ ((Fun 𝐹 ∧ Fun 𝐺 ∧ dom 𝐹 = dom 𝐺) → (𝐹 = 𝐺 ↔ 𝐹 ⊆ 𝐺)) | ||
Theorem | fununiq 31667 | The uniqueness condition of functions. (Contributed by Scott Fenton, 18-Feb-2013.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ 𝐶 ∈ V ⇒ ⊢ (Fun 𝐹 → ((𝐴𝐹𝐵 ∧ 𝐴𝐹𝐶) → 𝐵 = 𝐶)) | ||
Theorem | funbreq 31668 | An equality condition for functions. (Contributed by Scott Fenton, 18-Feb-2013.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ 𝐶 ∈ V ⇒ ⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝐵) → (𝐴𝐹𝐶 ↔ 𝐵 = 𝐶)) | ||
Theorem | fprb 31669* | A condition for functionhood over a pair. (Contributed by Scott Fenton, 16-Sep-2013.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴 ≠ 𝐵 → (𝐹:{𝐴, 𝐵}⟶𝑅 ↔ ∃𝑥 ∈ 𝑅 ∃𝑦 ∈ 𝑅 𝐹 = {〈𝐴, 𝑥〉, 〈𝐵, 𝑦〉})) | ||
Theorem | br1steq 31670 | Uniqueness condition for the binary relation 1st. (Contributed by Scott Fenton, 11-Apr-2014.) (Proof shortened by Mario Carneiro, 3-May-2015.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (〈𝐴, 𝐵〉1st 𝐶 ↔ 𝐶 = 𝐴) | ||
Theorem | br2ndeq 31671 | Uniqueness condition for the binary relation 2nd. (Contributed by Scott Fenton, 11-Apr-2014.) (Proof shortened by Mario Carneiro, 3-May-2015.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (〈𝐴, 𝐵〉2nd 𝐶 ↔ 𝐶 = 𝐵) | ||
Theorem | 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)) | ||
Theorem | 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)) | ||
Theorem | br1steqgOLD 31674 | Obsolete version of br1steqg 31672 as of 9-Feb-2022. (Contributed by Scott Fenton, 2-Jul-2020.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (〈𝐴, 𝐵〉1st 𝐶 ↔ 𝐶 = 𝐴)) | ||
Theorem | br2ndeqgOLD 31675 | Obsolete version of br2ndeqg 31673 as of 9-Feb-2022. (Contributed by Scott Fenton, 2-Jul-2020.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (〈𝐴, 𝐵〉2nd 𝐶 ↔ 𝐶 = 𝐵)) | ||
Theorem | dfdm5 31676 | Definition of domain in terms of 1st and image. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
⊢ dom 𝐴 = ((1st ↾ (V × V)) “ 𝐴) | ||
Theorem | dfrn5 31677 | Definition of range in terms of 2nd and image. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
⊢ ran 𝐴 = ((2nd ↾ (V × V)) “ 𝐴) | ||
Theorem | opelco3 31678 | Alternate way of saying that an ordered pair is in a composition. (Contributed by Scott Fenton, 6-May-2018.) |
⊢ (〈𝐴, 𝐵〉 ∈ (𝐶 ∘ 𝐷) ↔ 𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴}))) | ||
Theorem | elima4 31679 | Quantifier-free expression saying that a class is a member of an image. (Contributed by Scott Fenton, 8-May-2018.) |
⊢ (𝐴 ∈ (𝑅 “ 𝐵) ↔ (𝑅 ∩ (𝐵 × {𝐴})) ≠ ∅) | ||
Theorem | fv1stcnv 31680 | The value of the converse of 1st restricted to a singleton. (Contributed by Scott Fenton, 2-Jul-2020.) |
⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑉) → (◡(1st ↾ (𝐴 × {𝑌}))‘𝑋) = 〈𝑋, 𝑌〉) | ||
Theorem | fv2ndcnv 31681 | The value of the converse of 2nd restricted to a singleton. (Contributed by Scott Fenton, 2-Jul-2020.) |
⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → (◡(2nd ↾ ({𝑋} × 𝐴))‘𝑌) = 〈𝑋, 𝑌〉) | ||
Theorem | imaindm 31682 | The image is unaffected by intersection with the domain. (Contributed by Scott Fenton, 17-Dec-2021.) |
⊢ (𝑅 “ 𝐴) = (𝑅 “ (𝐴 ∩ dom 𝑅)) | ||
Theorem | setinds 31683* | Principle of E induction (set induction). If a property passes from all elements of 𝑥 to 𝑥 itself, then it holds for all 𝑥. (Contributed by Scott Fenton, 10-Mar-2011.) |
⊢ (∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝜑 → 𝜑) ⇒ ⊢ 𝜑 | ||
Theorem | setinds2f 31684* | E induction schema, using implicit substitution. (Contributed by Scott Fenton, 10-Mar-2011.) (Revised by Mario Carneiro, 11-Dec-2016.) |
⊢ Ⅎ𝑥𝜓 & ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) & ⊢ (∀𝑦 ∈ 𝑥 𝜓 → 𝜑) ⇒ ⊢ 𝜑 | ||
Theorem | setinds2 31685* | E induction schema, using implicit substitution. (Contributed by Scott Fenton, 10-Mar-2011.) |
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) & ⊢ (∀𝑦 ∈ 𝑥 𝜓 → 𝜑) ⇒ ⊢ 𝜑 | ||
Theorem | elpotr 31686* | A class of transitive sets is partially ordered by E. (Contributed by Scott Fenton, 15-Oct-2010.) |
⊢ (∀𝑧 ∈ 𝐴 Tr 𝑧 → E Po 𝐴) | ||
Theorem | dford5reg 31687 | Given ax-reg 8497, an ordinal is a transitive class totally ordered by epsilon. (Contributed by Scott Fenton, 28-Jan-2011.) |
⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ E Or 𝐴)) | ||
Theorem | dfon2lem1 31688 | Lemma for dfon2 31697. (Contributed by Scott Fenton, 28-Feb-2011.) |
⊢ Tr ∪ {𝑥 ∣ (𝜑 ∧ Tr 𝑥 ∧ 𝜓)} | ||
Theorem | dfon2lem2 31689* | Lemma for dfon2 31697. (Contributed by Scott Fenton, 28-Feb-2011.) |
⊢ ∪ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝜑 ∧ 𝜓)} ⊆ 𝐴 | ||
Theorem | dfon2lem3 31690* | Lemma for dfon2 31697. All sets satisfying the new definition are transitive and untangled. (Contributed by Scott Fenton, 25-Feb-2011.) |
⊢ (𝐴 ∈ 𝑉 → (∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) → (Tr 𝐴 ∧ ∀𝑧 ∈ 𝐴 ¬ 𝑧 ∈ 𝑧))) | ||
Theorem | dfon2lem4 31691* | Lemma for dfon2 31697. If two sets satisfy the new definition, then one is a subset of the other. (Contributed by Scott Fenton, 25-Feb-2011.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ ((∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) ∧ ∀𝑦((𝑦 ⊊ 𝐵 ∧ Tr 𝑦) → 𝑦 ∈ 𝐵)) → (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴)) | ||
Theorem | dfon2lem5 31692* | Lemma for dfon2 31697. Two sets satisfying the new definition also satisfy trichotomy with respect to ∈. (Contributed by Scott Fenton, 25-Feb-2011.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ ((∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) ∧ ∀𝑦((𝑦 ⊊ 𝐵 ∧ Tr 𝑦) → 𝑦 ∈ 𝐵)) → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴)) | ||
Theorem | dfon2lem6 31693* | Lemma for dfon2 31697. A transitive class of sets satisfying the new definition satisfies the new definition. (Contributed by Scott Fenton, 25-Feb-2011.) |
⊢ ((Tr 𝑆 ∧ ∀𝑥 ∈ 𝑆 ∀𝑧((𝑧 ⊊ 𝑥 ∧ Tr 𝑧) → 𝑧 ∈ 𝑥)) → ∀𝑦((𝑦 ⊊ 𝑆 ∧ Tr 𝑦) → 𝑦 ∈ 𝑆)) | ||
Theorem | dfon2lem7 31694* | Lemma for dfon2 31697. All elements of a new ordinal are new ordinals. (Contributed by Scott Fenton, 25-Feb-2011.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) → (𝐵 ∈ 𝐴 → ∀𝑦((𝑦 ⊊ 𝐵 ∧ Tr 𝑦) → 𝑦 ∈ 𝐵))) | ||
Theorem | dfon2lem8 31695* | Lemma for dfon2 31697. The intersection of a nonempty class 𝐴 of new ordinals is itself a new ordinal and is contained within 𝐴 (Contributed by Scott Fenton, 26-Feb-2011.) |
⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 ∀𝑦((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) → 𝑦 ∈ 𝑥)) → (∀𝑧((𝑧 ⊊ ∩ 𝐴 ∧ Tr 𝑧) → 𝑧 ∈ ∩ 𝐴) ∧ ∩ 𝐴 ∈ 𝐴)) | ||
Theorem | dfon2lem9 31696* | Lemma for dfon2 31697. A class of new ordinals is well-founded by E. (Contributed by Scott Fenton, 3-Mar-2011.) |
⊢ (∀𝑥 ∈ 𝐴 ∀𝑦((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) → 𝑦 ∈ 𝑥) → E Fr 𝐴) | ||
Theorem | dfon2 31697* | On consists of all sets that contain all its transitive proper subsets. This definition comes from J. R. Isbell, "A Definition of Ordinal Numbers," American Mathematical Monthly, vol 67 (1960), pp. 51-52. (Contributed by Scott Fenton, 20-Feb-2011.) |
⊢ On = {𝑥 ∣ ∀𝑦((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) → 𝑦 ∈ 𝑥)} | ||
Theorem | domep 31698 | The domain of the epsilon relation is the universe. (Contributed by Scott Fenton, 27-Oct-2010.) |
⊢ dom E = V | ||
Theorem | rdgprc0 31699 | The value of the recursive definition generator at ∅ when the base value is a proper class. (Contributed by Scott Fenton, 26-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
⊢ (¬ 𝐼 ∈ V → (rec(𝐹, 𝐼)‘∅) = ∅) | ||
Theorem | rdgprc 31700 | The value of the recursive definition generator when 𝐼 is a proper class. (Contributed by Scott Fenton, 26-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
⊢ (¬ 𝐼 ∈ V → rec(𝐹, 𝐼) = rec(𝐹, ∅)) |
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