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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | 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) | ||
Theorem | 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)) | ||
Theorem | 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))) | ||
Theorem | 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) | ||
Theorem | 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) | ||
Theorem | 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)) | ||
Theorem | 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)) | ||
Theorem | 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))) | ||
Theorem | 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) | ||
Theorem | 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))) | ||
Theorem | 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))) | ||
Theorem | 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))) | ||
Syntax | cvts 30713 | The Vinogradov trigonometric sums. |
class vts | ||
Definition | 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)))))) | ||
Theorem | 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))))) | ||
Theorem | 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) | ||
Theorem | 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))))) | ||
Theorem | 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) | ||
Theorem | 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 | ||
Theorem | 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) | ||
Theorem | 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) | ||
Axiom | 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)) | ||
Axiom | 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) | ||
Axiom | 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)) | ||
Theorem | 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)) | ||
Theorem | 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))) | ||
Theorem | 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)) | ||
Theorem | 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))) | ||
Theorem | 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)) | ||
Theorem | 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) | ||
Theorem | 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)))))) | ||
Theorem | 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))))) | ||
Theorem | 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) | ||
Theorem | 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)))) | ||
Theorem | 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)))))) | ||
Theorem | 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))) | ||
Theorem | 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) | ||
Theorem | 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)))))) | ||
Theorem | 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))) | ||
Theorem | 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))) | ||
Theorem | 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)) | ||
This definition has been superseded by DimTarskiG≥ and is no longer needed in the main part of set.mm. It is only kept here for reference. | ||
Syntax | cstrkg2d 30742 | Extends class notation with the class of geometries fulfilling the planarity axioms. |
class TarskiG2D | ||
Definition | df-trkg2d 30743* | Define the class of geometries fulfilling the lower dimension axiom, Axiom A8 of [Schwabhauser] p. 12, and the upper dimension axiom, Axiom A9 of [Schwabhauser] p. 13, for dimension 2. (Contributed by Thierry Arnoux, 14-Mar-2019.) (New usage is discouraged.) |
⊢ TarskiG2D = {𝑓 ∣ [(Base‘𝑓) / 𝑝][(dist‘𝑓) / 𝑑][(Itv‘𝑓) / 𝑖](∃𝑥 ∈ 𝑝 ∃𝑦 ∈ 𝑝 ∃𝑧 ∈ 𝑝 ¬ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧)) ∧ ∀𝑥 ∈ 𝑝 ∀𝑦 ∈ 𝑝 ∀𝑧 ∈ 𝑝 ∀𝑢 ∈ 𝑝 ∀𝑣 ∈ 𝑝 ((((𝑥𝑑𝑢) = (𝑥𝑑𝑣) ∧ (𝑦𝑑𝑢) = (𝑦𝑑𝑣) ∧ (𝑧𝑑𝑢) = (𝑧𝑑𝑣)) ∧ 𝑢 ≠ 𝑣) → (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))))} | ||
Theorem | istrkg2d 30744* | Property of fulfilling dimension 2 axiom. (Contributed by Thierry Arnoux, 29-May-2019.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) ⇒ ⊢ (𝐺 ∈ TarskiG2D ↔ (𝐺 ∈ V ∧ (∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 ∃𝑧 ∈ 𝑃 ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)) ∧ ∀𝑥 ∈ 𝑃 ∀𝑦 ∈ 𝑃 ∀𝑧 ∈ 𝑃 ∀𝑢 ∈ 𝑃 ∀𝑣 ∈ 𝑃 ((((𝑥 − 𝑢) = (𝑥 − 𝑣) ∧ (𝑦 − 𝑢) = (𝑦 − 𝑣) ∧ (𝑧 − 𝑢) = (𝑧 − 𝑣)) ∧ 𝑢 ≠ 𝑣) → (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)))))) | ||
Theorem | axtglowdim2OLD 30745* | Lower dimension axiom for dimension 2, Axiom A8 of [Schwabhauser] p. 13. There exist 3 non-colinear points. (Contributed by Thierry Arnoux, 29-May-2019.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG2D) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 ∃𝑧 ∈ 𝑃 ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))) | ||
Theorem | axtgupdim2OLD 30746 | Upper dimension axiom for dimension 2, Axiom A9 of [Schwabhauser] p. 13. (Contributed by Thierry Arnoux, 29-May-2019.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑌 ∈ 𝑃) & ⊢ (𝜑 → 𝑍 ∈ 𝑃) & ⊢ (𝜑 → 𝑈 ∈ 𝑃) & ⊢ (𝜑 → 𝑉 ∈ 𝑃) & ⊢ (𝜑 → 𝑈 ≠ 𝑉) & ⊢ (𝜑 → (𝑋 − 𝑈) = (𝑋 − 𝑉)) & ⊢ (𝜑 → (𝑌 − 𝑈) = (𝑌 − 𝑉)) & ⊢ (𝜑 → (𝑍 − 𝑈) = (𝑍 − 𝑉)) & ⊢ (𝜑 → 𝐺 ∈ TarskiG2D) ⇒ ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))) | ||
Syntax | cafs 30747 | Declare the syntax for the outer five segment configuration. |
class AFS | ||
Definition | df-afs 30748* | The outer five segment configuration is an abbreviation for the conditions of the Five Segment Axiom (axtg5seg 25364). See df-ofs 32090. Definition 2.10 of [Schwabhauser] p. 28. (Contributed by Scott Fenton, 21-Sep-2013.) (Modified by Thierry Arnoux, 15-Mar-2019.) |
⊢ AFS = (𝑔 ∈ TarskiG ↦ {〈𝑒, 𝑓〉 ∣ [(Base‘𝑔) / 𝑝][(dist‘𝑔) / ℎ][(Itv‘𝑔) / 𝑖]∃𝑎 ∈ 𝑝 ∃𝑏 ∈ 𝑝 ∃𝑐 ∈ 𝑝 ∃𝑑 ∈ 𝑝 ∃𝑥 ∈ 𝑝 ∃𝑦 ∈ 𝑝 ∃𝑧 ∈ 𝑝 ∃𝑤 ∈ 𝑝 (𝑒 = 〈〈𝑎, 𝑏〉, 〈𝑐, 𝑑〉〉 ∧ 𝑓 = 〈〈𝑥, 𝑦〉, 〈𝑧, 𝑤〉〉 ∧ ((𝑏 ∈ (𝑎𝑖𝑐) ∧ 𝑦 ∈ (𝑥𝑖𝑧)) ∧ ((𝑎ℎ𝑏) = (𝑥ℎ𝑦) ∧ (𝑏ℎ𝑐) = (𝑦ℎ𝑧)) ∧ ((𝑎ℎ𝑑) = (𝑥ℎ𝑤) ∧ (𝑏ℎ𝑑) = (𝑦ℎ𝑤))))}) | ||
Theorem | afsval 30749* | Value of the AFS relation for a given geometry structure. (Contributed by Thierry Arnoux, 20-Mar-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) ⇒ ⊢ (𝜑 → (AFS‘𝐺) = {〈𝑒, 𝑓〉 ∣ ∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 ∃𝑐 ∈ 𝑃 ∃𝑑 ∈ 𝑃 ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 ∃𝑧 ∈ 𝑃 ∃𝑤 ∈ 𝑃 (𝑒 = 〈〈𝑎, 𝑏〉, 〈𝑐, 𝑑〉〉 ∧ 𝑓 = 〈〈𝑥, 𝑦〉, 〈𝑧, 𝑤〉〉 ∧ ((𝑏 ∈ (𝑎𝐼𝑐) ∧ 𝑦 ∈ (𝑥𝐼𝑧)) ∧ ((𝑎 − 𝑏) = (𝑥 − 𝑦) ∧ (𝑏 − 𝑐) = (𝑦 − 𝑧)) ∧ ((𝑎 − 𝑑) = (𝑥 − 𝑤) ∧ (𝑏 − 𝑑) = (𝑦 − 𝑤))))}) | ||
Theorem | brafs 30750 | Binary relation form of the outer five segment predicate. (Contributed by Scott Fenton, 21-Sep-2013.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ 𝑂 = (AFS‘𝐺) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → 𝑋 ∈ 𝑃) & ⊢ (𝜑 → 𝑌 ∈ 𝑃) & ⊢ (𝜑 → 𝑍 ∈ 𝑃) & ⊢ (𝜑 → 𝑊 ∈ 𝑃) ⇒ ⊢ (𝜑 → (〈〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉〉𝑂〈〈𝑋, 𝑌〉, 〈𝑍, 𝑊〉〉 ↔ ((𝐵 ∈ (𝐴𝐼𝐶) ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ ((𝐴 − 𝐵) = (𝑋 − 𝑌) ∧ (𝐵 − 𝐶) = (𝑌 − 𝑍)) ∧ ((𝐴 − 𝐷) = (𝑋 − 𝑊) ∧ (𝐵 − 𝐷) = (𝑌 − 𝑊))))) | ||
Theorem | tg5segofs 30751 | Rephrase axtg5seg 25364 using the outer five segment predicate. Theorem 2.10 of [Schwabhauser] p. 28. (Contributed by Thierry Arnoux, 23-Mar-2019.) |
⊢ 𝑃 = (Base‘𝐺) & ⊢ − = (dist‘𝐺) & ⊢ 𝐼 = (Itv‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ TarskiG) & ⊢ (𝜑 → 𝐴 ∈ 𝑃) & ⊢ (𝜑 → 𝐵 ∈ 𝑃) & ⊢ (𝜑 → 𝐶 ∈ 𝑃) & ⊢ (𝜑 → 𝐷 ∈ 𝑃) & ⊢ (𝜑 → 𝐸 ∈ 𝑃) & ⊢ (𝜑 → 𝐹 ∈ 𝑃) & ⊢ 𝑂 = (AFS‘𝐺) & ⊢ (𝜑 → 𝐻 ∈ 𝑃) & ⊢ (𝜑 → 𝐼 ∈ 𝑃) & ⊢ (𝜑 → 〈〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉〉𝑂〈〈𝐸, 𝐹〉, 〈𝐻, 𝐼〉〉) & ⊢ (𝜑 → 𝐴 ≠ 𝐵) ⇒ ⊢ (𝜑 → (𝐶 − 𝐷) = (𝐻 − 𝐼)) | ||
Note: On 4-Sep-2016 and after, 745 unused theorems were deleted from this mathbox, and 359 theorems used only once or twice were merged into their referencing theorems. The originals can be recovered from set.mm versions prior to this date. | ||
Syntax | w-bnj17 30752 | Extend wff notation with the 4-way conjunction. (New usage is discouraged.) |
wff (𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) | ||
Definition | df-bnj17 30753 | Define the 4-way conjunction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ ((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃)) | ||
Syntax | c-bnj14 30754 | Extend class notation with the function giving: the class of all elements of 𝐴 that are "smaller" than 𝑋 according to 𝑅. (New usage is discouraged.) |
class pred(𝑋, 𝐴, 𝑅) | ||
Definition | df-bnj14 30755* | Define the function giving: the class of all elements of 𝐴 that are "smaller" than 𝑋 according to 𝑅. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ pred(𝑋, 𝐴, 𝑅) = {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋} | ||
Syntax | w-bnj13 30756 | Extend wff notation with the following predicate: 𝑅 is set-like on 𝐴. (New usage is discouraged.) |
wff 𝑅 Se 𝐴 | ||
Definition | df-bnj13 30757* | Define the following predicate: 𝑅 is set-like on 𝐴. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ (𝑅 Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 pred(𝑥, 𝐴, 𝑅) ∈ V) | ||
Syntax | w-bnj15 30758 | Extend wff notation with the following predicate: 𝑅 is both well-founded and set-like on 𝐴. (New usage is discouraged.) |
wff 𝑅 FrSe 𝐴 | ||
Definition | df-bnj15 30759 | Define the following predicate: 𝑅 is both well-founded and set-like on 𝐴. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ (𝑅 FrSe 𝐴 ↔ (𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴)) | ||
Syntax | c-bnj18 30760 | Extend class notation with the function giving: the transitive closure of 𝑋 in 𝐴 by 𝑅. (New usage is discouraged.) |
class trCl(𝑋, 𝐴, 𝑅) | ||
Definition | df-bnj18 30761* | Define the function giving: the transitive closure of 𝑋 in 𝐴 by 𝑅. This definition has been designed for facilitating verification that it is eliminable and that the $d restrictions are sound and complete. For a more readable definition see bnj882 30996. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ trCl(𝑋, 𝐴, 𝑅) = ∪ 𝑓 ∈ {𝑓 ∣ ∃𝑛 ∈ (ω ∖ {∅})(𝑓 Fn 𝑛 ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))}∪ 𝑖 ∈ dom 𝑓(𝑓‘𝑖) | ||
Syntax | w-bnj19 30762 | Extend wff notation with the following predicate: 𝐵 is transitive for 𝐴 and 𝑅. (New usage is discouraged.) |
wff TrFo(𝐵, 𝐴, 𝑅) | ||
Definition | df-bnj19 30763* | Define the following predicate: 𝐵 is transitive for 𝐴 and 𝑅. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ ( TrFo(𝐵, 𝐴, 𝑅) ↔ ∀𝑥 ∈ 𝐵 pred(𝑥, 𝐴, 𝑅) ⊆ 𝐵) | ||
Theorem | bnj170 30764 | ∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) (New usage is discouraged.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜓 ∧ 𝜒) ∧ 𝜑)) | ||
Theorem | bnj240 30765 | ∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ (𝜓 → 𝜓′) & ⊢ (𝜒 → 𝜒′) ⇒ ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜓′ ∧ 𝜒′)) | ||
Theorem | bnj248 30766 | ∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ (((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃)) | ||
Theorem | bnj250 30767 | ∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ (𝜑 ∧ ((𝜓 ∧ 𝜒) ∧ 𝜃))) | ||
Theorem | bnj251 30768 | ∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ (𝜑 ∧ (𝜓 ∧ (𝜒 ∧ 𝜃)))) | ||
Theorem | bnj252 30769 | ∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ (𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜃))) | ||
Theorem | bnj253 30770 | ∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ ((𝜑 ∧ 𝜓) ∧ 𝜒 ∧ 𝜃)) | ||
Theorem | bnj255 30771 | ∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ (𝜑 ∧ 𝜓 ∧ (𝜒 ∧ 𝜃))) | ||
Theorem | bnj256 30772 | ∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ ((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃))) | ||
Theorem | bnj257 30773 | ∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ (𝜑 ∧ 𝜓 ∧ 𝜃 ∧ 𝜒)) | ||
Theorem | bnj258 30774 | ∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ ((𝜑 ∧ 𝜓 ∧ 𝜃) ∧ 𝜒)) | ||
Theorem | bnj268 30775 | ∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ (𝜑 ∧ 𝜒 ∧ 𝜓 ∧ 𝜃)) | ||
Theorem | bnj290 30776 | ∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ (𝜑 ∧ 𝜒 ∧ 𝜃 ∧ 𝜓)) | ||
Theorem | bnj291 30777 | ∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ ((𝜑 ∧ 𝜒 ∧ 𝜃) ∧ 𝜓)) | ||
Theorem | bnj312 30778 | ∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ (𝜓 ∧ 𝜑 ∧ 𝜒 ∧ 𝜃)) | ||
Theorem | bnj334 30779 | ∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) (New usage is discouraged.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ (𝜒 ∧ 𝜑 ∧ 𝜓 ∧ 𝜃)) | ||
Theorem | bnj345 30780 | ∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) (New usage is discouraged.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ (𝜃 ∧ 𝜑 ∧ 𝜓 ∧ 𝜒)) | ||
Theorem | bnj422 30781 | ∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) (New usage is discouraged.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ (𝜒 ∧ 𝜃 ∧ 𝜑 ∧ 𝜓)) | ||
Theorem | bnj432 30782 | ∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ ((𝜒 ∧ 𝜃) ∧ (𝜑 ∧ 𝜓))) | ||
Theorem | bnj446 30783 | ∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ ((𝜓 ∧ 𝜒 ∧ 𝜃) ∧ 𝜑)) | ||
Theorem | bnj23 30784* | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Mario Carneiro, 22-Dec-2016.) (New usage is discouraged.) |
⊢ 𝐵 = {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ⇒ ⊢ (∀𝑧 ∈ 𝐵 ¬ 𝑧𝑅𝑦 → ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑦 → [𝑤 / 𝑥]𝜑)) | ||
Theorem | bnj31 30785 | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓) & ⊢ (𝜓 → 𝜒) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜒) | ||
Theorem | bnj62 30786* | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ ([𝑧 / 𝑥]𝑥 Fn 𝐴 ↔ 𝑧 Fn 𝐴) | ||
Theorem | bnj89 30787* | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ 𝑍 ∈ V ⇒ ⊢ ([𝑍 / 𝑦]∃!𝑥𝜑 ↔ ∃!𝑥[𝑍 / 𝑦]𝜑) | ||
Theorem | bnj90 30788* | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Mario Carneiro, 22-Dec-2016.) (New usage is discouraged.) |
⊢ 𝑌 ∈ V ⇒ ⊢ ([𝑌 / 𝑥]𝑧 Fn 𝑥 ↔ 𝑧 Fn 𝑌) | ||
Theorem | bnj101 30789 | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ ∃𝑥𝜑 & ⊢ (𝜑 → 𝜓) ⇒ ⊢ ∃𝑥𝜓 | ||
Theorem | bnj105 30790 | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ 1𝑜 ∈ V | ||
Theorem | bnj115 30791 | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ (𝜂 ↔ ∀𝑛 ∈ 𝐷 (𝜏 → 𝜃)) ⇒ ⊢ (𝜂 ↔ ∀𝑛((𝑛 ∈ 𝐷 ∧ 𝜏) → 𝜃)) | ||
Theorem | bnj132 30792* | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) (New usage is discouraged.) |
⊢ (𝜑 ↔ ∃𝑥(𝜓 → 𝜒)) ⇒ ⊢ (𝜑 ↔ (𝜓 → ∃𝑥𝜒)) | ||
Theorem | bnj133 30793 | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ (𝜑 ↔ ∃𝑥𝜓) & ⊢ (𝜒 ↔ 𝜓) ⇒ ⊢ (𝜑 ↔ ∃𝑥𝜒) | ||
Theorem | bnj142OLD 30794 | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Mario Carneiro, 22-Dec-2016.) (New usage is discouraged.) (Proof modification is discouraged.) Obsolete as of 29-Dec-2018. This is now incorporated into the proof of fnsnb 6432. |
⊢ (𝐹 Fn {𝐴} → (𝑢 ∈ 𝐹 → 𝑢 = 〈𝐴, (𝐹‘𝐴)〉)) | ||
Theorem | bnj145OLD 30795 | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) (Proof modification is discouraged.) Obsolete as of 29-Dec-2018. This is now incorporated into the proof of fnsnb 6432. |
⊢ 𝐴 ∈ V & ⊢ (𝐹‘𝐴) ∈ V ⇒ ⊢ (𝐹 Fn {𝐴} → 𝐹 = {〈𝐴, (𝐹‘𝐴)〉}) | ||
Theorem | bnj156 30796 | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ (𝜁0 ↔ (𝑓 Fn 1𝑜 ∧ 𝜑′ ∧ 𝜓′)) & ⊢ (𝜁1 ↔ [𝑔 / 𝑓]𝜁0) & ⊢ (𝜑1 ↔ [𝑔 / 𝑓]𝜑′) & ⊢ (𝜓1 ↔ [𝑔 / 𝑓]𝜓′) ⇒ ⊢ (𝜁1 ↔ (𝑔 Fn 1𝑜 ∧ 𝜑1 ∧ 𝜓1)) | ||
Theorem | bnj158 30797* | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ 𝐷 = (ω ∖ {∅}) ⇒ ⊢ (𝑚 ∈ 𝐷 → ∃𝑝 ∈ ω 𝑚 = suc 𝑝) | ||
Theorem | bnj168 30798* | First-order logic and set theory. Revised to remove dependence on ax-reg 8497. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Revised by NM, 21-Dec-2016.) (New usage is discouraged.) |
⊢ 𝐷 = (ω ∖ {∅}) ⇒ ⊢ ((𝑛 ≠ 1𝑜 ∧ 𝑛 ∈ 𝐷) → ∃𝑚 ∈ 𝐷 𝑛 = suc 𝑚) | ||
Theorem | bnj206 30799 | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ (𝜑′ ↔ [𝑀 / 𝑛]𝜑) & ⊢ (𝜓′ ↔ [𝑀 / 𝑛]𝜓) & ⊢ (𝜒′ ↔ [𝑀 / 𝑛]𝜒) & ⊢ 𝑀 ∈ V ⇒ ⊢ ([𝑀 / 𝑛](𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (𝜑′ ∧ 𝜓′ ∧ 𝜒′)) | ||
Theorem | bnj216 30800 | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴 = suc 𝐵 → 𝐵 ∈ 𝐴) |
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