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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | breprexplema 31901* | Lemma for breprexp 31904 (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 31902 | Lemma for breprexp 31904 (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 31903* | Lemma for breprexp 31904 (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 31904* | 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 31905 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 31905* | 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 31906 | The Vinogradov trigonometric sums. |
class vts | ||
Definition | df-vts 31907* | 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 31908* | 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 31909 | 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 31910* | 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 31911* | 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 31912* | 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 31913* | 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 31914* | 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 31915* | 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 31916 | Theorem 12. of [RosserSchoenfeld] p. 71. Theorem chpo1ubb 26057 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 31917 | Theorem 13. of [RosserSchoenfeld] p. 71. Theorem chpchtlim 26055 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 31918* | 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 31919* | 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 31920* | A deduction version of ax-hgt749 31915. (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 31921 | 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 31922 | Lemma for tgoldbachgtd 31933. (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 31923 | 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 31924* | 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 31925* | 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 31926* | 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 31927* | 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 31928* | 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 31929* | 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 31930* | Lemma for tgoldbachgtd 31933. (Contributed by Thierry Arnoux, 15-Dec-2021.) |
⊢ 𝑂 = {𝑧 ∈ ℤ ∣ ¬ 2 ∥ 𝑧} & ⊢ (𝜑 → 𝑁 ∈ 𝑂) & ⊢ (𝜑 → (;10↑;27) ≤ 𝑁) ⇒ ⊢ (𝜑 → 𝑁 ∈ ℕ) | ||
Theorem | tgoldbachgtde 31931* | Lemma for tgoldbachgtd 31933. (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 31932* | Lemma for tgoldbachgtd 31933. (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 31933* | 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 31934* | 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 31935 | Extends class notation with the class of geometries fulfilling the planarity axioms. |
class TarskiG2D | ||
Definition | df-trkg2d 31936* | 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 31937* | 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 | axtglowdim2ALTV 31938* | Alternate version of axtglowdim2 26256. (Contributed by Thierry Arnoux, 29-May-2019.) (New usage 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 | axtgupdim2ALTV 31939 | Alternate version of axtgupdim2 26257. (Contributed by Thierry Arnoux, 29-May-2019.) (New usage 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 31940 | Declare the syntax for the outer five segment configuration. |
class AFS | ||
Definition | df-afs 31941* | The outer five segment configuration is an abbreviation for the conditions of the Five Segment Axiom (axtg5seg 26251). See df-ofs 33444. Definition 2.10 of [Schwabhauser] p. 28. (Contributed by Scott Fenton, 21-Sep-2013.) (Revised 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 31942* | 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 31943 | 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 31944 | Rephrase axtg5seg 26251 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)) | ||
Syntax | clpad 31945 | Extend class notation with the leftpad function. |
class leftpad | ||
Definition | df-lpad 31946* | Define the leftpad function. (Contributed by Thierry Arnoux, 7-Aug-2023.) |
⊢ leftpad = (𝑐 ∈ V, 𝑤 ∈ V ↦ (𝑙 ∈ ℕ0 ↦ (((0..^(𝑙 − (♯‘𝑤))) × {𝑐}) ++ 𝑤))) | ||
Theorem | lpadval 31947 | Value of the leftpad function. (Contributed by Thierry Arnoux, 7-Aug-2023.) |
⊢ (𝜑 → 𝐿 ∈ ℕ0) & ⊢ (𝜑 → 𝑊 ∈ Word 𝑆) & ⊢ (𝜑 → 𝐶 ∈ 𝑆) ⇒ ⊢ (𝜑 → ((𝐶 leftpad 𝑊)‘𝐿) = (((0..^(𝐿 − (♯‘𝑊))) × {𝐶}) ++ 𝑊)) | ||
Theorem | lpadlem1 31948 | Lemma for the leftpad theorems. (Contributed by Thierry Arnoux, 7-Aug-2023.) |
⊢ (𝜑 → 𝐶 ∈ 𝑆) ⇒ ⊢ (𝜑 → ((0..^(𝐿 − (♯‘𝑊))) × {𝐶}) ∈ Word 𝑆) | ||
Theorem | lpadlem3 31949 | Lemma for lpadlen1 31950 (Contributed by Thierry Arnoux, 7-Aug-2023.) |
⊢ (𝜑 → 𝐿 ∈ ℕ0) & ⊢ (𝜑 → 𝑊 ∈ Word 𝑆) & ⊢ (𝜑 → 𝐶 ∈ 𝑆) & ⊢ (𝜑 → 𝐿 ≤ (♯‘𝑊)) ⇒ ⊢ (𝜑 → ((0..^(𝐿 − (♯‘𝑊))) × {𝐶}) = ∅) | ||
Theorem | lpadlen1 31950 | Length of a left-padded word, in the case the length of the given word 𝑊 is at least the desired length. (Contributed by Thierry Arnoux, 7-Aug-2023.) |
⊢ (𝜑 → 𝐿 ∈ ℕ0) & ⊢ (𝜑 → 𝑊 ∈ Word 𝑆) & ⊢ (𝜑 → 𝐶 ∈ 𝑆) & ⊢ (𝜑 → 𝐿 ≤ (♯‘𝑊)) ⇒ ⊢ (𝜑 → (♯‘((𝐶 leftpad 𝑊)‘𝐿)) = (♯‘𝑊)) | ||
Theorem | lpadlem2 31951 | Lemma for the leftpad theorems. (Contributed by Thierry Arnoux, 7-Aug-2023.) |
⊢ (𝜑 → 𝐿 ∈ ℕ0) & ⊢ (𝜑 → 𝑊 ∈ Word 𝑆) & ⊢ (𝜑 → 𝐶 ∈ 𝑆) & ⊢ (𝜑 → (♯‘𝑊) ≤ 𝐿) ⇒ ⊢ (𝜑 → (♯‘((0..^(𝐿 − (♯‘𝑊))) × {𝐶})) = (𝐿 − (♯‘𝑊))) | ||
Theorem | lpadlen2 31952 | Length of a left-padded word, in the case the given word 𝑊 is shorter than the desired length. (Contributed by Thierry Arnoux, 7-Aug-2023.) |
⊢ (𝜑 → 𝐿 ∈ ℕ0) & ⊢ (𝜑 → 𝑊 ∈ Word 𝑆) & ⊢ (𝜑 → 𝐶 ∈ 𝑆) & ⊢ (𝜑 → (♯‘𝑊) ≤ 𝐿) ⇒ ⊢ (𝜑 → (♯‘((𝐶 leftpad 𝑊)‘𝐿)) = 𝐿) | ||
Theorem | lpadmax 31953 | Length of a left-padded word, in the general case, expressed with an if statement. (Contributed by Thierry Arnoux, 7-Aug-2023.) |
⊢ (𝜑 → 𝐿 ∈ ℕ0) & ⊢ (𝜑 → 𝑊 ∈ Word 𝑆) & ⊢ (𝜑 → 𝐶 ∈ 𝑆) ⇒ ⊢ (𝜑 → (♯‘((𝐶 leftpad 𝑊)‘𝐿)) = if(𝐿 ≤ (♯‘𝑊), (♯‘𝑊), 𝐿)) | ||
Theorem | lpadleft 31954 | The contents of prefix of a left-padded word is always the letter 𝐶. (Contributed by Thierry Arnoux, 7-Aug-2023.) |
⊢ (𝜑 → 𝐿 ∈ ℕ0) & ⊢ (𝜑 → 𝑊 ∈ Word 𝑆) & ⊢ (𝜑 → 𝐶 ∈ 𝑆) & ⊢ (𝜑 → 𝑁 ∈ (0..^(𝐿 − (♯‘𝑊)))) ⇒ ⊢ (𝜑 → (((𝐶 leftpad 𝑊)‘𝐿)‘𝑁) = 𝐶) | ||
Theorem | lpadright 31955 | The suffix of a left-padded word the original word 𝑊. (Contributed by Thierry Arnoux, 7-Aug-2023.) |
⊢ (𝜑 → 𝐿 ∈ ℕ0) & ⊢ (𝜑 → 𝑊 ∈ Word 𝑆) & ⊢ (𝜑 → 𝐶 ∈ 𝑆) & ⊢ (𝜑 → 𝑀 = if(𝐿 ≤ (♯‘𝑊), 0, (𝐿 − (♯‘𝑊)))) & ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝑊))) ⇒ ⊢ (𝜑 → (((𝐶 leftpad 𝑊)‘𝐿)‘(𝑁 + 𝑀)) = (𝑊‘𝑁)) | ||
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 31956 | Extend wff notation with the 4-way conjunction. (New usage is discouraged.) |
wff (𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) | ||
Definition | df-bnj17 31957 | 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 31958 | 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 31959* | 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 31960 | Extend wff notation with the following predicate: 𝑅 is set-like on 𝐴. (New usage is discouraged.) |
wff 𝑅 Se 𝐴 | ||
Definition | df-bnj13 31961* | 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 31962 | 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 31963 | 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 31964 | 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 31965* | 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 32198. (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 31966 | 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 31967* | 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 31968 | ∧-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 31969 | ∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ (𝜓 → 𝜓′) & ⊢ (𝜒 → 𝜒′) ⇒ ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜓′ ∧ 𝜒′)) | ||
Theorem | bnj248 31970 | ∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ (((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃)) | ||
Theorem | bnj250 31971 | ∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ (𝜑 ∧ ((𝜓 ∧ 𝜒) ∧ 𝜃))) | ||
Theorem | bnj251 31972 | ∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ (𝜑 ∧ (𝜓 ∧ (𝜒 ∧ 𝜃)))) | ||
Theorem | bnj252 31973 | ∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ (𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜃))) | ||
Theorem | bnj253 31974 | ∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ ((𝜑 ∧ 𝜓) ∧ 𝜒 ∧ 𝜃)) | ||
Theorem | bnj255 31975 | ∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ (𝜑 ∧ 𝜓 ∧ (𝜒 ∧ 𝜃))) | ||
Theorem | bnj256 31976 | ∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ ((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃))) | ||
Theorem | bnj257 31977 | ∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ (𝜑 ∧ 𝜓 ∧ 𝜃 ∧ 𝜒)) | ||
Theorem | bnj258 31978 | ∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ ((𝜑 ∧ 𝜓 ∧ 𝜃) ∧ 𝜒)) | ||
Theorem | bnj268 31979 | ∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ (𝜑 ∧ 𝜒 ∧ 𝜓 ∧ 𝜃)) | ||
Theorem | bnj290 31980 | ∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ (𝜑 ∧ 𝜒 ∧ 𝜃 ∧ 𝜓)) | ||
Theorem | bnj291 31981 | ∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ ((𝜑 ∧ 𝜒 ∧ 𝜃) ∧ 𝜓)) | ||
Theorem | bnj312 31982 | ∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ (𝜓 ∧ 𝜑 ∧ 𝜒 ∧ 𝜃)) | ||
Theorem | bnj334 31983 | ∧-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 31984 | ∧-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 31985 | ∧-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 31986 | ∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ ((𝜒 ∧ 𝜃) ∧ (𝜑 ∧ 𝜓))) | ||
Theorem | bnj446 31987 | ∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ ((𝜓 ∧ 𝜒 ∧ 𝜃) ∧ 𝜑)) | ||
Theorem | bnj23 31988* | 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 31989 | 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 31990* | 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 31991* | 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 31992* | 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 31993 | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ ∃𝑥𝜑 & ⊢ (𝜑 → 𝜓) ⇒ ⊢ ∃𝑥𝜓 | ||
Theorem | bnj105 31994 | First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ 1o ∈ V | ||
Theorem | bnj115 31995 | 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 31996* | 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 31997 | 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 | bnj156 31998 | 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 31999* | 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 32000* | First-order logic and set theory. Revised to remove dependence on ax-reg 9056. (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) |
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