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Type | Label | Description |
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Statement | ||
Theorem | fdvneggt 34101* | 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)) | ||
Theorem | fdvposle 34102* | 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)) | ||
Theorem | fdvnegge 34103* | Functions with a nonpositive 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)) | ||
Theorem | prodfzo03 34104* | 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))) | ||
Theorem | actfunsnf1o 34105* | 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) | ||
Theorem | actfunsnrndisj 34106* | 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) | ||
Theorem | itgexpif 34107* | 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)) | ||
Theorem | fsum2dsub 34108* | Lemma for breprexp 34134- 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) | ||
Syntax | crepr 34109 | Representations of a number as a sum of nonnegative integers. |
class repr | ||
Definition | df-repr 34110* | 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})) | ||
Theorem | reprval 34111* | 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}) | ||
Theorem | repr0 34112 | 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, {∅}, ∅)) | ||
Theorem | reprf 34113 | 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) | ||
Theorem | reprsum 34114* | 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) | ||
Theorem | reprle 34115 | 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) | ||
Theorem | reprsuc 34116* | 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) | ||
Theorem | reprfi 34117 | 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) | ||
Theorem | reprss 34118 | 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)) | ||
Theorem | reprinrn 34119* | 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))) | ||
Theorem | reprlt 34120 | 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) = ∅) | ||
Theorem | hashreprin 34121* | 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))) | ||
Theorem | reprgt 34122 | 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) = ∅) | ||
Theorem | reprinfz1 34123 | 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)) | ||
Theorem | reprfi2 34124 | Corollary of reprinfz1 34123. (Contributed by Thierry Arnoux, 15-Dec-2021.) |
⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → 𝑆 ∈ ℕ0) & ⊢ (𝜑 → 𝐴 ⊆ ℕ) ⇒ ⊢ (𝜑 → (𝐴(repr‘𝑆)𝑁) ∈ Fin) | ||
Theorem | reprfz1 34125 | Corollary of reprinfz1 34123. (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 34126* | 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 34127* | 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 34128* | 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 34129* | 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 34130* | 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 34131* | Lemma for breprexp 34134 (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 34132 | Lemma for breprexp 34134 (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 34133* | Lemma for breprexp 34134 (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 34134* | 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 34135 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 34135* | 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 34136 | The Vinogradov trigonometric sums. |
class vts | ||
Definition | df-vts 34137* | 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 34138* | 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 34139 | 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 34140* | 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 34141* | 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 34142* | 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 34143* | 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 34144* | 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 34145* | 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 34146 | Theorem 12. of [RosserSchoenfeld] p. 71. Theorem chpo1ubb 27330 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 34147 | Theorem 13. of [RosserSchoenfeld] p. 71. Theorem chpchtlim 27328 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 34148* | 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 34149* | 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 34150* | A deduction version of ax-hgt749 34145. (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 34151 | 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 34152 | Lemma for tgoldbachgtd 34163. (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 34153 | 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 34154* | 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 34155* | 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 34156* | 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 34157* | 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 34158* | 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 34159* | 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 34160* | Lemma for tgoldbachgtd 34163. (Contributed by Thierry Arnoux, 15-Dec-2021.) |
⊢ 𝑂 = {𝑧 ∈ ℤ ∣ ¬ 2 ∥ 𝑧} & ⊢ (𝜑 → 𝑁 ∈ 𝑂) & ⊢ (𝜑 → (;10↑;27) ≤ 𝑁) ⇒ ⊢ (𝜑 → 𝑁 ∈ ℕ) | ||
Theorem | tgoldbachgtde 34161* | Lemma for tgoldbachgtd 34163. (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 34162* | Lemma for tgoldbachgtd 34163. (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 34163* | 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 34164* | 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 34165 | Extends class notation with the class of geometries fulfilling the planarity axioms. |
class TarskiG2D | ||
Definition | df-trkg2d 34166* | 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 34167* | 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 34168* | Alternate version of axtglowdim2 28190. (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 34169 | Alternate version of axtgupdim2 28191. (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 34170 | Declare the syntax for the outer five segment configuration. |
class AFS | ||
Definition | df-afs 34171* | The outer five segment configuration is an abbreviation for the conditions of the Five Segment Axiom (axtg5seg 28185). See df-ofs 35450. 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 34172* | 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 34173 | 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 34174 | Rephrase axtg5seg 28185 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 34175 | Extend class notation with the leftpad function. |
class leftpad | ||
Definition | df-lpad 34176* | 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 34177 | 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 34178 | Lemma for the leftpad theorems. (Contributed by Thierry Arnoux, 7-Aug-2023.) |
⊢ (𝜑 → 𝐶 ∈ 𝑆) ⇒ ⊢ (𝜑 → ((0..^(𝐿 − (♯‘𝑊))) × {𝐶}) ∈ Word 𝑆) | ||
Theorem | lpadlem3 34179 | Lemma for lpadlen1 34180. (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 34180 | 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 34181 | 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 34182 | 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 34183 | 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 34184 | 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 34185 | 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 34186 | Extend wff notation with the 4-way conjunction. (New usage is discouraged.) |
wff (𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) | ||
Definition | df-bnj17 34187 | 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 34188 | 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 34189* | 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 34190 | Extend wff notation with the following predicate: 𝑅 is set-like on 𝐴. (New usage is discouraged.) |
wff 𝑅 Se 𝐴 | ||
Definition | df-bnj13 34191* | 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 34192 | 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 34193 | 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 34194 | 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 34195* | 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 34426. (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 34196 | 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 34197* | 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 34198 | ∧-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 34199 | ∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ (𝜓 → 𝜓′) & ⊢ (𝜒 → 𝜒′) ⇒ ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜓′ ∧ 𝜒′)) | ||
Theorem | bnj248 34200 | ∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ (((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃)) |
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