 Home Metamath Proof ExplorerTheorem List (p. 309 of 429) < Previous  Next > Bad symbols? Try the GIF version. Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

 Color key: Metamath Proof Explorer (1-27903) Hilbert Space Explorer (27904-29428) Users' Mathboxes (29429-42879)

Theorem List for Metamath Proof Explorer - 30801-30900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcxpcncf1 30801* The power function on complex numbers, for fixed exponent A, is continuous. Similar to cxpcn 24531. (Contributed by Thierry Arnoux, 20-Dec-2021.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐷 ⊆ (ℂ ∖ (-∞(,]0)))       (𝜑 → (𝑥𝐷 ↦ (𝑥𝑐𝐴)) ∈ (𝐷cn→ℂ))

Theoremefmul2picn 30802* Multiplying by (i · (2 · π)) and taking the exponential preserves continuity. (Contributed by Thierry Arnoux, 13-Dec-2021.)
(𝜑 → (𝑥𝐴𝐵) ∈ (𝐴cn→ℂ))       (𝜑 → (𝑥𝐴 ↦ (exp‘((i · (2 · π)) · 𝐵))) ∈ (𝐴cn→ℂ))

Theoremfct2relem 30803 Lemma for ftc2re 30804. (Contributed by Thierry Arnoux, 20-Dec-2021.)
𝐸 = (𝐶(,)𝐷)    &   (𝜑𝐴𝐸)    &   (𝜑𝐵𝐸)       (𝜑 → (𝐴[,]𝐵) ⊆ 𝐸)

Theoremftc2re 30804* The Fundamental Theorem of Calculus, part two, for functions continuous on 𝐷. (Contributed by Thierry Arnoux, 1-Dec-2021.)
𝐸 = (𝐶(,)𝐷)    &   (𝜑𝐴𝐸)    &   (𝜑𝐵𝐸)    &   (𝜑𝐴𝐵)    &   (𝜑𝐹:𝐸⟶ℂ)    &   (𝜑 → (ℝ D 𝐹) ∈ (𝐸cn→ℂ))       (𝜑 → ∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 = ((𝐹𝐵) − (𝐹𝐴)))

Theoremfdvposlt 30805* Functions with a positive derivative, i.e. monotonously growing functions, preserve strict ordering. (Contributed by Thierry Arnoux, 20-Dec-2021.)
𝐸 = (𝐶(,)𝐷)    &   (𝜑𝐴𝐸)    &   (𝜑𝐵𝐸)    &   (𝜑𝐹:𝐸⟶ℝ)    &   (𝜑 → (ℝ D 𝐹) ∈ (𝐸cn→ℝ))    &   (𝜑𝐴 < 𝐵)    &   ((𝜑𝑥 ∈ (𝐴(,)𝐵)) → 0 < ((ℝ D 𝐹)‘𝑥))       (𝜑 → (𝐹𝐴) < (𝐹𝐵))

Theoremfdvneggt 30806* Functions with a negative derivative, i.e. monotonously decreasing functions, inverse strict ordering. (Contributed by Thierry Arnoux, 20-Dec-2021.)
𝐸 = (𝐶(,)𝐷)    &   (𝜑𝐴𝐸)    &   (𝜑𝐵𝐸)    &   (𝜑𝐹:𝐸⟶ℝ)    &   (𝜑 → (ℝ D 𝐹) ∈ (𝐸cn→ℝ))    &   (𝜑𝐴 < 𝐵)    &   ((𝜑𝑥 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑥) < 0)       (𝜑 → (𝐹𝐵) < (𝐹𝐴))

Theoremfdvposle 30807* Functions with a nonnegative derivative, i.e. monotonously growing functions, preserve ordering. (Contributed by Thierry Arnoux, 20-Dec-2021.)
𝐸 = (𝐶(,)𝐷)    &   (𝜑𝐴𝐸)    &   (𝜑𝐵𝐸)    &   (𝜑𝐹:𝐸⟶ℝ)    &   (𝜑 → (ℝ D 𝐹) ∈ (𝐸cn→ℝ))    &   (𝜑𝐴𝐵)    &   ((𝜑𝑥 ∈ (𝐴(,)𝐵)) → 0 ≤ ((ℝ D 𝐹)‘𝑥))       (𝜑 → (𝐹𝐴) ≤ (𝐹𝐵))

Theoremfdvnegge 30808* Functions with a non-positive derivative, i.e. decreasing functions, preserve ordering. (Contributed by Thierry Arnoux, 20-Dec-2021.)
𝐸 = (𝐶(,)𝐷)    &   (𝜑𝐴𝐸)    &   (𝜑𝐵𝐸)    &   (𝜑𝐹:𝐸⟶ℝ)    &   (𝜑 → (ℝ D 𝐹) ∈ (𝐸cn→ℝ))    &   (𝜑𝐴𝐵)    &   ((𝜑𝑥 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑥) ≤ 0)       (𝜑 → (𝐹𝐵) ≤ (𝐹𝐴))

Theoremprodfzo03 30809* 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)    &   ((𝜑𝑘 ∈ (0..^3)) → 𝐷 ∈ ℂ)       (𝜑 → ∏𝑘 ∈ (0..^3)𝐷 = (𝐴 · (𝐵 · 𝐶)))

Theoremactfunsnf1o 30810* 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.)
((𝜑𝑘𝐶) → 𝐴 ⊆ (𝐶𝑚 𝐵))    &   (𝜑𝐶 ∈ V)    &   (𝜑𝐼𝑉)    &   (𝜑 → ¬ 𝐼𝐵)    &   𝐹 = (𝑥𝐴 ↦ (𝑥 ∪ {⟨𝐼, 𝑘⟩}))       ((𝜑𝑘𝐶) → 𝐹:𝐴1-1-onto→ran 𝐹)

Theoremactfunsnrndisj 30811* 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.)
((𝜑𝑘𝐶) → 𝐴 ⊆ (𝐶𝑚 𝐵))    &   (𝜑𝐶 ∈ V)    &   (𝜑𝐼𝑉)    &   (𝜑 → ¬ 𝐼𝐵)    &   𝐹 = (𝑥𝐴 ↦ (𝑥 ∪ {⟨𝐼, 𝑘⟩}))       (𝜑Disj 𝑘𝐶 ran 𝐹)

Theoremitgexpif 30812* 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))

Theoremfsum2dsub 30813* Lemma for breprexp 30839- Re-index a double sum, using difference of the initial indices. (Contributed by Thierry Arnoux, 7-Dec-2021.)
(𝜑𝑀 ∈ ℕ0)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝑖 = (𝑘𝑗) → 𝐴 = 𝐵)    &   ((𝜑𝑖 ∈ (ℤ‘-𝑗) ∧ 𝑗 ∈ (1...𝑁)) → 𝐴 ∈ ℂ)    &   (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ (((𝑀 + 𝑗) + 1)...(𝑀 + 𝑁))) → 𝐵 = 0)    &   (((𝜑𝑗 ∈ (1...𝑁)) ∧ 𝑘 ∈ (0..^𝑗)) → 𝐵 = 0)       (𝜑 → Σ𝑖 ∈ (0...𝑀𝑗 ∈ (1...𝑁)𝐴 = Σ𝑘 ∈ (0...(𝑀 + 𝑁))Σ𝑗 ∈ (1...𝑁)𝐵)

20.3.26.1  Representations of a number as sums of integers

Syntaxcrepr 30814 Representations of a number as a sum of nonnegative integers.
class repr

Definitiondf-repr 30815* 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 ∈ (𝑏𝑚 (0..^𝑠)) ∣ Σ𝑎 ∈ (0..^𝑠)(𝑐𝑎) = 𝑚}))

Theoremreprval 30816* 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.)
(𝜑𝐴 ⊆ ℕ)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑆 ∈ ℕ0)       (𝜑 → (𝐴(repr‘𝑆)𝑀) = {𝑐 ∈ (𝐴𝑚 (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐𝑎) = 𝑀})

Theoremrepr0 30817 There is exactly one representation with no elements (an empty sum), only for 𝑀 = 0. (Contributed by Thierry Arnoux, 2-Dec-2021.)
(𝜑𝐴 ⊆ ℕ)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑆 ∈ ℕ0)       (𝜑 → (𝐴(repr‘0)𝑀) = if(𝑀 = 0, {∅}, ∅))

Theoremreprf 30818 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.)
(𝜑𝐴 ⊆ ℕ)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑆 ∈ ℕ0)    &   (𝜑𝐶 ∈ (𝐴(repr‘𝑆)𝑀))       (𝜑𝐶:(0..^𝑆)⟶𝐴)

Theoremreprsum 30819* Sums of values of the members of the representation of 𝑀 equal 𝑀. (Contributed by Thierry Arnoux, 5-Dec-2021.)
(𝜑𝐴 ⊆ ℕ)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑆 ∈ ℕ0)    &   (𝜑𝐶 ∈ (𝐴(repr‘𝑆)𝑀))       (𝜑 → Σ𝑎 ∈ (0..^𝑆)(𝐶𝑎) = 𝑀)

Theoremreprle 30820 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.)
(𝜑𝐴 ⊆ ℕ)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑆 ∈ ℕ0)    &   (𝜑𝐶 ∈ (𝐴(repr‘𝑆)𝑀))    &   (𝜑𝑋 ∈ (0..^𝑆))       (𝜑 → (𝐶𝑋) ≤ 𝑀)

Theoremreprsuc 30821* Express the representations recursively. (Contributed by Thierry Arnoux, 5-Dec-2021.)
(𝜑𝐴 ⊆ ℕ)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑆 ∈ ℕ0)    &   𝐹 = (𝑐 ∈ (𝐴(repr‘𝑆)(𝑀𝑏)) ↦ (𝑐 ∪ {⟨𝑆, 𝑏⟩}))       (𝜑 → (𝐴(repr‘(𝑆 + 1))𝑀) = 𝑏𝐴 ran 𝐹)

Theoremreprfi 30822 Bounded representations are finite sets. (Contributed by Thierry Arnoux, 7-Dec-2021.)
(𝜑𝐴 ⊆ ℕ)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑆 ∈ ℕ0)    &   (𝜑𝐴 ∈ Fin)       (𝜑 → (𝐴(repr‘𝑆)𝑀) ∈ Fin)

Theoremreprss 30823 Representations with terms in a subset. (Contributed by Thierry Arnoux, 11-Dec-2021.)
(𝜑𝐴 ⊆ ℕ)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑆 ∈ ℕ0)    &   (𝜑𝐵𝐴)       (𝜑 → (𝐵(repr‘𝑆)𝑀) ⊆ (𝐴(repr‘𝑆)𝑀))

Theoremreprinrn 30824* Representations with term in an intersection. (Contributed by Thierry Arnoux, 11-Dec-2021.)
(𝜑𝐴 ⊆ ℕ)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑆 ∈ ℕ0)       (𝜑 → (𝑐 ∈ ((𝐴𝐵)(repr‘𝑆)𝑀) ↔ (𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ran 𝑐𝐵)))

Theoremreprlt 30825 There are no representations of 𝑀 with more than 𝑀 terms. Remark of [Nathanson] p. 123 (Contributed by Thierry Arnoux, 7-Dec-2021.)
(𝜑𝐴 ⊆ ℕ)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑆 ∈ ℕ0)    &   (𝜑𝑀 < 𝑆)       (𝜑 → (𝐴(repr‘𝑆)𝑀) = ∅)

Theoremhashreprin 30826* Express a sum of representations over an intersection using a product of the indicator function (Contributed by Thierry Arnoux, 11-Dec-2021.)
(𝜑𝐴 ⊆ ℕ)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑆 ∈ ℕ0)    &   (𝜑𝐵 ∈ Fin)    &   (𝜑𝐵 ⊆ ℕ)       (𝜑 → (#‘((𝐴𝐵)(repr‘𝑆)𝑀)) = Σ𝑐 ∈ (𝐵(repr‘𝑆)𝑀)∏𝑎 ∈ (0..^𝑆)(((𝟭‘ℕ)‘𝐴)‘(𝑐𝑎)))

Theoremreprgt 30827 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.)
(𝜑𝑁 ∈ ℕ0)    &   (𝜑𝐴 ⊆ (1...𝑁))    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝑆 ∈ ℕ0)    &   (𝜑 → (𝑆 · 𝑁) < 𝑀)       (𝜑 → (𝐴(repr‘𝑆)𝑀) = ∅)

Theoremreprinfz1 30828 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.)
(𝜑𝑁 ∈ ℕ0)    &   (𝜑𝑆 ∈ ℕ0)    &   (𝜑𝐴 ⊆ ℕ)       (𝜑 → (𝐴(repr‘𝑆)𝑁) = ((𝐴 ∩ (1...𝑁))(repr‘𝑆)𝑁))

Theoremreprfi2 30829 Corollary of reprinfz1 30828. (Contributed by Thierry Arnoux, 15-Dec-2021.)
(𝜑𝑁 ∈ ℕ0)    &   (𝜑𝑆 ∈ ℕ0)    &   (𝜑𝐴 ⊆ ℕ)       (𝜑 → (𝐴(repr‘𝑆)𝑁) ∈ Fin)

Theoremreprfz1 30830 Corollary of reprinfz1 30828. (Contributed by Thierry Arnoux, 14-Dec-2021.)
(𝜑𝑁 ∈ ℕ0)    &   (𝜑𝑆 ∈ ℕ0)       (𝜑 → (ℕ(repr‘𝑆)𝑁) = ((1...𝑁)(repr‘𝑆)𝑁))

Theoremhashrepr 30831* 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.)
(𝜑𝐴 ⊆ ℕ)    &   (𝜑𝑀 ∈ ℕ0)    &   (𝜑𝑆 ∈ ℕ0)       (𝜑 → (#‘(𝐴(repr‘𝑆)𝑀)) = Σ𝑐 ∈ (ℕ(repr‘𝑆)𝑀)∏𝑎 ∈ (0..^𝑆)(((𝟭‘ℕ)‘𝐴)‘(𝑐𝑎)))

Theoremreprpmtf1o 30832* 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.)
(𝜑𝑆 ∈ ℕ)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐴 ⊆ ℕ)    &   (𝜑𝑋 ∈ (0..^𝑆))    &   𝑂 = {𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∣ ¬ (𝑐‘0) ∈ 𝐵}    &   𝑃 = {𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∣ ¬ (𝑐𝑋) ∈ 𝐵}    &   𝑇 = if(𝑋 = 0, ( I ↾ (0..^𝑆)), ((pmTrsp‘(0..^𝑆))‘{𝑋, 0}))    &   𝐹 = (𝑐𝑃 ↦ (𝑐𝑇))       (𝜑𝐹:𝑃1-1-onto𝑂)

Theoremreprdifc 30833* 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) ∣ ¬ (𝑐𝑥) ∈ 𝐵}    &   (𝜑𝐴 ⊆ ℕ)    &   (𝜑𝐵 ⊆ ℕ)    &   (𝜑𝑀 ∈ ℕ0)    &   (𝜑𝑆 ∈ ℕ0)       (𝜑 → ((𝐴(repr‘𝑆)𝑀) ∖ (𝐵(repr‘𝑆)𝑀)) = 𝑥 ∈ (0..^𝑆)𝐶)

Theoremchpvalz 30834* 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))

Theoremchtvalz 30835* Value of the Chebyshev function for integers. (Contributed by Thierry Arnoux, 28-Dec-2021.)
(𝑁 ∈ ℤ → (θ‘𝑁) = Σ𝑛 ∈ ((1...𝑁) ∩ ℙ)(log‘𝑛))

Theorembreprexplema 30836* Lemma for breprexp 30839 (induction step for weighted sums over representations) (Contributed by Thierry Arnoux, 7-Dec-2021.)
(𝜑𝑁 ∈ ℕ0)    &   (𝜑𝑆 ∈ ℕ0)    &   (𝜑𝑀 ∈ ℕ0)    &   (𝜑𝑀 ≤ ((𝑆 + 1) · 𝑁))    &   (((𝜑𝑥 ∈ (0..^(𝑆 + 1))) ∧ 𝑦 ∈ ℕ) → ((𝐿𝑥)‘𝑦) ∈ ℂ)       (𝜑 → Σ𝑑 ∈ ((1...𝑁)(repr‘(𝑆 + 1))𝑀)∏𝑎 ∈ (0..^(𝑆 + 1))((𝐿𝑎)‘(𝑑𝑎)) = Σ𝑏 ∈ (1...𝑁𝑑 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑑𝑎)) · ((𝐿𝑆)‘𝑏)))

Theorembreprexplemb 30837 Lemma for breprexp 30839 (closure) (Contributed by Thierry Arnoux, 7-Dec-2021.)
(𝜑𝑁 ∈ ℕ0)    &   (𝜑𝑆 ∈ ℕ0)    &   (𝜑𝑍 ∈ ℂ)    &   (𝜑𝐿:(0..^𝑆)⟶(ℂ ↑𝑚 ℕ))    &   (𝜑𝑋 ∈ (0..^𝑆))    &   (𝜑𝑌 ∈ ℕ)       (𝜑 → ((𝐿𝑋)‘𝑌) ∈ ℂ)

Theorembreprexplemc 30838* Lemma for breprexp 30839 (induction step) (Contributed by Thierry Arnoux, 6-Dec-2021.)
(𝜑𝑁 ∈ ℕ0)    &   (𝜑𝑆 ∈ ℕ0)    &   (𝜑𝑍 ∈ ℂ)    &   (𝜑𝐿:(0..^𝑆)⟶(ℂ ↑𝑚 ℕ))    &   (𝜑𝑇 ∈ ℕ0)    &   (𝜑 → (𝑇 + 1) ≤ 𝑆)    &   (𝜑 → ∏𝑎 ∈ (0..^𝑇𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)) = Σ𝑚 ∈ (0...(𝑇 · 𝑁))Σ𝑑 ∈ ((1...𝑁)(repr‘𝑇)𝑚)(∏𝑎 ∈ (0..^𝑇)((𝐿𝑎)‘(𝑑𝑎)) · (𝑍𝑚)))       (𝜑 → ∏𝑎 ∈ (0..^(𝑇 + 1))Σ𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)) = Σ𝑚 ∈ (0...((𝑇 + 1) · 𝑁))Σ𝑑 ∈ ((1...𝑁)(repr‘(𝑇 + 1))𝑚)(∏𝑎 ∈ (0..^(𝑇 + 1))((𝐿𝑎)‘(𝑑𝑎)) · (𝑍𝑚)))

Theorembreprexp 30839* 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 30840 for the simple case presented in the proposition of [Nathanson] p. 123. (Contributed by Thierry Arnoux, 6-Dec-2021.)
(𝜑𝑁 ∈ ℕ0)    &   (𝜑𝑆 ∈ ℕ0)    &   (𝜑𝑍 ∈ ℂ)    &   (𝜑𝐿:(0..^𝑆)⟶(ℂ ↑𝑚 ℕ))       (𝜑 → ∏𝑎 ∈ (0..^𝑆𝑏 ∈ (1...𝑁)(((𝐿𝑎)‘𝑏) · (𝑍𝑏)) = Σ𝑚 ∈ (0...(𝑆 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (𝑍𝑚)))

Theorembreprexpnat 30840* 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.)
(𝜑𝑁 ∈ ℕ0)    &   (𝜑𝑆 ∈ ℕ0)    &   (𝜑𝑍 ∈ ℂ)    &   (𝜑𝐴 ⊆ ℕ)    &   𝑃 = Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍𝑏)    &   𝑅 = (#‘((𝐴 ∩ (1...𝑁))(repr‘𝑆)𝑚))       (𝜑 → (𝑃𝑆) = Σ𝑚 ∈ (0...(𝑆 · 𝑁))(𝑅 · (𝑍𝑚)))

20.3.26.2  Vinogradov Trigonometric Sums and the Circle Method

Syntaxcvts 30841 The Vinogradov trigonometric sums.
class vts

Definitiondf-vts 30842* Define the Vinogradov trigonometric sums. (Contributed by Thierry Arnoux, 1-Dec-2021.)
vts = (𝑙 ∈ (ℂ ↑𝑚 ℕ), 𝑛 ∈ ℕ0 ↦ (𝑥 ∈ ℂ ↦ Σ𝑎 ∈ (1...𝑛)((𝑙𝑎) · (exp‘((i · (2 · π)) · (𝑎 · 𝑥))))))

Theoremvtsval 30843* Value of the Vinogradov trigonometric sums (Contributed by Thierry Arnoux, 1-Dec-2021.)
(𝜑𝑁 ∈ ℕ0)    &   (𝜑𝑋 ∈ ℂ)    &   (𝜑𝐿:ℕ⟶ℂ)       (𝜑 → ((𝐿vts𝑁)‘𝑋) = Σ𝑎 ∈ (1...𝑁)((𝐿𝑎) · (exp‘((i · (2 · π)) · (𝑎 · 𝑋)))))

Theoremvtscl 30844 Closure of the Vinogradov trigonometric sums (Contributed by Thierry Arnoux, 14-Dec-2021.)
(𝜑𝑁 ∈ ℕ0)    &   (𝜑𝑋 ∈ ℂ)    &   (𝜑𝐿:ℕ⟶ℂ)       (𝜑 → ((𝐿vts𝑁)‘𝑋) ∈ ℂ)

Theoremvtsprod 30845* Express the Vinogradov trigonometric sums to the power of 𝑆 (Contributed by Thierry Arnoux, 12-Dec-2021.)
(𝜑𝑁 ∈ ℕ0)    &   (𝜑𝑋 ∈ ℂ)    &   (𝜑𝑆 ∈ ℕ0)    &   (𝜑𝐿:(0..^𝑆)⟶(ℂ ↑𝑚 ℕ))       (𝜑 → ∏𝑎 ∈ (0..^𝑆)(((𝐿𝑎)vts𝑁)‘𝑋) = Σ𝑚 ∈ (0...(𝑆 · 𝑁))Σ𝑐 ∈ ((1...𝑁)(repr‘𝑆)𝑚)(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) · (exp‘((i · (2 · π)) · (𝑚 · 𝑋)))))

Theoremcirclemeth 30846* The Hardy, Littlewood and Ramanujan Circle Method, in a generic form, with different weighting / smoothing functions. (Contributed by Thierry Arnoux, 13-Dec-2021.)
(𝜑𝑁 ∈ ℕ0)    &   (𝜑𝑆 ∈ ℕ)    &   (𝜑𝐿:(0..^𝑆)⟶(ℂ ↑𝑚 ℕ))       (𝜑 → Σ𝑐 ∈ (ℕ(repr‘𝑆)𝑁)∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑐𝑎)) = ∫(0(,)1)(∏𝑎 ∈ (0..^𝑆)(((𝐿𝑎)vts𝑁)‘𝑥) · (exp‘((i · (2 · π)) · (-𝑁 · 𝑥)))) d𝑥)

Theoremcirclemethnat 30847* 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)((𝐹𝑆) · (exp‘((i · (2 · π)) · (-𝑁 · 𝑥)))) d𝑥

Theoremcirclevma 30848* 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.)
(𝜑𝑁 ∈ 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)

Theoremcirclemethhgt 30849* 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.)
(𝜑𝐻:ℕ⟶ℝ)    &   (𝜑𝐾:ℕ⟶ℝ)    &   (𝜑𝑁 ∈ 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)

20.3.26.3  The Ternary Goldbach Conjecture: Final Statement

Axiomax-hgt749 30850* Statement 7.49 of [Helfgott] p. 70. For a sufficiently big odd 𝑁, this postulates the existence of smoothing functions (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 → ∃ ∈ ((0[,)+∞) ↑𝑚 ℕ)∃𝑘 ∈ ((0[,)+∞) ↑𝑚 ℕ)(∀𝑚 ∈ ℕ (𝑘𝑚) ≤ (1.079955) ∧ ∀𝑚 ∈ ℕ (𝑚) ≤ (1.414) ∧ ((0.00042248) · (𝑛↑2)) ≤ ∫(0(,)1)(((((Λ ∘𝑓 · )vts𝑛)‘𝑥) · ((((Λ ∘𝑓 · 𝑘)vts𝑛)‘𝑥)↑2)) · (exp‘((i · (2 · π)) · (-𝑛 · 𝑥)))) d𝑥))

Axiomax-ros335 30851 Theorem 12. of [RosserSchoenfeld] p. 71. Theorem chpo1ubb 25215 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) < ((1.03883) · 𝑥)

Axiomax-ros336 30852 Theorem 13. of [RosserSchoenfeld] p. 71. Theorem chpchtlim 25213 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)) < ((1.4262) · (√‘𝑥))

Theoremhgt750lemc 30853* An upper bound to the summatory function of the von Mangoldt function. (Contributed by Thierry Arnoux, 29-Dec-2021.)
(𝜑𝑁 ∈ ℕ)       (𝜑 → Σ𝑗 ∈ (1...𝑁)(Λ‘𝑗) < ((1.03883) · 𝑁))

Theoremhgt750lemd 30854* An upper bound to the summatory function of the von Mangoldt function on non-primes. (Contributed by Thierry Arnoux, 29-Dec-2021.)
(𝜑𝑁 ∈ ℕ)    &   (𝜑 → (10↑27) ≤ 𝑁)       (𝜑 → Σ𝑖 ∈ (((1...𝑁) ∖ ℙ) ∪ {2})(Λ‘𝑖) < ((1.4263) · (√‘𝑁)))

Theoremhgt749d 30855* A deduction version of ax-hgt749 30850. (Contributed by Thierry Arnoux, 15-Dec-2021.)
𝑂 = {𝑧 ∈ ℤ ∣ ¬ 2 ∥ 𝑧}    &   (𝜑𝑁𝑂)    &   (𝜑 → (10↑27) ≤ 𝑁)       (𝜑 → ∃ ∈ ((0[,)+∞) ↑𝑚 ℕ)∃𝑘 ∈ ((0[,)+∞) ↑𝑚 ℕ)(∀𝑚 ∈ ℕ (𝑘𝑚) ≤ (1.079955) ∧ ∀𝑚 ∈ ℕ (𝑚) ≤ (1.414) ∧ ((0.00042248) · (𝑁↑2)) ≤ ∫(0(,)1)(((((Λ ∘𝑓 · )vts𝑁)‘𝑥) · ((((Λ ∘𝑓 · 𝑘)vts𝑁)‘𝑥)↑2)) · (exp‘((i · (2 · π)) · (-𝑁 · 𝑥)))) d𝑥))

Theoremlogdivsqrle 30856 Conditions for ((log x ) / ( sqrt 𝑥)) to be decreasing. (Contributed by Thierry Arnoux, 20-Dec-2021.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ+)    &   (𝜑 → (exp‘2) ≤ 𝐴)    &   (𝜑𝐴𝐵)       (𝜑 → ((log‘𝐵) / (√‘𝐵)) ≤ ((log‘𝐴) / (√‘𝐴)))

Theoremhgt750lem 30857 Lemma for tgoldbachgtd 30868. (Contributed by Thierry Arnoux, 17-Dec-2021.)
((𝑁 ∈ ℕ0 ∧ (10↑27) ≤ 𝑁) → ((7.348) · ((log‘𝑁) / (√‘𝑁))) < (0.00042248))

Theoremhgt750lem2 30858 Decimal multiplication galore! (Contributed by Thierry Arnoux, 26-Dec-2021.)
(3 · ((((1.079955)↑2) · (1.414)) · ((1.4263) · (1.03883)))) < (7.348)

Theoremhgt750lemf 30859* Lemma for the statement 7.50 of [Helfgott] p. 69. (Contributed by Thierry Arnoux, 1-Jan-2022.)
(𝜑𝐴 ∈ Fin)    &   (𝜑𝑃 ∈ ℝ)    &   (𝜑𝑄 ∈ ℝ)    &   (𝜑𝐻:ℕ⟶(0[,)+∞))    &   (𝜑𝐾:ℕ⟶(0[,)+∞))    &   ((𝜑𝑛𝐴) → (𝑛‘0) ∈ ℕ)    &   ((𝜑𝑛𝐴) → (𝑛‘1) ∈ ℕ)    &   ((𝜑𝑛𝐴) → (𝑛‘2) ∈ ℕ)    &   ((𝜑𝑚 ∈ ℕ) → (𝐾𝑚) ≤ 𝑃)    &   ((𝜑𝑚 ∈ ℕ) → (𝐻𝑚) ≤ 𝑄)       (𝜑 → Σ𝑛𝐴 (((Λ‘(𝑛‘0)) · (𝐻‘(𝑛‘0))) · (((Λ‘(𝑛‘1)) · (𝐾‘(𝑛‘1))) · ((Λ‘(𝑛‘2)) · (𝐾‘(𝑛‘2))))) ≤ (((𝑃↑2) · 𝑄) · Σ𝑛𝐴 ((Λ‘(𝑛‘0)) · ((Λ‘(𝑛‘1)) · (Λ‘(𝑛‘2))))))

Theoremhgt750lemg 30860* 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 = (𝑐𝑅 ↦ (𝑐𝑇))    &   (𝜑𝑇:(0..^3)–1-1-onto→(0..^3))    &   (𝜑𝑁:(0..^3)⟶ℕ)    &   (𝜑𝐿:ℕ⟶ℝ)    &   (𝜑𝑁𝑅)       (𝜑 → ((𝐿‘((𝐹𝑁)‘0)) · ((𝐿‘((𝐹𝑁)‘1)) · (𝐿‘((𝐹𝑁)‘2)))) = ((𝐿‘(𝑁‘0)) · ((𝐿‘(𝑁‘1)) · (𝐿‘(𝑁‘2)))))

Theoremoddprm2 30861* 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)

Theoremhgt750lemb 30862* 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}    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑 → 2 ≤ 𝑁)    &   𝐴 = {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ)}       (𝜑 → Σ𝑛𝐴 ((Λ‘(𝑛‘0)) · ((Λ‘(𝑛‘1)) · (Λ‘(𝑛‘2)))) ≤ ((log‘𝑁) · (Σ𝑖 ∈ (((1...𝑁) ∖ ℙ) ∪ {2})(Λ‘𝑖) · Σ𝑗 ∈ (1...𝑁)(Λ‘𝑗))))

Theoremhgt750lema 30863* 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}    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑 → 2 ≤ 𝑁)    &   𝐴 = {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐‘0) ∈ (𝑂 ∩ ℙ)}    &   𝐹 = (𝑑 ∈ {𝑐 ∈ (ℕ(repr‘3)𝑁) ∣ ¬ (𝑐𝑎) ∈ (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 · Σ𝑛𝐴 ((Λ‘(𝑛‘0)) · ((Λ‘(𝑛‘1)) · (Λ‘(𝑛‘2))))))

Theoremhgt750leme 30864* 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}    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑 → (10↑27) ≤ 𝑁)    &   (𝜑𝐻:ℕ⟶(0[,)+∞))    &   (𝜑𝐾:ℕ⟶(0[,)+∞))    &   ((𝜑𝑚 ∈ ℕ) → (𝐾𝑚) ≤ (1.079955))    &   ((𝜑𝑚 ∈ ℕ) → (𝐻𝑚) ≤ (1.414))       (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))))) ≤ (((7.348) · ((log‘𝑁) / (√‘𝑁))) · (𝑁↑2)))

Theoremtgoldbachgnn 30865* Lemma for tgoldbachgtd 30868. (Contributed by Thierry Arnoux, 15-Dec-2021.)
𝑂 = {𝑧 ∈ ℤ ∣ ¬ 2 ∥ 𝑧}    &   (𝜑𝑁𝑂)    &   (𝜑 → (10↑27) ≤ 𝑁)       (𝜑𝑁 ∈ ℕ)

Theoremtgoldbachgtde 30866* Lemma for tgoldbachgtd 30868. (Contributed by Thierry Arnoux, 15-Dec-2021.)
𝑂 = {𝑧 ∈ ℤ ∣ ¬ 2 ∥ 𝑧}    &   (𝜑𝑁𝑂)    &   (𝜑 → (10↑27) ≤ 𝑁)    &   (𝜑𝐻:ℕ⟶(0[,)+∞))    &   (𝜑𝐾:ℕ⟶(0[,)+∞))    &   ((𝜑𝑚 ∈ ℕ) → (𝐾𝑚) ≤ (1.079955))    &   ((𝜑𝑚 ∈ ℕ) → (𝐻𝑚) ≤ (1.414))    &   (𝜑 → ((0.00042248) · (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))))))

Theoremtgoldbachgtda 30867* Lemma for tgoldbachgtd 30868. (Contributed by Thierry Arnoux, 15-Dec-2021.)
𝑂 = {𝑧 ∈ ℤ ∣ ¬ 2 ∥ 𝑧}    &   (𝜑𝑁𝑂)    &   (𝜑 → (10↑27) ≤ 𝑁)    &   (𝜑𝐻:ℕ⟶(0[,)+∞))    &   (𝜑𝐾:ℕ⟶(0[,)+∞))    &   ((𝜑𝑚 ∈ ℕ) → (𝐾𝑚) ≤ (1.079955))    &   ((𝜑𝑚 ∈ ℕ) → (𝐻𝑚) ≤ (1.414))    &   (𝜑 → ((0.00042248) · (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)))

Theoremtgoldbachgtd 30868* 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}    &   (𝜑𝑁𝑂)    &   (𝜑 → (10↑27) ≤ 𝑁)       (𝜑 → 0 < (#‘((𝑂 ∩ ℙ)(repr‘3)𝑁)))

Theoremtgoldbachgt 30869* 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 = {𝑧𝑂 ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ ((𝑝𝑂𝑞𝑂𝑟𝑂) ∧ 𝑧 = ((𝑝 + 𝑞) + 𝑟))}       𝑚 ∈ ℕ (𝑚 ≤ (10↑27) ∧ ∀𝑛𝑂 (𝑚 < 𝑛𝑛𝐺))

20.3.27  Elementary Geometry

20.3.27.1  Two-dimension geometry

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.

Syntaxcstrkg2d 30870 Extends class notation with the class of geometries fulfilling the planarity axioms.
class TarskiG2D

Definitiondf-trkg2d 30871* 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](∃𝑥𝑝𝑦𝑝𝑧𝑝 ¬ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧)) ∧ ∀𝑥𝑝𝑦𝑝𝑧𝑝𝑢𝑝𝑣𝑝 ((((𝑥𝑑𝑢) = (𝑥𝑑𝑣) ∧ (𝑦𝑑𝑢) = (𝑦𝑑𝑣) ∧ (𝑧𝑑𝑢) = (𝑧𝑑𝑣)) ∧ 𝑢𝑣) → (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))))}

Theoremistrkg2d 30872* 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 ∧ (∃𝑥𝑃𝑦𝑃𝑧𝑃 ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)) ∧ ∀𝑥𝑃𝑦𝑃𝑧𝑃𝑢𝑃𝑣𝑃 ((((𝑥 𝑢) = (𝑥 𝑣) ∧ (𝑦 𝑢) = (𝑦 𝑣) ∧ (𝑧 𝑢) = (𝑧 𝑣)) ∧ 𝑢𝑣) → (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))))))

Theoremaxtglowdim2OLD 30873* 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)    &   (𝜑𝐺 ∈ TarskiG2D)       (𝜑 → ∃𝑥𝑃𝑦𝑃𝑧𝑃 ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)))

Theoremaxtgupdim2OLD 30874 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)    &   (𝜑𝑋𝑃)    &   (𝜑𝑌𝑃)    &   (𝜑𝑍𝑃)    &   (𝜑𝑈𝑃)    &   (𝜑𝑉𝑃)    &   (𝜑𝑈𝑉)    &   (𝜑 → (𝑋 𝑈) = (𝑋 𝑉))    &   (𝜑 → (𝑌 𝑈) = (𝑌 𝑉))    &   (𝜑 → (𝑍 𝑈) = (𝑍 𝑉))    &   (𝜑𝐺 ∈ TarskiG2D)       (𝜑 → (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)))

20.3.27.2  Outer Five Segment (not used, no need to move to main)

Syntaxcafs 30875 Declare the syntax for the outer five segment configuration.
class AFS

Definitiondf-afs 30876* The outer five segment configuration is an abbreviation for the conditions of the Five Segment Axiom (axtg5seg 25409). See df-ofs 32215. 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) / ][(Itv‘𝑔) / 𝑖]𝑎𝑝𝑏𝑝𝑐𝑝𝑑𝑝𝑥𝑝𝑦𝑝𝑧𝑝𝑤𝑝 (𝑒 = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ 𝑓 = ⟨⟨𝑥, 𝑦⟩, ⟨𝑧, 𝑤⟩⟩ ∧ ((𝑏 ∈ (𝑎𝑖𝑐) ∧ 𝑦 ∈ (𝑥𝑖𝑧)) ∧ ((𝑎𝑏) = (𝑥𝑦) ∧ (𝑏𝑐) = (𝑦𝑧)) ∧ ((𝑎𝑑) = (𝑥𝑤) ∧ (𝑏𝑑) = (𝑦𝑤))))})

Theoremafsval 30877* 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)    &   (𝜑𝐺 ∈ TarskiG)       (𝜑 → (AFS‘𝐺) = {⟨𝑒, 𝑓⟩ ∣ ∃𝑎𝑃𝑏𝑃𝑐𝑃𝑑𝑃𝑥𝑃𝑦𝑃𝑧𝑃𝑤𝑃 (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))))})

Theorembrafs 30878 Binary relation form of the outer five segment predicate. (Contributed by Scott Fenton, 21-Sep-2013.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG)    &   𝑂 = (AFS‘𝐺)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝑋𝑃)    &   (𝜑𝑌𝑃)    &   (𝜑𝑍𝑃)    &   (𝜑𝑊𝑃)       (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)))))

Theoremtg5segofs 30879 Rephrase axtg5seg 25409 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)    &   (𝜑𝐺 ∈ TarskiG)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   𝑂 = (AFS‘𝐺)    &   (𝜑𝐻𝑃)    &   (𝜑𝐼𝑃)    &   (𝜑 → ⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩𝑂⟨⟨𝐸, 𝐹⟩, ⟨𝐻, 𝐼⟩⟩)    &   (𝜑𝐴𝐵)       (𝜑 → (𝐶 𝐷) = (𝐻 𝐼))

20.4  Mathbox for Jonathan Ben-Naim

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.

Syntaxw-bnj17 30880 Extend wff notation with the 4-way conjunction. (New usage is discouraged.)
wff (𝜑𝜓𝜒𝜃)

Definitiondf-bnj17 30881 Define the 4-way conjunction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝜑𝜓𝜒𝜃) ↔ ((𝜑𝜓𝜒) ∧ 𝜃))

Syntaxc-bnj14 30882 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)

Definitiondf-bnj14 30883* 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) = {𝑦𝐴𝑦𝑅𝑋}

Syntaxw-bnj13 30884 Extend wff notation with the following predicate: 𝑅 is set-like on 𝐴. (New usage is discouraged.)
wff 𝑅 Se 𝐴

Definitiondf-bnj13 30885* 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 ↔ ∀𝑥𝐴 pred(𝑥, 𝐴, 𝑅) ∈ V)

Syntaxw-bnj15 30886 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

Definitiondf-bnj15 30887 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 𝐴𝑅 Se 𝐴))

Syntaxc-bnj18 30888 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)

Definitiondf-bnj18 30889* 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 31122. (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 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))} 𝑖 ∈ dom 𝑓(𝑓𝑖)

Syntaxw-bnj19 30890 Extend wff notation with the following predicate: 𝐵 is transitive for 𝐴 and 𝑅. (New usage is discouraged.)
wff TrFo(𝐵, 𝐴, 𝑅)

Definitiondf-bnj19 30891* 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) ↔ ∀𝑥𝐵 pred(𝑥, 𝐴, 𝑅) ⊆ 𝐵)

20.4.1  First-order logic and set theory

Theorembnj170 30892 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) (New usage is discouraged.)
((𝜑𝜓𝜒) ↔ ((𝜓𝜒) ∧ 𝜑))

Theorembnj240 30893 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜓𝜓′)    &   (𝜒𝜒′)       ((𝜑𝜓𝜒) → (𝜓′𝜒′))

Theorembnj248 30894 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝜑𝜓𝜒𝜃) ↔ (((𝜑𝜓) ∧ 𝜒) ∧ 𝜃))

Theorembnj250 30895 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝜑𝜓𝜒𝜃) ↔ (𝜑 ∧ ((𝜓𝜒) ∧ 𝜃)))

Theorembnj251 30896 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝜑𝜓𝜒𝜃) ↔ (𝜑 ∧ (𝜓 ∧ (𝜒𝜃))))

Theorembnj252 30897 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝜑𝜓𝜒𝜃) ↔ (𝜑 ∧ (𝜓𝜒𝜃)))

Theorembnj253 30898 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝜑𝜓𝜒𝜃) ↔ ((𝜑𝜓) ∧ 𝜒𝜃))

Theorembnj255 30899 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝜑𝜓𝜒𝜃) ↔ (𝜑𝜓 ∧ (𝜒𝜃)))

Theorembnj256 30900 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝜑𝜓𝜒𝜃) ↔ ((𝜑𝜓) ∧ (𝜒𝜃)))

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42879
 Copyright terms: Public domain < Previous  Next >