P. 347 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 34601-34700   *Has distinct variable group(s)

Type	Label	Description
Statement
21.3.27.1  Representations of a number as sums of integers
Syntax	crepr 34601	Representations of a number as a sum of nonnegative integers.
class repr
Definition	df-repr 34602*	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 34603*	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 34604	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 34605	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 34606*	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 34607	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 34608*	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 34609	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 34610	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 34611*	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 34612	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 34613*	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 34614	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 34615	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 34616	Corollary of reprinfz1 34615. (Contributed by Thierry Arnoux, 15-Dec-2021.)
⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → 𝑆 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐴 ⊆ ℕ)    ⇒   ⊢ (𝜑 → (𝐴(repr‘𝑆)𝑁) ∈ Fin)
Theorem	reprfz1 34617	Corollary of reprinfz1 34615. (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 34618*	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 34619*	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 34620*	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 34621*	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 34622*	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 34623*	Lemma for breprexp 34626 (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 34624	Lemma for breprexp 34626 (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 34625*	Lemma for breprexp 34626 (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 34626*	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 34627 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 34627*	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)))
21.3.27.2  Vinogradov Trigonometric Sums and the Circle Method
Syntax	cvts 34628	The Vinogradov trigonometric sums.
class vts
Definition	df-vts 34629*	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 34630*	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 34631	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 34632*	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 34633*	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 34634*	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 34635*	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 34636*	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)
21.3.27.3  The Ternary Goldbach Conjecture: Final Statement
Axiom	ax-hgt749 34637*	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 34638	Theorem 12. of [RosserSchoenfeld] p. 71. Theorem chpo1ubb 27539 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 34639	Theorem 13. of [RosserSchoenfeld] p. 71. Theorem chpchtlim 27537 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 34640*	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 34641*	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 34642*	A deduction version of ax-hgt749 34637. (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 34643	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 34644	Lemma for tgoldbachgtd 34655. (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 34645	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 34646*	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 34647*	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 34648*	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 34649*	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 34650*	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 34651*	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 34652*	Lemma for tgoldbachgtd 34655. (Contributed by Thierry Arnoux, 15-Dec-2021.)
⊢ 𝑂 = {𝑧 ∈ ℤ ∣ ¬ 2 ∥ 𝑧}    &   ⊢ (𝜑 → 𝑁 ∈ 𝑂)    &   ⊢ (𝜑 → (;10↑;27) ≤ 𝑁)    ⇒   ⊢ (𝜑 → 𝑁 ∈ ℕ)
Theorem	tgoldbachgtde 34653*	Lemma for tgoldbachgtd 34655. (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 34654*	Lemma for tgoldbachgtd 34655. (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 34655*	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 34656*	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))
21.3.28  Elementary Geometry
21.3.28.1  Two-dimensional 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.
Syntax	cstrkg2d 34657	Extends class notation with the class of geometries fulfilling the planarity axioms.
class TarskiG2D
Definition	df-trkg2d 34658*	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 34659*	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 34660*	Alternate version of axtglowdim2 28492. (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 34661	Alternate version of axtgupdim2 28493. (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)))
21.3.28.2  Outer Five Segment (not used, no need to move to main)
Syntax	cafs 34662	Declare the syntax for the outer five segment configuration.
class AFS
Definition	df-afs 34663*	The outer five segment configuration is an abbreviation for the conditions of the Five Segment Axiom (axtg5seg 28487). See df-ofs 35964. 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 34664*	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 34665	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 34666	Rephrase axtg5seg 28487 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))
21.3.29  LeftPad Project See https://github.com/tirix/lets-prove-leftpad.
Syntax	clpad 34667	Extend class notation with the leftpad function.
class leftpad
Definition	df-lpad 34668*	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 34669	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 34670	Lemma for the leftpad theorems. (Contributed by Thierry Arnoux, 7-Aug-2023.)
⊢ (𝜑 → 𝐶 ∈ 𝑆)    ⇒   ⊢ (𝜑 → ((0..^(𝐿 − (♯‘𝑊))) × {𝐶}) ∈ Word 𝑆)
Theorem	lpadlem3 34671	Lemma for lpadlen1 34672. (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 34672	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 34673	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 34674	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 34675	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 34676	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 34677	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))
21.4  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.
Syntax	w-bnj17 34678	Extend wff notation with the 4-way conjunction. (New usage is discouraged.)
wff (𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃)
Definition	df-bnj17 34679	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 34680	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 34681*	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 34682	Extend wff notation with the following predicate: 𝑅 is set-like on 𝐴. (New usage is discouraged.)
wff 𝑅 Se 𝐴
Definition	df-bnj13 34683*	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 34684	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 34685	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 34686	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 34687*	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 34918. (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 34688	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 34689*	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)
21.4.1  First-order logic and set theory
Theorem	bnj170 34690	∧-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 34691	∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜓 → 𝜓′)    &   ⊢ (𝜒 → 𝜒′)    ⇒   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜓′ ∧ 𝜒′))
Theorem	bnj248 34692	∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ (((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃))
Theorem	bnj250 34693	∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ (𝜑 ∧ ((𝜓 ∧ 𝜒) ∧ 𝜃)))
Theorem	bnj251 34694	∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ (𝜑 ∧ (𝜓 ∧ (𝜒 ∧ 𝜃))))
Theorem	bnj252 34695	∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ (𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜃)))
Theorem	bnj253 34696	∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ ((𝜑 ∧ 𝜓) ∧ 𝜒 ∧ 𝜃))
Theorem	bnj255 34697	∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ (𝜑 ∧ 𝜓 ∧ (𝜒 ∧ 𝜃)))
Theorem	bnj256 34698	∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ ((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)))
Theorem	bnj257 34699	∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ (𝜑 ∧ 𝜓 ∧ 𝜃 ∧ 𝜒))
Theorem	bnj258 34700	∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ ((𝜑 ∧ 𝜓 ∧ 𝜃) ∧ 𝜒))