P. 344 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 34301-34400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	fdvposlt 34301*	Functions with a positive derivative, i.e. monotonously growing functions, preserve 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)) → 0 < ((ℝ D 𝐹)‘𝑥))    ⇒   ⊢ (𝜑 → (𝐹‘𝐴) < (𝐹‘𝐵))
Theorem	fdvneggt 34302*	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 34303*	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 34304*	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 34305*	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 34306*	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 34307*	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 34308*	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 34309*	Lemma for breprexp 34335- 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)
21.3.26.1  Representations of a number as sums of integers
Syntax	crepr 34310	Representations of a number as a sum of nonnegative integers.
class repr
Definition	df-repr 34311*	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 34312*	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 34313	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 34314	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 34315*	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 34316	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 34317*	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 34318	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 34319	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 34320*	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 34321	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 34322*	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 34323	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 34324	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 34325	Corollary of reprinfz1 34324. (Contributed by Thierry Arnoux, 15-Dec-2021.)
⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → 𝑆 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐴 ⊆ ℕ)    ⇒   ⊢ (𝜑 → (𝐴(repr‘𝑆)𝑁) ∈ Fin)
Theorem	reprfz1 34326	Corollary of reprinfz1 34324. (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 34327*	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 34328*	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 34329*	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 34330*	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 34331*	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 34332*	Lemma for breprexp 34335 (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 34333	Lemma for breprexp 34335 (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 34334*	Lemma for breprexp 34335 (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 34335*	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 34336 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 34336*	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)))
21.3.26.2  Vinogradov Trigonometric Sums and the Circle Method
Syntax	cvts 34337	The Vinogradov trigonometric sums.
class vts
Definition	df-vts 34338*	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 34339*	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 34340	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 34341*	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 34342*	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 34343*	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 34344*	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 34345*	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)
21.3.26.3  The Ternary Goldbach Conjecture: Final Statement
Axiom	ax-hgt749 34346*	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 34347	Theorem 12. of [RosserSchoenfeld] p. 71. Theorem chpo1ubb 27444 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 34348	Theorem 13. of [RosserSchoenfeld] p. 71. Theorem chpchtlim 27442 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 34349*	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 34350*	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 34351*	A deduction version of ax-hgt749 34346. (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 34352	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 34353	Lemma for tgoldbachgtd 34364. (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 34354	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 34355*	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 34356*	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 34357*	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 34358*	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 34359*	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 34360*	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 34361*	Lemma for tgoldbachgtd 34364. (Contributed by Thierry Arnoux, 15-Dec-2021.)
⊢ 𝑂 = {𝑧 ∈ ℤ ∣ ¬ 2 ∥ 𝑧}    &   ⊢ (𝜑 → 𝑁 ∈ 𝑂)    &   ⊢ (𝜑 → (;10↑;27) ≤ 𝑁)    ⇒   ⊢ (𝜑 → 𝑁 ∈ ℕ)
Theorem	tgoldbachgtde 34362*	Lemma for tgoldbachgtd 34364. (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 34363*	Lemma for tgoldbachgtd 34364. (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 34364*	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 34365*	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))
21.3.27  Elementary Geometry
21.3.27.1  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 34366	Extends class notation with the class of geometries fulfilling the planarity axioms.
class TarskiG2D
Definition	df-trkg2d 34367*	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 34368*	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 34369*	Alternate version of axtglowdim2 28330. (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 34370	Alternate version of axtgupdim2 28331. (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)))
21.3.27.2  Outer Five Segment (not used, no need to move to main)
Syntax	cafs 34371	Declare the syntax for the outer five segment configuration.
class AFS
Definition	df-afs 34372*	The outer five segment configuration is an abbreviation for the conditions of the Five Segment Axiom (axtg5seg 28325). See df-ofs 35649. 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 34373*	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 34374	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 34375	Rephrase axtg5seg 28325 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))
21.3.28  LeftPad Project See https://github.com/tirix/lets-prove-leftpad.
Syntax	clpad 34376	Extend class notation with the leftpad function.
class leftpad
Definition	df-lpad 34377*	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 34378	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 34379	Lemma for the leftpad theorems. (Contributed by Thierry Arnoux, 7-Aug-2023.)
⊢ (𝜑 → 𝐶 ∈ 𝑆)    ⇒   ⊢ (𝜑 → ((0..^(𝐿 − (♯‘𝑊))) × {𝐶}) ∈ Word 𝑆)
Theorem	lpadlem3 34380	Lemma for lpadlen1 34381. (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 34381	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 34382	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 34383	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 34384	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 34385	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 34386	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 34387	Extend wff notation with the 4-way conjunction. (New usage is discouraged.)
wff (𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃)
Definition	df-bnj17 34388	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 34389	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 34390*	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 34391	Extend wff notation with the following predicate: 𝑅 is set-like on 𝐴. (New usage is discouraged.)
wff 𝑅 Se 𝐴
Definition	df-bnj13 34392*	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 34393	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 34394	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 34395	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 34396*	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 34627. (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 34397	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 34398*	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 34399	∧-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 34400	∧-manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
⊢ (𝜓 → 𝜓′)    &   ⊢ (𝜒 → 𝜒′)    ⇒   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → (𝜓′ ∧ 𝜒′))