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Theorem List for Metamath Proof Explorer - 31801-31900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorembreprexplema 31801* Lemma for breprexp 31804 (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 31802 Lemma for breprexp 31804 (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 31803* Lemma for breprexp 31804 (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 31804* 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 31805 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 31805* 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 31806 The Vinogradov trigonometric sums.
class vts
 
Definitiondf-vts 31807* 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 31808* 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 31809 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 31810* 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 31811* 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 31812* 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 31813* 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 31814* 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 31815* 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 31816 Theorem 12. of [RosserSchoenfeld] p. 71. Theorem chpo1ubb 25985 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 31817 Theorem 13. of [RosserSchoenfeld] p. 71. Theorem chpchtlim 25983 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 31818* 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 31819* 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 31820* A deduction version of ax-hgt749 31815. (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 31821 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 31822 Lemma for tgoldbachgtd 31833. (Contributed by Thierry Arnoux, 17-Dec-2021.)
((𝑁 ∈ ℕ0 ∧ (10↑27) ≤ 𝑁) → ((7.348) · ((log‘𝑁) / (√‘𝑁))) < (0.00042248))
 
Theoremhgt750lem2 31823 Decimal multiplication galore! (Contributed by Thierry Arnoux, 26-Dec-2021.)
(3 · ((((1.079955)↑2) · (1.414)) · ((1.4263) · (1.03883)))) < (7.348)
 
Theoremhgt750lemf 31824* 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 31825* 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 31826* 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 31827* 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 31828* 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 31829* 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 31830* Lemma for tgoldbachgtd 31833. (Contributed by Thierry Arnoux, 15-Dec-2021.)
𝑂 = {𝑧 ∈ ℤ ∣ ¬ 2 ∥ 𝑧}    &   (𝜑𝑁𝑂)    &   (𝜑 → (10↑27) ≤ 𝑁)       (𝜑𝑁 ∈ ℕ)
 
Theoremtgoldbachgtde 31831* Lemma for tgoldbachgtd 31833. (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 31832* Lemma for tgoldbachgtd 31833. (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 31833* 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 31834* 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-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.

 
Syntaxcstrkg2d 31835 Extends class notation with the class of geometries fulfilling the planarity axioms.
class TarskiG2D
 
Definitiondf-trkg2d 31836* 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 31837* 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))))))
 
Theoremaxtglowdim2ALTV 31838* Alternate version of axtglowdim2 26184. (Contributed by Thierry Arnoux, 29-May-2019.) (New usage is discouraged.)
𝑃 = (Base‘𝐺)    &    = (dist‘𝐺)    &   𝐼 = (Itv‘𝐺)    &   (𝜑𝐺 ∈ TarskiG2D)       (𝜑 → ∃𝑥𝑃𝑦𝑃𝑧𝑃 ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)))
 
Theoremaxtgupdim2ALTV 31839 Alternate version of axtgupdim2 26185. (Contributed by Thierry Arnoux, 29-May-2019.) (New usage 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 31840 Declare the syntax for the outer five segment configuration.
class AFS
 
Definitiondf-afs 31841* The outer five segment configuration is an abbreviation for the conditions of the Five Segment Axiom (axtg5seg 26179). See df-ofs 33342. 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) / ][(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 31842* 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 31843 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 31844 Rephrase axtg5seg 26179 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))
 
20.3.28  LeftPad Project

See https://github.com/tirix/lets-prove-leftpad.

 
Syntaxclpad 31845 Extend class notation with the leftpad function.
class leftpad
 
Definitiondf-lpad 31846* Define the leftpad function. (Contributed by Thierry Arnoux, 7-Aug-2023.)
leftpad = (𝑐 ∈ V, 𝑤 ∈ V ↦ (𝑙 ∈ ℕ0 ↦ (((0..^(𝑙 − (♯‘𝑤))) × {𝑐}) ++ 𝑤)))
 
Theoremlpadval 31847 Value of the leftpad function. (Contributed by Thierry Arnoux, 7-Aug-2023.)
(𝜑𝐿 ∈ ℕ0)    &   (𝜑𝑊 ∈ Word 𝑆)    &   (𝜑𝐶𝑆)       (𝜑 → ((𝐶 leftpad 𝑊)‘𝐿) = (((0..^(𝐿 − (♯‘𝑊))) × {𝐶}) ++ 𝑊))
 
Theoremlpadlem1 31848 Lemma for the leftpad theorems. (Contributed by Thierry Arnoux, 7-Aug-2023.)
(𝜑𝐶𝑆)       (𝜑 → ((0..^(𝐿 − (♯‘𝑊))) × {𝐶}) ∈ Word 𝑆)
 
Theoremlpadlem3 31849 Lemma for lpadlen1 31850 (Contributed by Thierry Arnoux, 7-Aug-2023.)
(𝜑𝐿 ∈ ℕ0)    &   (𝜑𝑊 ∈ Word 𝑆)    &   (𝜑𝐶𝑆)    &   (𝜑𝐿 ≤ (♯‘𝑊))       (𝜑 → ((0..^(𝐿 − (♯‘𝑊))) × {𝐶}) = ∅)
 
Theoremlpadlen1 31850 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.)
(𝜑𝐿 ∈ ℕ0)    &   (𝜑𝑊 ∈ Word 𝑆)    &   (𝜑𝐶𝑆)    &   (𝜑𝐿 ≤ (♯‘𝑊))       (𝜑 → (♯‘((𝐶 leftpad 𝑊)‘𝐿)) = (♯‘𝑊))
 
Theoremlpadlem2 31851 Lemma for the leftpad theorems. (Contributed by Thierry Arnoux, 7-Aug-2023.)
(𝜑𝐿 ∈ ℕ0)    &   (𝜑𝑊 ∈ Word 𝑆)    &   (𝜑𝐶𝑆)    &   (𝜑 → (♯‘𝑊) ≤ 𝐿)       (𝜑 → (♯‘((0..^(𝐿 − (♯‘𝑊))) × {𝐶})) = (𝐿 − (♯‘𝑊)))
 
Theoremlpadlen2 31852 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.)
(𝜑𝐿 ∈ ℕ0)    &   (𝜑𝑊 ∈ Word 𝑆)    &   (𝜑𝐶𝑆)    &   (𝜑 → (♯‘𝑊) ≤ 𝐿)       (𝜑 → (♯‘((𝐶 leftpad 𝑊)‘𝐿)) = 𝐿)
 
Theoremlpadmax 31853 Length of a left-padded word, in the general case, expressed with an if statement. (Contributed by Thierry Arnoux, 7-Aug-2023.)
(𝜑𝐿 ∈ ℕ0)    &   (𝜑𝑊 ∈ Word 𝑆)    &   (𝜑𝐶𝑆)       (𝜑 → (♯‘((𝐶 leftpad 𝑊)‘𝐿)) = if(𝐿 ≤ (♯‘𝑊), (♯‘𝑊), 𝐿))
 
Theoremlpadleft 31854 The contents of prefix of a left-padded word is always the letter 𝐶. (Contributed by Thierry Arnoux, 7-Aug-2023.)
(𝜑𝐿 ∈ ℕ0)    &   (𝜑𝑊 ∈ Word 𝑆)    &   (𝜑𝐶𝑆)    &   (𝜑𝑁 ∈ (0..^(𝐿 − (♯‘𝑊))))       (𝜑 → (((𝐶 leftpad 𝑊)‘𝐿)‘𝑁) = 𝐶)
 
Theoremlpadright 31855 The suffix of a left-padded word the original word 𝑊. (Contributed by Thierry Arnoux, 7-Aug-2023.)
(𝜑𝐿 ∈ ℕ0)    &   (𝜑𝑊 ∈ Word 𝑆)    &   (𝜑𝐶𝑆)    &   (𝜑𝑀 = if(𝐿 ≤ (♯‘𝑊), 0, (𝐿 − (♯‘𝑊))))    &   (𝜑𝑁 ∈ (0..^(♯‘𝑊)))       (𝜑 → (((𝐶 leftpad 𝑊)‘𝐿)‘(𝑁 + 𝑀)) = (𝑊𝑁))
 
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 31856 Extend wff notation with the 4-way conjunction. (New usage is discouraged.)
wff (𝜑𝜓𝜒𝜃)
 
Definitiondf-bnj17 31857 Define the 4-way conjunction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝜑𝜓𝜒𝜃) ↔ ((𝜑𝜓𝜒) ∧ 𝜃))
 
Syntaxc-bnj14 31858 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 31859* 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 31860 Extend wff notation with the following predicate: 𝑅 is set-like on 𝐴. (New usage is discouraged.)
wff 𝑅 Se 𝐴
 
Definitiondf-bnj13 31861* 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 31862 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 31863 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 31864 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 31865* 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 32098. (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 31866 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 31867* 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 31868 -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 31869 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜓𝜓′)    &   (𝜒𝜒′)       ((𝜑𝜓𝜒) → (𝜓′𝜒′))
 
Theorembnj248 31870 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝜑𝜓𝜒𝜃) ↔ (((𝜑𝜓) ∧ 𝜒) ∧ 𝜃))
 
Theorembnj250 31871 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝜑𝜓𝜒𝜃) ↔ (𝜑 ∧ ((𝜓𝜒) ∧ 𝜃)))
 
Theorembnj251 31872 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝜑𝜓𝜒𝜃) ↔ (𝜑 ∧ (𝜓 ∧ (𝜒𝜃))))
 
Theorembnj252 31873 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝜑𝜓𝜒𝜃) ↔ (𝜑 ∧ (𝜓𝜒𝜃)))
 
Theorembnj253 31874 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝜑𝜓𝜒𝜃) ↔ ((𝜑𝜓) ∧ 𝜒𝜃))
 
Theorembnj255 31875 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝜑𝜓𝜒𝜃) ↔ (𝜑𝜓 ∧ (𝜒𝜃)))
 
Theorembnj256 31876 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝜑𝜓𝜒𝜃) ↔ ((𝜑𝜓) ∧ (𝜒𝜃)))
 
Theorembnj257 31877 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝜑𝜓𝜒𝜃) ↔ (𝜑𝜓𝜃𝜒))
 
Theorembnj258 31878 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝜑𝜓𝜒𝜃) ↔ ((𝜑𝜓𝜃) ∧ 𝜒))
 
Theorembnj268 31879 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝜑𝜓𝜒𝜃) ↔ (𝜑𝜒𝜓𝜃))
 
Theorembnj290 31880 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝜑𝜓𝜒𝜃) ↔ (𝜑𝜒𝜃𝜓))
 
Theorembnj291 31881 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝜑𝜓𝜒𝜃) ↔ ((𝜑𝜒𝜃) ∧ 𝜓))
 
Theorembnj312 31882 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝜑𝜓𝜒𝜃) ↔ (𝜓𝜑𝜒𝜃))
 
Theorembnj334 31883 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) (New usage is discouraged.)
((𝜑𝜓𝜒𝜃) ↔ (𝜒𝜑𝜓𝜃))
 
Theorembnj345 31884 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) (New usage is discouraged.)
((𝜑𝜓𝜒𝜃) ↔ (𝜃𝜑𝜓𝜒))
 
Theorembnj422 31885 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) (New usage is discouraged.)
((𝜑𝜓𝜒𝜃) ↔ (𝜒𝜃𝜑𝜓))
 
Theorembnj432 31886 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝜑𝜓𝜒𝜃) ↔ ((𝜒𝜃) ∧ (𝜑𝜓)))
 
Theorembnj446 31887 -manipulation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
((𝜑𝜓𝜒𝜃) ↔ ((𝜓𝜒𝜃) ∧ 𝜑))
 
Theorembnj23 31888* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Mario Carneiro, 22-Dec-2016.) (New usage is discouraged.)
𝐵 = {𝑥𝐴 ∣ ¬ 𝜑}       (∀𝑧𝐵 ¬ 𝑧𝑅𝑦 → ∀𝑤𝐴 (𝑤𝑅𝑦[𝑤 / 𝑥]𝜑))
 
Theorembnj31 31889 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 → ∃𝑥𝐴 𝜓)    &   (𝜓𝜒)       (𝜑 → ∃𝑥𝐴 𝜒)
 
Theorembnj62 31890* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
([𝑧 / 𝑥]𝑥 Fn 𝐴𝑧 Fn 𝐴)
 
Theorembnj89 31891* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝑍 ∈ V       ([𝑍 / 𝑦]∃!𝑥𝜑 ↔ ∃!𝑥[𝑍 / 𝑦]𝜑)
 
Theorembnj90 31892* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Mario Carneiro, 22-Dec-2016.) (New usage is discouraged.)
𝑌 ∈ V       ([𝑌 / 𝑥]𝑧 Fn 𝑥𝑧 Fn 𝑌)
 
Theorembnj101 31893 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝑥𝜑    &   (𝜑𝜓)       𝑥𝜓
 
Theorembnj105 31894 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
1o ∈ V
 
Theorembnj115 31895 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜂 ↔ ∀𝑛𝐷 (𝜏𝜃))       (𝜂 ↔ ∀𝑛((𝑛𝐷𝜏) → 𝜃))
 
Theorembnj132 31896* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) (New usage is discouraged.)
(𝜑 ↔ ∃𝑥(𝜓𝜒))       (𝜑 ↔ (𝜓 → ∃𝑥𝜒))
 
Theorembnj133 31897 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜑 ↔ ∃𝑥𝜓)    &   (𝜒𝜓)       (𝜑 ↔ ∃𝑥𝜒)
 
Theorembnj156 31898 First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
(𝜁0 ↔ (𝑓 Fn 1o𝜑′𝜓′))    &   (𝜁1[𝑔 / 𝑓]𝜁0)    &   (𝜑1[𝑔 / 𝑓]𝜑′)    &   (𝜓1[𝑔 / 𝑓]𝜓′)       (𝜁1 ↔ (𝑔 Fn 1o𝜑1𝜓1))
 
Theorembnj158 31899* First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
𝐷 = (ω ∖ {∅})       (𝑚𝐷 → ∃𝑝 ∈ ω 𝑚 = suc 𝑝)
 
Theorembnj168 31900* First-order logic and set theory. Revised to remove dependence on ax-reg 9045. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Revised by NM, 21-Dec-2016.) (New usage is discouraged.)
𝐷 = (ω ∖ {∅})       ((𝑛 ≠ 1o𝑛𝐷) → ∃𝑚𝐷 𝑛 = suc 𝑚)
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