P. 488 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 48701-48800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	mogoldbb 48701*	If the modern version of the original formulation of the Goldbach conjecture is valid, the (weak) binary Goldbach conjecture also holds. (Contributed by AV, 26-Dec-2021.)
⊢ (∀𝑛 ∈ (ℤ≥‘6)∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟) → ∀𝑛 ∈ Even (2 < 𝑛 → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞)))
Theorem	sbgoldbmb 48702*	The strong binary Goldbach conjecture and the modern version of the original formulation of the Goldbach conjecture are equivalent. (Contributed by AV, 26-Dec-2021.)
⊢ (∀𝑛 ∈ Even (4 < 𝑛 → 𝑛 ∈ GoldbachEven ) ↔ ∀𝑛 ∈ (ℤ≥‘6)∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟))
Theorem	sbgoldbo 48703*	If the strong binary Goldbach conjecture is valid, the original formulation of the Goldbach conjecture also holds: Every integer greater than 2 can be expressed as the sum of three "primes" with regarding 1 to be a prime (as Goldbach did). Original text: "Es scheint wenigstens, dass eine jede Zahl, die groesser ist als 2, ein aggregatum trium numerorum primorum sey." (Goldbach, 1742). (Contributed by AV, 25-Dec-2021.)
⊢ 𝑃 = ({1} ∪ ℙ)    ⇒   ⊢ (∀𝑛 ∈ Even (4 < 𝑛 → 𝑛 ∈ GoldbachEven ) → ∀𝑛 ∈ (ℤ≥‘3)∃𝑝 ∈ 𝑃 ∃𝑞 ∈ 𝑃 ∃𝑟 ∈ 𝑃 𝑛 = ((𝑝 + 𝑞) + 𝑟))
Theorem	nnsum3primes4 48704*	4 is the sum of at most 3 (actually 2) primes. (Contributed by AV, 2-Aug-2020.)
⊢ ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑m (1...𝑑))(𝑑 ≤ 3 ∧ 4 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘))
Theorem	nnsum4primes4 48705*	4 is the sum of at most 4 (actually 2) primes. (Contributed by AV, 23-Jul-2020.) (Proof shortened by AV, 2-Aug-2020.)
⊢ ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑m (1...𝑑))(𝑑 ≤ 4 ∧ 4 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘))
Theorem	nnsum3primesprm 48706*	Every prime is "the sum of at most 3" (actually one - the prime itself) primes. (Contributed by AV, 2-Aug-2020.) (Proof shortened by AV, 17-Apr-2021.)
⊢ (𝑃 ∈ ℙ → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑m (1...𝑑))(𝑑 ≤ 3 ∧ 𝑃 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘)))
Theorem	nnsum4primesprm 48707*	Every prime is "the sum of at most 4" (actually one - the prime itself) primes. (Contributed by AV, 23-Jul-2020.) (Proof shortened by AV, 2-Aug-2020.)
⊢ (𝑃 ∈ ℙ → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑m (1...𝑑))(𝑑 ≤ 4 ∧ 𝑃 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘)))
Theorem	nnsum3primesgbe 48708*	Any even Goldbach number is the sum of at most 3 (actually 2) primes. (Contributed by AV, 2-Aug-2020.)
⊢ (𝑁 ∈ GoldbachEven → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑m (1...𝑑))(𝑑 ≤ 3 ∧ 𝑁 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘)))
Theorem	nnsum4primesgbe 48709*	Any even Goldbach number is the sum of at most 4 (actually 2) primes. (Contributed by AV, 23-Jul-2020.) (Proof shortened by AV, 2-Aug-2020.)
⊢ (𝑁 ∈ GoldbachEven → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑m (1...𝑑))(𝑑 ≤ 4 ∧ 𝑁 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘)))
Theorem	nnsum3primesle9 48710*	Every integer greater than 1 and less than or equal to 8 is the sum of at most 3 primes. (Contributed by AV, 2-Aug-2020.)
⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑁 ≤ 8) → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑m (1...𝑑))(𝑑 ≤ 3 ∧ 𝑁 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘)))
Theorem	nnsum4primesle9 48711*	Every integer greater than 1 and less than or equal to 8 is the sum of at most 4 primes. (Contributed by AV, 24-Jul-2020.) (Proof shortened by AV, 2-Aug-2020.)
⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑁 ≤ 8) → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑m (1...𝑑))(𝑑 ≤ 4 ∧ 𝑁 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘)))
Theorem	nnsum4primesodd 48712*	If the (weak) ternary Goldbach conjecture is valid, then every odd integer greater than 5 is the sum of 3 primes. (Contributed by AV, 2-Jul-2020.)
⊢ (∀𝑚 ∈ Odd (5 < 𝑚 → 𝑚 ∈ GoldbachOddW ) → ((𝑁 ∈ (ℤ≥‘6) ∧ 𝑁 ∈ Odd ) → ∃𝑓 ∈ (ℙ ↑m (1...3))𝑁 = Σ𝑘 ∈ (1...3)(𝑓‘𝑘)))
Theorem	nnsum4primesoddALTV 48713*	If the (strong) ternary Goldbach conjecture is valid, then every odd integer greater than 7 is the sum of 3 primes. (Contributed by AV, 26-Jul-2020.)
⊢ (∀𝑚 ∈ Odd (7 < 𝑚 → 𝑚 ∈ GoldbachOdd ) → ((𝑁 ∈ (ℤ≥‘8) ∧ 𝑁 ∈ Odd ) → ∃𝑓 ∈ (ℙ ↑m (1...3))𝑁 = Σ𝑘 ∈ (1...3)(𝑓‘𝑘)))
Theorem	evengpop3 48714*	If the (weak) ternary Goldbach conjecture is valid, then every even integer greater than 8 is the sum of an odd Goldbach number and 3. (Contributed by AV, 24-Jul-2020.)
⊢ (∀𝑚 ∈ Odd (5 < 𝑚 → 𝑚 ∈ GoldbachOddW ) → ((𝑁 ∈ (ℤ≥‘9) ∧ 𝑁 ∈ Even ) → ∃𝑜 ∈ GoldbachOddW 𝑁 = (𝑜 + 3)))
Theorem	evengpoap3 48715*	If the (strong) ternary Goldbach conjecture is valid, then every even integer greater than 10 is the sum of an odd Goldbach number and 3. (Contributed by AV, 27-Jul-2020.) (Proof shortened by AV, 15-Sep-2021.)
⊢ (∀𝑚 ∈ Odd (7 < 𝑚 → 𝑚 ∈ GoldbachOdd ) → ((𝑁 ∈ (ℤ≥‘;12) ∧ 𝑁 ∈ Even ) → ∃𝑜 ∈ GoldbachOdd 𝑁 = (𝑜 + 3)))
Theorem	nnsum4primeseven 48716*	If the (weak) ternary Goldbach conjecture is valid, then every even integer greater than 8 is the sum of 4 primes. (Contributed by AV, 25-Jul-2020.)
⊢ (∀𝑚 ∈ Odd (5 < 𝑚 → 𝑚 ∈ GoldbachOddW ) → ((𝑁 ∈ (ℤ≥‘9) ∧ 𝑁 ∈ Even ) → ∃𝑓 ∈ (ℙ ↑m (1...4))𝑁 = Σ𝑘 ∈ (1...4)(𝑓‘𝑘)))
Theorem	nnsum4primesevenALTV 48717*	If the (strong) ternary Goldbach conjecture is valid, then every even integer greater than 10 is the sum of 4 primes. (Contributed by AV, 27-Jul-2020.)
⊢ (∀𝑚 ∈ Odd (7 < 𝑚 → 𝑚 ∈ GoldbachOdd ) → ((𝑁 ∈ (ℤ≥‘;12) ∧ 𝑁 ∈ Even ) → ∃𝑓 ∈ (ℙ ↑m (1...4))𝑁 = Σ𝑘 ∈ (1...4)(𝑓‘𝑘)))
Theorem	wtgoldbnnsum4prm 48718*	If the (weak) ternary Goldbach conjecture is valid, then every integer greater than 1 is the sum of at most 4 primes, showing that Schnirelmann's constant would be less than or equal to 4. See corollary 1.1 in [Helfgott] p. 4. (Contributed by AV, 25-Jul-2020.)
⊢ (∀𝑚 ∈ Odd (5 < 𝑚 → 𝑚 ∈ GoldbachOddW ) → ∀𝑛 ∈ (ℤ≥‘2)∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑m (1...𝑑))(𝑑 ≤ 4 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘)))
Theorem	stgoldbnnsum4prm 48719*	If the (strong) ternary Goldbach conjecture is valid, then every integer greater than 1 is the sum of at most 4 primes. (Contributed by AV, 27-Jul-2020.)
⊢ (∀𝑚 ∈ Odd (7 < 𝑚 → 𝑚 ∈ GoldbachOdd ) → ∀𝑛 ∈ (ℤ≥‘2)∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑m (1...𝑑))(𝑑 ≤ 4 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘)))
Theorem	bgoldbnnsum3prm 48720*	If the binary Goldbach conjecture is valid, then every integer greater than 1 is the sum of at most 3 primes, showing that Schnirelmann's constant would be equal to 3. (Contributed by AV, 2-Aug-2020.)
⊢ (∀𝑚 ∈ Even (4 < 𝑚 → 𝑚 ∈ GoldbachEven ) → ∀𝑛 ∈ (ℤ≥‘2)∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑m (1...𝑑))(𝑑 ≤ 3 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘)))
Theorem	bgoldbtbndlem1 48721	Lemma 1 for bgoldbtbnd 48725: the odd numbers between 7 and 13 (exclusive) are odd Goldbach numbers. (Contributed by AV, 29-Jul-2020.)
⊢ ((𝑁 ∈ Odd ∧ 7 < 𝑁 ∧ 𝑁 ∈ (7[,);13)) → 𝑁 ∈ GoldbachOdd )
Theorem	bgoldbtbndlem2 48722*	Lemma 2 for bgoldbtbnd 48725. (Contributed by AV, 1-Aug-2020.)
⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘;11))    &   ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘;11))    &   ⊢ (𝜑 → ∀𝑛 ∈ Even ((4 < 𝑛 ∧ 𝑛 < 𝑁) → 𝑛 ∈ GoldbachEven ))    &   ⊢ (𝜑 → 𝐷 ∈ (ℤ≥‘3))    &   ⊢ (𝜑 → 𝐹 ∈ (RePart‘𝐷))    &   ⊢ (𝜑 → ∀𝑖 ∈ (0..^𝐷)((𝐹‘𝑖) ∈ (ℙ ∖ {2}) ∧ ((𝐹‘(𝑖 + 1)) − (𝐹‘𝑖)) < (𝑁 − 4) ∧ 4 < ((𝐹‘(𝑖 + 1)) − (𝐹‘𝑖))))    &   ⊢ (𝜑 → (𝐹‘0) = 7)    &   ⊢ (𝜑 → (𝐹‘1) = ;13)    &   ⊢ (𝜑 → 𝑀 < (𝐹‘𝐷))    &   ⊢ 𝑆 = (𝑋 − (𝐹‘(𝐼 − 1)))    ⇒   ⊢ ((𝜑 ∧ 𝑋 ∈ Odd ∧ 𝐼 ∈ (1..^𝐷)) → ((𝑋 ∈ ((𝐹‘𝐼)[,)(𝐹‘(𝐼 + 1))) ∧ (𝑋 − (𝐹‘𝐼)) ≤ 4) → (𝑆 ∈ Even ∧ 𝑆 < 𝑁 ∧ 4 < 𝑆)))
Theorem	bgoldbtbndlem3 48723*	Lemma 3 for bgoldbtbnd 48725. (Contributed by AV, 1-Aug-2020.)
⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘;11))    &   ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘;11))    &   ⊢ (𝜑 → ∀𝑛 ∈ Even ((4 < 𝑛 ∧ 𝑛 < 𝑁) → 𝑛 ∈ GoldbachEven ))    &   ⊢ (𝜑 → 𝐷 ∈ (ℤ≥‘3))    &   ⊢ (𝜑 → 𝐹 ∈ (RePart‘𝐷))    &   ⊢ (𝜑 → ∀𝑖 ∈ (0..^𝐷)((𝐹‘𝑖) ∈ (ℙ ∖ {2}) ∧ ((𝐹‘(𝑖 + 1)) − (𝐹‘𝑖)) < (𝑁 − 4) ∧ 4 < ((𝐹‘(𝑖 + 1)) − (𝐹‘𝑖))))    &   ⊢ (𝜑 → (𝐹‘0) = 7)    &   ⊢ (𝜑 → (𝐹‘1) = ;13)    &   ⊢ (𝜑 → 𝑀 < (𝐹‘𝐷))    &   ⊢ (𝜑 → (𝐹‘𝐷) ∈ ℝ)    &   ⊢ 𝑆 = (𝑋 − (𝐹‘𝐼))    ⇒   ⊢ ((𝜑 ∧ 𝑋 ∈ Odd ∧ 𝐼 ∈ (1..^𝐷)) → ((𝑋 ∈ ((𝐹‘𝐼)[,)(𝐹‘(𝐼 + 1))) ∧ 4 < 𝑆) → (𝑆 ∈ Even ∧ 𝑆 < 𝑁 ∧ 4 < 𝑆)))
Theorem	bgoldbtbndlem4 48724*	Lemma 4 for bgoldbtbnd 48725. (Contributed by AV, 1-Aug-2020.)
⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘;11))    &   ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘;11))    &   ⊢ (𝜑 → ∀𝑛 ∈ Even ((4 < 𝑛 ∧ 𝑛 < 𝑁) → 𝑛 ∈ GoldbachEven ))    &   ⊢ (𝜑 → 𝐷 ∈ (ℤ≥‘3))    &   ⊢ (𝜑 → 𝐹 ∈ (RePart‘𝐷))    &   ⊢ (𝜑 → ∀𝑖 ∈ (0..^𝐷)((𝐹‘𝑖) ∈ (ℙ ∖ {2}) ∧ ((𝐹‘(𝑖 + 1)) − (𝐹‘𝑖)) < (𝑁 − 4) ∧ 4 < ((𝐹‘(𝑖 + 1)) − (𝐹‘𝑖))))    &   ⊢ (𝜑 → (𝐹‘0) = 7)    &   ⊢ (𝜑 → (𝐹‘1) = ;13)    &   ⊢ (𝜑 → 𝑀 < (𝐹‘𝐷))    &   ⊢ (𝜑 → (𝐹‘𝐷) ∈ ℝ)    ⇒   ⊢ (((𝜑 ∧ 𝐼 ∈ (1..^𝐷)) ∧ 𝑋 ∈ Odd ) → ((𝑋 ∈ ((𝐹‘𝐼)[,)(𝐹‘(𝐼 + 1))) ∧ (𝑋 − (𝐹‘𝐼)) ≤ 4) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ ((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑟 ∈ Odd ) ∧ 𝑋 = ((𝑝 + 𝑞) + 𝑟))))
Theorem	bgoldbtbnd 48725*	If the binary Goldbach conjecture is valid up to an integer 𝑁, and there is a series ("ladder") of primes with a difference of at most 𝑁 up to an integer 𝑀, then the strong ternary Goldbach conjecture is valid up to 𝑀, see section 1.2.2 in [Helfgott] p. 4 with N = 4 x 10^18, taken from [OeSilva], and M = 8.875 x 10^30. (Contributed by AV, 1-Aug-2020.)
⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘;11))    &   ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘;11))    &   ⊢ (𝜑 → ∀𝑛 ∈ Even ((4 < 𝑛 ∧ 𝑛 < 𝑁) → 𝑛 ∈ GoldbachEven ))    &   ⊢ (𝜑 → 𝐷 ∈ (ℤ≥‘3))    &   ⊢ (𝜑 → 𝐹 ∈ (RePart‘𝐷))    &   ⊢ (𝜑 → ∀𝑖 ∈ (0..^𝐷)((𝐹‘𝑖) ∈ (ℙ ∖ {2}) ∧ ((𝐹‘(𝑖 + 1)) − (𝐹‘𝑖)) < (𝑁 − 4) ∧ 4 < ((𝐹‘(𝑖 + 1)) − (𝐹‘𝑖))))    &   ⊢ (𝜑 → (𝐹‘0) = 7)    &   ⊢ (𝜑 → (𝐹‘1) = ;13)    &   ⊢ (𝜑 → 𝑀 < (𝐹‘𝐷))    &   ⊢ (𝜑 → (𝐹‘𝐷) ∈ ℝ)    ⇒   ⊢ (𝜑 → ∀𝑛 ∈ Odd ((7 < 𝑛 ∧ 𝑛 < 𝑀) → 𝑛 ∈ GoldbachOdd ))
Axiom	ax-bgbltosilva 48726	The binary Goldbach conjecture is valid for all even numbers less than or equal to 4x10^18, see section 2 in [OeSilva] p. 2042. Temporarily provided as "axiom". (Contributed by AV, 3-Aug-2020.) (Revised by AV, 9-Sep-2021.)
⊢ ((𝑁 ∈ Even ∧ 4 < 𝑁 ∧ 𝑁 ≤ (4 · (;10↑;18))) → 𝑁 ∈ GoldbachEven )
Axiom	ax-tgoldbachgt 48727*	Temporary duplicate of tgoldbachgt 35171, provided as "axiom" as long as this theorem is in the mathbox of Thierry Arnoux: 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))
Theorem	tgoldbachgtALTV 48728*	Variant of Thierry Arnoux's tgoldbachgt 35171 using the symbols Odd and GoldbachOdd: The ternary Goldbach conjecture is valid for large odd numbers (i.e. for all odd numbers greater than a fixed 𝑚). This is proven by Helfgott (see section 7.4 in [Helfgott] p. 70) for 𝑚 = 10^27. (Contributed by AV, 2-Aug-2020.) (Revised by AV, 15-Jan-2022.)
⊢ ∃𝑚 ∈ ℕ (𝑚 ≤ (;10↑;27) ∧ ∀𝑛 ∈ Odd (𝑚 < 𝑛 → 𝑛 ∈ GoldbachOdd ))
Theorem	bgoldbachlt 48729*	The binary Goldbach conjecture is valid for small even numbers (i.e. for all even numbers less than or equal to a fixed big 𝑚). This is verified for m = 4 x 10^18 by Oliveira e Silva, see ax-bgbltosilva 48726. (Contributed by AV, 3-Aug-2020.) (Revised by AV, 9-Sep-2021.)
⊢ ∃𝑚 ∈ ℕ ((4 · (;10↑;18)) ≤ 𝑚 ∧ ∀𝑛 ∈ Even ((4 < 𝑛 ∧ 𝑛 < 𝑚) → 𝑛 ∈ GoldbachEven ))
Axiom	ax-hgprmladder 48730	There is a partition ("ladder") of primes from 7 to 8.8 x 10^30 with parts ("rungs") having lengths of at least 4 and at most N - 4, see section 1.2.2 in [Helfgott] p. 4. Temporarily provided as "axiom". (Contributed by AV, 3-Aug-2020.) (Revised by AV, 9-Sep-2021.)
⊢ ∃𝑑 ∈ (ℤ≥‘3)∃𝑓 ∈ (RePart‘𝑑)(((𝑓‘0) = 7 ∧ (𝑓‘1) = ;13 ∧ (𝑓‘𝑑) = (;89 · (;10↑;29))) ∧ ∀𝑖 ∈ (0..^𝑑)((𝑓‘𝑖) ∈ (ℙ ∖ {2}) ∧ ((𝑓‘(𝑖 + 1)) − (𝑓‘𝑖)) < ((4 · (;10↑;18)) − 4) ∧ 4 < ((𝑓‘(𝑖 + 1)) − (𝑓‘𝑖))))
Theorem	tgblthelfgott 48731	The ternary Goldbach conjecture is valid for all odd numbers less than 8.8 x 10^30 (actually 8.875694 x 10^30, see section 1.2.2 in [Helfgott] p. 4, using bgoldbachlt 48729, ax-hgprmladder 48730 and bgoldbtbnd 48725. (Contributed by AV, 4-Aug-2020.) (Revised by AV, 9-Sep-2021.)
⊢ ((𝑁 ∈ Odd ∧ 7 < 𝑁 ∧ 𝑁 < (;88 · (;10↑;29))) → 𝑁 ∈ GoldbachOdd )
Theorem	tgoldbachlt 48732*	The ternary Goldbach conjecture is valid for small odd numbers (i.e. for all odd numbers less than a fixed big 𝑚 greater than 8 x 10^30). This is verified for m = 8.875694 x 10^30 by Helfgott, see tgblthelfgott 48731. (Contributed by AV, 4-Aug-2020.) (Revised by AV, 9-Sep-2021.)
⊢ ∃𝑚 ∈ ℕ ((8 · (;10↑;30)) < 𝑚 ∧ ∀𝑛 ∈ Odd ((7 < 𝑛 ∧ 𝑛 < 𝑚) → 𝑛 ∈ GoldbachOdd ))
Theorem	tgoldbach 48733	The ternary Goldbach conjecture is valid. Main theorem in [Helfgott] p. 2. This follows from tgoldbachlt 48732 and ax-tgoldbachgt 48727. (Contributed by AV, 2-Aug-2020.) (Revised by AV, 9-Sep-2021.)
⊢ ∀𝑛 ∈ Odd (7 < 𝑛 → 𝑛 ∈ GoldbachOdd )
21.50.15  Graph theory (extension)
21.50.15.1  Closed neighborhood of a vertex
Syntax	cclnbgr 48734	Extend class notation with closed neighborhoods (of a vertex in a graph).
class ClNeighbVtx
Definition	df-clnbgr 48735*	Define the closed neighborhood resp. the class of all neighbors of a vertex (in a graph) and the vertex itself, see definition in section I.1 of [Bollobas] p. 3. The closed neighborhood of a vertex is the set of all vertices which are connected with this vertex by an edge and the vertex itself (in contrast to an open neighborhood, see df-nbgr 29793). Alternatively, a closed neighborhood of a vertex could have been defined as its open neighborhood enhanced by the vertex itself, see dfclnbgr4 48740. This definition is applicable even for arbitrary hypergraphs. (Contributed by AV, 7-May-2025.)
⊢ ClNeighbVtx = (𝑔 ∈ V, 𝑣 ∈ (Vtx‘𝑔) ↦ ({𝑣} ∪ {𝑛 ∈ (Vtx‘𝑔) ∣ ∃𝑒 ∈ (Edg‘𝑔){𝑣, 𝑛} ⊆ 𝑒}))
Theorem	clnbgrprc0 48736	The closed neighborhood is empty if the graph 𝐺 or the vertex 𝑁 are proper classes. (Contributed by AV, 7-May-2025.)
⊢ (¬ (𝐺 ∈ V ∧ 𝑁 ∈ V) → (𝐺 ClNeighbVtx 𝑁) = ∅)
Theorem	clnbgrcl 48737	If a class 𝑋 has at least one element in its closed neighborhood, this class must be a vertex. (Contributed by AV, 7-May-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (𝑁 ∈ (𝐺 ClNeighbVtx 𝑋) → 𝑋 ∈ 𝑉)
Theorem	clnbgrval 48738*	The closed neighborhood of a vertex 𝑉 in a graph 𝐺. (Contributed by AV, 7-May-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ (𝑁 ∈ 𝑉 → (𝐺 ClNeighbVtx 𝑁) = ({𝑁} ∪ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒}))
Theorem	dfclnbgr2 48739*	Alternate definition of the closed neighborhood of a vertex breaking up the subset relationship of an unordered pair. (Contributed by AV, 7-May-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ (𝑁 ∈ 𝑉 → (𝐺 ClNeighbVtx 𝑁) = ({𝑁} ∪ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)}))
Theorem	dfclnbgr4 48740	Alternate definition of the closed neighborhood of a vertex as union of the vertex with its open neighborhood. (Contributed by AV, 8-May-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (𝑁 ∈ 𝑉 → (𝐺 ClNeighbVtx 𝑁) = ({𝑁} ∪ (𝐺 NeighbVtx 𝑁)))
Theorem	elclnbgrelnbgr 48741	An element of the closed neighborhood of a vertex which is not the vertex itself is an element of the open neighborhood of the vertex. (Contributed by AV, 24-Sep-2025.)
⊢ ((𝑋 ∈ (𝐺 ClNeighbVtx 𝑁) ∧ 𝑋 ≠ 𝑁) → 𝑋 ∈ (𝐺 NeighbVtx 𝑁))
Theorem	dfclnbgr3 48742*	Alternate definition of the closed neighborhood of a vertex using the edge function instead of the edges themselves (see also clnbgrval 48738). (Contributed by AV, 8-May-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    ⇒   ⊢ ((𝑁 ∈ 𝑉 ∧ Fun 𝐼) → (𝐺 ClNeighbVtx 𝑁) = ({𝑁} ∪ {𝑛 ∈ 𝑉 ∣ ∃𝑖 ∈ dom 𝐼{𝑁, 𝑛} ⊆ (𝐼‘𝑖)}))
Theorem	clnbgrnvtx0 48743	If a class 𝑋 is not a vertex of a graph 𝐺, then it has an empty closed neighborhood in 𝐺. (Contributed by AV, 8-May-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (𝑋 ∉ 𝑉 → (𝐺 ClNeighbVtx 𝑋) = ∅)
Theorem	clnbgrel 48744*	Characterization of a member 𝑁 of the closed neighborhood of a vertex 𝑋 in a graph 𝐺. (Contributed by AV, 9-May-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ (𝑁 ∈ (𝐺 ClNeighbVtx 𝑋) ↔ ((𝑁 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉) ∧ (𝑁 = 𝑋 ∨ ∃𝑒 ∈ 𝐸 {𝑋, 𝑁} ⊆ 𝑒)))
Theorem	clnbgrvtxel 48745	Every vertex 𝐾 is a member of its closed neighborhood. (Contributed by AV, 10-May-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (𝐾 ∈ 𝑉 → 𝐾 ∈ (𝐺 ClNeighbVtx 𝐾))
Theorem	clnbgrisvtx 48746	Every member 𝑁 of the closed neighborhood of a vertex 𝐾 is a vertex. (Contributed by AV, 9-May-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (𝑁 ∈ (𝐺 ClNeighbVtx 𝐾) → 𝑁 ∈ 𝑉)
Theorem	clnbgrssvtx 48747	The closed neighborhood of a vertex 𝐾 in a graph is a subset of all vertices of the graph. (Contributed by AV, 9-May-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (𝐺 ClNeighbVtx 𝐾) ⊆ 𝑉
Theorem	clnbgrn0 48748	The closed neighborhood of a vertex is never empty. (Contributed by AV, 16-May-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (𝑁 ∈ 𝑉 → (𝐺 ClNeighbVtx 𝑁) ≠ ∅)
Theorem	clnbupgr 48749*	The closed neighborhood of a vertex in a pseudograph. (Contributed by AV, 10-May-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → (𝐺 ClNeighbVtx 𝑁) = ({𝑁} ∪ {𝑛 ∈ 𝑉 ∣ {𝑁, 𝑛} ∈ 𝐸}))
Theorem	clnbupgrel 48750	A member of the closed neighborhood of a vertex in a pseudograph. (Contributed by AV, 10-May-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ UPGraph ∧ 𝐾 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉) → (𝑁 ∈ (𝐺 ClNeighbVtx 𝐾) ↔ (𝑁 = 𝐾 ∨ {𝑁, 𝐾} ∈ 𝐸)))
Theorem	clnbupgreli 48751	A member of the closed neighborhood of a vertex in a pseudograph. (Contributed by AV, 28-Dec-2025.)
⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ (𝐺 ClNeighbVtx 𝐾)) → (𝑁 = 𝐾 ∨ {𝑁, 𝐾} ∈ 𝐸))
Theorem	clnbgr0vtx 48752	In a null graph (with no vertices), all closed neighborhoods are empty. (Contributed by AV, 15-Nov-2020.)
⊢ ((Vtx‘𝐺) = ∅ → (𝐺 ClNeighbVtx 𝐾) = ∅)
Theorem	clnbgr0edg 48753	In an empty graph (with no edges), all closed neighborhoods consists of a single vertex. (Contributed by AV, 10-May-2025.)
⊢ (((Edg‘𝐺) = ∅ ∧ 𝐾 ∈ (Vtx‘𝐺)) → (𝐺 ClNeighbVtx 𝐾) = {𝐾})
Theorem	clnbgrsym 48754	In a graph, the closed neighborhood relation is symmetric: a vertex 𝑁 in a graph 𝐺 is a neighbor of a second vertex 𝐾 iff the second vertex 𝐾 is a neighbor of the first vertex 𝑁. (Contributed by AV, 10-May-2025.)
⊢ (𝑁 ∈ (𝐺 ClNeighbVtx 𝐾) ↔ 𝐾 ∈ (𝐺 ClNeighbVtx 𝑁))
Theorem	predgclnbgrel 48755	If a (not necessarily proper) unordered pair containing a vertex is an edge, the other vertex is in the closed neighborhood of the first vertex. (Contributed by AV, 23-Aug-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉 ∧ {𝑋, 𝑁} ∈ 𝐸) → 𝑁 ∈ (𝐺 ClNeighbVtx 𝑋))
Theorem	clnbgredg 48756	A vertex connected by an edge with another vertex is a neighbor of that vertex. (Contributed by AV, 24-Aug-2025.)
⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝑁 = (𝐺 ClNeighbVtx 𝑋)    ⇒   ⊢ ((𝐺 ∈ UHGraph ∧ (𝐾 ∈ 𝐸 ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐾)) → 𝑌 ∈ 𝑁)
Theorem	clnbgrssedg 48757	The vertices connected by an edge are a subset of the neighborhood of each of these vertices. (Contributed by AV, 26-May-2025.) (Proof shortened by AV, 24-Aug-2025.)
⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝑁 = (𝐺 ClNeighbVtx 𝑋)    ⇒   ⊢ ((𝐺 ∈ UHGraph ∧ 𝐾 ∈ 𝐸 ∧ 𝑋 ∈ 𝐾) → 𝐾 ⊆ 𝑁)
Theorem	edgusgrclnbfin 48758*	The size of the closed neighborhood of a vertex in a simple graph is finite iff the number of edges having this vertex as endpoint is finite. (Contributed by AV, 10-May-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ USGraph ∧ 𝑈 ∈ 𝑉) → ((𝐺 ClNeighbVtx 𝑈) ∈ Fin ↔ {𝑒 ∈ 𝐸 ∣ 𝑈 ∈ 𝑒} ∈ Fin))
Theorem	clnbusgrfi 48759	The closed neighborhood of a vertex in a simple graph with a finite number of edges is a finite set. (Contributed by AV, 10-May-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ USGraph ∧ 𝐸 ∈ Fin ∧ 𝑈 ∈ 𝑉) → (𝐺 ClNeighbVtx 𝑈) ∈ Fin)
Theorem	clnbfiusgrfi 48760	The closed neighborhood of a vertex in a finite simple graph is a finite set. (Contributed by AV, 10-May-2025.)
⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ (Vtx‘𝐺)) → (𝐺 ClNeighbVtx 𝑁) ∈ Fin)
Theorem	clnbgrlevtx 48761	The size of the closed neighborhood of a vertex is at most the number of vertices of a graph. (Contributed by AV, 10-May-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (♯‘(𝐺 ClNeighbVtx 𝑈)) ≤ (♯‘𝑉)
21.50.15.2  Semiclosed and semiopen neighborhoods (experimental) We have already definitions for open and closed neighborhoods of a vertex, which differs only in the fact that the first never contains the vertex, and the latter always contains the vertex. One of these definitions, however, cannot be simply derived from the other. This would be possible if a definition of a semiclosed neighborhood was available, see dfsclnbgr2 48762. The definitions for open and closed neighborhoods could be derived from such a more simple, but otherwise probably useless definition, see dfnbgr5 48767 and dfclnbgr5 48766. Depending on the existence of certain edges, a vertex belongs to its semiclosed neighborhood or not. An alternate approach is to introduce semiopen neighborhoods, see dfvopnbgr2 48769. The definitions for open and closed neighborhoods could also be derived from such a definition, see dfnbgr6 48773 and dfclnbgr6 48772. Like with semiclosed neighborhood, depending on the existence of certain edges, a vertex belongs to its semiopen neighborhood or not. It is unclear if either definition is/will be useful, and in contrast to dfsclnbgr2 48762, the definition of semiopen neighborhoods is much more complex.
Theorem	dfsclnbgr2 48762*	Alternate definition of the semiclosed neighborhood of a vertex breaking up the subset relationship of an unordered pair. A semiclosed neighborhood 𝑆 of a vertex 𝑁 is the set of all vertices incident with edges which join the vertex 𝑁 with a vertex. Therefore, a vertex is contained in its semiclosed neighborhood if it is connected with any vertex by an edge (see sclnbgrelself 48764), even only with itself (i.e., by a loop). (Contributed by AV, 16-May-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝑆 = {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒}    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ (𝑁 ∈ 𝑉 → 𝑆 = {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)})
Theorem	sclnbgrel 48763*	Characterization of a member 𝑋 of the semiclosed neighborhood of a vertex 𝑁 in a graph 𝐺. (Contributed by AV, 16-May-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝑆 = {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒}    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ (𝑋 ∈ 𝑆 ↔ (𝑋 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 {𝑁, 𝑋} ⊆ 𝑒))
Theorem	sclnbgrelself 48764*	A vertex 𝑁 is a member of its semiclosed neighborhood iff there is an edge joining the vertex with a vertex. (Contributed by AV, 16-May-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝑆 = {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒}    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ (𝑁 ∈ 𝑆 ↔ (𝑁 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 𝑁 ∈ 𝑒))
Theorem	sclnbgrisvtx 48765*	Every member 𝑋 of the semiclosed neighborhood of a vertex 𝑁 is a vertex. (Contributed by AV, 16-May-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝑆 = {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒}    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ (𝑋 ∈ 𝑆 → 𝑋 ∈ 𝑉)
Theorem	dfclnbgr5 48766*	Alternate definition of the closed neighborhood of a vertex as union of the vertex with its semiclosed neighborhood. (Contributed by AV, 16-May-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝑆 = {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒}    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ (𝑁 ∈ 𝑉 → (𝐺 ClNeighbVtx 𝑁) = ({𝑁} ∪ 𝑆))
Theorem	dfnbgr5 48767*	Alternate definition of the (open) neighborhood of a vertex as a semiclosed neighborhood without itself. (Contributed by AV, 16-May-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝑆 = {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒}    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ (𝑁 ∈ 𝑉 → (𝐺 NeighbVtx 𝑁) = (𝑆 ∖ {𝑁}))
Theorem	dfnbgrss 48768*	Subset chain for different kinds of neighborhoods of a vertex. (Contributed by AV, 16-May-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝑆 = {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒}    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ (𝑁 ∈ 𝑉 → ((𝐺 NeighbVtx 𝑁) ⊆ 𝑆 ∧ 𝑆 ⊆ (𝐺 ClNeighbVtx 𝑁)))
Theorem	dfvopnbgr2 48769*	Alternate definition of the semiopen neighborhood of a vertex breaking up the subset relationship of an unordered pair. A semiopen neighborhood 𝑈 of a vertex 𝑁 is its open neighborhood together with itself if there is a loop at this vertex. (Contributed by AV, 15-May-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝑈 = {𝑛 ∈ 𝑉 ∣ (𝑛 ∈ (𝐺 NeighbVtx 𝑁) ∨ ∃𝑒 ∈ 𝐸 (𝑁 = 𝑛 ∧ 𝑒 = {𝑁}))}    ⇒   ⊢ (𝑁 ∈ 𝑉 → 𝑈 = {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 ((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛}))})
Theorem	vopnbgrel 48770*	Characterization of a member 𝑋 of the semiopen neighborhood of a vertex 𝑁 in a graph 𝐺. (Contributed by AV, 16-May-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝑈 = {𝑛 ∈ 𝑉 ∣ (𝑛 ∈ (𝐺 NeighbVtx 𝑁) ∨ ∃𝑒 ∈ 𝐸 (𝑁 = 𝑛 ∧ 𝑒 = {𝑁}))}    ⇒   ⊢ (𝑁 ∈ 𝑉 → (𝑋 ∈ 𝑈 ↔ (𝑋 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 ((𝑋 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑋 ∈ 𝑒) ∨ (𝑋 = 𝑁 ∧ 𝑒 = {𝑋})))))
Theorem	vopnbgrelself 48771*	A vertex 𝑁 is a member of its semiopen neighborhood iff there is a loop joining the vertex with itself. (Contributed by AV, 16-May-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝑈 = {𝑛 ∈ 𝑉 ∣ (𝑛 ∈ (𝐺 NeighbVtx 𝑁) ∨ ∃𝑒 ∈ 𝐸 (𝑁 = 𝑛 ∧ 𝑒 = {𝑁}))}    ⇒   ⊢ (𝑁 ∈ 𝑉 → (𝑁 ∈ 𝑈 ↔ ∃𝑒 ∈ 𝐸 𝑒 = {𝑁}))
Theorem	dfclnbgr6 48772*	Alternate definition of the closed neighborhood of a vertex as union of the vertex with its semiopen neighborhood. (Contributed by AV, 17-May-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝑈 = {𝑛 ∈ 𝑉 ∣ (𝑛 ∈ (𝐺 NeighbVtx 𝑁) ∨ ∃𝑒 ∈ 𝐸 (𝑁 = 𝑛 ∧ 𝑒 = {𝑁}))}    ⇒   ⊢ (𝑁 ∈ 𝑉 → (𝐺 ClNeighbVtx 𝑁) = ({𝑁} ∪ 𝑈))
Theorem	dfnbgr6 48773*	Alternate definition of the (open) neighborhood of a vertex as a difference of its semiopen neighborhood and the singleton of itself. (Contributed by AV, 17-May-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝑈 = {𝑛 ∈ 𝑉 ∣ (𝑛 ∈ (𝐺 NeighbVtx 𝑁) ∨ ∃𝑒 ∈ 𝐸 (𝑁 = 𝑛 ∧ 𝑒 = {𝑁}))}    ⇒   ⊢ (𝑁 ∈ 𝑉 → (𝐺 NeighbVtx 𝑁) = (𝑈 ∖ {𝑁}))
Theorem	dfsclnbgr6 48774*	Alternate definition of a semiclosed neighborhood of a vertex as a union of a semiopen neighborhood and the vertex itself if there is a loop at this vertex. (Contributed by AV, 17-May-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝑈 = {𝑛 ∈ 𝑉 ∣ (𝑛 ∈ (𝐺 NeighbVtx 𝑁) ∨ ∃𝑒 ∈ 𝐸 (𝑁 = 𝑛 ∧ 𝑒 = {𝑁}))}    &   ⊢ 𝑆 = {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒}    ⇒   ⊢ (𝑁 ∈ 𝑉 → 𝑆 = (𝑈 ∪ {𝑛 ∈ {𝑁} ∣ ∃𝑒 ∈ 𝐸 𝑛 ∈ 𝑒}))
Theorem	dfnbgrss2 48775*	Subset chain for different kinds of neighborhoods of a vertex. (Contributed by AV, 16-May-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝑈 = {𝑛 ∈ 𝑉 ∣ (𝑛 ∈ (𝐺 NeighbVtx 𝑁) ∨ ∃𝑒 ∈ 𝐸 (𝑁 = 𝑛 ∧ 𝑒 = {𝑁}))}    &   ⊢ 𝑆 = {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒}    ⇒   ⊢ (𝑁 ∈ 𝑉 → ((𝐺 NeighbVtx 𝑁) ⊆ 𝑈 ∧ 𝑈 ⊆ 𝑆 ∧ 𝑆 ⊆ (𝐺 ClNeighbVtx 𝑁)))
21.50.15.3  Induced subgraphs
Syntax	cisubgr 48776	Extend class notation with induced subgraphs.
class ISubGr
Definition	df-isubgr 48777*	Define the function mapping graphs and subsets of their vertices to their induced subgraphs. A subgraph induced by a subset of vertices of a graph is a subgraph of the graph which contains all edges of the graph that join vertices of the subgraph (see section I.1 in [Bollobas] p. 2 or section 1.1 in [Diestel] p. 4). Although a graph may be given in any meaningful representation, its induced subgraphs are always ordered pairs of vertices and edges. (Contributed by AV, 27-Apr-2025.)
⊢ ISubGr = (𝑔 ∈ V, 𝑣 ∈ 𝒫 (Vtx‘𝑔) ↦ ⟨𝑣, ⦋(iEdg‘𝑔) / 𝑒⦌(𝑒 ↾ {𝑥 ∈ dom 𝑒 ∣ (𝑒‘𝑥) ⊆ 𝑣})⟩)
Theorem	isisubgr 48778*	The subgraph induced by a subset of vertices. (Contributed by AV, 12-May-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (iEdg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉) → (𝐺 ISubGr 𝑆) = ⟨𝑆, (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) ⊆ 𝑆})⟩)
Theorem	isubgriedg 48779*	The edges of an induced subgraph. (Contributed by AV, 12-May-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (iEdg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉) → (iEdg‘(𝐺 ISubGr 𝑆)) = (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) ⊆ 𝑆}))
Theorem	isubgrvtxuhgr 48780	The subgraph induced by the full set of vertices of a hypergraph. (Contributed by AV, 12-May-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (iEdg‘𝐺)    ⇒   ⊢ (𝐺 ∈ UHGraph → (𝐺 ISubGr 𝑉) = ⟨𝑉, 𝐸⟩)
Theorem	isubgredgss 48781	The edges of an induced subgraph of a graph are edges of the graph. (Contributed by AV, 24-Sep-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝐻 = (𝐺 ISubGr 𝑆)    &   ⊢ 𝐼 = (Edg‘𝐻)    ⇒   ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉) → 𝐼 ⊆ 𝐸)
Theorem	isubgredg 48782	An edge of an induced subgraph of a hypergraph is an edge of the hypergraph connecting vertices of the subgraph. (Contributed by AV, 24-Sep-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝐻 = (𝐺 ISubGr 𝑆)    &   ⊢ 𝐼 = (Edg‘𝐻)    ⇒   ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → (𝐾 ∈ 𝐼 ↔ (𝐾 ∈ 𝐸 ∧ 𝐾 ⊆ 𝑆)))
Theorem	isubgrvtx 48783	The vertices of an induced subgraph. (Contributed by AV, 12-May-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉) → (Vtx‘(𝐺 ISubGr 𝑆)) = 𝑆)
Theorem	isubgruhgr 48784	An induced subgraph of a hypergraph is a hypergraph. (Contributed by AV, 13-May-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → (𝐺 ISubGr 𝑆) ∈ UHGraph)
Theorem	isubgrsubgr 48785	An induced subgraph of a hypergraph is a subgraph of the hypergraph. (Contributed by AV, 14-May-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → (𝐺 ISubGr 𝑆) SubGraph 𝐺)
Theorem	isubgrupgr 48786	An induced subgraph of a pseudograph is a pseudograph. (Contributed by AV, 14-May-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ ((𝐺 ∈ UPGraph ∧ 𝑆 ⊆ 𝑉) → (𝐺 ISubGr 𝑆) ∈ UPGraph)
Theorem	isubgrumgr 48787	An induced subgraph of a multigraph is a multigraph. (Contributed by AV, 15-May-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ ((𝐺 ∈ UMGraph ∧ 𝑆 ⊆ 𝑉) → (𝐺 ISubGr 𝑆) ∈ UMGraph)
Theorem	isubgrusgr 48788	An induced subgraph of a simple graph is a simple graph. (Contributed by AV, 15-May-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ ((𝐺 ∈ USGraph ∧ 𝑆 ⊆ 𝑉) → (𝐺 ISubGr 𝑆) ∈ USGraph)
Theorem	isubgr0uhgr 48789	The subgraph induced by an empty set of vertices of a hypergraph. (Contributed by AV, 13-May-2025.)
⊢ (𝐺 ∈ UHGraph → (𝐺 ISubGr ∅) = ⟨∅, ∅⟩)
21.50.15.4  Isomorphisms of graphs This section is about isomorphisms of graphs, whereby the term "isomorphism" is used in both of its meanings (according to the Meriam-Webster dictionary, see https://www.merriam-webster.com/dictionary/isomorphism): "1: the quality or state of being isomorphic." and "2: a one-to-one correspondence between two mathematical sets". At first, an operation GraphIso is defined (see df-grim 48794) which provides the graph isomorphisms (as "one-to-one correspondence") between two given graphs. This definition, however, is applicable for any two sets, but is meaningful only if these sets have "vertices" and "edges". Afterwards, a binary relation ≃𝑔𝑟 is defined (see df-gric 48797) which is true for two graphs iff there is a graph isomorphisms between these graphs. Then these graphs are called "isomorphic". Therefore, this relation is also called "is isomorphic to" relation. More formally, 𝐴 ≃𝑔𝑟 𝐵 ↔ ∃𝑓𝑓 ∈ (𝐴 GraphIso 𝐵) resp. 𝐴 ≃𝑔𝑟 𝐵 ↔ (𝐴 GraphIso 𝐵) ≠ ∅. Notice that there can be multiple isomorphisms between two graphs. For example, let ⟨{𝐴, 𝐵}, {{𝐴, 𝐵}}⟩ and ⟨{{𝑀, 𝑁}, {{𝑀, 𝑁}}⟩ be two graphs with two vertices and one edge, then 𝐴 ↦ 𝑀, 𝐵 ↦ 𝑁 and 𝐴 ↦ 𝑁, 𝐵 ↦ 𝑀 are two different isomorphisms between these graphs. The names and symbols are chosen analogously to group isomorphisms GrpIso (see df-gim 19386) resp. isomorphism between groups ≃𝑔 (see df-gic 19387). The general definition of graph isomorphisms and the relation "is isomorphic to" for graphs is specialized for simple hypergraphs (gricushgr 48833) and simple pseudographs (gricuspgr 48834). The latter corresponds to the definition in [Bollobas] p. 3. It is shown that the relation "is isomorphic to" for graphs is an equivalence relation, see gricer 48840. Finally, isomorphic graphs with different representations are studied (opstrgric 48842, ushggricedg 48843). Another approach could be to define a category of graphs (there are maybe multiple ones), where graph morphisms are couples consisting of a function on vertices and a function on edges with required compatibilities, as used in the definition of GraphIso. And then, a graph isomorphism is defined as an isomorphism in the category of graphs (something like "GraphIsom = ( Iso ` GraphCat )" ). Then general category theory theorems could be used, e.g., to show that graph isomorphism is an equivalence relation.
Syntax	cgrisom 48790	Extend class notation to include the graph ispmorphisms as pair.
class GraphIsom
Syntax	cgrim 48791	Extend class notation to include the graph ispmorphisms.
class GraphIso
Syntax	cgric 48792	Extend class notation to include the "is isomorphic to" relation for graphs.
class ≃𝑔𝑟
Definition	df-grisom 48793*	Define the class of all isomorphisms between two graphs. In contrast to (𝐹 GraphIso 𝐻), which is a set of functions between the vertices, (𝐹 GraphIsom 𝐻) is a set of pairs of functions: a function between the vertices, and a function between the (indices of the) edges. It is not clear if such a definition is useful. In the definition by [Diestel] p. 3, for example, the bijection between the vertices is called an isomorphism, as formalized in df-grim 48794. (Contributed by AV, 11-Dec-2022.) (New usage is discouraged.)
⊢ GraphIsom = (𝑥 ∈ V, 𝑦 ∈ V ↦ {⟨𝑓, 𝑔⟩ ∣ (𝑓:(Vtx‘𝑥)–1-1-onto→(Vtx‘𝑦) ∧ 𝑔:dom (iEdg‘𝑥)–1-1-onto→dom (iEdg‘𝑦) ∧ ∀𝑖 ∈ dom (iEdg‘𝑥)(𝑓 “ ((iEdg‘𝑥)‘𝑖)) = ((iEdg‘𝑦)‘(𝑔‘𝑖)))})
Definition	df-grim 48794*	An isomorphism between two graphs is a bijection between the sets of vertices of the two graphs that preserves adjacency, see definition in [Diestel] p. 3. (Contributed by AV, 19-Apr-2025.)
⊢ GraphIso = (𝑔 ∈ V, ℎ ∈ V ↦ {𝑓 ∣ (𝑓:(Vtx‘𝑔)–1-1-onto→(Vtx‘ℎ) ∧ ∃𝑗[(iEdg‘𝑔) / 𝑒][(iEdg‘ℎ) / 𝑑](𝑗:dom 𝑒–1-1-onto→dom 𝑑 ∧ ∀𝑖 ∈ dom 𝑒(𝑑‘(𝑗‘𝑖)) = (𝑓 “ (𝑒‘𝑖))))})
Theorem	grimfn 48795	The graph isomorphism function is a well-defined function. (Contributed by AV, 28-Apr-2025.)
⊢ GraphIso Fn (V × V)
Theorem	grimdmrel 48796	The domain of the graph isomorphism function is a relation. (Contributed by AV, 28-Apr-2025.)
⊢ Rel dom GraphIso
Definition	df-gric 48797	Two graphs are said to be isomorphic iff they are connected by at least one isomorphism, see definition in [Diestel] p. 3 and definition in [Bollobas] p. 3. Isomorphic graphs share all global graph properties like order and size. (Contributed by AV, 11-Nov-2022.) (Revised by AV, 19-Apr-2025.)
⊢ ≃𝑔𝑟 = (◡ GraphIso “ (V ∖ 1o))
Theorem	isgrim 48798*	An isomorphism of graphs is a bijection between their vertices that preserves adjacency. (Contributed by AV, 19-Apr-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝑊 = (Vtx‘𝐻)    &   ⊢ 𝐸 = (iEdg‘𝐺)    &   ⊢ 𝐷 = (iEdg‘𝐻)    ⇒   ⊢ ((𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌 ∧ 𝐹 ∈ 𝑍) → (𝐹 ∈ (𝐺 GraphIso 𝐻) ↔ (𝐹:𝑉–1-1-onto→𝑊 ∧ ∃𝑗(𝑗:dom 𝐸–1-1-onto→dom 𝐷 ∧ ∀𝑖 ∈ dom 𝐸(𝐷‘(𝑗‘𝑖)) = (𝐹 “ (𝐸‘𝑖))))))
Theorem	grimprop 48799*	Properties of an isomorphism of graphs. (Contributed by AV, 29-Apr-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝑊 = (Vtx‘𝐻)    &   ⊢ 𝐸 = (iEdg‘𝐺)    &   ⊢ 𝐷 = (iEdg‘𝐻)    ⇒   ⊢ (𝐹 ∈ (𝐺 GraphIso 𝐻) → (𝐹:𝑉–1-1-onto→𝑊 ∧ ∃𝑗(𝑗:dom 𝐸–1-1-onto→dom 𝐷 ∧ ∀𝑖 ∈ dom 𝐸(𝐷‘(𝑗‘𝑖)) = (𝐹 “ (𝐸‘𝑖)))))
Theorem	grimf1o 48800	An isomorphism of graphs is a bijection between their vertices. (Contributed by AV, 29-Apr-2025.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝑊 = (Vtx‘𝐻)    ⇒   ⊢ (𝐹 ∈ (𝐺 GraphIso 𝐻) → 𝐹:𝑉–1-1-onto→𝑊)