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Type | Label | Description |
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Statement | ||
Theorem | sbgoldbwt 47701* | If the strong binary Goldbach conjecture is valid, then the (weak) ternary Goldbach conjecture holds, too. (Contributed by AV, 20-Jul-2020.) |
⊢ (∀𝑛 ∈ Even (4 < 𝑛 → 𝑛 ∈ GoldbachEven ) → ∀𝑚 ∈ Odd (5 < 𝑚 → 𝑚 ∈ GoldbachOddW )) | ||
Theorem | sbgoldbst 47702* | If the strong binary Goldbach conjecture is valid, then the (strong) ternary Goldbach conjecture holds, too. (Contributed by AV, 26-Jul-2020.) |
⊢ (∀𝑛 ∈ Even (4 < 𝑛 → 𝑛 ∈ GoldbachEven ) → ∀𝑚 ∈ Odd (7 < 𝑚 → 𝑚 ∈ GoldbachOdd )) | ||
Theorem | sbgoldbaltlem1 47703 | Lemma 1 for sbgoldbalt 47705: If an even number greater than 4 is the sum of two primes, one of the prime summands must be odd, i.e. not 2. (Contributed by AV, 22-Jul-2020.) |
⊢ ((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) → ((𝑁 ∈ Even ∧ 4 < 𝑁 ∧ 𝑁 = (𝑃 + 𝑄)) → 𝑄 ∈ Odd )) | ||
Theorem | sbgoldbaltlem2 47704 | Lemma 2 for sbgoldbalt 47705: If an even number greater than 4 is the sum of two primes, the primes must be odd, i.e. not 2. (Contributed by AV, 22-Jul-2020.) |
⊢ ((𝑃 ∈ ℙ ∧ 𝑄 ∈ ℙ) → ((𝑁 ∈ Even ∧ 4 < 𝑁 ∧ 𝑁 = (𝑃 + 𝑄)) → (𝑃 ∈ Odd ∧ 𝑄 ∈ Odd ))) | ||
Theorem | sbgoldbalt 47705* | An alternate (related to the original) formulation of the binary Goldbach conjecture: Every even integer greater than 2 can be expressed as the sum of two primes. (Contributed by AV, 22-Jul-2020.) |
⊢ (∀𝑛 ∈ Even (4 < 𝑛 → 𝑛 ∈ GoldbachEven ) ↔ ∀𝑛 ∈ Even (2 < 𝑛 → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))) | ||
Theorem | sbgoldbb 47706* | If the strong binary Goldbach conjecture is valid, the binary Goldbach conjecture is valid. (Contributed by AV, 23-Dec-2021.) |
⊢ (∀𝑛 ∈ Even (4 < 𝑛 → 𝑛 ∈ GoldbachEven ) → ∀𝑛 ∈ Even (2 < 𝑛 → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞))) | ||
Theorem | sgoldbeven3prm 47707* | If the binary Goldbach conjecture is valid, then an even integer greater than 5 can be expressed as the sum of three primes: Since (𝑁 − 2) is even iff 𝑁 is even, there would be primes 𝑝 and 𝑞 with (𝑁 − 2) = (𝑝 + 𝑞), and therefore 𝑁 = ((𝑝 + 𝑞) + 2). (Contributed by AV, 24-Dec-2021.) |
⊢ (∀𝑛 ∈ Even (4 < 𝑛 → 𝑛 ∈ GoldbachEven ) → ((𝑁 ∈ Even ∧ 6 ≤ 𝑁) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑁 = ((𝑝 + 𝑞) + 𝑟))) | ||
Theorem | sbgoldbm 47708* | If the strong binary Goldbach conjecture is valid, the modern version of the original formulation of the Goldbach conjecture also holds: Every integer greater than 5 can be expressed as the sum of three primes. (Contributed by AV, 24-Dec-2021.) |
⊢ (∀𝑛 ∈ Even (4 < 𝑛 → 𝑛 ∈ GoldbachEven ) → ∀𝑛 ∈ (ℤ≥‘6)∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟)) | ||
Theorem | mogoldbb 47709* | 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 47710* | 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 47711* | 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 47712* | 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 47713* | 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 47714* | 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 47715* | 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 47716* | 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 47717* | 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 47718* | 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 47719* | 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 47720* | 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 47721* | 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 47722* | 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 47723* | 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 47724* | 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 47725* | 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 47726* | 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 47727* | 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 47728* | 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 47729 | Lemma 1 for bgoldbtbnd 47733: 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 47730* | Lemma 2 for bgoldbtbnd 47733. (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 47731* | Lemma 3 for bgoldbtbnd 47733. (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 47732* | Lemma 4 for bgoldbtbnd 47733. (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 47733* | 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 47734 | 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 47735* | Temporary duplicate of tgoldbachgt 34656, 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 47736* | Variant of Thierry Arnoux's tgoldbachgt 34656 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 47737* | 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 47734. (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 47738 | 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 47739 | 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 47737, ax-hgprmladder 47738 and bgoldbtbnd 47733. (Contributed by AV, 4-Aug-2020.) (Revised by AV, 9-Sep-2021.) |
⊢ ((𝑁 ∈ Odd ∧ 7 < 𝑁 ∧ 𝑁 < (;88 · (;10↑;29))) → 𝑁 ∈ GoldbachOdd ) | ||
Theorem | tgoldbachlt 47740* | 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 47739. (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 47741 | The ternary Goldbach conjecture is valid. Main theorem in [Helfgott] p. 2. This follows from tgoldbachlt 47740 and ax-tgoldbachgt 47735. (Contributed by AV, 2-Aug-2020.) (Revised by AV, 9-Sep-2021.) |
⊢ ∀𝑛 ∈ Odd (7 < 𝑛 → 𝑛 ∈ GoldbachOdd ) | ||
Syntax | cclnbgr 47742 | Extend class notation with closed neighborhoods (of a vertex in a graph). |
class ClNeighbVtx | ||
Definition | df-clnbgr 47743* | 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 are all vertices which are connected with this vertex by an edge and the vertex itself (in contrast to an open neighborhood, see df-nbgr 29364). Alternatively, a closed neighborhood of a vertex could have been defined as its open neighborhood enhanced by the vertex itself, see dfclnbgr4 47748. 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 47744 | 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 47745 | 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 47746* | 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 47747* | 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 47748 | 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 47749 | 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 47750* | Alternate definition of the closed neighborhood of a vertex using the edge function instead of the edges themselves (see also clnbgrval 47746). (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 47751 | 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 47752* | 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 47753 | 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 47754 | 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 47755 | 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 47756 | 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 47757* | 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 47758 | 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 | clnbgr0vtx 47759 | 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 47760 | 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 47761 | 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 47762 | 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 47763 | 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 47764 | 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 47765* | 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 47766 | 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 47767 | 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 47768 | 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) | ||
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 47769. The definitions for open and closed neighborhoods could be derived from such a more simple, but otherwise probably useless definition, see dfnbgr5 47774 and dfclnbgr5 47773. 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 47776. The definitions for open and closed neighborhoods could also be derived from such a definition, see dfnbgr6 47780 and dfclnbgr6 47779. 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 47769, the definition of semiopen neighborhoods is much more complex. | ||
Theorem | dfsclnbgr2 47769* | 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 47771), 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 47770* | 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 47771* | 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 47772* | 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 47773* | 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 47774* | 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 47775* | 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 47776* | 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 47777* | 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 47778* | 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 47779* | 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 47780* | 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 47781* | 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 47782* | 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))) | ||
Syntax | cisubgr 47783 | Extend class notation with induced subgraphs. |
class ISubGr | ||
Definition | df-isubgr 47784* | 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 47785* | 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 47786* | 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 47787 | 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 47788 | 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 47789 | 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 47790 | 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 47791 | 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 47792 | 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 47793 | 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 47794 | 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 47795 | 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 47796 | The subgraph induced by an empty set of vertices of a hypergraph. (Contributed by AV, 13-May-2025.) |
⊢ (𝐺 ∈ UHGraph → (𝐺 ISubGr ∅) = 〈∅, ∅〉) | ||
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 47801) 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 47804) 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 19289) resp. isomorphism between groups ≃𝑔 (see df-gic 19290). The general definition of graph isomorphisms and the relation "is isomorphic to" for graphs is specialized for simple hypergraphs (gricushgr 47823) and simple pseudographs (gricuspgr 47824). 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 47830. Finally, isomorphic graphs with different representations are studied (opstrgric 47832, ushggricedg 47833). 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 47797 | Extend class notation to include the graph ispmorphisms as pair. |
class GraphIsom | ||
Syntax | cgrim 47798 | Extend class notation to include the graph ispmorphisms. |
class GraphIso | ||
Syntax | cgric 47799 | Extend class notation to include the "is isomorphic to" relation for graphs. |
class ≃𝑔𝑟 | ||
Definition | df-grisom 47800* |
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 47801. (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)))}) |
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