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Type | Label | Description |
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Statement | ||
Theorem | sgoldbeven3prm 44301* | 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 44302* | 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 44303* | 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 44304* | 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 44305* | 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 44306* | 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 44307* | 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 44308* | 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 44309* | 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 44310* | 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 44311* | 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 44312* | 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 44313* | 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 44314* | 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 44315* | 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 44316* | 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 44317* | 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 44318* | 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 44319* | 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 44320* | 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 44321* | 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 44322* | 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 44323 | Lemma 1 for bgoldbtbnd 44327: 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 44324* | Lemma 2 for bgoldbtbnd 44327. (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 44325* | Lemma 3 for bgoldbtbnd 44327. (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 44326* | Lemma 4 for bgoldbtbnd 44327. (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 44327* | 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 44328 | 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 44329* | Temporary duplicate of tgoldbachgt 32044, 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 44330* | Variant of Thierry Arnoux's tgoldbachgt 32044 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 44331* | 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 44328. (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 44332 | 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 44333 | 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 44331, ax-hgprmladder 44332 and bgoldbtbnd 44327. (Contributed by AV, 4-Aug-2020.) (Revised by AV, 9-Sep-2021.) |
⊢ ((𝑁 ∈ Odd ∧ 7 < 𝑁 ∧ 𝑁 < (;88 · (;10↑;29))) → 𝑁 ∈ GoldbachOdd ) | ||
Theorem | tgoldbachlt 44334* | 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 44333. (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 44335 | The ternary Goldbach conjecture is valid. Main theorem in [Helfgott] p. 2. This follows from tgoldbachlt 44334 and ax-tgoldbachgt 44329. (Contributed by AV, 2-Aug-2020.) (Revised by AV, 9-Sep-2021.) |
⊢ ∀𝑛 ∈ Odd (7 < 𝑛 → 𝑛 ∈ GoldbachOdd ) | ||
In the following, a general definition of the isomorphy relation for graphs and specializations for simple hypergraphs (isomushgr 44344) and simple pseudographs (isomuspgr 44352) are provided. The latter corresponds to the definition in [Bollobas] p. 3). It is shown that the isomorphy relation for graphs is an equivalence relation (isomgrref 44353, isomgrsym 44354, isomgrtr 44357). Fianlly, isomorphic graphs with different representations are studied (strisomgrop 44358, ushrisomgr 44359). Maybe more important than graph isomorphy is the notion of graph isomorphism, which can be defined as in df-grisom 44338. Then 𝐴 IsomGr 𝐵 ↔ ∃𝑓𝑓 ∈ (𝐴 GrIsom 𝐵) resp. 𝐴 IsomGr 𝐵 ↔ (𝐴 GrIsom 𝐵) ≠ ∅. Notice that there can be multiple isomorphisms between two graphs (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). Another approach could be to define a category of graphs (there are maybe multiple ones), where graph morphisms are couples consisting in a function on vertices and a function on edges with required compatibilities, as used in the definition of GrIsom. 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 isomorphy is an equivalence relation. | ||
Syntax | cgrisom 44336 | Extend class notation to include the graph ispmorphisms. |
class GrIsom | ||
Syntax | cisomgr 44337 | Extend class notation to include the isomorphy relation for graphs. |
class IsomGr | ||
Definition | df-grisom 44338* | Define the class of all isomorphisms between two graphs. (Contributed by AV, 11-Dec-2022.) |
⊢ GrIsom = (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-isomgr 44339* | Define the isomorphy relation for graphs. (Contributed by AV, 11-Nov-2022.) |
⊢ IsomGr = {〈𝑥, 𝑦〉 ∣ ∃𝑓(𝑓:(Vtx‘𝑥)–1-1-onto→(Vtx‘𝑦) ∧ ∃𝑔(𝑔:dom (iEdg‘𝑥)–1-1-onto→dom (iEdg‘𝑦) ∧ ∀𝑖 ∈ dom (iEdg‘𝑥)(𝑓 “ ((iEdg‘𝑥)‘𝑖)) = ((iEdg‘𝑦)‘(𝑔‘𝑖))))} | ||
Theorem | isomgrrel 44340 | The isomorphy relation for graphs is a relation. (Contributed by AV, 11-Nov-2022.) |
⊢ Rel IsomGr | ||
Theorem | isomgr 44341* | The isomorphy relation for two graphs. (Contributed by AV, 11-Nov-2022.) |
⊢ 𝑉 = (Vtx‘𝐴) & ⊢ 𝑊 = (Vtx‘𝐵) & ⊢ 𝐼 = (iEdg‘𝐴) & ⊢ 𝐽 = (iEdg‘𝐵) ⇒ ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → (𝐴 IsomGr 𝐵 ↔ ∃𝑓(𝑓:𝑉–1-1-onto→𝑊 ∧ ∃𝑔(𝑔:dom 𝐼–1-1-onto→dom 𝐽 ∧ ∀𝑖 ∈ dom 𝐼(𝑓 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖)))))) | ||
Theorem | isisomgr 44342* | Implications of two graphs being isomorphic. (Contributed by AV, 11-Nov-2022.) |
⊢ 𝑉 = (Vtx‘𝐴) & ⊢ 𝑊 = (Vtx‘𝐵) & ⊢ 𝐼 = (iEdg‘𝐴) & ⊢ 𝐽 = (iEdg‘𝐵) ⇒ ⊢ (𝐴 IsomGr 𝐵 → ∃𝑓(𝑓:𝑉–1-1-onto→𝑊 ∧ ∃𝑔(𝑔:dom 𝐼–1-1-onto→dom 𝐽 ∧ ∀𝑖 ∈ dom 𝐼(𝑓 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖))))) | ||
Theorem | isomgreqve 44343 | A set is isomorphic to a hypergraph if it has the same vertices and the same edges. (Contributed by AV, 11-Nov-2022.) |
⊢ (((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) ∧ ((Vtx‘𝐴) = (Vtx‘𝐵) ∧ (iEdg‘𝐴) = (iEdg‘𝐵))) → 𝐴 IsomGr 𝐵) | ||
Theorem | isomushgr 44344* | The isomorphy relation for two simple hypergraphs. (Contributed by AV, 28-Nov-2022.) |
⊢ 𝑉 = (Vtx‘𝐴) & ⊢ 𝑊 = (Vtx‘𝐵) & ⊢ 𝐸 = (Edg‘𝐴) & ⊢ 𝐾 = (Edg‘𝐵) ⇒ ⊢ ((𝐴 ∈ USHGraph ∧ 𝐵 ∈ USHGraph) → (𝐴 IsomGr 𝐵 ↔ ∃𝑓(𝑓:𝑉–1-1-onto→𝑊 ∧ ∃𝑔(𝑔:𝐸–1-1-onto→𝐾 ∧ ∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒))))) | ||
Theorem | isomuspgrlem1 44345* | Lemma 1 for isomuspgr 44352. (Contributed by AV, 29-Nov-2022.) |
⊢ 𝑉 = (Vtx‘𝐴) & ⊢ 𝑊 = (Vtx‘𝐵) & ⊢ 𝐸 = (Edg‘𝐴) & ⊢ 𝐾 = (Edg‘𝐵) ⇒ ⊢ (((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊) ∧ (𝑔:𝐸–1-1-onto→𝐾 ∧ ∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒))) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ({(𝑓‘𝑎), (𝑓‘𝑏)} ∈ 𝐾 → {𝑎, 𝑏} ∈ 𝐸)) | ||
Theorem | isomuspgrlem2a 44346* | Lemma 1 for isomuspgrlem2 44351. (Contributed by AV, 29-Nov-2022.) |
⊢ 𝑉 = (Vtx‘𝐴) & ⊢ 𝑊 = (Vtx‘𝐵) & ⊢ 𝐸 = (Edg‘𝐴) & ⊢ 𝐾 = (Edg‘𝐵) & ⊢ 𝐺 = (𝑥 ∈ 𝐸 ↦ (𝐹 “ 𝑥)) ⇒ ⊢ (𝐹 ∈ 𝑋 → ∀𝑒 ∈ 𝐸 (𝐹 “ 𝑒) = (𝐺‘𝑒)) | ||
Theorem | isomuspgrlem2b 44347* | Lemma 2 for isomuspgrlem2 44351. (Contributed by AV, 29-Nov-2022.) |
⊢ 𝑉 = (Vtx‘𝐴) & ⊢ 𝑊 = (Vtx‘𝐵) & ⊢ 𝐸 = (Edg‘𝐴) & ⊢ 𝐾 = (Edg‘𝐵) & ⊢ 𝐺 = (𝑥 ∈ 𝐸 ↦ (𝐹 “ 𝑥)) & ⊢ (𝜑 → 𝐴 ∈ USPGraph) & ⊢ (𝜑 → 𝐹:𝑉–1-1-onto→𝑊) & ⊢ (𝜑 → ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ({𝑎, 𝑏} ∈ 𝐸 ↔ {(𝐹‘𝑎), (𝐹‘𝑏)} ∈ 𝐾)) ⇒ ⊢ (𝜑 → 𝐺:𝐸⟶𝐾) | ||
Theorem | isomuspgrlem2c 44348* | Lemma 3 for isomuspgrlem2 44351. (Contributed by AV, 29-Nov-2022.) |
⊢ 𝑉 = (Vtx‘𝐴) & ⊢ 𝑊 = (Vtx‘𝐵) & ⊢ 𝐸 = (Edg‘𝐴) & ⊢ 𝐾 = (Edg‘𝐵) & ⊢ 𝐺 = (𝑥 ∈ 𝐸 ↦ (𝐹 “ 𝑥)) & ⊢ (𝜑 → 𝐴 ∈ USPGraph) & ⊢ (𝜑 → 𝐹:𝑉–1-1-onto→𝑊) & ⊢ (𝜑 → ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ({𝑎, 𝑏} ∈ 𝐸 ↔ {(𝐹‘𝑎), (𝐹‘𝑏)} ∈ 𝐾)) & ⊢ (𝜑 → 𝐹 ∈ 𝑋) ⇒ ⊢ (𝜑 → 𝐺:𝐸–1-1→𝐾) | ||
Theorem | isomuspgrlem2d 44349* | Lemma 4 for isomuspgrlem2 44351. (Contributed by AV, 1-Dec-2022.) |
⊢ 𝑉 = (Vtx‘𝐴) & ⊢ 𝑊 = (Vtx‘𝐵) & ⊢ 𝐸 = (Edg‘𝐴) & ⊢ 𝐾 = (Edg‘𝐵) & ⊢ 𝐺 = (𝑥 ∈ 𝐸 ↦ (𝐹 “ 𝑥)) & ⊢ (𝜑 → 𝐴 ∈ USPGraph) & ⊢ (𝜑 → 𝐹:𝑉–1-1-onto→𝑊) & ⊢ (𝜑 → ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ({𝑎, 𝑏} ∈ 𝐸 ↔ {(𝐹‘𝑎), (𝐹‘𝑏)} ∈ 𝐾)) & ⊢ (𝜑 → 𝐹 ∈ 𝑋) & ⊢ (𝜑 → 𝐵 ∈ USPGraph) ⇒ ⊢ (𝜑 → 𝐺:𝐸–onto→𝐾) | ||
Theorem | isomuspgrlem2e 44350* | Lemma 5 for isomuspgrlem2 44351. (Contributed by AV, 1-Dec-2022.) |
⊢ 𝑉 = (Vtx‘𝐴) & ⊢ 𝑊 = (Vtx‘𝐵) & ⊢ 𝐸 = (Edg‘𝐴) & ⊢ 𝐾 = (Edg‘𝐵) & ⊢ 𝐺 = (𝑥 ∈ 𝐸 ↦ (𝐹 “ 𝑥)) & ⊢ (𝜑 → 𝐴 ∈ USPGraph) & ⊢ (𝜑 → 𝐹:𝑉–1-1-onto→𝑊) & ⊢ (𝜑 → ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ({𝑎, 𝑏} ∈ 𝐸 ↔ {(𝐹‘𝑎), (𝐹‘𝑏)} ∈ 𝐾)) & ⊢ (𝜑 → 𝐹 ∈ 𝑋) & ⊢ (𝜑 → 𝐵 ∈ USPGraph) ⇒ ⊢ (𝜑 → 𝐺:𝐸–1-1-onto→𝐾) | ||
Theorem | isomuspgrlem2 44351* | Lemma 2 for isomuspgr 44352. (Contributed by AV, 1-Dec-2022.) |
⊢ 𝑉 = (Vtx‘𝐴) & ⊢ 𝑊 = (Vtx‘𝐵) & ⊢ 𝐸 = (Edg‘𝐴) & ⊢ 𝐾 = (Edg‘𝐵) ⇒ ⊢ (((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉–1-1-onto→𝑊) → (∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ({𝑎, 𝑏} ∈ 𝐸 ↔ {(𝑓‘𝑎), (𝑓‘𝑏)} ∈ 𝐾) → ∃𝑔(𝑔:𝐸–1-1-onto→𝐾 ∧ ∀𝑒 ∈ 𝐸 (𝑓 “ 𝑒) = (𝑔‘𝑒)))) | ||
Theorem | isomuspgr 44352* | The isomorphy relation for two simple pseudographs. This corresponds to the definition in [Bollobas] p. 3. (Contributed by AV, 1-Dec-2022.) |
⊢ 𝑉 = (Vtx‘𝐴) & ⊢ 𝑊 = (Vtx‘𝐵) & ⊢ 𝐸 = (Edg‘𝐴) & ⊢ 𝐾 = (Edg‘𝐵) ⇒ ⊢ ((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) → (𝐴 IsomGr 𝐵 ↔ ∃𝑓(𝑓:𝑉–1-1-onto→𝑊 ∧ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ({𝑎, 𝑏} ∈ 𝐸 ↔ {(𝑓‘𝑎), (𝑓‘𝑏)} ∈ 𝐾)))) | ||
Theorem | isomgrref 44353 | The isomorphy relation is reflexive for hypergraphs. (Contributed by AV, 11-Nov-2022.) |
⊢ (𝐺 ∈ UHGraph → 𝐺 IsomGr 𝐺) | ||
Theorem | isomgrsym 44354 | The isomorphy relation is symmetric for hypergraphs. (Contributed by AV, 11-Nov-2022.) |
⊢ ((𝐴 ∈ UHGraph ∧ 𝐵 ∈ 𝑌) → (𝐴 IsomGr 𝐵 → 𝐵 IsomGr 𝐴)) | ||
Theorem | isomgrsymb 44355 | The isomorphy relation is symmetric for hypergraphs. (Contributed by AV, 11-Nov-2022.) |
⊢ ((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph) → (𝐴 IsomGr 𝐵 ↔ 𝐵 IsomGr 𝐴)) | ||
Theorem | isomgrtrlem 44356* | Lemma for isomgrtr 44357. (Contributed by AV, 5-Dec-2022.) |
⊢ (((((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph ∧ 𝐶 ∈ 𝑋) ∧ 𝑓:(Vtx‘𝐴)–1-1-onto→(Vtx‘𝐵) ∧ 𝑣:(Vtx‘𝐵)–1-1-onto→(Vtx‘𝐶)) ∧ (𝑔:dom (iEdg‘𝐴)–1-1-onto→dom (iEdg‘𝐵) ∧ ∀𝑖 ∈ dom (iEdg‘𝐴)(𝑓 “ ((iEdg‘𝐴)‘𝑖)) = ((iEdg‘𝐵)‘(𝑔‘𝑖)))) ∧ (𝑤:dom (iEdg‘𝐵)–1-1-onto→dom (iEdg‘𝐶) ∧ ∀𝑘 ∈ dom (iEdg‘𝐵)(𝑣 “ ((iEdg‘𝐵)‘𝑘)) = ((iEdg‘𝐶)‘(𝑤‘𝑘)))) → ∀𝑗 ∈ dom (iEdg‘𝐴)((𝑣 ∘ 𝑓) “ ((iEdg‘𝐴)‘𝑗)) = ((iEdg‘𝐶)‘((𝑤 ∘ 𝑔)‘𝑗))) | ||
Theorem | isomgrtr 44357 | The isomorphy relation is transitive for hypergraphs. (Contributed by AV, 5-Dec-2022.) |
⊢ ((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph ∧ 𝐶 ∈ 𝑋) → ((𝐴 IsomGr 𝐵 ∧ 𝐵 IsomGr 𝐶) → 𝐴 IsomGr 𝐶)) | ||
Theorem | strisomgrop 44358 | A graph represented as an extensible structure with vertices as base set and indexed edges is isomorphic to a hypergraph represented as ordered pair with the same vertices and edges. (Contributed by AV, 11-Nov-2022.) |
⊢ 𝐺 = 〈𝑉, 𝐸〉 & ⊢ 𝐻 = {〈(Base‘ndx), 𝑉〉, 〈(.ef‘ndx), 𝐸〉} ⇒ ⊢ ((𝐺 ∈ UHGraph ∧ 𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → 𝐺 IsomGr 𝐻) | ||
Theorem | ushrisomgr 44359 | A simple hypergraph (with arbitrarily indexed edges) is isomorphic to a graph with the same vertices and the same edges, indexed by the edges themselves. (Contributed by AV, 11-Nov-2022.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) & ⊢ 𝐻 = 〈𝑉, ( I ↾ 𝐸)〉 ⇒ ⊢ (𝐺 ∈ USHGraph → 𝐺 IsomGr 𝐻) | ||
Theorem | 1hegrlfgr 44360* | A graph 𝐺 with one hyperedge joining at least two vertices is a loop-free graph. (Contributed by AV, 23-Feb-2021.) |
⊢ (𝜑 → 𝐴 ∈ 𝑋) & ⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ≠ 𝐶) & ⊢ (𝜑 → 𝐸 ∈ 𝒫 𝑉) & ⊢ (𝜑 → (iEdg‘𝐺) = {〈𝐴, 𝐸〉}) & ⊢ (𝜑 → {𝐵, 𝐶} ⊆ 𝐸) ⇒ ⊢ (𝜑 → (iEdg‘𝐺):{𝐴}⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}) | ||
Syntax | cupwlks 44361 | Extend class notation with walks (of a pseudograph). |
class UPWalks | ||
Definition | df-upwlks 44362* |
Define the set of all walks (in a pseudograph), called "simple walks"
in
the following.
According to Wikipedia ("Path (graph theory)", https://en.wikipedia.org/wiki/Path_(graph_theory), 3-Oct-2017): "A walk of length k in a graph is an alternating sequence of vertices and edges, v0 , e0 , v1 , e1 , v2 , ... , v(k-1) , e(k-1) , v(k) which begins and ends with vertices. If the graph is undirected, then the endpoints of e(i) are v(i) and v(i+1)." According to Bollobas: " A walk W in a graph is an alternating sequence of vertices and edges x0 , e1 , x1 , e2 , ... , e(l) , x(l) where e(i) = x(i-1)x(i), 0<i<=l.", see Definition of [Bollobas] p. 4. Therefore, a walk can be represented by two mappings f from { 1 , ... , n } and p from { 0 , ... , n }, where f enumerates the (indices of the) edges, and p enumerates the vertices. So the walk is represented by the following sequence: p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n). Although this definition is also applicable for arbitrary hypergraphs, it allows only walks consisting of not proper hyperedges (i.e. edges connecting at most two vertices). Therefore, it should be used for pseudograhs only. (Contributed by Alexander van der Vekens and Mario Carneiro, 4-Oct-2017.) (Revised by AV, 28-Dec-2020.) |
⊢ UPWalks = (𝑔 ∈ V ↦ {〈𝑓, 𝑝〉 ∣ (𝑓 ∈ Word dom (iEdg‘𝑔) ∧ 𝑝:(0...(♯‘𝑓))⟶(Vtx‘𝑔) ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))((iEdg‘𝑔)‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))})}) | ||
Theorem | upwlksfval 44363* | The set of simple walks (in an undirected graph). (Contributed by Alexander van der Vekens, 19-Oct-2017.) (Revised by AV, 28-Dec-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ (𝐺 ∈ 𝑊 → (UPWalks‘𝐺) = {〈𝑓, 𝑝〉 ∣ (𝑓 ∈ Word dom 𝐼 ∧ 𝑝:(0...(♯‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))(𝐼‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))})}) | ||
Theorem | isupwlk 44364* | Properties of a pair of functions to be a simple walk. (Contributed by Alexander van der Vekens, 20-Oct-2017.) (Revised by AV, 28-Dec-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐹 ∈ 𝑈 ∧ 𝑃 ∈ 𝑍) → (𝐹(UPWalks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))) | ||
Theorem | isupwlkg 44365* | Generalization of isupwlk 44364: Conditions for two classes to represent a simple walk. (Contributed by AV, 5-Nov-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ (𝐺 ∈ 𝑊 → (𝐹(UPWalks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))) | ||
Theorem | upwlkbprop 44366 | Basic properties of a simple walk. (Contributed by Alexander van der Vekens, 31-Oct-2017.) (Revised by AV, 29-Dec-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ (𝐹(UPWalks‘𝐺)𝑃 → (𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V)) | ||
Theorem | upwlkwlk 44367 | A simple walk is a walk. (Contributed by AV, 30-Dec-2020.) (Proof shortened by AV, 27-Feb-2021.) |
⊢ (𝐹(UPWalks‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃) | ||
Theorem | upgrwlkupwlk 44368 | In a pseudograph, a walk is a simple walk. (Contributed by AV, 30-Dec-2020.) (Proof shortened by AV, 2-Jan-2021.) |
⊢ ((𝐺 ∈ UPGraph ∧ 𝐹(Walks‘𝐺)𝑃) → 𝐹(UPWalks‘𝐺)𝑃) | ||
Theorem | upgrwlkupwlkb 44369 | In a pseudograph, the definitions for a walk and a simple walk are equivalent. (Contributed by AV, 30-Dec-2020.) |
⊢ (𝐺 ∈ UPGraph → (𝐹(Walks‘𝐺)𝑃 ↔ 𝐹(UPWalks‘𝐺)𝑃)) | ||
Theorem | upgrisupwlkALT 44370* | Alternate proof of upgriswlk 27430 using the definition of UPGraph and related theorems. (Contributed by AV, 2-Jan-2021.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹 ∈ 𝑈 ∧ 𝑃 ∈ 𝑍) → (𝐹(Walks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))) | ||
Theorem | upgredgssspr 44371 | The set of edges of a pseudograph is a subset of the set of unordered pairs of vertices. (Contributed by AV, 24-Nov-2021.) |
⊢ (𝐺 ∈ UPGraph → (Edg‘𝐺) ⊆ (Pairs‘(Vtx‘𝐺))) | ||
Theorem | uspgropssxp 44372* | The set 𝐺 of "simple pseudographs" for a fixed set 𝑉 of vertices is a subset of a Cartesian product. For more details about the class 𝐺 of all "simple pseudographs" see comments on uspgrbisymrel 44382. (Contributed by AV, 24-Nov-2021.) |
⊢ 𝑃 = 𝒫 (Pairs‘𝑉) & ⊢ 𝐺 = {〈𝑣, 𝑒〉 ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))} ⇒ ⊢ (𝑉 ∈ 𝑊 → 𝐺 ⊆ (𝑊 × 𝑃)) | ||
Theorem | uspgrsprfv 44373* | The value of the function 𝐹 which maps a "simple pseudograph" for a fixed set 𝑉 of vertices to the set of edges (i.e. range of the edge function) of the graph. Solely for 𝐺 as defined here, the function 𝐹 is a bijection between the "simple pseudographs" and the subsets of the set of pairs 𝑃 over the fixed set 𝑉 of vertices, see uspgrbispr 44379. (Contributed by AV, 24-Nov-2021.) |
⊢ 𝑃 = 𝒫 (Pairs‘𝑉) & ⊢ 𝐺 = {〈𝑣, 𝑒〉 ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))} & ⊢ 𝐹 = (𝑔 ∈ 𝐺 ↦ (2nd ‘𝑔)) ⇒ ⊢ (𝑋 ∈ 𝐺 → (𝐹‘𝑋) = (2nd ‘𝑋)) | ||
Theorem | uspgrsprf 44374* | The mapping 𝐹 is a function from the "simple pseudographs" with a fixed set of vertices 𝑉 into the subsets of the set of pairs over the set 𝑉. (Contributed by AV, 24-Nov-2021.) |
⊢ 𝑃 = 𝒫 (Pairs‘𝑉) & ⊢ 𝐺 = {〈𝑣, 𝑒〉 ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))} & ⊢ 𝐹 = (𝑔 ∈ 𝐺 ↦ (2nd ‘𝑔)) ⇒ ⊢ 𝐹:𝐺⟶𝑃 | ||
Theorem | uspgrsprf1 44375* | The mapping 𝐹 is a one-to-one function from the "simple pseudographs" with a fixed set of vertices 𝑉 into the subsets of the set of pairs over the set 𝑉. (Contributed by AV, 25-Nov-2021.) |
⊢ 𝑃 = 𝒫 (Pairs‘𝑉) & ⊢ 𝐺 = {〈𝑣, 𝑒〉 ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))} & ⊢ 𝐹 = (𝑔 ∈ 𝐺 ↦ (2nd ‘𝑔)) ⇒ ⊢ 𝐹:𝐺–1-1→𝑃 | ||
Theorem | uspgrsprfo 44376* | The mapping 𝐹 is a function from the "simple pseudographs" with a fixed set of vertices 𝑉 onto the subsets of the set of pairs over the set 𝑉. (Contributed by AV, 25-Nov-2021.) |
⊢ 𝑃 = 𝒫 (Pairs‘𝑉) & ⊢ 𝐺 = {〈𝑣, 𝑒〉 ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))} & ⊢ 𝐹 = (𝑔 ∈ 𝐺 ↦ (2nd ‘𝑔)) ⇒ ⊢ (𝑉 ∈ 𝑊 → 𝐹:𝐺–onto→𝑃) | ||
Theorem | uspgrsprf1o 44377* | The mapping 𝐹 is a bijection between the "simple pseudographs" with a fixed set of vertices 𝑉 and the subsets of the set of pairs over the set 𝑉. See also the comments on uspgrbisymrel 44382. (Contributed by AV, 25-Nov-2021.) |
⊢ 𝑃 = 𝒫 (Pairs‘𝑉) & ⊢ 𝐺 = {〈𝑣, 𝑒〉 ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))} & ⊢ 𝐹 = (𝑔 ∈ 𝐺 ↦ (2nd ‘𝑔)) ⇒ ⊢ (𝑉 ∈ 𝑊 → 𝐹:𝐺–1-1-onto→𝑃) | ||
Theorem | uspgrex 44378* | The class 𝐺 of all "simple pseudographs" with a fixed set of vertices 𝑉 is a set. (Contributed by AV, 26-Nov-2021.) |
⊢ 𝑃 = 𝒫 (Pairs‘𝑉) & ⊢ 𝐺 = {〈𝑣, 𝑒〉 ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))} ⇒ ⊢ (𝑉 ∈ 𝑊 → 𝐺 ∈ V) | ||
Theorem | uspgrbispr 44379* | There is a bijection between the "simple pseudographs" with a fixed set of vertices 𝑉 and the subsets of the set of pairs over the set 𝑉. (Contributed by AV, 26-Nov-2021.) |
⊢ 𝑃 = 𝒫 (Pairs‘𝑉) & ⊢ 𝐺 = {〈𝑣, 𝑒〉 ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))} ⇒ ⊢ (𝑉 ∈ 𝑊 → ∃𝑓 𝑓:𝐺–1-1-onto→𝑃) | ||
Theorem | uspgrspren 44380* | The set 𝐺 of the "simple pseudographs" with a fixed set of vertices 𝑉 and the class 𝑃 of subsets of the set of pairs over the fixed set 𝑉 are equinumerous. (Contributed by AV, 27-Nov-2021.) |
⊢ 𝑃 = 𝒫 (Pairs‘𝑉) & ⊢ 𝐺 = {〈𝑣, 𝑒〉 ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))} ⇒ ⊢ (𝑉 ∈ 𝑊 → 𝐺 ≈ 𝑃) | ||
Theorem | uspgrymrelen 44381* | The set 𝐺 of the "simple pseudographs" with a fixed set of vertices 𝑉 and the class 𝑅 of the symmetric relations on the fixed set 𝑉 are equinumerous. For more details about the class 𝐺 of all "simple pseudographs" see comments on uspgrbisymrel 44382. (Contributed by AV, 27-Nov-2021.) |
⊢ 𝐺 = {〈𝑣, 𝑒〉 ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))} & ⊢ 𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑟𝑦 ↔ 𝑦𝑟𝑥)} ⇒ ⊢ (𝑉 ∈ 𝑊 → 𝐺 ≈ 𝑅) | ||
Theorem | uspgrbisymrel 44382* |
There is a bijection between the "simple pseudographs" for a fixed
set
𝑉 of vertices and the class 𝑅 of the
symmetric relations on the
fixed set 𝑉. The simple pseudographs, which are
graphs without
hyper- or multiedges, but which may contain loops, are expressed as
ordered pairs of the vertices and the edges (as proper or improper
unordered pairs of vertices, not as indexed edges!) in this theorem.
That class 𝐺 of such simple pseudographs is a set
(if 𝑉 is a
set, see uspgrex 44378) of equivalence classes of graphs
abstracting from
the index sets of their edge functions.
Solely for this abstraction, there is a bijection between the "simple pseudographs" as members of 𝐺 and the symmetric relations 𝑅 on the fixed set 𝑉 of vertices. This theorem would not hold for 𝐺 = {𝑔 ∈ USPGraph ∣ (Vtx‘𝑔) = 𝑉} and even not for 𝐺 = {〈𝑣, 𝑒〉 ∣ (𝑣 = 𝑉 ∧ 〈𝑣, 𝑒〉 ∈ USPGraph)}, because these are much bigger classes. (Proposed by Gerard Lang, 16-Nov-2021.) (Contributed by AV, 27-Nov-2021.) |
⊢ 𝐺 = {〈𝑣, 𝑒〉 ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))} & ⊢ 𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑟𝑦 ↔ 𝑦𝑟𝑥)} ⇒ ⊢ (𝑉 ∈ 𝑊 → ∃𝑓 𝑓:𝐺–1-1-onto→𝑅) | ||
Theorem | uspgrbisymrelALT 44383* | Alternate proof of uspgrbisymrel 44382 not using the definition of equinumerosity. (Contributed by AV, 26-Nov-2021.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ 𝐺 = {〈𝑣, 𝑒〉 ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))} & ⊢ 𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑟𝑦 ↔ 𝑦𝑟𝑥)} ⇒ ⊢ (𝑉 ∈ 𝑊 → ∃𝑓 𝑓:𝐺–1-1-onto→𝑅) | ||
Theorem | ovn0dmfun 44384 | If a class operation value for two operands is not the empty set, then the operands are contained in the domain of the class, and the class restricted to the operands is a function, analogous to fvfundmfvn0 6683. (Contributed by AV, 27-Jan-2020.) |
⊢ ((𝐴𝐹𝐵) ≠ ∅ → (〈𝐴, 𝐵〉 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {〈𝐴, 𝐵〉}))) | ||
Theorem | xpsnopab 44385* | A Cartesian product with a singleton expressed as ordered-pair class abstraction. (Contributed by AV, 27-Jan-2020.) |
⊢ ({𝑋} × 𝐶) = {〈𝑎, 𝑏〉 ∣ (𝑎 = 𝑋 ∧ 𝑏 ∈ 𝐶)} | ||
Theorem | xpiun 44386* | A Cartesian product expressed as indexed union of ordered-pair class abstractions. (Contributed by AV, 27-Jan-2020.) |
⊢ (𝐵 × 𝐶) = ∪ 𝑥 ∈ 𝐵 {〈𝑎, 𝑏〉 ∣ (𝑎 = 𝑥 ∧ 𝑏 ∈ 𝐶)} | ||
Theorem | ovn0ssdmfun 44387* | If a class' operation value for two operands is not the empty set, the operands are contained in the domain of the class, and the class restricted to the operands is a function, analogous to fvfundmfvn0 6683. (Contributed by AV, 27-Jan-2020.) |
⊢ (∀𝑎 ∈ 𝐷 ∀𝑏 ∈ 𝐸 (𝑎𝐹𝑏) ≠ ∅ → ((𝐷 × 𝐸) ⊆ dom 𝐹 ∧ Fun (𝐹 ↾ (𝐷 × 𝐸)))) | ||
Theorem | fnxpdmdm 44388 | The domain of the domain of a function over a Cartesian square. (Contributed by AV, 13-Jan-2020.) |
⊢ (𝐹 Fn (𝐴 × 𝐴) → dom dom 𝐹 = 𝐴) | ||
Theorem | cnfldsrngbas 44389 | The base set of a subring of the field of complex numbers. (Contributed by AV, 31-Jan-2020.) |
⊢ 𝑅 = (ℂfld ↾s 𝑆) ⇒ ⊢ (𝑆 ⊆ ℂ → 𝑆 = (Base‘𝑅)) | ||
Theorem | cnfldsrngadd 44390 | The group addition operation of a subring of the field of complex numbers. (Contributed by AV, 31-Jan-2020.) |
⊢ 𝑅 = (ℂfld ↾s 𝑆) ⇒ ⊢ (𝑆 ∈ 𝑉 → + = (+g‘𝑅)) | ||
Theorem | cnfldsrngmul 44391 | The ring multiplication operation of a subring of the field of complex numbers. (Contributed by AV, 31-Jan-2020.) |
⊢ 𝑅 = (ℂfld ↾s 𝑆) ⇒ ⊢ (𝑆 ∈ 𝑉 → · = (.r‘𝑅)) | ||
Theorem | plusfreseq 44392 | If the empty set is not contained in the range of the group addition function of an extensible structure (not necessarily a magma), the restriction of the addition operation to (the Cartesian square of) the base set is the functionalization of it. (Contributed by AV, 28-Jan-2020.) |
⊢ 𝐵 = (Base‘𝑀) & ⊢ + = (+g‘𝑀) & ⊢ ⨣ = (+𝑓‘𝑀) ⇒ ⊢ (∅ ∉ ran ⨣ → ( + ↾ (𝐵 × 𝐵)) = ⨣ ) | ||
Theorem | mgmplusfreseq 44393 | If the empty set is not contained in the base set of a magma, the restriction of the addition operation to (the Cartesian square of) the base set is the functionalization of it. (Contributed by AV, 28-Jan-2020.) |
⊢ 𝐵 = (Base‘𝑀) & ⊢ + = (+g‘𝑀) & ⊢ ⨣ = (+𝑓‘𝑀) ⇒ ⊢ ((𝑀 ∈ Mgm ∧ ∅ ∉ 𝐵) → ( + ↾ (𝐵 × 𝐵)) = ⨣ ) | ||
Theorem | 0mgm 44394 | A set with an empty base set is always a magma. (Contributed by AV, 25-Feb-2020.) |
⊢ (Base‘𝑀) = ∅ ⇒ ⊢ (𝑀 ∈ 𝑉 → 𝑀 ∈ Mgm) | ||
Theorem | mgmpropd 44395* | If two structures have the same (nonempty) base set, and the values of their group (addition) operations are equal for all pairs of elements of the base set, one is a magma iff the other one is. (Contributed by AV, 25-Feb-2020.) |
⊢ (𝜑 → 𝐵 = (Base‘𝐾)) & ⊢ (𝜑 → 𝐵 = (Base‘𝐿)) & ⊢ (𝜑 → 𝐵 ≠ ∅) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦)) ⇒ ⊢ (𝜑 → (𝐾 ∈ Mgm ↔ 𝐿 ∈ Mgm)) | ||
Theorem | ismgmd 44396* | Deduce a magma from its properties. (Contributed by AV, 25-Feb-2020.) |
⊢ (𝜑 → 𝐵 = (Base‘𝐺)) & ⊢ (𝜑 → 𝐺 ∈ 𝑉) & ⊢ (𝜑 → + = (+g‘𝐺)) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵) ⇒ ⊢ (𝜑 → 𝐺 ∈ Mgm) | ||
Syntax | cmgmhm 44397 | Hom-set generator class for magmas. |
class MgmHom | ||
Syntax | csubmgm 44398 | Class function taking a magma to its lattice of submagmas. |
class SubMgm | ||
Definition | df-mgmhm 44399* | A magma homomorphism is a function on the base sets which preserves the binary operation. (Contributed by AV, 24-Feb-2020.) |
⊢ MgmHom = (𝑠 ∈ Mgm, 𝑡 ∈ Mgm ↦ {𝑓 ∈ ((Base‘𝑡) ↑m (Base‘𝑠)) ∣ ∀𝑥 ∈ (Base‘𝑠)∀𝑦 ∈ (Base‘𝑠)(𝑓‘(𝑥(+g‘𝑠)𝑦)) = ((𝑓‘𝑥)(+g‘𝑡)(𝑓‘𝑦))}) | ||
Definition | df-submgm 44400* | A submagma is a subset of a magma which is closed under the operation. Such subsets are themselves magmas. (Contributed by AV, 24-Feb-2020.) |
⊢ SubMgm = (𝑠 ∈ Mgm ↦ {𝑡 ∈ 𝒫 (Base‘𝑠) ∣ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥(+g‘𝑠)𝑦) ∈ 𝑡}) |
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