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Theorem List for Metamath Proof Explorer - 42701-42800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorem9gbo 42701 9 is an odd Goldbach number. (Contributed by AV, 26-Jul-2020.)
9 ∈ GoldbachOdd

Theorem11gbo 42702 11 is an odd Goldbach number. (Contributed by AV, 29-Jul-2020.)
11 ∈ GoldbachOdd

Theoremstgoldbwt 42703 If the strong ternary Goldbach conjecture is valid, then the weak ternary Goldbach conjecture holds, too. (Contributed by AV, 27-Jul-2020.)
(∀𝑛 ∈ Odd (7 < 𝑛𝑛 ∈ GoldbachOdd ) → ∀𝑛 ∈ Odd (5 < 𝑛𝑛 ∈ GoldbachOddW ))

Theoremsbgoldbwt 42704* 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 < 𝑛𝑛 ∈ GoldbachEven ) → ∀𝑚 ∈ Odd (5 < 𝑚𝑚 ∈ GoldbachOddW ))

Theoremsbgoldbst 42705* 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 < 𝑛𝑛 ∈ GoldbachEven ) → ∀𝑚 ∈ Odd (7 < 𝑚𝑚 ∈ GoldbachOdd ))

Theoremsbgoldbaltlem1 42706 Lemma 1 for sbgoldbalt 42708: 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 < 𝑁𝑁 = (𝑃 + 𝑄)) → 𝑄 ∈ Odd ))

Theoremsbgoldbaltlem2 42707 Lemma 2 for sbgoldbalt 42708: 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 < 𝑁𝑁 = (𝑃 + 𝑄)) → (𝑃 ∈ Odd ∧ 𝑄 ∈ Odd )))

Theoremsbgoldbalt 42708* 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 < 𝑛𝑛 ∈ GoldbachEven ) ↔ ∀𝑛 ∈ Even (2 < 𝑛 → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞)))

Theoremsbgoldbb 42709* If the strong binary Goldbach conjecture is valid, the binary Goldbach conjecture is valid. (Contributed by AV, 23-Dec-2021.)
(∀𝑛 ∈ Even (4 < 𝑛𝑛 ∈ GoldbachEven ) → ∀𝑛 ∈ Even (2 < 𝑛 → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ 𝑛 = (𝑝 + 𝑞)))

Theoremsgoldbeven3prm 42710* 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 < 𝑛𝑛 ∈ GoldbachEven ) → ((𝑁 ∈ Even ∧ 6 ≤ 𝑁) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑁 = ((𝑝 + 𝑞) + 𝑟)))

Theoremsbgoldbm 42711* 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 < 𝑛𝑛 ∈ GoldbachEven ) → ∀𝑛 ∈ (ℤ‘6)∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟))

Theoremmogoldbb 42712* 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)))

Theoremsbgoldbmb 42713* 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 < 𝑛𝑛 ∈ GoldbachEven ) ↔ ∀𝑛 ∈ (ℤ‘6)∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟))

Theoremsbgoldbo 42714* 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 < 𝑛𝑛 ∈ GoldbachEven ) → ∀𝑛 ∈ (ℤ‘3)∃𝑝𝑃𝑞𝑃𝑟𝑃 𝑛 = ((𝑝 + 𝑞) + 𝑟))

Theoremnnsum3primes4 42715* 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)(𝑓𝑘))

Theoremnnsum4primes4 42716* 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)(𝑓𝑘))

Theoremnnsum3primesprm 42717* 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)(𝑓𝑘)))

Theoremnnsum4primesprm 42718* 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)(𝑓𝑘)))

Theoremnnsum3primesgbe 42719* 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)(𝑓𝑘)))

Theoremnnsum4primesgbe 42720* 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)(𝑓𝑘)))

Theoremnnsum3primesle9 42721* 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)(𝑓𝑘)))

Theoremnnsum4primesle9 42722* 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)(𝑓𝑘)))

Theoremnnsum4primesodd 42723* 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 < 𝑚𝑚 ∈ GoldbachOddW ) → ((𝑁 ∈ (ℤ‘6) ∧ 𝑁 ∈ Odd ) → ∃𝑓 ∈ (ℙ ↑𝑚 (1...3))𝑁 = Σ𝑘 ∈ (1...3)(𝑓𝑘)))

Theoremnnsum4primesoddALTV 42724* 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 < 𝑚𝑚 ∈ GoldbachOdd ) → ((𝑁 ∈ (ℤ‘8) ∧ 𝑁 ∈ Odd ) → ∃𝑓 ∈ (ℙ ↑𝑚 (1...3))𝑁 = Σ𝑘 ∈ (1...3)(𝑓𝑘)))

Theoremevengpop3 42725* 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 < 𝑚𝑚 ∈ GoldbachOddW ) → ((𝑁 ∈ (ℤ‘9) ∧ 𝑁 ∈ Even ) → ∃𝑜 ∈ GoldbachOddW 𝑁 = (𝑜 + 3)))

Theoremevengpoap3 42726* 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 < 𝑚𝑚 ∈ GoldbachOdd ) → ((𝑁 ∈ (ℤ12) ∧ 𝑁 ∈ Even ) → ∃𝑜 ∈ GoldbachOdd 𝑁 = (𝑜 + 3)))

Theoremnnsum4primeseven 42727* 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 < 𝑚𝑚 ∈ GoldbachOddW ) → ((𝑁 ∈ (ℤ‘9) ∧ 𝑁 ∈ Even ) → ∃𝑓 ∈ (ℙ ↑𝑚 (1...4))𝑁 = Σ𝑘 ∈ (1...4)(𝑓𝑘)))

Theoremnnsum4primesevenALTV 42728* 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 < 𝑚𝑚 ∈ GoldbachOdd ) → ((𝑁 ∈ (ℤ12) ∧ 𝑁 ∈ Even ) → ∃𝑓 ∈ (ℙ ↑𝑚 (1...4))𝑁 = Σ𝑘 ∈ (1...4)(𝑓𝑘)))

Theoremwtgoldbnnsum4prm 42729* 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 < 𝑚𝑚 ∈ GoldbachOddW ) → ∀𝑛 ∈ (ℤ‘2)∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑𝑚 (1...𝑑))(𝑑 ≤ 4 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓𝑘)))

Theoremstgoldbnnsum4prm 42730* 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 < 𝑚𝑚 ∈ GoldbachOdd ) → ∀𝑛 ∈ (ℤ‘2)∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑𝑚 (1...𝑑))(𝑑 ≤ 4 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓𝑘)))

Theorembgoldbnnsum3prm 42731* 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 < 𝑚𝑚 ∈ GoldbachEven ) → ∀𝑛 ∈ (ℤ‘2)∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑𝑚 (1...𝑑))(𝑑 ≤ 3 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓𝑘)))

Theorembgoldbtbndlem1 42732 Lemma 1 for bgoldbtbnd 42736: the odd numbers between 7 and 13 (exclusive) are odd Goldbach numbers. (Contributed by AV, 29-Jul-2020.)
((𝑁 ∈ Odd ∧ 7 < 𝑁𝑁 ∈ (7[,)13)) → 𝑁 ∈ GoldbachOdd )

Theorembgoldbtbndlem2 42733* Lemma 2 for bgoldbtbnd 42736. (Contributed by AV, 1-Aug-2020.)
(𝜑𝑀 ∈ (ℤ11))    &   (𝜑𝑁 ∈ (ℤ11))    &   (𝜑 → ∀𝑛 ∈ Even ((4 < 𝑛𝑛 < 𝑁) → 𝑛 ∈ GoldbachEven ))    &   (𝜑𝐷 ∈ (ℤ‘3))    &   (𝜑𝐹 ∈ (RePart‘𝐷))    &   (𝜑 → ∀𝑖 ∈ (0..^𝐷)((𝐹𝑖) ∈ (ℙ ∖ {2}) ∧ ((𝐹‘(𝑖 + 1)) − (𝐹𝑖)) < (𝑁 − 4) ∧ 4 < ((𝐹‘(𝑖 + 1)) − (𝐹𝑖))))    &   (𝜑 → (𝐹‘0) = 7)    &   (𝜑 → (𝐹‘1) = 13)    &   (𝜑𝑀 < (𝐹𝐷))    &   𝑆 = (𝑋 − (𝐹‘(𝐼 − 1)))       ((𝜑𝑋 ∈ Odd ∧ 𝐼 ∈ (1..^𝐷)) → ((𝑋 ∈ ((𝐹𝐼)[,)(𝐹‘(𝐼 + 1))) ∧ (𝑋 − (𝐹𝐼)) ≤ 4) → (𝑆 ∈ Even ∧ 𝑆 < 𝑁 ∧ 4 < 𝑆)))

Theorembgoldbtbndlem3 42734* Lemma 3 for bgoldbtbnd 42736. (Contributed by AV, 1-Aug-2020.)
(𝜑𝑀 ∈ (ℤ11))    &   (𝜑𝑁 ∈ (ℤ11))    &   (𝜑 → ∀𝑛 ∈ Even ((4 < 𝑛𝑛 < 𝑁) → 𝑛 ∈ GoldbachEven ))    &   (𝜑𝐷 ∈ (ℤ‘3))    &   (𝜑𝐹 ∈ (RePart‘𝐷))    &   (𝜑 → ∀𝑖 ∈ (0..^𝐷)((𝐹𝑖) ∈ (ℙ ∖ {2}) ∧ ((𝐹‘(𝑖 + 1)) − (𝐹𝑖)) < (𝑁 − 4) ∧ 4 < ((𝐹‘(𝑖 + 1)) − (𝐹𝑖))))    &   (𝜑 → (𝐹‘0) = 7)    &   (𝜑 → (𝐹‘1) = 13)    &   (𝜑𝑀 < (𝐹𝐷))    &   (𝜑 → (𝐹𝐷) ∈ ℝ)    &   𝑆 = (𝑋 − (𝐹𝐼))       ((𝜑𝑋 ∈ Odd ∧ 𝐼 ∈ (1..^𝐷)) → ((𝑋 ∈ ((𝐹𝐼)[,)(𝐹‘(𝐼 + 1))) ∧ 4 < 𝑆) → (𝑆 ∈ Even ∧ 𝑆 < 𝑁 ∧ 4 < 𝑆)))

Theorembgoldbtbndlem4 42735* Lemma 4 for bgoldbtbnd 42736. (Contributed by AV, 1-Aug-2020.)
(𝜑𝑀 ∈ (ℤ11))    &   (𝜑𝑁 ∈ (ℤ11))    &   (𝜑 → ∀𝑛 ∈ Even ((4 < 𝑛𝑛 < 𝑁) → 𝑛 ∈ GoldbachEven ))    &   (𝜑𝐷 ∈ (ℤ‘3))    &   (𝜑𝐹 ∈ (RePart‘𝐷))    &   (𝜑 → ∀𝑖 ∈ (0..^𝐷)((𝐹𝑖) ∈ (ℙ ∖ {2}) ∧ ((𝐹‘(𝑖 + 1)) − (𝐹𝑖)) < (𝑁 − 4) ∧ 4 < ((𝐹‘(𝑖 + 1)) − (𝐹𝑖))))    &   (𝜑 → (𝐹‘0) = 7)    &   (𝜑 → (𝐹‘1) = 13)    &   (𝜑𝑀 < (𝐹𝐷))    &   (𝜑 → (𝐹𝐷) ∈ ℝ)       (((𝜑𝐼 ∈ (1..^𝐷)) ∧ 𝑋 ∈ Odd ) → ((𝑋 ∈ ((𝐹𝐼)[,)(𝐹‘(𝐼 + 1))) ∧ (𝑋 − (𝐹𝐼)) ≤ 4) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ ((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑟 ∈ Odd ) ∧ 𝑋 = ((𝑝 + 𝑞) + 𝑟))))

Theorembgoldbtbnd 42736* 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.)
(𝜑𝑀 ∈ (ℤ11))    &   (𝜑𝑁 ∈ (ℤ11))    &   (𝜑 → ∀𝑛 ∈ Even ((4 < 𝑛𝑛 < 𝑁) → 𝑛 ∈ GoldbachEven ))    &   (𝜑𝐷 ∈ (ℤ‘3))    &   (𝜑𝐹 ∈ (RePart‘𝐷))    &   (𝜑 → ∀𝑖 ∈ (0..^𝐷)((𝐹𝑖) ∈ (ℙ ∖ {2}) ∧ ((𝐹‘(𝑖 + 1)) − (𝐹𝑖)) < (𝑁 − 4) ∧ 4 < ((𝐹‘(𝑖 + 1)) − (𝐹𝑖))))    &   (𝜑 → (𝐹‘0) = 7)    &   (𝜑 → (𝐹‘1) = 13)    &   (𝜑𝑀 < (𝐹𝐷))    &   (𝜑 → (𝐹𝐷) ∈ ℝ)       (𝜑 → ∀𝑛 ∈ Odd ((7 < 𝑛𝑛 < 𝑀) → 𝑛 ∈ GoldbachOdd ))

Axiomax-bgbltosilva 42737 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 < 𝑁𝑁 ≤ (4 · (10↑18))) → 𝑁 ∈ GoldbachEven )

Axiomax-tgoldbachgt 42738* Temporary duplicate of tgoldbachgt 31351, 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 = {𝑧𝑂 ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ ((𝑝𝑂𝑞𝑂𝑟𝑂) ∧ 𝑧 = ((𝑝 + 𝑞) + 𝑟))}       𝑚 ∈ ℕ (𝑚 ≤ (10↑27) ∧ ∀𝑛𝑂 (𝑚 < 𝑛𝑛𝐺))

TheoremtgoldbachgtALTV 42739* Variant of Thierry Arnoux's tgoldbachgt 31351 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 < 𝑛𝑛 ∈ GoldbachOdd ))

Theorembgoldbachlt 42740* 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 42737. (Contributed by AV, 3-Aug-2020.) (Revised by AV, 9-Sep-2021.)
𝑚 ∈ ℕ ((4 · (10↑18)) ≤ 𝑚 ∧ ∀𝑛 ∈ Even ((4 < 𝑛𝑛 < 𝑚) → 𝑛 ∈ GoldbachEven ))

Axiomax-hgprmladder 42741 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 ∧ (𝑓𝑑) = (89 · (10↑29))) ∧ ∀𝑖 ∈ (0..^𝑑)((𝑓𝑖) ∈ (ℙ ∖ {2}) ∧ ((𝑓‘(𝑖 + 1)) − (𝑓𝑖)) < ((4 · (10↑18)) − 4) ∧ 4 < ((𝑓‘(𝑖 + 1)) − (𝑓𝑖))))

Theoremtgblthelfgott 42742 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 42740, ax-hgprmladder 42741 and bgoldbtbnd 42736. (Contributed by AV, 4-Aug-2020.) (Revised by AV, 9-Sep-2021.)
((𝑁 ∈ Odd ∧ 7 < 𝑁𝑁 < (88 · (10↑29))) → 𝑁 ∈ GoldbachOdd )

Theoremtgoldbachlt 42743* 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 42742. (Contributed by AV, 4-Aug-2020.) (Revised by AV, 9-Sep-2021.)
𝑚 ∈ ℕ ((8 · (10↑30)) < 𝑚 ∧ ∀𝑛 ∈ Odd ((7 < 𝑛𝑛 < 𝑚) → 𝑛 ∈ GoldbachOdd ))

Theoremtgoldbach 42744 The ternary Goldbach conjecture is valid. Main theorem in [Helfgott] p. 2. This follows from tgoldbachlt 42743 and ax-tgoldbachgt 42738. (Contributed by AV, 2-Aug-2020.) (Revised by AV, 9-Sep-2021.)
𝑛 ∈ Odd (7 < 𝑛𝑛 ∈ GoldbachOdd )

20.36.13  Graph theory (extension)

20.36.13.1  Isomorphic graphs

In the following, a general definition of the isomorphy relation for graphs and specializations for simple hypergraphs (isomushgr 42753) and simple pseudographs (isomuspgr 42761) 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 42762, isomgrsym 42763, isomgrtr 42766). Fianlly, isomorphic graphs with different representations are studied (strisomgrop 42767, ushrisomgr 42768).

Maybe more important than graph isomorphy is the notion of graph isomorphism, which can be defined as in df-grisom 42747. 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 𝐴𝑀, 𝐵𝑁 and 𝐴𝑁, 𝐵𝑀 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.

Syntaxcgrisom 42745 Extend class notation to include the graph ispmorphisms.
class GrIsom

Syntaxcisomgr 42746 Extend class notation to include the isomorphy relation for graphs.
class IsomGr

Definitiondf-grisom 42747* 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)‘(𝑔𝑖)))})

Definitiondf-isomgr 42748* 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)‘(𝑔𝑖))))}

Theoremisomgrrel 42749 The isomorphy relation for graphs is a relation. (Contributed by AV, 11-Nov-2022.)
Rel IsomGr

Theoremisomgr 42750* The isomorphy relation for two graphs. (Contributed by AV, 11-Nov-2022.)
𝑉 = (Vtx‘𝐴)    &   𝑊 = (Vtx‘𝐵)    &   𝐼 = (iEdg‘𝐴)    &   𝐽 = (iEdg‘𝐵)       ((𝐴𝑋𝐵𝑌) → (𝐴 IsomGr 𝐵 ↔ ∃𝑓(𝑓:𝑉1-1-onto𝑊 ∧ ∃𝑔(𝑔:dom 𝐼1-1-onto→dom 𝐽 ∧ ∀𝑖 ∈ dom 𝐼(𝑓 “ (𝐼𝑖)) = (𝐽‘(𝑔𝑖))))))

Theoremisisomgr 42751* 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:𝑉1-1-onto𝑊 ∧ ∃𝑔(𝑔:dom 𝐼1-1-onto→dom 𝐽 ∧ ∀𝑖 ∈ dom 𝐼(𝑓 “ (𝐼𝑖)) = (𝐽‘(𝑔𝑖)))))

Theoremisomgreqve 42752 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 ∧ 𝐵𝑌) ∧ ((Vtx‘𝐴) = (Vtx‘𝐵) ∧ (iEdg‘𝐴) = (iEdg‘𝐵))) → 𝐴 IsomGr 𝐵)

Theoremisomushgr 42753* 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:𝑉1-1-onto𝑊 ∧ ∃𝑔(𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒)))))

Theoremisomuspgrlem1 42754* Lemma 1 for isomuspgr 42761. (Contributed by AV, 29-Nov-2022.)
𝑉 = (Vtx‘𝐴)    &   𝑊 = (Vtx‘𝐵)    &   𝐸 = (Edg‘𝐴)    &   𝐾 = (Edg‘𝐵)       (((((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) ∧ (𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))) ∧ (𝑎𝑉𝑏𝑉)) → ({(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾 → {𝑎, 𝑏} ∈ 𝐸))

Theoremisomuspgrlem2a 42755* Lemma 1 for isomuspgrlem2 42760. (Contributed by AV, 29-Nov-2022.)
𝑉 = (Vtx‘𝐴)    &   𝑊 = (Vtx‘𝐵)    &   𝐸 = (Edg‘𝐴)    &   𝐾 = (Edg‘𝐵)    &   𝐺 = (𝑥𝐸 ↦ (𝐹𝑥))       (𝐹𝑋 → ∀𝑒𝐸 (𝐹𝑒) = (𝐺𝑒))

Theoremisomuspgrlem2b 42756* Lemma 2 for isomuspgrlem2 42760. (Contributed by AV, 29-Nov-2022.)
𝑉 = (Vtx‘𝐴)    &   𝑊 = (Vtx‘𝐵)    &   𝐸 = (Edg‘𝐴)    &   𝐾 = (Edg‘𝐵)    &   𝐺 = (𝑥𝐸 ↦ (𝐹𝑥))    &   (𝜑𝐴 ∈ USPGraph)    &   (𝜑𝐹:𝑉1-1-onto𝑊)    &   (𝜑 → ∀𝑎𝑉𝑏𝑉 ({𝑎, 𝑏} ∈ 𝐸 ↔ {(𝐹𝑎), (𝐹𝑏)} ∈ 𝐾))       (𝜑𝐺:𝐸𝐾)

Theoremisomuspgrlem2c 42757* Lemma 3 for isomuspgrlem2 42760. (Contributed by AV, 29-Nov-2022.)
𝑉 = (Vtx‘𝐴)    &   𝑊 = (Vtx‘𝐵)    &   𝐸 = (Edg‘𝐴)    &   𝐾 = (Edg‘𝐵)    &   𝐺 = (𝑥𝐸 ↦ (𝐹𝑥))    &   (𝜑𝐴 ∈ USPGraph)    &   (𝜑𝐹:𝑉1-1-onto𝑊)    &   (𝜑 → ∀𝑎𝑉𝑏𝑉 ({𝑎, 𝑏} ∈ 𝐸 ↔ {(𝐹𝑎), (𝐹𝑏)} ∈ 𝐾))    &   (𝜑𝐹𝑋)       (𝜑𝐺:𝐸1-1𝐾)

Theoremisomuspgrlem2d 42758* Lemma 4 for isomuspgrlem2 42760. (Contributed by AV, 1-Dec-2022.)
𝑉 = (Vtx‘𝐴)    &   𝑊 = (Vtx‘𝐵)    &   𝐸 = (Edg‘𝐴)    &   𝐾 = (Edg‘𝐵)    &   𝐺 = (𝑥𝐸 ↦ (𝐹𝑥))    &   (𝜑𝐴 ∈ USPGraph)    &   (𝜑𝐹:𝑉1-1-onto𝑊)    &   (𝜑 → ∀𝑎𝑉𝑏𝑉 ({𝑎, 𝑏} ∈ 𝐸 ↔ {(𝐹𝑎), (𝐹𝑏)} ∈ 𝐾))    &   (𝜑𝐹𝑋)    &   (𝜑𝐵 ∈ USPGraph)       (𝜑𝐺:𝐸onto𝐾)

Theoremisomuspgrlem2e 42759* Lemma 5 for isomuspgrlem2 42760. (Contributed by AV, 1-Dec-2022.)
𝑉 = (Vtx‘𝐴)    &   𝑊 = (Vtx‘𝐵)    &   𝐸 = (Edg‘𝐴)    &   𝐾 = (Edg‘𝐵)    &   𝐺 = (𝑥𝐸 ↦ (𝐹𝑥))    &   (𝜑𝐴 ∈ USPGraph)    &   (𝜑𝐹:𝑉1-1-onto𝑊)    &   (𝜑 → ∀𝑎𝑉𝑏𝑉 ({𝑎, 𝑏} ∈ 𝐸 ↔ {(𝐹𝑎), (𝐹𝑏)} ∈ 𝐾))    &   (𝜑𝐹𝑋)    &   (𝜑𝐵 ∈ USPGraph)       (𝜑𝐺:𝐸1-1-onto𝐾)

Theoremisomuspgrlem2 42760* Lemma 2 for isomuspgr 42761. (Contributed by AV, 1-Dec-2022.)
𝑉 = (Vtx‘𝐴)    &   𝑊 = (Vtx‘𝐵)    &   𝐸 = (Edg‘𝐴)    &   𝐾 = (Edg‘𝐵)       (((𝐴 ∈ USPGraph ∧ 𝐵 ∈ USPGraph) ∧ 𝑓:𝑉1-1-onto𝑊) → (∀𝑎𝑉𝑏𝑉 ({𝑎, 𝑏} ∈ 𝐸 ↔ {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾) → ∃𝑔(𝑔:𝐸1-1-onto𝐾 ∧ ∀𝑒𝐸 (𝑓𝑒) = (𝑔𝑒))))

Theoremisomuspgr 42761* 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:𝑉1-1-onto𝑊 ∧ ∀𝑎𝑉𝑏𝑉 ({𝑎, 𝑏} ∈ 𝐸 ↔ {(𝑓𝑎), (𝑓𝑏)} ∈ 𝐾))))

Theoremisomgrref 42762 The isomorphy relation is reflexive for hypergraphs. (Contributed by AV, 11-Nov-2022.)
(𝐺 ∈ UHGraph → 𝐺 IsomGr 𝐺)

Theoremisomgrsym 42763 The isomorphy relation is symmetric for hypergraphs. (Contributed by AV, 11-Nov-2022.)
((𝐴 ∈ UHGraph ∧ 𝐵𝑌) → (𝐴 IsomGr 𝐵𝐵 IsomGr 𝐴))

Theoremisomgrsymb 42764 The isomorphy relation is symmetric for hypergraphs. (Contributed by AV, 11-Nov-2022.)
((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph) → (𝐴 IsomGr 𝐵𝐵 IsomGr 𝐴))

Theoremisomgrtrlem 42765* Lemma for isomgrtr 42766. (Contributed by AV, 5-Dec-2022.)
(((((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph ∧ 𝐶𝑋) ∧ 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)‘(𝑔𝑖)))) ∧ (𝑤:dom (iEdg‘𝐵)–1-1-onto→dom (iEdg‘𝐶) ∧ ∀𝑘 ∈ dom (iEdg‘𝐵)(𝑣 “ ((iEdg‘𝐵)‘𝑘)) = ((iEdg‘𝐶)‘(𝑤𝑘)))) → ∀𝑗 ∈ dom (iEdg‘𝐴)((𝑣𝑓) “ ((iEdg‘𝐴)‘𝑗)) = ((iEdg‘𝐶)‘((𝑤𝑔)‘𝑗)))

Theoremisomgrtr 42766 The isomorphy relation is transitive for hypergraphs. (Contributed by AV, 5-Dec-2022.)
((𝐴 ∈ UHGraph ∧ 𝐵 ∈ UHGraph ∧ 𝐶𝑋) → ((𝐴 IsomGr 𝐵𝐵 IsomGr 𝐶) → 𝐴 IsomGr 𝐶))

Theoremstrisomgrop 42767 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 ∧ 𝑉𝑋𝐸𝑌) → 𝐺 IsomGr 𝐻)

Theoremushrisomgr 42768 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)

20.36.13.2  Loop-free graphs - extension

Theorem1hegrlfgr 42769* A graph 𝐺 with one hyperedge joining at least two vertices is a loop-free graph. (Contributed by AV, 23-Feb-2021.)
(𝜑𝐴𝑋)    &   (𝜑𝐵𝑉)    &   (𝜑𝐶𝑉)    &   (𝜑𝐵𝐶)    &   (𝜑𝐸 ∈ 𝒫 𝑉)    &   (𝜑 → (iEdg‘𝐺) = {⟨𝐴, 𝐸⟩})    &   (𝜑 → {𝐵, 𝐶} ⊆ 𝐸)       (𝜑 → (iEdg‘𝐺):{𝐴}⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)})

20.36.13.3  Walks - extension

Syntaxcupwlks 42770 Extend class notation with walks (of a pseudograph).
class UPWalks

Definitiondf-upwlks 42771* 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)‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))})})

Theoremupwlksfval 42772* 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)       (𝐺𝑊 → (UPWalks‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom 𝐼𝑝:(0...(♯‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))(𝐼‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))})})

Theoremisupwlk 42773* 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)       ((𝐺𝑊𝐹𝑈𝑃𝑍) → (𝐹(UPWalks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom 𝐼𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})))

Theoremisupwlkg 42774* Generalization of isupwlk 42773: Conditions for two classes to represent a simple walk. (Contributed by AV, 5-Nov-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)       (𝐺𝑊 → (𝐹(UPWalks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom 𝐼𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})))

Theoremupwlkbprop 42775 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))

Theoremupwlkwlk 42776 A simple walk is a walk. (Contributed by AV, 30-Dec-2020.) (Proof shortened by AV, 27-Feb-2021.)
(𝐹(UPWalks‘𝐺)𝑃𝐹(Walks‘𝐺)𝑃)

Theoremupgrwlkupwlk 42777 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)

Theoremupgrwlkupwlkb 42778 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)𝑃𝐹(UPWalks‘𝐺)𝑃))

TheoremupgrisupwlkALT 42779* Alternate proof of upgriswlk 27005 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 ∧ 𝐹𝑈𝑃𝑍) → (𝐹(Walks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom 𝐼𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})))

20.36.13.4  Edges of graphs expressed as sets of unordered pairs

Theoremupgredgssspr 42780 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)))

Theoremuspgropssxp 42781* 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 42791. (Contributed by AV, 24-Nov-2021.)
𝑃 = 𝒫 (Pairs‘𝑉)    &   𝐺 = {⟨𝑣, 𝑒⟩ ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))}       (𝑉𝑊𝐺 ⊆ (𝑊 × 𝑃))

Theoremuspgrsprfv 42782* 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 42788. (Contributed by AV, 24-Nov-2021.)
𝑃 = 𝒫 (Pairs‘𝑉)    &   𝐺 = {⟨𝑣, 𝑒⟩ ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))}    &   𝐹 = (𝑔𝐺 ↦ (2nd𝑔))       (𝑋𝐺 → (𝐹𝑋) = (2nd𝑋))

Theoremuspgrsprf 42783* 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 = (𝑔𝐺 ↦ (2nd𝑔))       𝐹:𝐺𝑃

Theoremuspgrsprf1 42784* 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 = (𝑔𝐺 ↦ (2nd𝑔))       𝐹:𝐺1-1𝑃

Theoremuspgrsprfo 42785* 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 = (𝑔𝐺 ↦ (2nd𝑔))       (𝑉𝑊𝐹:𝐺onto𝑃)

Theoremuspgrsprf1o 42786* 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 42791. (Contributed by AV, 25-Nov-2021.)
𝑃 = 𝒫 (Pairs‘𝑉)    &   𝐺 = {⟨𝑣, 𝑒⟩ ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))}    &   𝐹 = (𝑔𝐺 ↦ (2nd𝑔))       (𝑉𝑊𝐹:𝐺1-1-onto𝑃)

Theoremuspgrex 42787* 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))}       (𝑉𝑊𝐺 ∈ V)

Theoremuspgrbispr 42788* 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))}       (𝑉𝑊 → ∃𝑓 𝑓:𝐺1-1-onto𝑃)

Theoremuspgrspren 42789* 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))}       (𝑉𝑊𝐺𝑃)

Theoremuspgrymrelen 42790* 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 42791. (Contributed by AV, 27-Nov-2021.)
𝐺 = {⟨𝑣, 𝑒⟩ ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))}    &   𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥𝑉𝑦𝑉 (𝑥𝑟𝑦𝑦𝑟𝑥)}       (𝑉𝑊𝐺𝑅)

Theoremuspgrbisymrel 42791* 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 42787) 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) ∣ ∀𝑥𝑉𝑦𝑉 (𝑥𝑟𝑦𝑦𝑟𝑥)}       (𝑉𝑊 → ∃𝑓 𝑓:𝐺1-1-onto𝑅)

TheoremuspgrbisymrelALT 42792* Alternate proof of uspgrbisymrel 42791 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) ∣ ∀𝑥𝑉𝑦𝑉 (𝑥𝑟𝑦𝑦𝑟𝑥)}       (𝑉𝑊 → ∃𝑓 𝑓:𝐺1-1-onto𝑅)

20.36.14  Monoids (extension)

20.36.14.1  Auxiliary theorems

Theoremovn0dmfun 42793 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 6487. (Contributed by AV, 27-Jan-2020.)
((𝐴𝐹𝐵) ≠ ∅ → (⟨𝐴, 𝐵⟩ ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {⟨𝐴, 𝐵⟩})))

Theoremxpsnopab 42794* A Cartesian product with a singleton expressed as ordered-pair class abstraction. (Contributed by AV, 27-Jan-2020.)
({𝑋} × 𝐶) = {⟨𝑎, 𝑏⟩ ∣ (𝑎 = 𝑋𝑏𝐶)}

Theoremxpiun 42795* A Cartesian product expressed as indexed union of ordered-pair class abstractions. (Contributed by AV, 27-Jan-2020.)
(𝐵 × 𝐶) = 𝑥𝐵 {⟨𝑎, 𝑏⟩ ∣ (𝑎 = 𝑥𝑏𝐶)}

Theoremovn0ssdmfun 42796* 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 6487. (Contributed by AV, 27-Jan-2020.)
(∀𝑎𝐷𝑏𝐸 (𝑎𝐹𝑏) ≠ ∅ → ((𝐷 × 𝐸) ⊆ dom 𝐹 ∧ Fun (𝐹 ↾ (𝐷 × 𝐸))))

Theoremfnxpdmdm 42797 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)

Theoremcnfldsrngbas 42798 The base set of a subring of the field of complex numbers. (Contributed by AV, 31-Jan-2020.)
𝑅 = (ℂflds 𝑆)       (𝑆 ⊆ ℂ → 𝑆 = (Base‘𝑅))

Theoremcnfldsrngadd 42799 The group addition operation of a subring of the field of complex numbers. (Contributed by AV, 31-Jan-2020.)
𝑅 = (ℂflds 𝑆)       (𝑆𝑉 → + = (+g𝑅))

Theoremcnfldsrngmul 42800 The ring multiplication operation of a subring of the field of complex numbers. (Contributed by AV, 31-Jan-2020.)
𝑅 = (ℂflds 𝑆)       (𝑆𝑉 → · = (.r𝑅))

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