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Theorem List for Metamath Proof Explorer - 42001-42100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremnnsum3primes4 42001* 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 42002* 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 42003* 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 42004* 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 42005* 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 42006* 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 42007* 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 42008* 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 42009* 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 42010* 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 42011* 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 42012* 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 42013* 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 42014* 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 42015* 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 42016* 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 42017* 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 42018 Lemma 1 for bgoldbtbnd 42022: 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 42019* Lemma 2 for bgoldbtbnd 42022. (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 42020* Lemma 3 for bgoldbtbnd 42022. (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 42021* Lemma 4 for bgoldbtbnd 42022. (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 42022* 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 42023 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 42024* Temporary duplicate of tgoldbachgt 30869, 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 42025* Variant of Thierry Arnoux's tgoldbachgt 30869 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 42026* 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 42023. (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 42027 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 42028 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 42026, ax-hgprmladder 42027 and bgoldbtbnd 42022. (Contributed by AV, 4-Aug-2020.) (Revised by AV, 9-Sep-2021.)
((𝑁 ∈ Odd ∧ 7 < 𝑁𝑁 < (88 · (10↑29))) → 𝑁 ∈ GoldbachOdd )

Theoremtgoldbachlt 42029* 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 42028. (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 42030 The ternary Goldbach conjecture is valid. Main theorem in [Helfgott] p. 2. This follows from tgoldbachlt 42029 and ax-tgoldbachgt 42024. (Contributed by AV, 2-Aug-2020.) (Revised by AV, 9-Sep-2021.)
𝑛 ∈ Odd (7 < 𝑛𝑛 ∈ GoldbachOdd )

Axiomax-bgbltosilvaOLD 42031 Obsolete version of ax-bgbltosilva 42023 as of 9-Sep-2021. (Contributed by AV, 3-Aug-2020.) (New usage is discouraged.)
((𝑁 ∈ Even ∧ 4 < 𝑁𝑁 ≤ (4 · (10↑18))) → 𝑁 ∈ GoldbachEven )

TheorembgoldbachltOLD 42032* Obsolete version of bgoldbachlt 42026 as of 9-Sep-2021. (Contributed by AV, 3-Aug-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑚 ∈ ℕ ((4 · (10↑18)) ≤ 𝑚 ∧ ∀𝑛 ∈ Even ((4 < 𝑛𝑛 < 𝑚) → 𝑛 ∈ GoldbachEven ))

Axiomax-hgprmladderOLD 42033 Obsolete version of ax-hgprmladder 42027 as of 9-Sep-2021. (Contributed by AV, 3-Aug-2020.) (New usage is discouraged.)
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))))

TheoremtgblthelfgottOLD 42034 Obsolete version of tgblthelfgott 42028 as of 9-Sep-2021. (Contributed by AV, 4-Aug-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝑁 ∈ Odd ∧ 7 < 𝑁𝑁 < (88 · (10↑29))) → 𝑁 ∈ GoldbachOdd )

TheoremtgoldbachltOLD 42035* Obsolete version of tgoldbachlt 42029 as of 9-Sep-2021. (Contributed by AV, 4-Aug-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑚 ∈ ℕ ((8 · (10↑30)) < 𝑚 ∧ ∀𝑛 ∈ Odd ((7 < 𝑛𝑛 < 𝑚) → 𝑛 ∈ GoldbachOdd ))

Axiomax-tgoldbachgtOLD 42036* Obsolete version of ax-tgoldbachgt 42024 as of 9-Sep-2021. (Contributed by AV, 2-Aug-2020.) (New usage is discouraged.)
𝑚 ∈ ℕ (𝑚 ≤ (10↑27) ∧ ∀𝑛 ∈ Odd (𝑚 < 𝑛𝑛 ∈ GoldbachOdd ))

TheoremtgoldbachOLD 42037 Obsolete version of tgoldbach 42030 as of 9-Sep-2021. (Contributed by AV, 2-Aug-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑛 ∈ Odd (7 < 𝑛𝑛 ∈ GoldbachOdd )

20.35.9  Graph theory (extension)

20.35.9.1  Loop-free graphs - extension

Theorem1hegrlfgr 42038* 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)})

20.35.9.2  Walks - extension

Syntaxcupwlks 42039 Extend class notation with walks (of a pseudograph).
class UPWalks

Definitiondf-upwlks 42040* 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 42041* 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 42042* 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 42043* Generalisation of isupwlk 42042: 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 42044 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 42045 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 42046 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 42047 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 42048* Alternate proof of upgriswlk 26593 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))})))

20.35.10  Set of unordered pairs

Theoremsprid 42049 Two identical representations of the class of all unordered pairs. (Contributed by AV, 21-Nov-2021.)
{𝑝 ∣ ∃𝑎 ∈ V ∃𝑏 ∈ V 𝑝 = {𝑎, 𝑏}} = {𝑝 ∣ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}}

Theoremelsprel 42050* An unordered pair is an element of all unordered pairs. At least one of the two elements of the unordered pair must be a set. Otherwise, the unordered pair would be the empty set, see prprc 4334, which is not an element of all unordered pairs, see spr0nelg 42051. (Contributed by AV, 21-Nov-2021.)
((𝐴𝑉𝐵𝑊) → {𝐴, 𝐵} ∈ {𝑝 ∣ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}})

Theoremspr0nelg 42051* The empty set is not an element of all unordered pairs. (Contributed by AV, 21-Nov-2021.)
∅ ∉ {𝑝 ∣ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}}

Syntaxcspr 42052 Extend class notation with set of pairs.
class Pairs

Definitiondf-spr 42053* Define the function which maps a set 𝑣 to the set of pairs consisting of elements of the set 𝑣. (Contributed by AV, 21-Nov-2021.)
Pairs = (𝑣 ∈ V ↦ {𝑝 ∣ ∃𝑎𝑣𝑏𝑣 𝑝 = {𝑎, 𝑏}})

Theoremsprval 42054* The set of all unordered pairs over a given set 𝑉. (Contributed by AV, 21-Nov-2021.)
(𝑉𝑊 → (Pairs‘𝑉) = {𝑝 ∣ ∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏}})

Theoremsprvalpw 42055* The set of all unordered pairs over a given set 𝑉, expressed by a restricted class abstraction. (Contributed by AV, 21-Nov-2021.)
(𝑉𝑊 → (Pairs‘𝑉) = {𝑝 ∈ 𝒫 𝑉 ∣ ∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏}})

Theoremsprssspr 42056* The set of all unordered pairs over a given set 𝑉 is a subset of the set of all unordered pairs. (Contributed by AV, 21-Nov-2021.)
(Pairs‘𝑉) ⊆ {𝑝 ∣ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}}

Theoremspr0el 42057 The empty set is not an unordered pair over any set 𝑉. (Contributed by AV, 21-Nov-2021.)
∅ ∉ (Pairs‘𝑉)

Theoremsprvalpwn0 42058* The set of all unordered pairs over a given set 𝑉, expressed by a restricted class abstraction. (Contributed by AV, 21-Nov-2021.)
(𝑉𝑊 → (Pairs‘𝑉) = {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ ∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏}})

Theoremsprel 42059* An element of the set of all unordered pairs over a given set 𝑉 is a pair of elements of the set 𝑉. (Contributed by AV, 22-Nov-2021.)
(𝑋 ∈ (Pairs‘𝑉) → ∃𝑎𝑉𝑏𝑉 𝑋 = {𝑎, 𝑏})

Theoremprssspr 42060* An element of a subset of the set of all unordered pairs over a given set 𝑉, is a pair of elements of the set 𝑉. (Contributed by AV, 22-Nov-2021.)
((𝑃 ⊆ (Pairs‘𝑉) ∧ 𝑋𝑃) → ∃𝑎𝑉𝑏𝑉 𝑋 = {𝑎, 𝑏})

Theoremprelspr 42061 An unordered pair of elements of a fixed set 𝑉 belongs to the set of all unordered pairs over the set 𝑉. (Contributed by AV, 21-Nov-2021.)
((𝑉𝑊 ∧ (𝑋𝑉𝑌𝑉)) → {𝑋, 𝑌} ∈ (Pairs‘𝑉))

Theoremprsprel 42062 The elements of a pair from the set of all unordered pairs over a given set 𝑉 are elements of the set 𝑉. (Contributed by AV, 22-Nov-2021.)
(({𝑋, 𝑌} ∈ (Pairs‘𝑉) ∧ (𝑋𝑈𝑌𝑊)) → (𝑋𝑉𝑌𝑉))

Theoremprsssprel 42063 The elements of a pair from a subset of the set of all unordered pairs over a given set 𝑉 are elements of the set 𝑉. (Contributed by AV, 21-Nov-2021.)
((𝑃 ⊆ (Pairs‘𝑉) ∧ {𝑋, 𝑌} ∈ 𝑃 ∧ (𝑋𝑈𝑌𝑊)) → (𝑋𝑉𝑌𝑉))

Theoremsprvalpwle2 42064* The set of all unordered pairs over a given set 𝑉, expressed by a restricted class abstraction. (Contributed by AV, 24-Nov-2021.)
(𝑉𝑊 → (Pairs‘𝑉) = {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑝) ≤ 2})

Theoremsprsymrelfvlem 42065* Lemma for sprsymrelf 42070 and sprsymrelfv 42069. (Contributed by AV, 19-Nov-2021.)
(𝑃 ⊆ (Pairs‘𝑉) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑃 𝑐 = {𝑥, 𝑦}} ∈ 𝒫 (𝑉 × 𝑉))

Theoremsprsymrelf1lem 42066* Lemma for sprsymrelf1 42071. (Contributed by AV, 22-Nov-2021.)
((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑏 ⊆ (Pairs‘𝑉)) → ({⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑎 𝑐 = {𝑥, 𝑦}} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑏 𝑐 = {𝑥, 𝑦}} → 𝑎𝑏))

Theoremsprsymrelfolem1 42067* Lemma 1 for sprsymrelfo 42072. (Contributed by AV, 22-Nov-2021.)
𝑄 = {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎𝑉𝑏𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑅𝑏)}       𝑄 ∈ 𝒫 (Pairs‘𝑉)

Theoremsprsymrelfolem2 42068* Lemma 2 for sprsymrelfo 42072. (Contributed by AV, 23-Nov-2021.)
𝑄 = {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎𝑉𝑏𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑅𝑏)}       ((𝑉𝑊𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥𝑉𝑦𝑉 (𝑥𝑅𝑦𝑦𝑅𝑥)) → (𝑥𝑅𝑦 ↔ ∃𝑐𝑄 𝑐 = {𝑥, 𝑦}))

Theoremsprsymrelfv 42069* The value of the function 𝐹 which maps a subset of the set of pairs over a fixed set 𝑉 to the relation relating two elements of the set 𝑉 iff they are in a pair of the subset. (Contributed by AV, 19-Nov-2021.)
𝑃 = 𝒫 (Pairs‘𝑉)    &   𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥𝑉𝑦𝑉 (𝑥𝑟𝑦𝑦𝑟𝑥)}    &   𝐹 = (𝑝𝑃 ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}})       (𝑋𝑃 → (𝐹𝑋) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑋 𝑐 = {𝑥, 𝑦}})

Theoremsprsymrelf 42070* The mapping 𝐹 is a function from the subsets of the set of pairs over a fixed set 𝑉 into the symmetric relations 𝑅 on the fixed set 𝑉. (Contributed by AV, 19-Nov-2021.)
𝑃 = 𝒫 (Pairs‘𝑉)    &   𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥𝑉𝑦𝑉 (𝑥𝑟𝑦𝑦𝑟𝑥)}    &   𝐹 = (𝑝𝑃 ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}})       𝐹:𝑃𝑅

Theoremsprsymrelf1 42071* The mapping 𝐹 is a one-to-one function from the subsets of the set of pairs over a fixed set 𝑉 into the symmetric relations 𝑅 on the fixed set 𝑉. (Contributed by AV, 19-Nov-2021.)
𝑃 = 𝒫 (Pairs‘𝑉)    &   𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥𝑉𝑦𝑉 (𝑥𝑟𝑦𝑦𝑟𝑥)}    &   𝐹 = (𝑝𝑃 ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}})       𝐹:𝑃1-1𝑅

Theoremsprsymrelfo 42072* The mapping 𝐹 is a function from the subsets of the set of pairs over a fixed set 𝑉 onto the symmetric relations 𝑅 on the fixed set 𝑉. (Contributed by AV, 23-Nov-2021.)
𝑃 = 𝒫 (Pairs‘𝑉)    &   𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥𝑉𝑦𝑉 (𝑥𝑟𝑦𝑦𝑟𝑥)}    &   𝐹 = (𝑝𝑃 ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}})       (𝑉𝑊𝐹:𝑃onto𝑅)

Theoremsprsymrelf1o 42073* The mapping 𝐹 is a bijection between the subsets of the set of pairs over a fixed set 𝑉 into the symmetric relations 𝑅 on the fixed set 𝑉. (Contributed by AV, 23-Nov-2021.)
𝑃 = 𝒫 (Pairs‘𝑉)    &   𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥𝑉𝑦𝑉 (𝑥𝑟𝑦𝑦𝑟𝑥)}    &   𝐹 = (𝑝𝑃 ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}})       (𝑉𝑊𝐹:𝑃1-1-onto𝑅)

Theoremsprbisymrel 42074* There is a bijection between the subsets of the set of pairs over a fixed set 𝑉 and the symmetric relations 𝑅 on the fixed set 𝑉. (Contributed by AV, 23-Nov-2021.)
𝑃 = 𝒫 (Pairs‘𝑉)    &   𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥𝑉𝑦𝑉 (𝑥𝑟𝑦𝑦𝑟𝑥)}       (𝑉𝑊 → ∃𝑓 𝑓:𝑃1-1-onto𝑅)

Theoremsprsymrelen 42075* The class 𝑃 of subsets of the set of pairs over a fixed set 𝑉 and the class 𝑅 of symmetric relations on the fixed set 𝑉 are equinumerous. (Contributed by AV, 27-Nov-2021.)
𝑃 = 𝒫 (Pairs‘𝑉)    &   𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥𝑉𝑦𝑉 (𝑥𝑟𝑦𝑦𝑟𝑥)}       (𝑉𝑊𝑃𝑅)

Theoremupgredgssspr 42076 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 42077* The set 𝐺 of "simple pseudographs" for a fixed set 𝑉 of vertices is a subset of an Cartesian product. For more details about the class 𝐺 of all "simple pseudographs" see comments on uspgrbisymrel 42087. (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 42078* 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 42084. (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 42079* 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 42080* 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 42081* 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 42082* 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 42087. (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 42083* 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 42084* 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 42085* 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 42086* 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 42087. (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 42087* 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 42083) 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 42088* Alternate proof of uspgrbisymrel 42087 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)

20.35.11  Monoids (extension)

20.35.11.1  Auxiliary theorems

Theoremovn0dmfun 42089 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 6264. (Contributed by AV, 27-Jan-2020.)
((𝐴𝐹𝐵) ≠ ∅ → (⟨𝐴, 𝐵⟩ ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {⟨𝐴, 𝐵⟩})))

Theoremxpsnopab 42090* 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 42091* 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 42092* 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 6264. (Contributed by AV, 27-Jan-2020.)
(∀𝑎𝐷𝑏𝐸 (𝑎𝐹𝑏) ≠ ∅ → ((𝐷 × 𝐸) ⊆ dom 𝐹 ∧ Fun (𝐹 ↾ (𝐷 × 𝐸))))

Theoremfnxpdmdm 42093 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 42094 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 42095 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 42096 The ring multiplication operation of a subring of the field of complex numbers. (Contributed by AV, 31-Jan-2020.)
𝑅 = (ℂflds 𝑆)       (𝑆𝑉 → · = (.r𝑅))

20.35.11.2  Magmas and Semigroups (extension)

Theoremplusfreseq 42097 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)    &    + = (+g𝑀)    &    = (+𝑓𝑀)       (∅ ∉ ran → ( + ↾ (𝐵 × 𝐵)) = )

Theoremmgmplusfreseq 42098 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)    &    + = (+g𝑀)    &    = (+𝑓𝑀)       ((𝑀 ∈ Mgm ∧ ∅ ∉ 𝐵) → ( + ↾ (𝐵 × 𝐵)) = )

Theorem0mgm 42099 A set with an empty base set is always a magma". (Contributed by AV, 25-Feb-2020.)
(Base‘𝑀) = ∅       (𝑀𝑉𝑀 ∈ Mgm)

Theoremmgmpropd 42100* 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.)
(𝜑𝐵 = (Base‘𝐾))    &   (𝜑𝐵 = (Base‘𝐿))    &   (𝜑𝐵 ≠ ∅)    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))       (𝜑 → (𝐾 ∈ Mgm ↔ 𝐿 ∈ Mgm))

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