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Theorem List for Metamath Proof Explorer - 41401-41500   *Has distinct variable group(s)
TypeLabelDescription
Statement

20.35.8.9  Perfect Number Theorem (revised)

TheoremperfectALTVlem1 41401 Lemma for perfectALTV 41403. (Contributed by Mario Carneiro, 7-Jun-2016.) (Revised by AV, 1-Jul-2020.)
(𝜑𝐴 ∈ ℕ)    &   (𝜑𝐵 ∈ ℕ)    &   (𝜑𝐵 ∈ Odd )    &   (𝜑 → (1 σ ((2↑𝐴) · 𝐵)) = (2 · ((2↑𝐴) · 𝐵)))       (𝜑 → ((2↑(𝐴 + 1)) ∈ ℕ ∧ ((2↑(𝐴 + 1)) − 1) ∈ ℕ ∧ (𝐵 / ((2↑(𝐴 + 1)) − 1)) ∈ ℕ))

TheoremperfectALTVlem2 41402 Lemma for perfectALTV 41403. (Contributed by Mario Carneiro, 17-May-2016.) (Revised by AV, 1-Jul-2020.)
(𝜑𝐴 ∈ ℕ)    &   (𝜑𝐵 ∈ ℕ)    &   (𝜑𝐵 ∈ Odd )    &   (𝜑 → (1 σ ((2↑𝐴) · 𝐵)) = (2 · ((2↑𝐴) · 𝐵)))       (𝜑 → (𝐵 ∈ ℙ ∧ 𝐵 = ((2↑(𝐴 + 1)) − 1)))

TheoremperfectALTV 41403* The Euclid-Euler theorem, or Perfect Number theorem. A positive even integer 𝑁 is a perfect number (that is, its divisor sum is 2𝑁) if and only if it is of the form 2↑(𝑝 − 1) · (2↑𝑝 − 1), where 2↑𝑝 − 1 is prime (a Mersenne prime). (It follows from this that 𝑝 is also prime.) This is Metamath 100 proof #70. (Contributed by Mario Carneiro, 17-May-2016.) (Revised by AV, 1-Jul-2020.) (Proof modification is discouraged.)
((𝑁 ∈ ℕ ∧ 𝑁 ∈ Even ) → ((1 σ 𝑁) = (2 · 𝑁) ↔ ∃𝑝 ∈ ℤ (((2↑𝑝) − 1) ∈ ℙ ∧ 𝑁 = ((2↑(𝑝 − 1)) · ((2↑𝑝) − 1)))))

20.35.8.10  Goldbach's conjectures

According to Wikipedia ("Goldbach's conjecture", 20-Jul-2020, https://en.wikipedia.org/wiki/Goldbach's_conjecture) "Goldbach's conjecture ... states: Every even integer greater than 2 can be expressed as the sum of two primes." "It is also known as strong, even or binary Goldbach conjecture, to distinguish it from a weaker conjecture, known ... as the Goldbach's weak conjecture, the odd Goldbach conjecture, or the ternary Goldbach conjecture. This weak conjecture asserts that all odd numbers greater than 7 are the sum of three odd primes.". In the following, the terms "binary Goldbach conjecture" resp. "ternary Goldbach conjecture" will be used (following the terminology used in [Helfgott] p. 2), because there are a strong and a weak version of the ternary Goldbach conjecture. The term Goldbach partition is used for a sum of two resp. three (odd) primes resulting in an even resp. odd number without further specialization.

Using the definition of a Goldbach number, which is "a positive even integer that can be expressed as the sum of two odd primes." (see df-gbe 41407), "another form of the statement of Goldbach's conjecture is that all even integers greater than 4 are Goldbach numbers.". 4 is not a Goldbach number, but it is the sum of two primes (2 and 2) nevertheless. sbgoldbalt 41440 shows that both forms are equivalent.

Hint (see Wikipedia, ("Goldbach's weak conjecture", 26-Jul-2020, https://en.wikipedia.org/wiki/Goldbach's_weak_conjecture): "Some state the [weak] conjecture as 'Every odd number greater than 7 can be expressed as the sum of three odd primes.' This version excludes 7 = 2+2+3 because this requires the even prime 2. On odd numbers larger than 7 it is slightly stronger as it also excludes sums like 17 = 2+2+13, which are allowed in the other formulation. Helfgott's proof [see below] covers both versions of the conjecture. Like the other formulation, this one also immediately follows from Goldbach's strong conjecture." The definition of "weak odd Goldbach numbers", see df-gbow 41408, is the basis for "the other formulation", to formulate the weak ternary Goldbach conjecture. Alternatively, df-gbo 41409 provides a definition of "(strong) odd Goldbach numbers" allowing for stating the strong ternary Goldbach conjecture. In literature, the term "Goldbach number" is used for "even Goldbach numbers" (according to definition df-gbe 41407), whereas there seems to be no explicit names and definitions for "odd Goldbach numbers". Since there are more theorems for "strong odd Goldbach numbers", "odd Goldbach numbers" refers to "strong odd Goldbach numbers" in the following. Otherwise, the term "weak odd Goldbach numbers" is explicitly used.

In contrast to the two versions of the binary Goldbach conjecture, the two versions of the ternary Goldbach conjecture are different not only for small numbers, but the strong version excludes cases like a=2+2+b in general, e.g. 23=2+2+19. Therefore, it seems to be more difficult to prove the strong ternary Goldbach conjecture than the weak version, because there are fewer possible partitions available.

Although the binary Goldbach conjecture is not proven yet, the ternary Goldbach conjecture was proven by Harald Helfgott in 2014 (the weak as well as the strong version, see Main theorem in [Helfgott] p. 2). It would be great if this proof can be formalized with Metamath (although it is not in the Metamath 100 list). This section should be a starting point for this.

The main problem will be to provide means to express the results from checking "small" numbers (performed with a computer): numbers up to about 4 x 10^18 for the binary Goldbach conjecture (see section 2 in [OeSilva] p. 2042, called "even Goldbach conjecture" here) resp. about 9 x 10^30 for the ternary Goldbach conjecture (see section 1.2.2 in [Helfgott] p. 4) or 8 x 10^26 (see theorem 2.1 in [OeSilva] p. 2057, called "odd Goldbach conjecture" here). Maybe each of the results must be provided as theorem, like 6gbe 41430, which would be quite a lot...

As proposed in the Google group discussion https://groups.google.com/g/metamath/c/DOXS4pg0h8w , this problem could be solved by using a reflective verifier or adding a concept of verification certificates that can be added into the metamath databases as a reference. To sidestep the computation problem for now, the corresponding theorems are temporarily provided as axioms, see ax-bgbltosilva 41469, ax-hgprmladder 41473 and ax-tgoldbachgt 41470.

# Summary/glossary:

TermSynonymsLabel fragment Definition/TheoremRemarks
binary Goldbach partition simply "Goldbach partition" A pair of primes (p,q) that sum to an even integer 2n=p+q See https://mathworld.wolfram.com/GoldbachPartition.html
weak Goldbach partition gbpart A sum of two resp. three primes resulting in an even resp. odd number without further specialization.
Goldbach partition gbpart A sum of two resp. three odd primes resulting in an even resp. odd number without further specialization.
even Goldbach number simply "Goldbach number" gbe df-gbe 41407 A positive even integer that can be expressed as the sum of two odd primes. See https://mathworld.wolfram.com/GoldbachNumber.html
weak odd Goldbach number gbow df-gbow 41408 A positive odd integer that can be expressed as the sum of three primes.
odd Goldbach number strong odd Goldbach number gbo df-gbo 41409 A positive odd integer that can be expressed as the sum of three odd primes.
strong binary Goldbach conjecture "the" Goldbach conjecture" [*1], even Goldbach conjecture [*2] sbgoldb Every even integer greater than 4 can be expressed as the sum of two odd primes. [*1] Equation (1) in [ApostolNT] p. 304 or [*2] introduction of [OeSilva] p. 2033.
binary Goldbach conjecture[*1][*3] strong Goldbach conjecture [*1], even Goldbach conjecture [*1], or simply "the Goldbach conjecture" [*1][*2] bgoldb, b sbgoldbb 41441 Every even integer greater than 2 can be expressed as the sum of two primes. See [*1] https://en.wikipedia.org/wiki/Goldbach's_conjecture, [*2] statement in [ApostolNT] p. 9 or [*3] section 1.1 in [Helfgott] p. 2.
weak ternary Goldbach conjecture Goldbach's weak conjecture [*1], odd Goldbach conjecture [*1][*3], ternary Goldbach conjecture [*2], ternary Goldbach problem[*1], three-primes problem [*1][*2] wtgoldb, wt stgoldbwt 41435, sbgoldbwt 41436 Every odd number greater than 5 can be expressed as the sum of three primes. See [*1] https://en.wikipedia.org/wiki/Goldbach's_weak_conjecture, [*2] section 1.1 in [Helfgott] p. 2 or [*3] section 2.4 in [OeSilva] p. 2057.
ternary Goldbach conjecture strong ternary Goldbach conjecture, the "weak" Goldbach conjecture tgoldb, stgoldb, st sbgoldbst 41437 Every odd number greater than 7 can be expressed as the sum of three odd primes. See https://en.wikipedia.org/wiki/Goldbach's_weak_conjecture, https://mathworld.wolfram.com/GoldbachConjecture.html or section 7.4 in [Helfgott] p. 71.
Goldbach's original conjecture (modern version) the "ternary" Goldbach conjecture mogoldb, m sbgoldbm 41443 Every integer greater than 5 can be written as the sum of three primes. See https://en.wikipedia.org/wiki/Goldbach's_weak_conjecture, and https://mathworld.wolfram.com/GoldbachConjecture.html
Goldbach's original conjecture (original version) ogoldb, o sbgoldbo 41446 Every integer greater than 2 can be written as the sum of three "primes" (considered the number 1 to be a "prime"). See https://en.wikipedia.org/wiki/Goldbach's_weak_conjecture, and https://mathworld.wolfram.com/GoldbachConjecture.html

Syntaxcgbe 41404 Extend the definition of a class to include the set of even numbers which have a Goldbach partition.
class GoldbachEven

Syntaxcgbow 41405 Extend the definition of a class to include the set of odd numbers which can be written as a sum of three primes.
class GoldbachOddW

Syntaxcgbo 41406 Extend the definition of a class to include the set of odd numbers which can be written as a sum of three odd primes.
class GoldbachOdd

Definitiondf-gbe 41407* Define the set of (even) Goldbach numbers, which are positive even integers that can be expressed as the sum of two odd primes. By this definition, the binary Goldbach conjecture can be expressed as 𝑛 ∈ Even (4 < 𝑛𝑛 ∈ GoldbachEven ). (Contributed by AV, 14-Jun-2020.)
GoldbachEven = {𝑧 ∈ Even ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑧 = (𝑝 + 𝑞))}

Definitiondf-gbow 41408* Define the set of weak odd Goldbach numbers, which are positive odd integers that can be expressed as the sum of three primes. By this definition, the weak ternary Goldbach conjecture can be expressed as 𝑚 ∈ Odd (5 < 𝑚𝑚 ∈ GoldbachOddW ). (Contributed by AV, 14-Jun-2020.)
GoldbachOddW = {𝑧 ∈ Odd ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑧 = ((𝑝 + 𝑞) + 𝑟)}

Definitiondf-gbo 41409* Define the set of (strong) odd Goldbach numbers, which are positive odd integers that can be expressed as the sum of three odd primes. By this definition, the strong ternary Goldbach conjecture can be expressed as 𝑚 ∈ Odd (7 < 𝑚𝑚 ∈ GoldbachOdd ). (Contributed by AV, 26-Jul-2020.)
GoldbachOdd = {𝑧 ∈ Odd ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ ((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑟 ∈ Odd ) ∧ 𝑧 = ((𝑝 + 𝑞) + 𝑟))}

Theoremisgbe 41410* The predicate "is an even Goldbach number". An even Goldbach number is an even integer having a Goldbach partition, i.e. which can be written as a sum of two odd primes. (Contributed by AV, 20-Jul-2020.)
(𝑍 ∈ GoldbachEven ↔ (𝑍 ∈ Even ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑍 = (𝑝 + 𝑞))))

Theoremisgbow 41411* The predicate "is a weak odd Goldbach number". A weak odd Goldbach number is an odd integer having a Goldbach partition, i.e. which can be written as a sum of three primes. (Contributed by AV, 20-Jul-2020.)
(𝑍 ∈ GoldbachOddW ↔ (𝑍 ∈ Odd ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑍 = ((𝑝 + 𝑞) + 𝑟)))

Theoremisgbo 41412* The predicate "is an odd Goldbach number". An odd Goldbach number is an odd integer having a Goldbach partition, i.e. which can be written as sum of three odd primes. (Contributed by AV, 26-Jul-2020.)
(𝑍 ∈ GoldbachOdd ↔ (𝑍 ∈ Odd ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ ((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑟 ∈ Odd ) ∧ 𝑍 = ((𝑝 + 𝑞) + 𝑟))))

Theoremgbeeven 41413 An even Goldbach number is even. (Contributed by AV, 25-Jul-2020.)
(𝑍 ∈ GoldbachEven → 𝑍 ∈ Even )

Theoremgbowodd 41414 A weak odd Goldbach number is odd. (Contributed by AV, 25-Jul-2020.)
(𝑍 ∈ GoldbachOddW → 𝑍 ∈ Odd )

Theoremgbogbow 41415 A (strong) odd Goldbach number is a weak Goldbach number. (Contributed by AV, 26-Jul-2020.)
(𝑍 ∈ GoldbachOdd → 𝑍 ∈ GoldbachOddW )

Theoremgboodd 41416 An odd Goldbach number is odd. (Contributed by AV, 26-Jul-2020.)
(𝑍 ∈ GoldbachOdd → 𝑍 ∈ Odd )

Theoremgbepos 41417 Any even Goldbach number is positive. (Contributed by AV, 20-Jul-2020.)
(𝑍 ∈ GoldbachEven → 𝑍 ∈ ℕ)

Theoremgbowpos 41418 Any weak odd Goldbach number is positive. (Contributed by AV, 20-Jul-2020.)
(𝑍 ∈ GoldbachOddW → 𝑍 ∈ ℕ)

Theoremgbopos 41419 Any odd Goldbach number is positive. (Contributed by AV, 26-Jul-2020.)
(𝑍 ∈ GoldbachOdd → 𝑍 ∈ ℕ)

Theoremgbegt5 41420 Any even Goldbach number is greater than 5. (Contributed by AV, 20-Jul-2020.)
(𝑍 ∈ GoldbachEven → 5 < 𝑍)

Theoremgbowgt5 41421 Any weak odd Goldbach number is greater than 5. (Contributed by AV, 20-Jul-2020.)
(𝑍 ∈ GoldbachOddW → 5 < 𝑍)

Theoremgbowge7 41422 Any weak odd Goldbach number is greater than or equal to 7. Because of 7gbow 41431, this bound is strict. (Contributed by AV, 20-Jul-2020.)
(𝑍 ∈ GoldbachOddW → 7 ≤ 𝑍)

Theoremgboge9 41423 Any odd Goldbach number is greater than or equal to 9. Because of 9gbo 41433, this bound is strict. (Contributed by AV, 26-Jul-2020.)
(𝑍 ∈ GoldbachOdd → 9 ≤ 𝑍)

Theoremgbege6 41424 Any even Goldbach number is greater than or equal to 6. Because of 6gbe 41430, this bound is strict. (Contributed by AV, 20-Jul-2020.)
(𝑍 ∈ GoldbachEven → 6 ≤ 𝑍)

Theoremgbpart6 41425 The Goldbach partition of 6. (Contributed by AV, 20-Jul-2020.)
6 = (3 + 3)

Theoremgbpart7 41426 The (weak) Goldbach partition of 7. (Contributed by AV, 20-Jul-2020.)
7 = ((2 + 2) + 3)

Theoremgbpart8 41427 The Goldbach partition of 8. (Contributed by AV, 20-Jul-2020.)
8 = (3 + 5)

Theoremgbpart9 41428 The (strong) Goldbach partition of 9. (Contributed by AV, 26-Jul-2020.)
9 = ((3 + 3) + 3)

Theoremgbpart11 41429 The (strong) Goldbach partition of 11. (Contributed by AV, 29-Jul-2020.)
11 = ((3 + 3) + 5)

Theorem6gbe 41430 6 is an even Goldbach number. (Contributed by AV, 20-Jul-2020.)
6 ∈ GoldbachEven

Theorem7gbow 41431 7 is a weak odd Goldbach number. (Contributed by AV, 20-Jul-2020.)
7 ∈ GoldbachOddW

Theorem8gbe 41432 8 is an even Goldbach number. (Contributed by AV, 20-Jul-2020.)
8 ∈ GoldbachEven

Theorem9gbo 41433 9 is an odd Goldbach number. (Contributed by AV, 26-Jul-2020.)
9 ∈ GoldbachOdd

Theorem11gbo 41434 11 is an odd Goldbach number. (Contributed by AV, 29-Jul-2020.)
11 ∈ GoldbachOdd

Theoremstgoldbwt 41435 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 41436* 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 41437* 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 41438 Lemma 1 for sbgoldbalt 41440: 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 41439 Lemma 2 for sbgoldbalt 41440: 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 41440* 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 41441* 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 41442* 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 41443* 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 41444* 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 41445* 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 41446* 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 41447* 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 41448* 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 41449* 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 41450* 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 41451* 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 41452* 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 41453* 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 41454* 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 41455* 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 41456* 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 41457* 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 41458* 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 41459* 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 41460* 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 41461* 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 41462* 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 41463* 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 41464 Lemma 1 for bgoldbtbnd 41468: 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 41465* Lemma 2 for bgoldbtbnd 41468. (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 41466* Lemma 3 for bgoldbtbnd 41468. (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 41467* Lemma 4 for bgoldbtbnd 41468. (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 41468* 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 41469 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 41470* Temporary duplicate of tgoldbachgt 30726, 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 41471* Variant of Thierry Arnoux's tgoldbachgt 30726 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 41472* 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 41469. (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 41473 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 41474 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 41472, ax-hgprmladder 41473 and bgoldbtbnd 41468. (Contributed by AV, 4-Aug-2020.) (Revised by AV, 9-Sep-2021.)
((𝑁 ∈ Odd ∧ 7 < 𝑁𝑁 < (88 · (10↑29))) → 𝑁 ∈ GoldbachOdd )

Theoremtgoldbachlt 41475* 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 41474. (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 41476 The ternary Goldbach conjecture is valid. Main theorem in [Helfgott] p. 2. This follows from tgoldbachlt 41475 and ax-tgoldbachgt 41470. (Contributed by AV, 2-Aug-2020.) (Revised by AV, 9-Sep-2021.)
𝑛 ∈ Odd (7 < 𝑛𝑛 ∈ GoldbachOdd )

Axiomax-bgbltosilvaOLD 41477 Obsolete version of ax-bgbltosilva 41469 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 41478* Obsolete version of bgoldbachlt 41472 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 41479 Obsolete version of ax-hgprmladder 41473 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 41480 Obsolete version of tgblthelfgott 41474 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 41481* Obsolete version of tgoldbachlt 41475 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 41482* Obsolete version of ax-tgoldbachgt 41470 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 41483 Obsolete version of tgoldbach 41476 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 41484* 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 41485 Extend class notation with walks (of a pseudograph).
class UPWalks

Definitiondf-upwlks 41486* 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 41487* 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 41488* 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 41489* Generalisation of isupwlk 41488: 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 41490 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 41491 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 41492 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 41493 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 41494* Alternate proof of upgriswlk 26531 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 41495 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 41496* 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 4300, which is not an element of all unordered pairs, see spr0nelg 41497. (Contributed by AV, 21-Nov-2021.)
((𝐴𝑉𝐵𝑊) → {𝐴, 𝐵} ∈ {𝑝 ∣ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}})

Theoremspr0nelg 41497* The empty set is not an element of all unordered pairs. (Contributed by AV, 21-Nov-2021.)
∅ ∉ {𝑝 ∣ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}}

Syntaxcspr 41498 Extend class notation with set of pairs.
class Pairs

Definitiondf-spr 41499* 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 41500* 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}})

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