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Theorem List for Metamath Proof Explorer - 40101-40200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdfodd3 40101 Alternate definition for odd numbers. (Contributed by AV, 18-Jun-2020.)
Odd = {𝑧 ∈ ℤ ∣ ¬ 2 ∥ 𝑧}

Theoremiseven2 40102 The predicate "is an even number". An even number is an integer which is divisible by 2. (Contributed by AV, 18-Jun-2020.)
(𝑍 ∈ Even ↔ (𝑍 ∈ ℤ ∧ 2 ∥ 𝑍))

Theoremisodd3 40103 The predicate "is an odd number". An odd number is an integer which is not divisible by 2. (Contributed by AV, 18-Jun-2020.)
(𝑍 ∈ Odd ↔ (𝑍 ∈ ℤ ∧ ¬ 2 ∥ 𝑍))

Theorem2dvdseven 40104 2 divides an even number. (Contributed by AV, 18-Jun-2020.)
(𝑍 ∈ Even → 2 ∥ 𝑍)

Theorem2ndvdsodd 40105 2 does not divide an odd number. (Contributed by AV, 18-Jun-2020.)
(𝑍 ∈ Odd → ¬ 2 ∥ 𝑍)

Theorem2dvdsoddp1 40106 2 divides an odd number increased by 1. (Contributed by AV, 18-Jun-2020.)
(𝑍 ∈ Odd → 2 ∥ (𝑍 + 1))

Theorem2dvdsoddm1 40107 2 divides an odd number decreased by 1. (Contributed by AV, 18-Jun-2020.)
(𝑍 ∈ Odd → 2 ∥ (𝑍 − 1))

21.34.5.3  Alternate definitions using the "modulo" operation

Theoremdfeven3 40108 Alternate definition for even numbers. (Contributed by AV, 18-Jun-2020.)
Even = {𝑧 ∈ ℤ ∣ (𝑧 mod 2) = 0}

Theoremdfodd4 40109 Alternate definition for odd numbers. (Contributed by AV, 18-Jun-2020.)
Odd = {𝑧 ∈ ℤ ∣ (𝑧 mod 2) = 1}

Theoremdfodd5 40110 Alternate definition for odd numbers. (Contributed by AV, 18-Jun-2020.)
Odd = {𝑧 ∈ ℤ ∣ (𝑧 mod 2) ≠ 0}

Theoremzefldiv2ALTV 40111 The floor of an even number divided by 2 is equal to the even number divided by 2. (Contributed by AV, 7-Jun-2020.) (Revised by AV, 18-Jun-2020.)
(𝑁 ∈ Even → (⌊‘(𝑁 / 2)) = (𝑁 / 2))

Theoremzofldiv2ALTV 40112 The floor of an odd numer divided by 2 is equal to the odd number first decreased by 1 and then divided by 2. (Contributed by AV, 7-Jun-2020.) (Revised by AV, 18-Jun-2020.)
(𝑁 ∈ Odd → (⌊‘(𝑁 / 2)) = ((𝑁 − 1) / 2))

TheoremoddflALTV 40113 Odd number representation by using the floor function. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Revised by AV, 18-Jun-2020.)
(𝐾 ∈ Odd → 𝐾 = ((2 · (⌊‘(𝐾 / 2))) + 1))

21.34.5.4  Alternate definitions using the "gcd" operation

Theoremiseven5 40114 The predicate "is an even number". An even number and 2 have 2 as greatest common divisor. (Contributed by AV, 1-Jul-2020.)
(𝑍 ∈ Even ↔ (𝑍 ∈ ℤ ∧ (2 gcd 𝑍) = 2))

Theoremisodd7 40115 The predicate "is an odd number". An odd number and 2 have 1 as greatest common divisor. (Contributed by AV, 1-Jul-2020.)
(𝑍 ∈ Odd ↔ (𝑍 ∈ ℤ ∧ (2 gcd 𝑍) = 1))

Theoremdfeven5 40116 Alternate definition for even numbers. (Contributed by AV, 1-Jul-2020.)
Even = {𝑧 ∈ ℤ ∣ (2 gcd 𝑧) = 2}

Theoremdfodd7 40117 Alternate definition for odd numbers. (Contributed by AV, 1-Jul-2020.)
Odd = {𝑧 ∈ ℤ ∣ (2 gcd 𝑧) = 1}

21.34.5.5  Theorems of part 5 revised

TheoremzneoALTV 40118 No even integer equals an odd integer (i.e. no integer can be both even and odd). Exercise 10(a) of [Apostol] p. 28. (Contributed by NM, 31-Jul-2004.) (Revised by AV, 16-Jun-2020.)
((𝐴 ∈ Even ∧ 𝐵 ∈ Odd ) → 𝐴𝐵)

TheoremzeoALTV 40119 An integer is even or odd. (Contributed by NM, 1-Jan-2006.) (Revised by AV, 16-Jun-2020.)
(𝑍 ∈ ℤ → (𝑍 ∈ Even ∨ 𝑍 ∈ Odd ))

Theoremzeo2ALTV 40120 An integer is even or odd but not both. (Contributed by Mario Carneiro, 12-Sep-2015.) (Revised by AV, 16-Jun-2020.)
(𝑍 ∈ ℤ → (𝑍 ∈ Even ↔ ¬ 𝑍 ∈ Odd ))

TheoremnneoALTV 40121 A positive integer is even or odd but not both. (Contributed by NM, 1-Jan-2006.) (Revised by AV, 19-Jun-2020.)
(𝑁 ∈ ℕ → (𝑁 ∈ Even ↔ ¬ 𝑁 ∈ Odd ))

TheoremnneoiALTV 40122 A positive integer is even or odd but not both. (Contributed by NM, 20-Aug-2001.) (Revised by AV, 19-Jun-2020.)
𝑁 ∈ ℕ       (𝑁 ∈ Even ↔ ¬ 𝑁 ∈ Odd )

21.34.5.6  Theorems of part 6 revised

Theoremodd2np1ALTV 40123* An integer is odd iff it is one plus twice another integer. (Contributed by Scott Fenton, 3-Apr-2014.) (Revised by AV, 19-Jun-2020.)
(𝑁 ∈ ℤ → (𝑁 ∈ Odd ↔ ∃𝑛 ∈ ℤ ((2 · 𝑛) + 1) = 𝑁))

Theoremoddm1evenALTV 40124 An integer is odd iff its predecessor is even. (Contributed by Mario Carneiro, 5-Sep-2016.) (Revised by AV, 19-Jun-2020.)
(𝑁 ∈ ℤ → (𝑁 ∈ Odd ↔ (𝑁 − 1) ∈ Even ))

Theoremoddp1evenALTV 40125 An integer is odd iff its successor is even. (Contributed by Mario Carneiro, 5-Sep-2016.) (Revised by AV, 19-Jun-2020.)
(𝑁 ∈ ℤ → (𝑁 ∈ Odd ↔ (𝑁 + 1) ∈ Even ))

TheoremoexpnegALTV 40126 The exponential of the negative of a number, when the exponent is odd. (Contributed by Mario Carneiro, 25-Apr-2015.) (Revised by AV, 19-Jun-2020.)
((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∈ Odd ) → (-𝐴𝑁) = -(𝐴𝑁))

Theoremoexpnegnz 40127 The exponential of the negative of a number not being 0, when the exponent is odd. (Contributed by AV, 19-Jun-2020.)
((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ Odd ) → (-𝐴𝑁) = -(𝐴𝑁))

Theorembits0ALTV 40128 Value of the zeroth bit. (Contributed by Mario Carneiro, 5-Sep-2016.) (Revised by AV, 19-Jun-2020.)
(𝑁 ∈ ℤ → (0 ∈ (bits‘𝑁) ↔ 𝑁 ∈ Odd ))

Theorembits0eALTV 40129 The zeroth bit of an even number is zero. (Contributed by Mario Carneiro, 5-Sep-2016.) (Revised by AV, 19-Jun-2020.)
(𝑁 ∈ Even → ¬ 0 ∈ (bits‘𝑁))

Theorembits0oALTV 40130 The zeroth bit of an odd number is zero. (Contributed by Mario Carneiro, 5-Sep-2016.) (Revised by AV, 19-Jun-2020.)
(𝑁 ∈ Odd → 0 ∈ (bits‘𝑁))

TheoremdivgcdoddALTV 40131 Either 𝐴 / (𝐴 gcd 𝐵) is odd or 𝐵 / (𝐴 gcd 𝐵) is odd. (Contributed by Scott Fenton, 19-Apr-2014.) (Revised by AV, 21-Jun-2020.)
((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝐴 / (𝐴 gcd 𝐵)) ∈ Odd ∨ (𝐵 / (𝐴 gcd 𝐵)) ∈ Odd ))

TheoremopoeALTV 40132 The sum of two odds is even. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by AV, 20-Jun-2020.)
((𝐴 ∈ Odd ∧ 𝐵 ∈ Odd ) → (𝐴 + 𝐵) ∈ Even )

TheoremopeoALTV 40133 The sum of an odd and an even is odd. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by AV, 20-Jun-2020.)
((𝐴 ∈ Odd ∧ 𝐵 ∈ Even ) → (𝐴 + 𝐵) ∈ Odd )

TheoremomoeALTV 40134 The difference of two odds is even. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by AV, 20-Jun-2020.)
((𝐴 ∈ Odd ∧ 𝐵 ∈ Odd ) → (𝐴𝐵) ∈ Even )

TheoremomeoALTV 40135 The difference of an odd and an even is odd. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by AV, 20-Jun-2020.)
((𝐴 ∈ Odd ∧ 𝐵 ∈ Even ) → (𝐴𝐵) ∈ Odd )

TheoremoddprmALTV 40136 A prime not equal to 2 is odd. (Contributed by Mario Carneiro, 4-Feb-2015.) (Revised by AV, 21-Jun-2020.)
(𝑁 ∈ (ℙ ∖ {2}) → 𝑁 ∈ Odd )

21.34.5.7  Theorems of AV's mathbox revised

Theorem0evenALTV 40137 0 is an even number. (Contributed by AV, 11-Feb-2020.) (Revised by AV, 17-Jun-2020.)
0 ∈ Even

Theorem0noddALTV 40138 0 is not an odd number. (Contributed by AV, 3-Feb-2020.) (Revised by AV, 17-Jun-2020.)
0 ∉ Odd

Theorem1oddALTV 40139 1 is an odd number. (Contributed by AV, 3-Feb-2020.) (Revised by AV, 18-Jun-2020.)
1 ∈ Odd

Theorem1nevenALTV 40140 1 is not an even number. (Contributed by AV, 12-Feb-2020.) (Revised by AV, 18-Jun-2020.)
1 ∉ Even

Theorem2evenALTV 40141 2 is an even number. (Contributed by AV, 12-Feb-2020.) (Revised by AV, 18-Jun-2020.)
2 ∈ Even

Theorem2noddALTV 40142 2 is not an odd number. (Contributed by AV, 3-Feb-2020.) (Revised by AV, 18-Jun-2020.)
2 ∉ Odd

Theoremnn0o1gt2ALTV 40143 An odd nonnegative integer is either 1 or greater than 2. (Contributed by AV, 2-Jun-2020.) (Revised by AV, 21-Jun-2020.)
((𝑁 ∈ ℕ0𝑁 ∈ Odd ) → (𝑁 = 1 ∨ 2 < 𝑁))

TheoremnnoALTV 40144 An alternate characterization of an odd number greater than 1. (Contributed by AV, 2-Jun-2020.) (Revised by AV, 21-Jun-2020.)
((𝑁 ∈ (ℤ‘2) ∧ 𝑁 ∈ Odd ) → ((𝑁 − 1) / 2) ∈ ℕ)

Theoremnn0oALTV 40145 An alternate characterization of an odd nonnegative integer. (Contributed by AV, 28-May-2020.) (Revised by AV, 21-Jun-2020.)
((𝑁 ∈ ℕ0𝑁 ∈ Odd ) → ((𝑁 − 1) / 2) ∈ ℕ0)

Theoremnn0e 40146 An alternate characterization of an even nonnegative integer. (Contributed by AV, 22-Jun-2020.)
((𝑁 ∈ ℕ0𝑁 ∈ Even ) → (𝑁 / 2) ∈ ℕ0)

Theoremnn0onn0exALTV 40147* For each odd nonnegative integer there is a nonnegative integer which, multiplied by 2 and increased by 1, results in the odd nonnegative integer. (Contributed by AV, 30-May-2020.) (Revised by AV, 22-Jun-2020.)
((𝑁 ∈ ℕ0𝑁 ∈ Odd ) → ∃𝑚 ∈ ℕ0 𝑁 = ((2 · 𝑚) + 1))

Theoremnn0enn0exALTV 40148* For each even nonnegative integer there is a nonnegative integer which, multiplied by 2, results in the even nonnegative integer. (Contributed by AV, 30-May-2020.) (Revised by AV, 22-Jun-2020.)
((𝑁 ∈ ℕ0𝑁 ∈ Even ) → ∃𝑚 ∈ ℕ0 𝑁 = (2 · 𝑚))

Theoremnnpw2evenALTV 40149 2 to the power of a positive integer is even. (Contributed by AV, 2-Jun-2020.) (Revised by AV, 20-Jun-2020.)
(𝑁 ∈ ℕ → (2↑𝑁) ∈ Even )

Theoremepoo 40150 The sum of an even and an odd is odd. (Contributed by AV, 24-Jul-2020.)
((𝐴 ∈ Even ∧ 𝐵 ∈ Odd ) → (𝐴 + 𝐵) ∈ Odd )

Theorememoo 40151 The difference of an even and an odd is odd. (Contributed by AV, 24-Jul-2020.)
((𝐴 ∈ Even ∧ 𝐵 ∈ Odd ) → (𝐴𝐵) ∈ Odd )

Theoremepee 40152 The sum of two even numbers is even. (Contributed by AV, 21-Jul-2020.)
((𝐴 ∈ Even ∧ 𝐵 ∈ Even ) → (𝐴 + 𝐵) ∈ Even )

Theorememee 40153 The difference of two even numbers is even. (Contributed by AV, 21-Jul-2020.)
((𝐴 ∈ Even ∧ 𝐵 ∈ Even ) → (𝐴𝐵) ∈ Even )

Theoremevensumeven 40154 If a summand is even, the other summand is even iff the sum is even. (Contributed by AV, 21-Jul-2020.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ Even ) → (𝐴 ∈ Even ↔ (𝐴 + 𝐵) ∈ Even ))

Theorem3odd 40155 3 is an odd number. (Contributed by AV, 20-Jul-2020.)
3 ∈ Odd

Theorem4even 40156 4 is an even number. (Contributed by AV, 23-Jul-2020.)
4 ∈ Even

Theorem5odd 40157 5 is an odd number. (Contributed by AV, 23-Jul-2020.)
5 ∈ Odd

Theorem6even 40158 6 is an even number. (Contributed by AV, 20-Jul-2020.)
6 ∈ Even

Theorem7odd 40159 7 is an odd number. (Contributed by AV, 20-Jul-2020.)
7 ∈ Odd

Theorem8even 40160 8 is an even number. (Contributed by AV, 23-Jul-2020.)
8 ∈ Even

Theoremevenprm2 40161 A prime number is even iff it is 2. (Contributed by AV, 21-Jul-2020.)
(𝑃 ∈ ℙ → (𝑃 ∈ Even ↔ 𝑃 = 2))

Theoremoddprmne2 40162 Every prime number not being 2 is an odd prime number. (Contributed by AV, 21-Aug-2021.)
((𝑃 ∈ ℙ ∧ 𝑃 ∈ Odd ) ↔ 𝑃 ∈ (ℙ ∖ {2}))

Theoremoddprmuzge3 40163 A prime number which is odd is an integer greater than or equal to 3. (Contributed by AV, 20-Jul-2020.) (Proof shortened by AV, 21-Aug-2021.)
((𝑃 ∈ ℙ ∧ 𝑃 ∈ Odd ) → 𝑃 ∈ (ℤ‘3))

21.34.5.9  Perfect Number Theorem (revised)

TheoremperfectALTVlem1 40164 Lemma for perfectALTV 40166. (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 40165 Lemma for perfectALTV 40166. (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 40166* 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)))))

21.34.5.10  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, because there are are a strong and a weak version of the ternary Goldbach conjecture. The term Goldbach partion 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 40170), "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. sgoldbalt 40203 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." Our definition of "odd Goldbach Numbers", see df-gbo 40171, is the basis for "the other formulation", to formulate the weak ternary Goldbach conjecture. Alternatively, df-gboa 40172 provides a definition allowing for stating the strong ternaty Goldbach conjecture. 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 seems to be 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), providing the still missing official acceptance (usually obtained by a publication in a peer-reviewed journal). 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 strong Goldbach conjecture (see result of [OeSilva] p. ?) resp. about 9 x 10^30 for the weak Goldbach conjecture (see section 1.2.2 in [Helfgott] p. 4). Maybe each of the results must be provided as theorem, like 6gbe 40193, which would be quite a lot...

As proposed in the Google group discussion https://groups.google.com/g/metamath/c/DOXS4pg0h8w/m/O3oBPuzhBAAJ , 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 40226, ax-hgprmladder 40228 and ax-tgoldbachgt 40231.

Syntaxcgbe 40167 Extend the definition of a class to include the set of even numbers which have a Goldbach partition.
class GoldbachEven

Syntaxcgbo 40168 Extend the definition of a class to include the set of odd numbers which can be written as sum of three primes.
class GoldbachOdd

Syntaxcgboa 40169 Extend the definition of a class to include the set of odd numbers which can be written as sum of three odd primes.
class GoldbachOddALTV

Definitiondf-gbe 40170* 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-gbo 40171* Define the set of 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 ∈ GoldbachOdd ). (Contributed by AV, 14-Jun-2020.)
GoldbachOdd = {𝑧 ∈ Odd ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑧 = ((𝑝 + 𝑞) + 𝑟)}

Definitiondf-gboa 40172* Define the set of 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 ∈ GoldbachOddALTV ). (Contributed by AV, 26-Jul-2020.)
GoldbachOddALTV = {𝑧 ∈ Odd ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ ((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑟 ∈ Odd ) ∧ 𝑧 = ((𝑝 + 𝑞) + 𝑟))}

Theoremisgbe 40173* The predicate "is an even Goldbach number". An even Goldbach number is an even number having a Goldbach partition, i.e. which can be written as 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))))

Theoremisgbo 40174* The predicate "is an odd Goldbach number". An odd Goldbach number is an odd number integer having a Goldbach partition, i.e. which which can be written as sum of three primes. (Contributed by AV, 20-Jul-2020.)
(𝑍 ∈ GoldbachOdd ↔ (𝑍 ∈ Odd ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑍 = ((𝑝 + 𝑞) + 𝑟)))

Theoremisgboa 40175* The predicate "is an odd Goldbach number". An odd Goldbach number is an odd number integer having a Goldbach partition, i.e. which can be written as sum of three odd primes. (Contributed by AV, 26-Jul-2020.)
(𝑍 ∈ GoldbachOddALTV ↔ (𝑍 ∈ Odd ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ ((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑟 ∈ Odd ) ∧ 𝑍 = ((𝑝 + 𝑞) + 𝑟))))

Theoremgbeeven 40176 An even Goldbach number is even. (Contributed by AV, 25-Jul-2020.)
(𝑍 ∈ GoldbachEven → 𝑍 ∈ Even )

Theoremgboodd 40177 An odd Goldbach number is odd. (Contributed by AV, 25-Jul-2020.)
(𝑍 ∈ GoldbachOdd → 𝑍 ∈ Odd )

Theoremgboagbo 40178 An odd Goldbach number (strong version) is an odd Goldbach number (weak version). (Contributed by AV, 26-Jul-2020.)
(𝑍 ∈ GoldbachOddALTV → 𝑍 ∈ GoldbachOdd )

Theoremgboaodd 40179 An odd Goldbach number is odd. (Contributed by AV, 26-Jul-2020.)
(𝑍 ∈ GoldbachOddALTV → 𝑍 ∈ Odd )

Theoremgbepos 40180 Any even Goldbach number is positive. (Contributed by AV, 20-Jul-2020.)
(𝑍 ∈ GoldbachEven → 𝑍 ∈ ℕ)

Theoremgbopos 40181 Any odd Goldbach number is positive. (Contributed by AV, 20-Jul-2020.)
(𝑍 ∈ GoldbachOdd → 𝑍 ∈ ℕ)

Theoremgboapos 40182 Any odd Goldbach number is positive. (Contributed by AV, 26-Jul-2020.)
(𝑍 ∈ GoldbachOddALTV → 𝑍 ∈ ℕ)

Theoremgbegt5 40183 Any even Goldbach number is greater than 5. (Contributed by AV, 20-Jul-2020.)
(𝑍 ∈ GoldbachEven → 5 < 𝑍)

Theoremgbogt5 40184 Any odd Goldbach number is greater than 5. (Contributed by AV, 20-Jul-2020.)
(𝑍 ∈ GoldbachOdd → 5 < 𝑍)

Theoremgboge7 40185 Any odd Goldbach number is greater than or equal to 7. Because of 7gbo 40194, this bound is strict. (Contributed by AV, 20-Jul-2020.)
(𝑍 ∈ GoldbachOdd → 7 ≤ 𝑍)

Theoremgboage9 40186 Any odd Goldbach number (strong version) is greater than or equal to 9. Because of 9gboa 40196, this bound is strict. (Contributed by AV, 26-Jul-2020.)
(𝑍 ∈ GoldbachOddALTV → 9 ≤ 𝑍)

Theoremgbege6 40187 Any even Goldbach number is greater than or equal to 6. Because of 6gbe 40193, this bound is strict. (Contributed by AV, 20-Jul-2020.)
(𝑍 ∈ GoldbachEven → 6 ≤ 𝑍)

Theoremgbpart6 40188 The Goldbach partition of 6. (Contributed by AV, 20-Jul-2020.)
6 = (3 + 3)

Theoremgbpart7 40189 The (weak) Goldbach partition of 7. (Contributed by AV, 20-Jul-2020.)
7 = ((2 + 2) + 3)

Theoremgbpart8 40190 The Goldbach partition of 8. (Contributed by AV, 20-Jul-2020.)
8 = (3 + 5)

Theoremgbpart9 40191 The (strong) Goldbach partition of 9. (Contributed by AV, 26-Jul-2020.)
9 = ((3 + 3) + 3)

Theoremgbpart11 40192 The (strong) Goldbach partition of 11. (Contributed by AV, 29-Jul-2020.)
11 = ((3 + 3) + 5)

Theorem6gbe 40193 6 is an even Goldbach number. (Contributed by AV, 20-Jul-2020.)
6 ∈ GoldbachEven

Theorem7gbo 40194 7 is an odd Goldbach number. (Contributed by AV, 20-Jul-2020.)
7 ∈ GoldbachOdd

Theorem8gbe 40195 8 is an even Goldbach number. (Contributed by AV, 20-Jul-2020.)
8 ∈ GoldbachEven

Theorem9gboa 40196 9 is an odd Goldbach number. (Contributed by AV, 26-Jul-2020.)
9 ∈ GoldbachOddALTV

Theorem11gboa 40197 11 is an odd Goldbach number. (Contributed by AV, 29-Jul-2020.)
11 ∈ GoldbachOddALTV

Theoremstgoldbwt 40198 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 ∈ GoldbachOddALTV ) → ∀𝑛 ∈ Odd (5 < 𝑛𝑛 ∈ GoldbachOdd ))

Theorembgoldbwt 40199* If the 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 ∈ GoldbachOdd ))

Theorembgoldbst 40200* If the 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 ∈ GoldbachOddALTV ))

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