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Definition df-gbo 40171
Description: 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.)
Assertion
Ref Expression
df-gbo GoldbachOdd = {𝑧 ∈ Odd ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑧 = ((𝑝 + 𝑞) + 𝑟)}
Distinct variable group:   𝑧,𝑝,𝑞,𝑟

Detailed syntax breakdown of Definition df-gbo
StepHypRef Expression
1 cgbo 40168 . 2 class GoldbachOdd
2 vz . . . . . . . 8 setvar 𝑧
32cv 1474 . . . . . . 7 class 𝑧
4 vp . . . . . . . . . 10 setvar 𝑝
54cv 1474 . . . . . . . . 9 class 𝑝
6 vq . . . . . . . . . 10 setvar 𝑞
76cv 1474 . . . . . . . . 9 class 𝑞
8 caddc 9818 . . . . . . . . 9 class +
95, 7, 8co 6549 . . . . . . . 8 class (𝑝 + 𝑞)
10 vr . . . . . . . . 9 setvar 𝑟
1110cv 1474 . . . . . . . 8 class 𝑟
129, 11, 8co 6549 . . . . . . 7 class ((𝑝 + 𝑞) + 𝑟)
133, 12wceq 1475 . . . . . 6 wff 𝑧 = ((𝑝 + 𝑞) + 𝑟)
14 cprime 15223 . . . . . 6 class
1513, 10, 14wrex 2897 . . . . 5 wff 𝑟 ∈ ℙ 𝑧 = ((𝑝 + 𝑞) + 𝑟)
1615, 6, 14wrex 2897 . . . 4 wff 𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑧 = ((𝑝 + 𝑞) + 𝑟)
1716, 4, 14wrex 2897 . . 3 wff 𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑧 = ((𝑝 + 𝑞) + 𝑟)
18 codd 40076 . . 3 class Odd
1917, 2, 18crab 2900 . 2 class {𝑧 ∈ Odd ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑧 = ((𝑝 + 𝑞) + 𝑟)}
201, 19wceq 1475 1 wff GoldbachOdd = {𝑧 ∈ Odd ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑧 = ((𝑝 + 𝑞) + 𝑟)}
Colors of variables: wff setvar class
This definition is referenced by:  isgbo  40174
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