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Definition df-gbow 42419
Description: 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.)
Assertion
Ref Expression
df-gbow GoldbachOddW = {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-gbow
StepHypRef Expression
1 cgbow 42416 . 2 class GoldbachOddW
2 vz . . . . . . . 8 setvar 𝑧
32cv 1652 . . . . . . 7 class 𝑧
4 vp . . . . . . . . . 10 setvar 𝑝
54cv 1652 . . . . . . . . 9 class 𝑝
6 vq . . . . . . . . . 10 setvar 𝑞
76cv 1652 . . . . . . . . 9 class 𝑞
8 caddc 10227 . . . . . . . . 9 class +
95, 7, 8co 6878 . . . . . . . 8 class (𝑝 + 𝑞)
10 vr . . . . . . . . 9 setvar 𝑟
1110cv 1652 . . . . . . . 8 class 𝑟
129, 11, 8co 6878 . . . . . . 7 class ((𝑝 + 𝑞) + 𝑟)
133, 12wceq 1653 . . . . . 6 wff 𝑧 = ((𝑝 + 𝑞) + 𝑟)
14 cprime 15719 . . . . . 6 class
1513, 10, 14wrex 3090 . . . . 5 wff 𝑟 ∈ ℙ 𝑧 = ((𝑝 + 𝑞) + 𝑟)
1615, 6, 14wrex 3090 . . . 4 wff 𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑧 = ((𝑝 + 𝑞) + 𝑟)
1716, 4, 14wrex 3090 . . 3 wff 𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑧 = ((𝑝 + 𝑞) + 𝑟)
18 codd 42320 . . 3 class Odd
1917, 2, 18crab 3093 . 2 class {𝑧 ∈ Odd ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑧 = ((𝑝 + 𝑞) + 𝑟)}
201, 19wceq 1653 1 wff GoldbachOddW = {𝑧 ∈ Odd ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑧 = ((𝑝 + 𝑞) + 𝑟)}
Colors of variables: wff setvar class
This definition is referenced by:  isgbow  42422
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