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Definition df-gbow 46031
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 46028 . 2 class GoldbachOddW
2 vz . . . . . . . 8 setvar 𝑧
32cv 1541 . . . . . . 7 class 𝑧
4 vp . . . . . . . . . 10 setvar 𝑝
54cv 1541 . . . . . . . . 9 class 𝑝
6 vq . . . . . . . . . 10 setvar 𝑞
76cv 1541 . . . . . . . . 9 class 𝑞
8 caddc 11062 . . . . . . . . 9 class +
95, 7, 8co 7361 . . . . . . . 8 class (𝑝 + 𝑞)
10 vr . . . . . . . . 9 setvar 𝑟
1110cv 1541 . . . . . . . 8 class 𝑟
129, 11, 8co 7361 . . . . . . 7 class ((𝑝 + 𝑞) + 𝑟)
133, 12wceq 1542 . . . . . 6 wff 𝑧 = ((𝑝 + 𝑞) + 𝑟)
14 cprime 16555 . . . . . 6 class
1513, 10, 14wrex 3070 . . . . 5 wff 𝑟 ∈ ℙ 𝑧 = ((𝑝 + 𝑞) + 𝑟)
1615, 6, 14wrex 3070 . . . 4 wff 𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑧 = ((𝑝 + 𝑞) + 𝑟)
1716, 4, 14wrex 3070 . . 3 wff 𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑧 = ((𝑝 + 𝑞) + 𝑟)
18 codd 45907 . . 3 class Odd
1917, 2, 18crab 3406 . 2 class {𝑧 ∈ Odd ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑧 = ((𝑝 + 𝑞) + 𝑟)}
201, 19wceq 1542 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  46034
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