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Definition df-gbow 42158
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 42155 . 2 class GoldbachOddW
2 vz . . . . . . . 8 setvar 𝑧
32cv 1630 . . . . . . 7 class 𝑧
4 vp . . . . . . . . . 10 setvar 𝑝
54cv 1630 . . . . . . . . 9 class 𝑝
6 vq . . . . . . . . . 10 setvar 𝑞
76cv 1630 . . . . . . . . 9 class 𝑞
8 caddc 10139 . . . . . . . . 9 class +
95, 7, 8co 6791 . . . . . . . 8 class (𝑝 + 𝑞)
10 vr . . . . . . . . 9 setvar 𝑟
1110cv 1630 . . . . . . . 8 class 𝑟
129, 11, 8co 6791 . . . . . . 7 class ((𝑝 + 𝑞) + 𝑟)
133, 12wceq 1631 . . . . . 6 wff 𝑧 = ((𝑝 + 𝑞) + 𝑟)
14 cprime 15585 . . . . . 6 class
1513, 10, 14wrex 3062 . . . . 5 wff 𝑟 ∈ ℙ 𝑧 = ((𝑝 + 𝑞) + 𝑟)
1615, 6, 14wrex 3062 . . . 4 wff 𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑧 = ((𝑝 + 𝑞) + 𝑟)
1716, 4, 14wrex 3062 . . 3 wff 𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑧 = ((𝑝 + 𝑞) + 𝑟)
18 codd 42059 . . 3 class Odd
1917, 2, 18crab 3065 . 2 class {𝑧 ∈ Odd ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑧 = ((𝑝 + 𝑞) + 𝑟)}
201, 19wceq 1631 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  42161
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