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Definition df-gbo 47358
Description: Define the set of (strong) 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 ∈ GoldbachOdd ). (Contributed by AV, 26-Jul-2020.)
Assertion
Ref Expression
df-gbo GoldbachOdd = {𝑧 ∈ Odd ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ ((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑟 ∈ Odd ) ∧ 𝑧 = ((𝑝 + 𝑞) + 𝑟))}
Distinct variable group:   𝑧,𝑝,𝑞,𝑟

Detailed syntax breakdown of Definition df-gbo
StepHypRef Expression
1 cgbo 47355 . 2 class GoldbachOdd
2 vp . . . . . . . . . 10 setvar 𝑝
32cv 1533 . . . . . . . . 9 class 𝑝
4 codd 47233 . . . . . . . . 9 class Odd
53, 4wcel 2099 . . . . . . . 8 wff 𝑝 ∈ Odd
6 vq . . . . . . . . . 10 setvar 𝑞
76cv 1533 . . . . . . . . 9 class 𝑞
87, 4wcel 2099 . . . . . . . 8 wff 𝑞 ∈ Odd
9 vr . . . . . . . . . 10 setvar 𝑟
109cv 1533 . . . . . . . . 9 class 𝑟
1110, 4wcel 2099 . . . . . . . 8 wff 𝑟 ∈ Odd
125, 8, 11w3a 1084 . . . . . . 7 wff (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑟 ∈ Odd )
13 vz . . . . . . . . 9 setvar 𝑧
1413cv 1533 . . . . . . . 8 class 𝑧
15 caddc 11152 . . . . . . . . . 10 class +
163, 7, 15co 7416 . . . . . . . . 9 class (𝑝 + 𝑞)
1716, 10, 15co 7416 . . . . . . . 8 class ((𝑝 + 𝑞) + 𝑟)
1814, 17wceq 1534 . . . . . . 7 wff 𝑧 = ((𝑝 + 𝑞) + 𝑟)
1912, 18wa 394 . . . . . 6 wff ((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑟 ∈ Odd ) ∧ 𝑧 = ((𝑝 + 𝑞) + 𝑟))
20 cprime 16667 . . . . . 6 class
2119, 9, 20wrex 3060 . . . . 5 wff 𝑟 ∈ ℙ ((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑟 ∈ Odd ) ∧ 𝑧 = ((𝑝 + 𝑞) + 𝑟))
2221, 6, 20wrex 3060 . . . 4 wff 𝑞 ∈ ℙ ∃𝑟 ∈ ℙ ((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑟 ∈ Odd ) ∧ 𝑧 = ((𝑝 + 𝑞) + 𝑟))
2322, 2, 20wrex 3060 . . 3 wff 𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ ((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑟 ∈ Odd ) ∧ 𝑧 = ((𝑝 + 𝑞) + 𝑟))
2423, 13, 4crab 3419 . 2 class {𝑧 ∈ Odd ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ ((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑟 ∈ Odd ) ∧ 𝑧 = ((𝑝 + 𝑞) + 𝑟))}
251, 24wceq 1534 1 wff GoldbachOdd = {𝑧 ∈ Odd ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ ((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑟 ∈ Odd ) ∧ 𝑧 = ((𝑝 + 𝑞) + 𝑟))}
Colors of variables: wff setvar class
This definition is referenced by:  isgbo  47361  tgoldbachgtALTV  47420
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