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Definition df-gbo 44908
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 44905 . 2 class GoldbachOdd
2 vp . . . . . . . . . 10 setvar 𝑝
32cv 1542 . . . . . . . . 9 class 𝑝
4 codd 44783 . . . . . . . . 9 class Odd
53, 4wcel 2112 . . . . . . . 8 wff 𝑝 ∈ Odd
6 vq . . . . . . . . . 10 setvar 𝑞
76cv 1542 . . . . . . . . 9 class 𝑞
87, 4wcel 2112 . . . . . . . 8 wff 𝑞 ∈ Odd
9 vr . . . . . . . . . 10 setvar 𝑟
109cv 1542 . . . . . . . . 9 class 𝑟
1110, 4wcel 2112 . . . . . . . 8 wff 𝑟 ∈ Odd
125, 8, 11w3a 1089 . . . . . . 7 wff (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑟 ∈ Odd )
13 vz . . . . . . . . 9 setvar 𝑧
1413cv 1542 . . . . . . . 8 class 𝑧
15 caddc 10759 . . . . . . . . . 10 class +
163, 7, 15co 7234 . . . . . . . . 9 class (𝑝 + 𝑞)
1716, 10, 15co 7234 . . . . . . . 8 class ((𝑝 + 𝑞) + 𝑟)
1814, 17wceq 1543 . . . . . . 7 wff 𝑧 = ((𝑝 + 𝑞) + 𝑟)
1912, 18wa 399 . . . . . 6 wff ((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑟 ∈ Odd ) ∧ 𝑧 = ((𝑝 + 𝑞) + 𝑟))
20 cprime 16258 . . . . . 6 class
2119, 9, 20wrex 3064 . . . . 5 wff 𝑟 ∈ ℙ ((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑟 ∈ Odd ) ∧ 𝑧 = ((𝑝 + 𝑞) + 𝑟))
2221, 6, 20wrex 3064 . . . 4 wff 𝑞 ∈ ℙ ∃𝑟 ∈ ℙ ((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑟 ∈ Odd ) ∧ 𝑧 = ((𝑝 + 𝑞) + 𝑟))
2322, 2, 20wrex 3064 . . 3 wff 𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ ((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑟 ∈ Odd ) ∧ 𝑧 = ((𝑝 + 𝑞) + 𝑟))
2423, 13, 4crab 3067 . 2 class {𝑧 ∈ Odd ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ ((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑟 ∈ Odd ) ∧ 𝑧 = ((𝑝 + 𝑞) + 𝑟))}
251, 24wceq 1543 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  44911  tgoldbachgtALTV  44970
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