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Definition df-gbo 42156
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 42153 . 2 class GoldbachOdd
2 vp . . . . . . . . . 10 setvar 𝑝
32cv 1629 . . . . . . . . 9 class 𝑝
4 codd 42056 . . . . . . . . 9 class Odd
53, 4wcel 2144 . . . . . . . 8 wff 𝑝 ∈ Odd
6 vq . . . . . . . . . 10 setvar 𝑞
76cv 1629 . . . . . . . . 9 class 𝑞
87, 4wcel 2144 . . . . . . . 8 wff 𝑞 ∈ Odd
9 vr . . . . . . . . . 10 setvar 𝑟
109cv 1629 . . . . . . . . 9 class 𝑟
1110, 4wcel 2144 . . . . . . . 8 wff 𝑟 ∈ Odd
125, 8, 11w3a 1070 . . . . . . 7 wff (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑟 ∈ Odd )
13 vz . . . . . . . . 9 setvar 𝑧
1413cv 1629 . . . . . . . 8 class 𝑧
15 caddc 10140 . . . . . . . . . 10 class +
163, 7, 15co 6792 . . . . . . . . 9 class (𝑝 + 𝑞)
1716, 10, 15co 6792 . . . . . . . 8 class ((𝑝 + 𝑞) + 𝑟)
1814, 17wceq 1630 . . . . . . 7 wff 𝑧 = ((𝑝 + 𝑞) + 𝑟)
1912, 18wa 382 . . . . . 6 wff ((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑟 ∈ Odd ) ∧ 𝑧 = ((𝑝 + 𝑞) + 𝑟))
20 cprime 15591 . . . . . 6 class
2119, 9, 20wrex 3061 . . . . 5 wff 𝑟 ∈ ℙ ((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑟 ∈ Odd ) ∧ 𝑧 = ((𝑝 + 𝑞) + 𝑟))
2221, 6, 20wrex 3061 . . . 4 wff 𝑞 ∈ ℙ ∃𝑟 ∈ ℙ ((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑟 ∈ Odd ) ∧ 𝑧 = ((𝑝 + 𝑞) + 𝑟))
2322, 2, 20wrex 3061 . . 3 wff 𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ ((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑟 ∈ Odd ) ∧ 𝑧 = ((𝑝 + 𝑞) + 𝑟))
2423, 13, 4crab 3064 . 2 class {𝑧 ∈ Odd ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ ((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑟 ∈ Odd ) ∧ 𝑧 = ((𝑝 + 𝑞) + 𝑟))}
251, 24wceq 1630 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  42159  tgoldbachgtALTV  42218
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