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Definition df-gboa 40172
 Description: Define the set of 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 < 𝑚 → 𝑚 ∈ GoldbachOddALTV ). (Contributed by AV, 26-Jul-2020.)
Assertion
Ref Expression
df-gboa GoldbachOddALTV = {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-gboa
StepHypRef Expression
1 cgboa 40169 . 2 class GoldbachOddALTV
2 vp . . . . . . . . . 10 setvar 𝑝
32cv 1474 . . . . . . . . 9 class 𝑝
4 codd 40076 . . . . . . . . 9 class Odd
53, 4wcel 1977 . . . . . . . 8 wff 𝑝 ∈ Odd
6 vq . . . . . . . . . 10 setvar 𝑞
76cv 1474 . . . . . . . . 9 class 𝑞
87, 4wcel 1977 . . . . . . . 8 wff 𝑞 ∈ Odd
9 vr . . . . . . . . . 10 setvar 𝑟
109cv 1474 . . . . . . . . 9 class 𝑟
1110, 4wcel 1977 . . . . . . . 8 wff 𝑟 ∈ Odd
125, 8, 11w3a 1031 . . . . . . 7 wff (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑟 ∈ Odd )
13 vz . . . . . . . . 9 setvar 𝑧
1413cv 1474 . . . . . . . 8 class 𝑧
15 caddc 9818 . . . . . . . . . 10 class +
163, 7, 15co 6549 . . . . . . . . 9 class (𝑝 + 𝑞)
1716, 10, 15co 6549 . . . . . . . 8 class ((𝑝 + 𝑞) + 𝑟)
1814, 17wceq 1475 . . . . . . 7 wff 𝑧 = ((𝑝 + 𝑞) + 𝑟)
1912, 18wa 383 . . . . . 6 wff ((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑟 ∈ Odd ) ∧ 𝑧 = ((𝑝 + 𝑞) + 𝑟))
20 cprime 15223 . . . . . 6 class
2119, 9, 20wrex 2897 . . . . 5 wff 𝑟 ∈ ℙ ((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑟 ∈ Odd ) ∧ 𝑧 = ((𝑝 + 𝑞) + 𝑟))
2221, 6, 20wrex 2897 . . . 4 wff 𝑞 ∈ ℙ ∃𝑟 ∈ ℙ ((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑟 ∈ Odd ) ∧ 𝑧 = ((𝑝 + 𝑞) + 𝑟))
2322, 2, 20wrex 2897 . . 3 wff 𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ ((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑟 ∈ Odd ) ∧ 𝑧 = ((𝑝 + 𝑞) + 𝑟))
2423, 13, 4crab 2900 . 2 class {𝑧 ∈ Odd ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ ((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑟 ∈ Odd ) ∧ 𝑧 = ((𝑝 + 𝑞) + 𝑟))}
251, 24wceq 1475 1 wff GoldbachOddALTV = {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:  isgboa  40175
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