Definition df-gbo 48984

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 < 𝑚 → 𝑚 ∈ 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

Step	Hyp	Ref	Expression
1		cgbo 48981	. 2 class GoldbachOdd
2		vp	. . . . . . . . . 10 setvar 𝑝
3	2	cv 1568	. . . . . . . . 9 class 𝑝
4		codd 48859	. . . . . . . . 9 class Odd
5	3, 4	wcel 2144	. . . . . . . 8 wff 𝑝 ∈ Odd
6		vq	. . . . . . . . . 10 setvar 𝑞
7	6	cv 1568	. . . . . . . . 9 class 𝑞
8	7, 4	wcel 2144	. . . . . . . 8 wff 𝑞 ∈ Odd
9		vr	. . . . . . . . . 10 setvar 𝑟
10	9	cv 1568	. . . . . . . . 9 class 𝑟
11	10, 4	wcel 2144	. . . . . . . 8 wff 𝑟 ∈ Odd
12	5, 8, 11	w3a 1103	. . . . . . 7 wff (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑟 ∈ Odd )
13		vz	. . . . . . . . 9 setvar 𝑧
14	13	cv 1568	. . . . . . . 8 class 𝑧
15		caddc 11210	. . . . . . . . . 10 class +
16	3, 7, 15	co 7424	. . . . . . . . 9 class (𝑝 + 𝑞)
17	16, 10, 15	co 7424	. . . . . . . 8 class ((𝑝 + 𝑞) + 𝑟)
18	14, 17	wceq 1569	. . . . . . 7 wff 𝑧 = ((𝑝 + 𝑞) + 𝑟)
19	12, 18	wa 401	. . . . . 6 wff ((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑟 ∈ Odd ) ∧ 𝑧 = ((𝑝 + 𝑞) + 𝑟))
20		cprime 16855	. . . . . 6 class ℙ
21	19, 9, 20	wrex 3086	. . . . 5 wff ∃𝑟 ∈ ℙ ((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑟 ∈ Odd ) ∧ 𝑧 = ((𝑝 + 𝑞) + 𝑟))
22	21, 6, 20	wrex 3086	. . . 4 wff ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ ((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑟 ∈ Odd ) ∧ 𝑧 = ((𝑝 + 𝑞) + 𝑟))
23	22, 2, 20	wrex 3086	. . 3 wff ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ ((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑟 ∈ Odd ) ∧ 𝑧 = ((𝑝 + 𝑞) + 𝑟))
24	23, 13, 4	crab 3412	. 2 class {𝑧 ∈ Odd ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ ((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑟 ∈ Odd ) ∧ 𝑧 = ((𝑝 + 𝑞) + 𝑟))}
25	1, 24	wceq 1569	1 wff GoldbachOdd = {𝑧 ∈ Odd ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ ((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑟 ∈ Odd ) ∧ 𝑧 = ((𝑝 + 𝑞) + 𝑟))}

This definition is used by:  isgbo  48987  tgoldbachgtALTV  49046