Definition df-gbo 48226

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 48223	. 2 class GoldbachOdd
2		vp	. . . . . . . . . 10 setvar 𝑝
3	2	cv 1541	. . . . . . . . 9 class 𝑝
4		codd 48101	. . . . . . . . 9 class Odd
5	3, 4	wcel 2114	. . . . . . . 8 wff 𝑝 ∈ Odd
6		vq	. . . . . . . . . 10 setvar 𝑞
7	6	cv 1541	. . . . . . . . 9 class 𝑞
8	7, 4	wcel 2114	. . . . . . . 8 wff 𝑞 ∈ Odd
9		vr	. . . . . . . . . 10 setvar 𝑟
10	9	cv 1541	. . . . . . . . 9 class 𝑟
11	10, 4	wcel 2114	. . . . . . . 8 wff 𝑟 ∈ Odd
12	5, 8, 11	w3a 1087	. . . . . . 7 wff (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑟 ∈ Odd )
13		vz	. . . . . . . . 9 setvar 𝑧
14	13	cv 1541	. . . . . . . 8 class 𝑧
15		caddc 11041	. . . . . . . . . 10 class +
16	3, 7, 15	co 7367	. . . . . . . . 9 class (𝑝 + 𝑞)
17	16, 10, 15	co 7367	. . . . . . . 8 class ((𝑝 + 𝑞) + 𝑟)
18	14, 17	wceq 1542	. . . . . . 7 wff 𝑧 = ((𝑝 + 𝑞) + 𝑟)
19	12, 18	wa 395	. . . . . 6 wff ((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑟 ∈ Odd ) ∧ 𝑧 = ((𝑝 + 𝑞) + 𝑟))
20		cprime 16640	. . . . . 6 class ℙ
21	19, 9, 20	wrex 3061	. . . . 5 wff ∃𝑟 ∈ ℙ ((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑟 ∈ Odd ) ∧ 𝑧 = ((𝑝 + 𝑞) + 𝑟))
22	21, 6, 20	wrex 3061	. . . 4 wff ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ ((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑟 ∈ Odd ) ∧ 𝑧 = ((𝑝 + 𝑞) + 𝑟))
23	22, 2, 20	wrex 3061	. . 3 wff ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ ((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑟 ∈ Odd ) ∧ 𝑧 = ((𝑝 + 𝑞) + 𝑟))
24	23, 13, 4	crab 3389	. 2 class {𝑧 ∈ Odd ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ ((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑟 ∈ Odd ) ∧ 𝑧 = ((𝑝 + 𝑞) + 𝑟))}
25	1, 24	wceq 1542	1 wff GoldbachOdd = {𝑧 ∈ Odd ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ ((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑟 ∈ Odd ) ∧ 𝑧 = ((𝑝 + 𝑞) + 𝑟))}

This definition is referenced by:  isgbo  48229  tgoldbachgtALTV  48288