Definition df-gbo 48310

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 48307	. 2 class GoldbachOdd
2		vp	. . . . . . . . . 10 setvar 𝑝
3	2	cv 1549	. . . . . . . . 9 class 𝑝
4		codd 48185	. . . . . . . . 9 class Odd
5	3, 4	wcel 2132	. . . . . . . 8 wff 𝑝 ∈ Odd
6		vq	. . . . . . . . . 10 setvar 𝑞
7	6	cv 1549	. . . . . . . . 9 class 𝑞
8	7, 4	wcel 2132	. . . . . . . 8 wff 𝑞 ∈ Odd
9		vr	. . . . . . . . . 10 setvar 𝑟
10	9	cv 1549	. . . . . . . . 9 class 𝑟
11	10, 4	wcel 2132	. . . . . . . 8 wff 𝑟 ∈ Odd
12	5, 8, 11	w3a 1095	. . . . . . 7 wff (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑟 ∈ Odd )
13		vz	. . . . . . . . 9 setvar 𝑧
14	13	cv 1549	. . . . . . . 8 class 𝑧
15		caddc 11062	. . . . . . . . . 10 class +
16	3, 7, 15	co 7381	. . . . . . . . 9 class (𝑝 + 𝑞)
17	16, 10, 15	co 7381	. . . . . . . 8 class ((𝑝 + 𝑞) + 𝑟)
18	14, 17	wceq 1550	. . . . . . 7 wff 𝑧 = ((𝑝 + 𝑞) + 𝑟)
19	12, 18	wa 398	. . . . . 6 wff ((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑟 ∈ Odd ) ∧ 𝑧 = ((𝑝 + 𝑞) + 𝑟))
20		cprime 16677	. . . . . 6 class ℙ
21	19, 9, 20	wrex 3076	. . . . 5 wff ∃𝑟 ∈ ℙ ((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑟 ∈ Odd ) ∧ 𝑧 = ((𝑝 + 𝑞) + 𝑟))
22	21, 6, 20	wrex 3076	. . . 4 wff ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ ((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑟 ∈ Odd ) ∧ 𝑧 = ((𝑝 + 𝑞) + 𝑟))
23	22, 2, 20	wrex 3076	. . 3 wff ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ ((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑟 ∈ Odd ) ∧ 𝑧 = ((𝑝 + 𝑞) + 𝑟))
24	23, 13, 4	crab 3404	. 2 class {𝑧 ∈ Odd ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ ((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑟 ∈ Odd ) ∧ 𝑧 = ((𝑝 + 𝑞) + 𝑟))}
25	1, 24	wceq 1550	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  48313  tgoldbachgtALTV  48372