Definition df-gbow 47736

Description: Define the set of weak odd Goldbach numbers, which are positive odd integers that can be expressed as the sum of three primes. By this definition, the weak ternary Goldbach conjecture can be expressed as ∀𝑚 ∈ Odd (5 < 𝑚 → 𝑚 ∈ GoldbachOddW ). (Contributed by AV, 14-Jun-2020.)

Assertion

Ref	Expression
df-gbow	⊢ GoldbachOddW = {𝑧 ∈ Odd ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑧 = ((𝑝 + 𝑞) + 𝑟)}

Distinct variable group:   𝑧,𝑝,𝑞,𝑟

Detailed syntax breakdown of Definition df-gbow

Step	Hyp	Ref	Expression
1		cgbow 47733	. 2 class GoldbachOddW
2		vz	. . . . . . . 8 setvar 𝑧
3	2	cv 1539	. . . . . . 7 class 𝑧
4		vp	. . . . . . . . . 10 setvar 𝑝
5	4	cv 1539	. . . . . . . . 9 class 𝑝
6		vq	. . . . . . . . . 10 setvar 𝑞
7	6	cv 1539	. . . . . . . . 9 class 𝑞
8		caddc 11158	. . . . . . . . 9 class +
9	5, 7, 8	co 7431	. . . . . . . 8 class (𝑝 + 𝑞)
10		vr	. . . . . . . . 9 setvar 𝑟
11	10	cv 1539	. . . . . . . 8 class 𝑟
12	9, 11, 8	co 7431	. . . . . . 7 class ((𝑝 + 𝑞) + 𝑟)
13	3, 12	wceq 1540	. . . . . 6 wff 𝑧 = ((𝑝 + 𝑞) + 𝑟)
14		cprime 16708	. . . . . 6 class ℙ
15	13, 10, 14	wrex 3070	. . . . 5 wff ∃𝑟 ∈ ℙ 𝑧 = ((𝑝 + 𝑞) + 𝑟)
16	15, 6, 14	wrex 3070	. . . 4 wff ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑧 = ((𝑝 + 𝑞) + 𝑟)
17	16, 4, 14	wrex 3070	. . 3 wff ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑧 = ((𝑝 + 𝑞) + 𝑟)
18		codd 47612	. . 3 class Odd
19	17, 2, 18	crab 3436	. 2 class {𝑧 ∈ Odd ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑧 = ((𝑝 + 𝑞) + 𝑟)}
20	1, 19	wceq 1540	1 wff GoldbachOddW = {𝑧 ∈ Odd ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑧 = ((𝑝 + 𝑞) + 𝑟)}

This definition is referenced by:  isgbow  47739