Definition df-gbow 48954

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 48951	. 2 class GoldbachOddW
2		vz	. . . . . . . 8 setvar 𝑧
3	2	cv 1568	. . . . . . 7 class 𝑧
4		vp	. . . . . . . . . 10 setvar 𝑝
5	4	cv 1568	. . . . . . . . 9 class 𝑝
6		vq	. . . . . . . . . 10 setvar 𝑞
7	6	cv 1568	. . . . . . . . 9 class 𝑞
8		caddc 11208	. . . . . . . . 9 class +
9	5, 7, 8	co 7422	. . . . . . . 8 class (𝑝 + 𝑞)
10		vr	. . . . . . . . 9 setvar 𝑟
11	10	cv 1568	. . . . . . . 8 class 𝑟
12	9, 11, 8	co 7422	. . . . . . 7 class ((𝑝 + 𝑞) + 𝑟)
13	3, 12	wceq 1569	. . . . . 6 wff 𝑧 = ((𝑝 + 𝑞) + 𝑟)
14		cprime 16853	. . . . . 6 class ℙ
15	13, 10, 14	wrex 3086	. . . . 5 wff ∃𝑟 ∈ ℙ 𝑧 = ((𝑝 + 𝑞) + 𝑟)
16	15, 6, 14	wrex 3086	. . . 4 wff ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑧 = ((𝑝 + 𝑞) + 𝑟)
17	16, 4, 14	wrex 3086	. . 3 wff ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑧 = ((𝑝 + 𝑞) + 𝑟)
18		codd 48830	. . 3 class Odd
19	17, 2, 18	crab 3412	. 2 class {𝑧 ∈ Odd ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑧 = ((𝑝 + 𝑞) + 𝑟)}
20	1, 19	wceq 1569	1 wff GoldbachOddW = {𝑧 ∈ Odd ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑧 = ((𝑝 + 𝑞) + 𝑟)}

This definition is used by:  isgbow  48957