Definition df-gbow 46970

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 46967	. 2 class GoldbachOddW
2		vz	. . . . . . . 8 setvar 𝑧
3	2	cv 1532	. . . . . . 7 class 𝑧
4		vp	. . . . . . . . . 10 setvar 𝑝
5	4	cv 1532	. . . . . . . . 9 class 𝑝
6		vq	. . . . . . . . . 10 setvar 𝑞
7	6	cv 1532	. . . . . . . . 9 class 𝑞
8		caddc 11112	. . . . . . . . 9 class +
9	5, 7, 8	co 7404	. . . . . . . 8 class (𝑝 + 𝑞)
10		vr	. . . . . . . . 9 setvar 𝑟
11	10	cv 1532	. . . . . . . 8 class 𝑟
12	9, 11, 8	co 7404	. . . . . . 7 class ((𝑝 + 𝑞) + 𝑟)
13	3, 12	wceq 1533	. . . . . 6 wff 𝑧 = ((𝑝 + 𝑞) + 𝑟)
14		cprime 16613	. . . . . 6 class ℙ
15	13, 10, 14	wrex 3064	. . . . 5 wff ∃𝑟 ∈ ℙ 𝑧 = ((𝑝 + 𝑞) + 𝑟)
16	15, 6, 14	wrex 3064	. . . 4 wff ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑧 = ((𝑝 + 𝑞) + 𝑟)
17	16, 4, 14	wrex 3064	. . 3 wff ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑧 = ((𝑝 + 𝑞) + 𝑟)
18		codd 46846	. . 3 class Odd
19	17, 2, 18	crab 3426	. 2 class {𝑧 ∈ Odd ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑧 = ((𝑝 + 𝑞) + 𝑟)}
20	1, 19	wceq 1533	1 wff GoldbachOddW = {𝑧 ∈ Odd ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑧 = ((𝑝 + 𝑞) + 𝑟)}

This definition is referenced by:  isgbow  46973