Definition df-gbow 43934

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 43931	. 2 class GoldbachOddW
2		vz	. . . . . . . 8 setvar 𝑧
3	2	cv 1536	. . . . . . 7 class 𝑧
4		vp	. . . . . . . . . 10 setvar 𝑝
5	4	cv 1536	. . . . . . . . 9 class 𝑝
6		vq	. . . . . . . . . 10 setvar 𝑞
7	6	cv 1536	. . . . . . . . 9 class 𝑞
8		caddc 10540	. . . . . . . . 9 class +
9	5, 7, 8	co 7156	. . . . . . . 8 class (𝑝 + 𝑞)
10		vr	. . . . . . . . 9 setvar 𝑟
11	10	cv 1536	. . . . . . . 8 class 𝑟
12	9, 11, 8	co 7156	. . . . . . 7 class ((𝑝 + 𝑞) + 𝑟)
13	3, 12	wceq 1537	. . . . . 6 wff 𝑧 = ((𝑝 + 𝑞) + 𝑟)
14		cprime 16015	. . . . . 6 class ℙ
15	13, 10, 14	wrex 3139	. . . . 5 wff ∃𝑟 ∈ ℙ 𝑧 = ((𝑝 + 𝑞) + 𝑟)
16	15, 6, 14	wrex 3139	. . . 4 wff ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑧 = ((𝑝 + 𝑞) + 𝑟)
17	16, 4, 14	wrex 3139	. . 3 wff ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑧 = ((𝑝 + 𝑞) + 𝑟)
18		codd 43810	. . 3 class Odd
19	17, 2, 18	crab 3142	. 2 class {𝑧 ∈ Odd ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑧 = ((𝑝 + 𝑞) + 𝑟)}
20	1, 19	wceq 1537	1 wff GoldbachOddW = {𝑧 ∈ Odd ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑧 = ((𝑝 + 𝑞) + 𝑟)}

This definition is referenced by:  isgbow  43937