Definition df-gbow 48237

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 48234	. 2 class GoldbachOddW
2		vz	. . . . . . . 8 setvar 𝑧
3	2	cv 1541	. . . . . . 7 class 𝑧
4		vp	. . . . . . . . . 10 setvar 𝑝
5	4	cv 1541	. . . . . . . . 9 class 𝑝
6		vq	. . . . . . . . . 10 setvar 𝑞
7	6	cv 1541	. . . . . . . . 9 class 𝑞
8		caddc 11032	. . . . . . . . 9 class +
9	5, 7, 8	co 7360	. . . . . . . 8 class (𝑝 + 𝑞)
10		vr	. . . . . . . . 9 setvar 𝑟
11	10	cv 1541	. . . . . . . 8 class 𝑟
12	9, 11, 8	co 7360	. . . . . . 7 class ((𝑝 + 𝑞) + 𝑟)
13	3, 12	wceq 1542	. . . . . 6 wff 𝑧 = ((𝑝 + 𝑞) + 𝑟)
14		cprime 16631	. . . . . 6 class ℙ
15	13, 10, 14	wrex 3062	. . . . 5 wff ∃𝑟 ∈ ℙ 𝑧 = ((𝑝 + 𝑞) + 𝑟)
16	15, 6, 14	wrex 3062	. . . 4 wff ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑧 = ((𝑝 + 𝑞) + 𝑟)
17	16, 4, 14	wrex 3062	. . 3 wff ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑧 = ((𝑝 + 𝑞) + 𝑟)
18		codd 48113	. . 3 class Odd
19	17, 2, 18	crab 3390	. 2 class {𝑧 ∈ Odd ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑧 = ((𝑝 + 𝑞) + 𝑟)}
20	1, 19	wceq 1542	1 wff GoldbachOddW = {𝑧 ∈ Odd ∣ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑧 = ((𝑝 + 𝑞) + 𝑟)}

This definition is referenced by:  isgbow  48240