Axiom ax-addass 8034

Description: Addition of complex numbers is associative. Axiom for real and complex numbers, justified by Theorem axaddass 7992. Proofs should normally use addass 8062 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)

Assertion

Ref	Expression
ax-addass	⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶)))

Detailed syntax breakdown of Axiom ax-addass

Step	Hyp	Ref	Expression
1		cA	. . . 4 class 𝐴
2		cc 7930	. . . 4 class ℂ
3	1, 2	wcel 2177	. . 3 wff 𝐴 ∈ ℂ
4		cB	. . . 4 class 𝐵
5	4, 2	wcel 2177	. . 3 wff 𝐵 ∈ ℂ
6		cC	. . . 4 class 𝐶
7	6, 2	wcel 2177	. . 3 wff 𝐶 ∈ ℂ
8	3, 5, 7	w3a 981	. 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)
9		caddc 7935	. . . . 5 class +
10	1, 4, 9	co 5951	. . . 4 class (𝐴 + 𝐵)
11	10, 6, 9	co 5951	. . 3 class ((𝐴 + 𝐵) + 𝐶)
12	4, 6, 9	co 5951	. . . 4 class (𝐵 + 𝐶)
13	1, 12, 9	co 5951	. . 3 class (𝐴 + (𝐵 + 𝐶))
14	11, 13	wceq 1373	. 2 wff ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶))
15	8, 14	wi 4	1 wff ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶)))

This axiom is referenced by:  addass  8062