Axiom ax-addass 7855

Description: Addition of complex numbers is associative. Axiom for real and complex numbers, justified by Theorem axaddass 7813. Proofs should normally use addass 7883 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 7751	. . . 4 class ℂ
3	1, 2	wcel 2136	. . 3 wff 𝐴 ∈ ℂ
4		cB	. . . 4 class 𝐵
5	4, 2	wcel 2136	. . 3 wff 𝐵 ∈ ℂ
6		cC	. . . 4 class 𝐶
7	6, 2	wcel 2136	. . 3 wff 𝐶 ∈ ℂ
8	3, 5, 7	w3a 968	. 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)
9		caddc 7756	. . . . 5 class +
10	1, 4, 9	co 5842	. . . 4 class (𝐴 + 𝐵)
11	10, 6, 9	co 5842	. . 3 class ((𝐴 + 𝐵) + 𝐶)
12	4, 6, 9	co 5842	. . . 4 class (𝐵 + 𝐶)
13	1, 12, 9	co 5842	. . 3 class (𝐴 + (𝐵 + 𝐶))
14	11, 13	wceq 1343	. 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  7883