Axiom ax-addass 8282

Description: Addition of complex numbers is associative. Axiom for real and complex numbers, justified by Theorem axaddass 8240. Proofs should normally use addass 8310 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 8178	. . . 4 class ℂ
3	1, 2	wcel 2209	. . 3 wff 𝐴 ∈ ℂ
4		cB	. . . 4 class 𝐵
5	4, 2	wcel 2209	. . 3 wff 𝐵 ∈ ℂ
6		cC	. . . 4 class 𝐶
7	6, 2	wcel 2209	. . 3 wff 𝐶 ∈ ℂ
8	3, 5, 7	w3a 1009	. 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)
9		caddc 8183	. . . . 5 class +
10	1, 4, 9	co 6085	. . . 4 class (𝐴 + 𝐵)
11	10, 6, 9	co 6085	. . 3 class ((𝐴 + 𝐵) + 𝐶)
12	4, 6, 9	co 6085	. . . 4 class (𝐵 + 𝐶)
13	1, 12, 9	co 6085	. . 3 class (𝐴 + (𝐵 + 𝐶))
14	11, 13	wceq 1402	. 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 used by:  addass  8310