Axiom ax-addass 8122

Description: Addition of complex numbers is associative. Axiom for real and complex numbers, justified by Theorem axaddass 8080. Proofs should normally use addass 8150 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 8018	. . . 4 class ℂ
3	1, 2	wcel 2200	. . 3 wff 𝐴 ∈ ℂ
4		cB	. . . 4 class 𝐵
5	4, 2	wcel 2200	. . 3 wff 𝐵 ∈ ℂ
6		cC	. . . 4 class 𝐶
7	6, 2	wcel 2200	. . 3 wff 𝐶 ∈ ℂ
8	3, 5, 7	w3a 1002	. 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)
9		caddc 8023	. . . . 5 class +
10	1, 4, 9	co 6011	. . . 4 class (𝐴 + 𝐵)
11	10, 6, 9	co 6011	. . 3 class ((𝐴 + 𝐵) + 𝐶)
12	4, 6, 9	co 6011	. . . 4 class (𝐵 + 𝐶)
13	1, 12, 9	co 6011	. . 3 class (𝐴 + (𝐵 + 𝐶))
14	11, 13	wceq 1395	. 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  8150