Axiom ax-addass 7876

Description: Addition of complex numbers is associative. Axiom for real and complex numbers, justified by Theorem axaddass 7834. Proofs should normally use addass 7904 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 7772	. . . 4 class ℂ
3	1, 2	wcel 2141	. . 3 wff 𝐴 ∈ ℂ
4		cB	. . . 4 class 𝐵
5	4, 2	wcel 2141	. . 3 wff 𝐵 ∈ ℂ
6		cC	. . . 4 class 𝐶
7	6, 2	wcel 2141	. . 3 wff 𝐶 ∈ ℂ
8	3, 5, 7	w3a 973	. 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)
9		caddc 7777	. . . . 5 class +
10	1, 4, 9	co 5853	. . . 4 class (𝐴 + 𝐵)
11	10, 6, 9	co 5853	. . 3 class ((𝐴 + 𝐵) + 𝐶)
12	4, 6, 9	co 5853	. . . 4 class (𝐵 + 𝐶)
13	1, 12, 9	co 5853	. . 3 class (𝐴 + (𝐵 + 𝐶))
14	11, 13	wceq 1348	. 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  7904