Axiom ax-addass 8133

Description: Addition of complex numbers is associative. Axiom for real and complex numbers, justified by Theorem axaddass 8091. Proofs should normally use addass 8161 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 8029	. . . 4 class ℂ
3	1, 2	wcel 2202	. . 3 wff 𝐴 ∈ ℂ
4		cB	. . . 4 class 𝐵
5	4, 2	wcel 2202	. . 3 wff 𝐵 ∈ ℂ
6		cC	. . . 4 class 𝐶
7	6, 2	wcel 2202	. . 3 wff 𝐶 ∈ ℂ
8	3, 5, 7	w3a 1004	. 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)
9		caddc 8034	. . . . 5 class +
10	1, 4, 9	co 6017	. . . 4 class (𝐴 + 𝐵)
11	10, 6, 9	co 6017	. . 3 class ((𝐴 + 𝐵) + 𝐶)
12	4, 6, 9	co 6017	. . . 4 class (𝐵 + 𝐶)
13	1, 12, 9	co 6017	. . 3 class (𝐴 + (𝐵 + 𝐶))
14	11, 13	wceq 1397	. 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  8161