Axiom ax-addass 8000

Description: Addition of complex numbers is associative. Axiom for real and complex numbers, justified by Theorem axaddass 7958. Proofs should normally use addass 8028 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 7896	. . . 4 class ℂ
3	1, 2	wcel 2167	. . 3 wff 𝐴 ∈ ℂ
4		cB	. . . 4 class 𝐵
5	4, 2	wcel 2167	. . 3 wff 𝐵 ∈ ℂ
6		cC	. . . 4 class 𝐶
7	6, 2	wcel 2167	. . 3 wff 𝐶 ∈ ℂ
8	3, 5, 7	w3a 980	. 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)
9		caddc 7901	. . . . 5 class +
10	1, 4, 9	co 5925	. . . 4 class (𝐴 + 𝐵)
11	10, 6, 9	co 5925	. . 3 class ((𝐴 + 𝐵) + 𝐶)
12	4, 6, 9	co 5925	. . . 4 class (𝐵 + 𝐶)
13	1, 12, 9	co 5925	. . 3 class (𝐴 + (𝐵 + 𝐶))
14	11, 13	wceq 1364	. 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  8028