Axiom ax-addass 6986

Description: Addition of complex numbers is associative. Axiom for real and complex numbers, justified by theorem axaddass 6946. Proofs should normally use addass 7011 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 6887	. . . 4 class ℂ
3	1, 2	wcel 1393	. . 3 wff 𝐴 ∈ ℂ
4		cB	. . . 4 class 𝐵
5	4, 2	wcel 1393	. . 3 wff 𝐵 ∈ ℂ
6		cC	. . . 4 class 𝐶
7	6, 2	wcel 1393	. . 3 wff 𝐶 ∈ ℂ
8	3, 5, 7	w3a 885	. 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)
9		caddc 6892	. . . . 5 class +
10	1, 4, 9	co 5512	. . . 4 class (𝐴 + 𝐵)
11	10, 6, 9	co 5512	. . 3 class ((𝐴 + 𝐵) + 𝐶)
12	4, 6, 9	co 5512	. . . 4 class (𝐵 + 𝐶)
13	1, 12, 9	co 5512	. . 3 class (𝐴 + (𝐵 + 𝐶))
14	11, 13	wceq 1243	. 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  7011