Axiom ax-addass 7915

Description: Addition of complex numbers is associative. Axiom for real and complex numbers, justified by Theorem axaddass 7873. Proofs should normally use addass 7943 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 7811	. . . 4 class ℂ
3	1, 2	wcel 2148	. . 3 wff 𝐴 ∈ ℂ
4		cB	. . . 4 class 𝐵
5	4, 2	wcel 2148	. . 3 wff 𝐵 ∈ ℂ
6		cC	. . . 4 class 𝐶
7	6, 2	wcel 2148	. . 3 wff 𝐶 ∈ ℂ
8	3, 5, 7	w3a 978	. 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)
9		caddc 7816	. . . . 5 class +
10	1, 4, 9	co 5877	. . . 4 class (𝐴 + 𝐵)
11	10, 6, 9	co 5877	. . 3 class ((𝐴 + 𝐵) + 𝐶)
12	4, 6, 9	co 5877	. . . 4 class (𝐵 + 𝐶)
13	1, 12, 9	co 5877	. . 3 class (𝐴 + (𝐵 + 𝐶))
14	11, 13	wceq 1353	. 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  7943