Axiom ax-addass 8194

Description: Addition of complex numbers is associative. Axiom for real and complex numbers, justified by Theorem axaddass 8152. Proofs should normally use addass 8222 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 8090	. . . 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 1005	. 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)
9		caddc 8095	. . . . 5 class +
10	1, 4, 9	co 6028	. . . 4 class (𝐴 + 𝐵)
11	10, 6, 9	co 6028	. . 3 class ((𝐴 + 𝐵) + 𝐶)
12	4, 6, 9	co 6028	. . . 4 class (𝐵 + 𝐶)
13	1, 12, 9	co 6028	. . 3 class (𝐴 + (𝐵 + 𝐶))
14	11, 13	wceq 1398	. 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  8222