Axiom ax-addass 8274

Description: Addition of complex numbers is associative. Axiom for real and complex numbers, justified by Theorem axaddass 8232. Proofs should normally use addass 8302 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 8170	. . . 4 class ℂ
3	1, 2	wcel 2209	. . 3 wff 𝐴 ∈ ℂ
4		cB	. . . 4 class 𝐵
5	4, 2	wcel 2209	. . 3 wff 𝐵 ∈ ℂ
6		cC	. . . 4 class 𝐶
7	6, 2	wcel 2209	. . 3 wff 𝐶 ∈ ℂ
8	3, 5, 7	w3a 1009	. 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)
9		caddc 8175	. . . . 5 class +
10	1, 4, 9	co 6078	. . . 4 class (𝐴 + 𝐵)
11	10, 6, 9	co 6078	. . 3 class ((𝐴 + 𝐵) + 𝐶)
12	4, 6, 9	co 6078	. . . 4 class (𝐵 + 𝐶)
13	1, 12, 9	co 6078	. . 3 class (𝐴 + (𝐵 + 𝐶))
14	11, 13	wceq 1402	. 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  8302