Axiom ax-addass 8231

Description: Addition of complex numbers is associative. Axiom for real and complex numbers, justified by Theorem axaddass 8189. Proofs should normally use addass 8259 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 8127	. . . 4 class ℂ
3	1, 2	wcel 2205	. . 3 wff 𝐴 ∈ ℂ
4		cB	. . . 4 class 𝐵
5	4, 2	wcel 2205	. . 3 wff 𝐵 ∈ ℂ
6		cC	. . . 4 class 𝐶
7	6, 2	wcel 2205	. . 3 wff 𝐶 ∈ ℂ
8	3, 5, 7	w3a 1005	. 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)
9		caddc 8132	. . . . 5 class +
10	1, 4, 9	co 6052	. . . 4 class (𝐴 + 𝐵)
11	10, 6, 9	co 6052	. . 3 class ((𝐴 + 𝐵) + 𝐶)
12	4, 6, 9	co 6052	. . . 4 class (𝐵 + 𝐶)
13	1, 12, 9	co 6052	. . 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  8259