Axiom ax-addass 8069

Description: Addition of complex numbers is associative. Axiom for real and complex numbers, justified by Theorem axaddass 8027. Proofs should normally use addass 8097 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 7965	. . . 4 class ℂ
3	1, 2	wcel 2180	. . 3 wff 𝐴 ∈ ℂ
4		cB	. . . 4 class 𝐵
5	4, 2	wcel 2180	. . 3 wff 𝐵 ∈ ℂ
6		cC	. . . 4 class 𝐶
7	6, 2	wcel 2180	. . 3 wff 𝐶 ∈ ℂ
8	3, 5, 7	w3a 983	. 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)
9		caddc 7970	. . . . 5 class +
10	1, 4, 9	co 5974	. . . 4 class (𝐴 + 𝐵)
11	10, 6, 9	co 5974	. . 3 class ((𝐴 + 𝐵) + 𝐶)
12	4, 6, 9	co 5974	. . . 4 class (𝐵 + 𝐶)
13	1, 12, 9	co 5974	. . 3 class (𝐴 + (𝐵 + 𝐶))
14	11, 13	wceq 1375	. 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  8097