Axiom ax-addass 7597

Description: Addition of complex numbers is associative. Axiom for real and complex numbers, justified by theorem axaddass 7557. Proofs should normally use addass 7622 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 7498	. . . 4 class ℂ
3	1, 2	wcel 1448	. . 3 wff 𝐴 ∈ ℂ
4		cB	. . . 4 class 𝐵
5	4, 2	wcel 1448	. . 3 wff 𝐵 ∈ ℂ
6		cC	. . . 4 class 𝐶
7	6, 2	wcel 1448	. . 3 wff 𝐶 ∈ ℂ
8	3, 5, 7	w3a 930	. 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)
9		caddc 7503	. . . . 5 class +
10	1, 4, 9	co 5706	. . . 4 class (𝐴 + 𝐵)
11	10, 6, 9	co 5706	. . 3 class ((𝐴 + 𝐵) + 𝐶)
12	4, 6, 9	co 5706	. . . 4 class (𝐵 + 𝐶)
13	1, 12, 9	co 5706	. . 3 class (𝐴 + (𝐵 + 𝐶))
14	11, 13	wceq 1299	. 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  7622