Axiom ax-mulcom 7875

Description: Multiplication of complex numbers is commutative. Axiom for real and complex numbers, justified by Theorem axmulcom 7833. Proofs should normally use mulcom 7903 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)

Assertion

Ref	Expression
ax-mulcom	⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) = (𝐵 · 𝐴))

Detailed syntax breakdown of Axiom ax-mulcom

Step	Hyp	Ref	Expression
1		cA	. . . 4 class 𝐴
2		cc 7772	. . . 4 class ℂ
3	1, 2	wcel 2141	. . 3 wff 𝐴 ∈ ℂ
4		cB	. . . 4 class 𝐵
5	4, 2	wcel 2141	. . 3 wff 𝐵 ∈ ℂ
6	3, 5	wa 103	. 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ)
7		cmul 7779	. . . 4 class ·
8	1, 4, 7	co 5853	. . 3 class (𝐴 · 𝐵)
9	4, 1, 7	co 5853	. . 3 class (𝐵 · 𝐴)
10	8, 9	wceq 1348	. 2 wff (𝐴 · 𝐵) = (𝐵 · 𝐴)
11	6, 10	wi 4	1 wff ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) = (𝐵 · 𝐴))

This axiom is referenced by:  mulcom  7903