Axiom ax-mulcom 8244

Description: Multiplication of complex numbers is commutative. Axiom for real and complex numbers, justified by Theorem axmulcom 8202. Proofs should normally use mulcom 8272 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 8141	. . . 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	3, 5	wa 104	. 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ)
7		cmul 8148	. . . 4 class ·
8	1, 4, 7	co 6058	. . 3 class (𝐴 · 𝐵)
9	4, 1, 7	co 6058	. . 3 class (𝐵 · 𝐴)
10	8, 9	wceq 1398	. 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  8272