Axiom ax-mulcom 8133

Description: Multiplication of complex numbers is commutative. Axiom for real and complex numbers, justified by Theorem axmulcom 8091. Proofs should normally use mulcom 8161 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 8030	. . . 4 class ℂ
3	1, 2	wcel 2202	. . 3 wff 𝐴 ∈ ℂ
4		cB	. . . 4 class 𝐵
5	4, 2	wcel 2202	. . 3 wff 𝐵 ∈ ℂ
6	3, 5	wa 104	. 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ)
7		cmul 8037	. . . 4 class ·
8	1, 4, 7	co 6018	. . 3 class (𝐴 · 𝐵)
9	4, 1, 7	co 6018	. . 3 class (𝐵 · 𝐴)
10	8, 9	wceq 1397	. 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  8161