Axiom ax-mulcom 8116

Description: Multiplication of complex numbers is commutative. Axiom for real and complex numbers, justified by Theorem axmulcom 8074. Proofs should normally use mulcom 8144 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 8013	. . . 4 class ℂ
3	1, 2	wcel 2200	. . 3 wff 𝐴 ∈ ℂ
4		cB	. . . 4 class 𝐵
5	4, 2	wcel 2200	. . 3 wff 𝐵 ∈ ℂ
6	3, 5	wa 104	. 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ)
7		cmul 8020	. . . 4 class ·
8	1, 4, 7	co 6010	. . 3 class (𝐴 · 𝐵)
9	4, 1, 7	co 6010	. . 3 class (𝐵 · 𝐴)
10	8, 9	wceq 1395	. 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  8144