Axiom ax-mulcom 7077

Description: Multiplication of complex numbers is commutative. Axiom for real and complex numbers, justified by theorem axmulcom 7037. Proofs should normally use mulcom 7102 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 6979	. . . 4 class ℂ
3	1, 2	wcel 1433	. . 3 wff 𝐴 ∈ ℂ
4		cB	. . . 4 class 𝐵
5	4, 2	wcel 1433	. . 3 wff 𝐵 ∈ ℂ
6	3, 5	wa 102	. 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ)
7		cmul 6986	. . . 4 class ·
8	1, 4, 7	co 5532	. . 3 class (𝐴 · 𝐵)
9	4, 1, 7	co 5532	. . 3 class (𝐵 · 𝐴)
10	8, 9	wceq 1284	. 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  7102