Theorem mulcomi 7125

Description: Commutative law for multiplication. (Contributed by NM, 23-Nov-1994.)

Hypotheses

Ref	Expression
axi.1	⊢ 𝐴 ∈ ℂ
axi.2	⊢ 𝐵 ∈ ℂ

Assertion

Ref	Expression
mulcomi	⊢ (𝐴 · 𝐵) = (𝐵 · 𝐴)

Proof of Theorem mulcomi

Step	Hyp	Ref	Expression
1		axi.1	. 2 ⊢ 𝐴 ∈ ℂ
2		axi.2	. 2 ⊢ 𝐵 ∈ ℂ
3		mulcom 7102	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) = (𝐵 · 𝐴))
4	1, 2, 3	mp2an 416	1 ⊢ (𝐴 · 𝐵) = (𝐵 · 𝐴)

Syntax hints:   = wceq 1284   ∈ wcel 1433  (class class class)co 5532  ℂcc 6979   · cmul 6986

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia3 106  ax-mulcom 7077

This theorem is referenced by:  mulcomli  7126  8th4div3  8250  numma2c  8522  nummul2c  8526  9t11e99  8606  binom2i  9583  fac3  9659