Theorem mulcomi 7905

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 7882	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) = (𝐵 · 𝐴))
4	1, 2, 3	mp2an 423	1 ⊢ (𝐴 · 𝐵) = (𝐵 · 𝐴)

Syntax hints:   = wceq 1343   ∈ wcel 2136  (class class class)co 5842  ℂcc 7751   · cmul 7758

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia3 107  ax-mulcom 7854

This theorem is referenced by:  mulcomli  7906  8th4div3  9076  numma2c  9367  nummul2c  9371  9t11e99  9451  binom2i  10563  fac3  10645  tanval2ap  11654  pockthi  12288  sincosq4sgn  13390  2logb9irrALT  13532