Theorem mulcomi 8120

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 8096	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) = (𝐵 · 𝐴))
4	1, 2, 3	mp2an 426	1 ⊢ (𝐴 · 𝐵) = (𝐵 · 𝐴)

Syntax hints:   = wceq 1375   ∈ wcel 2180  (class class class)co 5974  ℂcc 7965   · cmul 7972

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia3 108  ax-mulcom 8068

This theorem is referenced by:  mulcomli  8121  8th4div3  9298  numma2c  9591  nummul2c  9595  9t11e99  9675  binom2i  10837  fac3  10921  tanval2ap  12190  pockthi  12847  decsplit1  12917  decsplit  12918  sincosq4sgn  15468  2logb9irrALT  15613  2lgsoddprmlem2  15750  2lgsoddprmlem3d  15754