Theorem mulcomi 8332

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 8308	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) = (𝐵 · 𝐴))
4	1, 2, 3	mp2an 430	1 ⊢ (𝐴 · 𝐵) = (𝐵 · 𝐴)

This proof depends on syntax axioms:   = wceq 1402   ∈ wcel 2209  (class class class)co 6085  ℂcc 8177   · cmul 8184

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia3 108  ax-mulcom 8280

This theorem is used by:  mulcomli  8333  8th4div3  9524  numma2c  9822  nummul2c  9826  9t11e99  9906  binom2i  11085  fac3  11170  tanval2ap  12480  pockthi  13137  decsplit1  13207  decsplit  13208  sincosq4sgn  15931  2logb9irrALT  16080  log2ublem2  16088  log2ublem3  16089  log2ublog2  16090  2lgsoddprmlem2  16229  2lgsoddprmlem3d  16233