Theorem mulcomi 8190

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 8166	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) = (𝐵 · 𝐴))
4	1, 2, 3	mp2an 426	1 ⊢ (𝐴 · 𝐵) = (𝐵 · 𝐴)

Syntax hints:   = wceq 1397   ∈ wcel 2201  (class class class)co 6023  ℂcc 8035   · cmul 8042

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia3 108  ax-mulcom 8138

This theorem is referenced by:  mulcomli  8191  8th4div3  9368  numma2c  9661  nummul2c  9665  9t11e99  9745  binom2i  10916  fac3  11000  tanval2ap  12297  pockthi  12954  decsplit1  13024  decsplit  13025  sincosq4sgn  15582  2logb9irrALT  15727  2lgsoddprmlem2  15864  2lgsoddprmlem3d  15868