Theorem mulcomi 8168

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 8144	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) = (𝐵 · 𝐴))
4	1, 2, 3	mp2an 426	1 ⊢ (𝐴 · 𝐵) = (𝐵 · 𝐴)

Syntax hints:   = wceq 1395   ∈ wcel 2200  (class class class)co 6010  ℂcc 8013   · cmul 8020

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia3 108  ax-mulcom 8116

This theorem is referenced by:  mulcomli  8169  8th4div3  9346  numma2c  9639  nummul2c  9643  9t11e99  9723  binom2i  10887  fac3  10971  tanval2ap  12245  pockthi  12902  decsplit1  12972  decsplit  12973  sincosq4sgn  15524  2logb9irrALT  15669  2lgsoddprmlem2  15806  2lgsoddprmlem3d  15810