Theorem mulcomi 8085

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 8061	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) = (𝐵 · 𝐴))
4	1, 2, 3	mp2an 426	1 ⊢ (𝐴 · 𝐵) = (𝐵 · 𝐴)

Syntax hints:   = wceq 1373   ∈ wcel 2177  (class class class)co 5951  ℂcc 7930   · cmul 7937

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia3 108  ax-mulcom 8033

This theorem is referenced by:  mulcomli  8086  8th4div3  9263  numma2c  9556  nummul2c  9560  9t11e99  9640  binom2i  10800  fac3  10884  tanval2ap  12068  pockthi  12725  decsplit1  12795  decsplit  12796  sincosq4sgn  15345  2logb9irrALT  15490  2lgsoddprmlem2  15627  2lgsoddprmlem3d  15631