Theorem mulcomli 8033

Description: Commutative law for multiplication. (Contributed by NM, 23-Nov-1994.)

Hypotheses

Ref	Expression
axi.1	⊢ 𝐴 ∈ ℂ
axi.2	⊢ 𝐵 ∈ ℂ
mulcomli.3	⊢ (𝐴 · 𝐵) = 𝐶

Assertion

Ref	Expression
mulcomli	⊢ (𝐵 · 𝐴) = 𝐶

Proof of Theorem mulcomli

Step	Hyp	Ref	Expression
1		axi.2	. . 3 ⊢ 𝐵 ∈ ℂ
2		axi.1	. . 3 ⊢ 𝐴 ∈ ℂ
3	1, 2	mulcomi 8032	. 2 ⊢ (𝐵 · 𝐴) = (𝐴 · 𝐵)
4		mulcomli.3	. 2 ⊢ (𝐴 · 𝐵) = 𝐶
5	3, 4	eqtri 2217	1 ⊢ (𝐵 · 𝐴) = 𝐶

Syntax hints:   = wceq 1364   ∈ wcel 2167  (class class class)co 5922  ℂcc 7877   · cmul 7884

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1461  ax-gen 1463  ax-4 1524  ax-17 1540  ax-ext 2178  ax-mulcom 7980

This theorem depends on definitions:  df-bi 117  df-cleq 2189

This theorem is referenced by:  nummul2c  9506  halfthird  9599  5recm6rec  9600  sq4e2t8  10729  cos2bnd  11925  dec5nprm  12583  karatsuba  12599  2exp6  12602  2exp8  12604  2exp11  12605  2exp16  12606  2lgslem3a  15334  2lgsoddprmlem3c  15350  2lgsoddprmlem3d  15351  ex-exp  15373  ex-fac  15374