Theorem mulcomli 8026

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 8025	. 2 ⊢ (𝐵 · 𝐴) = (𝐴 · 𝐵)
4		mulcomli.3	. 2 ⊢ (𝐴 · 𝐵) = 𝐶
5	3, 4	eqtri 2214	1 ⊢ (𝐵 · 𝐴) = 𝐶

Syntax hints:   = wceq 1364   ∈ wcel 2164  (class class class)co 5918  ℂcc 7870   · cmul 7877

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 1458  ax-gen 1460  ax-4 1521  ax-17 1537  ax-ext 2175  ax-mulcom 7973

This theorem depends on definitions:  df-bi 117  df-cleq 2186

This theorem is referenced by:  nummul2c  9497  halfthird  9590  5recm6rec  9591  sq4e2t8  10708  cos2bnd  11903  2lgsoddprmlem3c  15197  2lgsoddprmlem3d  15198  ex-exp  15219  ex-fac  15220