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Theorem mulcomli 7780
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
StepHypRef Expression
1 axi.2 . . 3 𝐵 ∈ ℂ
2 axi.1 . . 3 𝐴 ∈ ℂ
31, 2mulcomi 7779 . 2 (𝐵 · 𝐴) = (𝐴 · 𝐵)
4 mulcomli.3 . 2 (𝐴 · 𝐵) = 𝐶
53, 4eqtri 2160 1 (𝐵 · 𝐴) = 𝐶
Colors of variables: wff set class
Syntax hints:   = wceq 1331  wcel 1480  (class class class)co 5774  cc 7625   · cmul 7632
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425  ax-4 1487  ax-17 1506  ax-ext 2121  ax-mulcom 7728
This theorem depends on definitions:  df-bi 116  df-cleq 2132
This theorem is referenced by:  nummul2c  9238  halfthird  9331  5recm6rec  9332  sq4e2t8  10397  cos2bnd  11473  ex-exp  12992  ex-fac  12993
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