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Theorem int-mulcomd 44182
Description: MultiplicationCommutativity generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
Hypotheses
Ref Expression
int-mulcomd.1 (𝜑𝐵 ∈ ℝ)
int-mulcomd.2 (𝜑𝐶 ∈ ℝ)
int-mulcomd.3 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
int-mulcomd (𝜑 → (𝐵 · 𝐶) = (𝐶 · 𝐴))

Proof of Theorem int-mulcomd
StepHypRef Expression
1 int-mulcomd.1 . . . 4 (𝜑𝐵 ∈ ℝ)
21recnd 11296 . . 3 (𝜑𝐵 ∈ ℂ)
3 int-mulcomd.2 . . . 4 (𝜑𝐶 ∈ ℝ)
43recnd 11296 . . 3 (𝜑𝐶 ∈ ℂ)
52, 4mulcomd 11289 . 2 (𝜑 → (𝐵 · 𝐶) = (𝐶 · 𝐵))
6 int-mulcomd.3 . . . 4 (𝜑𝐴 = 𝐵)
76eqcomd 2743 . . 3 (𝜑𝐵 = 𝐴)
87oveq2d 7454 . 2 (𝜑 → (𝐶 · 𝐵) = (𝐶 · 𝐴))
95, 8eqtrd 2777 1 (𝜑 → (𝐵 · 𝐶) = (𝐶 · 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2108  (class class class)co 7438  cr 11161   · cmul 11167
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-resscn 11219  ax-mulcom 11226
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3483  df-dif 3969  df-un 3971  df-ss 3983  df-nul 4343  df-if 4535  df-sn 4635  df-pr 4637  df-op 4641  df-uni 4916  df-br 5152  df-iota 6522  df-fv 6577  df-ov 7441
This theorem is referenced by: (None)
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