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Theorem int-mulcomd 44078
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 11314 . . 3 (𝜑𝐵 ∈ ℂ)
3 int-mulcomd.2 . . . 4 (𝜑𝐶 ∈ ℝ)
43recnd 11314 . . 3 (𝜑𝐶 ∈ ℂ)
52, 4mulcomd 11307 . 2 (𝜑 → (𝐵 · 𝐶) = (𝐶 · 𝐵))
6 int-mulcomd.3 . . . 4 (𝜑𝐴 = 𝐵)
76eqcomd 2740 . . 3 (𝜑𝐵 = 𝐴)
87oveq2d 7461 . 2 (𝜑 → (𝐶 · 𝐵) = (𝐶 · 𝐴))
95, 8eqtrd 2774 1 (𝜑 → (𝐵 · 𝐶) = (𝐶 · 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2103  (class class class)co 7445  cr 11179   · cmul 11185
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-ext 2705  ax-resscn 11237  ax-mulcom 11244
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2712  df-cleq 2726  df-clel 2813  df-rab 3439  df-v 3484  df-dif 3973  df-un 3975  df-ss 3987  df-nul 4348  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5170  df-iota 6524  df-fv 6580  df-ov 7448
This theorem is referenced by: (None)
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