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Theorem mul13d 41773
Description: Commutative/associative law that swaps the first and the third factor in a triple product. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypotheses
Ref Expression
mul13d.1 (𝜑𝐴 ∈ ℂ)
mul13d.2 (𝜑𝐵 ∈ ℂ)
mul13d.3 (𝜑𝐶 ∈ ℂ)
Assertion
Ref Expression
mul13d (𝜑 → (𝐴 · (𝐵 · 𝐶)) = (𝐶 · (𝐵 · 𝐴)))

Proof of Theorem mul13d
StepHypRef Expression
1 mul13d.1 . . 3 (𝜑𝐴 ∈ ℂ)
2 mul13d.2 . . 3 (𝜑𝐵 ∈ ℂ)
3 mul13d.3 . . 3 (𝜑𝐶 ∈ ℂ)
41, 2, 3mul12d 10841 . 2 (𝜑 → (𝐴 · (𝐵 · 𝐶)) = (𝐵 · (𝐴 · 𝐶)))
52, 1, 3mulassd 10656 . 2 (𝜑 → ((𝐵 · 𝐴) · 𝐶) = (𝐵 · (𝐴 · 𝐶)))
62, 1mulcld 10653 . . 3 (𝜑 → (𝐵 · 𝐴) ∈ ℂ)
76, 3mulcomd 10654 . 2 (𝜑 → ((𝐵 · 𝐴) · 𝐶) = (𝐶 · (𝐵 · 𝐴)))
84, 5, 73eqtr2d 2865 1 (𝜑 → (𝐴 · (𝐵 · 𝐶)) = (𝐶 · (𝐵 · 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2115  (class class class)co 7145  cc 10527   · cmul 10534
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-12 2179  ax-ext 2796  ax-mulcl 10591  ax-mulcom 10593  ax-mulass 10595
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-ex 1782  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-v 3482  df-un 3924  df-in 3926  df-ss 3936  df-sn 4550  df-pr 4552  df-op 4556  df-uni 4825  df-br 5053  df-iota 6302  df-fv 6351  df-ov 7148
This theorem is referenced by:  dirkertrigeqlem3  42605  fourierdlem83  42694
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