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Theorem mul13d 39805
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 10283 . 2 (𝜑 → (𝐴 · (𝐵 · 𝐶)) = (𝐵 · (𝐴 · 𝐶)))
52, 1, 3mulassd 10101 . 2 (𝜑 → ((𝐵 · 𝐴) · 𝐶) = (𝐵 · (𝐴 · 𝐶)))
62, 1mulcld 10098 . . 3 (𝜑 → (𝐵 · 𝐴) ∈ ℂ)
76, 3mulcomd 10099 . 2 (𝜑 → ((𝐵 · 𝐴) · 𝐶) = (𝐶 · (𝐵 · 𝐴)))
84, 5, 73eqtr2d 2691 1 (𝜑 → (𝐴 · (𝐵 · 𝐶)) = (𝐶 · (𝐵 · 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  wcel 2030  (class class class)co 6690  cc 9972   · cmul 9979
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-mulcl 10036  ax-mulcom 10038  ax-mulass 10040
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-iota 5889  df-fv 5934  df-ov 6693
This theorem is referenced by:  dirkertrigeqlem3  40635  fourierdlem83  40724
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