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Theorem mul13d 40009
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 10447 . 2 (𝜑 → (𝐴 · (𝐵 · 𝐶)) = (𝐵 · (𝐴 · 𝐶)))
52, 1, 3mulassd 10265 . 2 (𝜑 → ((𝐵 · 𝐴) · 𝐶) = (𝐵 · (𝐴 · 𝐶)))
62, 1mulcld 10262 . . 3 (𝜑 → (𝐵 · 𝐴) ∈ ℂ)
76, 3mulcomd 10263 . 2 (𝜑 → ((𝐵 · 𝐴) · 𝐶) = (𝐶 · (𝐵 · 𝐴)))
84, 5, 73eqtr2d 2811 1 (𝜑 → (𝐴 · (𝐵 · 𝐶)) = (𝐶 · (𝐵 · 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  wcel 2145  (class class class)co 6793  cc 10136   · cmul 10143
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-mulcl 10200  ax-mulcom 10202  ax-mulass 10204
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-iota 5994  df-fv 6039  df-ov 6796
This theorem is referenced by:  dirkertrigeqlem3  40834  fourierdlem83  40923
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