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Theorem mul12d 7578
Description: Commutative/associative law that swaps the first two factors in a triple product. (Contributed by Mario Carneiro, 27-May-2016.)
Hypotheses
Ref Expression
muld.1 (𝜑𝐴 ∈ ℂ)
addcomd.2 (𝜑𝐵 ∈ ℂ)
mul12d.3 (𝜑𝐶 ∈ ℂ)
Assertion
Ref Expression
mul12d (𝜑 → (𝐴 · (𝐵 · 𝐶)) = (𝐵 · (𝐴 · 𝐶)))

Proof of Theorem mul12d
StepHypRef Expression
1 muld.1 . 2 (𝜑𝐴 ∈ ℂ)
2 addcomd.2 . 2 (𝜑𝐵 ∈ ℂ)
3 mul12d.3 . 2 (𝜑𝐶 ∈ ℂ)
4 mul12 7555 . 2 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 · (𝐵 · 𝐶)) = (𝐵 · (𝐴 · 𝐶)))
51, 2, 3, 4syl3anc 1172 1 (𝜑 → (𝐴 · (𝐵 · 𝐶)) = (𝐵 · (𝐴 · 𝐶)))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1287  wcel 1436  (class class class)co 5613  cc 7292   · cmul 7299
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-mulcom 7390  ax-mulass 7392
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-rex 2361  df-v 2617  df-un 2992  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3637  df-br 3821  df-iota 4946  df-fv 4989  df-ov 5616
This theorem is referenced by:  mulreim  8022  divrecap  8094  remullem  10200  dvdscmulr  10700  bezoutlemnewy  10860  dvdsmulgcd  10889  lcmgcdlem  10934  cncongr1  10960
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