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Theorem assaassr 21793
Description: Right-associative property of an associative algebra. (Contributed by Mario Carneiro, 29-Dec-2014.)
Hypotheses
Ref Expression
isassa.v 𝑉 = (Base‘𝑊)
isassa.f 𝐹 = (Scalar‘𝑊)
isassa.b 𝐵 = (Base‘𝐹)
isassa.s · = ( ·𝑠𝑊)
isassa.t × = (.r𝑊)
Assertion
Ref Expression
assaassr ((𝑊 ∈ AssAlg ∧ (𝐴𝐵𝑋𝑉𝑌𝑉)) → (𝑋 × (𝐴 · 𝑌)) = (𝐴 · (𝑋 × 𝑌)))

Proof of Theorem assaassr
StepHypRef Expression
1 isassa.v . . 3 𝑉 = (Base‘𝑊)
2 isassa.f . . 3 𝐹 = (Scalar‘𝑊)
3 isassa.b . . 3 𝐵 = (Base‘𝐹)
4 isassa.s . . 3 · = ( ·𝑠𝑊)
5 isassa.t . . 3 × = (.r𝑊)
61, 2, 3, 4, 5assalem 21791 . 2 ((𝑊 ∈ AssAlg ∧ (𝐴𝐵𝑋𝑉𝑌𝑉)) → (((𝐴 · 𝑋) × 𝑌) = (𝐴 · (𝑋 × 𝑌)) ∧ (𝑋 × (𝐴 · 𝑌)) = (𝐴 · (𝑋 × 𝑌))))
76simprd 495 1 ((𝑊 ∈ AssAlg ∧ (𝐴𝐵𝑋𝑉𝑌𝑉)) → (𝑋 × (𝐴 · 𝑌)) = (𝐴 · (𝑋 × 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1085   = wceq 1534  wcel 2099  cfv 6548  (class class class)co 7420  Basecbs 17180  .rcmulr 17234  Scalarcsca 17236   ·𝑠 cvsca 17237  AssAlgcasa 21784
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-nul 5306
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rab 3430  df-v 3473  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-iota 6500  df-fv 6556  df-ov 7423  df-assa 21787
This theorem is referenced by:  assa2ass  21797  issubassa3  21799  sraassab  21801  asclmul2  21820  assamulgscmlem2  21833  mplmon2mul  22013  matinv  22592  cpmadugsumlemC  22790
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