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Theorem assaassr 20553
 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 20551 . 2 ((𝑊 ∈ AssAlg ∧ (𝐴𝐵𝑋𝑉𝑌𝑉)) → (((𝐴 · 𝑋) × 𝑌) = (𝐴 · (𝑋 × 𝑌)) ∧ (𝑋 × (𝐴 · 𝑌)) = (𝐴 · (𝑋 × 𝑌))))
76simprd 499 1 ((𝑊 ∈ AssAlg ∧ (𝐴𝐵𝑋𝑉𝑌𝑉)) → (𝑋 × (𝐴 · 𝑌)) = (𝐴 · (𝑋 × 𝑌)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ‘cfv 6325  (class class class)co 7136  Basecbs 16478  .rcmulr 16561  Scalarcsca 16563   ·𝑠 cvsca 16564  AssAlgcasa 20544 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-nul 5175 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5032  df-iota 6284  df-fv 6333  df-ov 7139  df-assa 20547 This theorem is referenced by:  assa2ass  20557  issubassa3  20559  asclmul2  20578  ascldimulOLD  20580  assamulgscmlem2  20592  mplmon2mul  20746  matinv  21292  cpmadugsumlemC  21490
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