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Theorem assaass 19811
Description: Left-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
assaass ((𝑊 ∈ AssAlg ∧ (𝐴𝐵𝑋𝑉𝑌𝑉)) → ((𝐴 · 𝑋) × 𝑌) = (𝐴 · (𝑋 × 𝑌)))

Proof of Theorem assaass
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 19810 . 2 ((𝑊 ∈ AssAlg ∧ (𝐴𝐵𝑋𝑉𝑌𝑉)) → (((𝐴 · 𝑋) × 𝑌) = (𝐴 · (𝑋 × 𝑌)) ∧ (𝑋 × (𝐴 · 𝑌)) = (𝐴 · (𝑋 × 𝑌))))
76simpld 487 1 ((𝑊 ∈ AssAlg ∧ (𝐴𝐵𝑋𝑉𝑌𝑉)) → ((𝐴 · 𝑋) × 𝑌) = (𝐴 · (𝑋 × 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387  w3a 1068   = wceq 1507  wcel 2050  cfv 6188  (class class class)co 6976  Basecbs 16339  .rcmulr 16422  Scalarcsca 16424   ·𝑠 cvsca 16425  AssAlgcasa 19803
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-ext 2751  ax-nul 5067
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2760  df-cleq 2772  df-clel 2847  df-nfc 2919  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3418  df-sbc 3683  df-dif 3833  df-un 3835  df-in 3837  df-ss 3844  df-nul 4180  df-if 4351  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-br 4930  df-iota 6152  df-fv 6196  df-ov 6979  df-assa 19806
This theorem is referenced by:  assa2ass  19816  issubassa  19818  asclmul1  19833  asclrhm  19836  assamulgscmlem2  19843  mplmon2mul  19994  matinv  20990
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