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Theorem assaass 21280
Description: Left-associative property of an associative algebra. (Contributed by Mario Carneiro, 29-Dec-2014.)
Hypotheses
Ref Expression
isassa.v 𝑉 = (Baseβ€˜π‘Š)
isassa.f 𝐹 = (Scalarβ€˜π‘Š)
isassa.b 𝐡 = (Baseβ€˜πΉ)
isassa.s Β· = ( ·𝑠 β€˜π‘Š)
isassa.t Γ— = (.rβ€˜π‘Š)
Assertion
Ref Expression
assaass ((π‘Š ∈ AssAlg ∧ (𝐴 ∈ 𝐡 ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉)) β†’ ((𝐴 Β· 𝑋) Γ— π‘Œ) = (𝐴 Β· (𝑋 Γ— π‘Œ)))

Proof of Theorem assaass
StepHypRef Expression
1 isassa.v . . 3 𝑉 = (Baseβ€˜π‘Š)
2 isassa.f . . 3 𝐹 = (Scalarβ€˜π‘Š)
3 isassa.b . . 3 𝐡 = (Baseβ€˜πΉ)
4 isassa.s . . 3 Β· = ( ·𝑠 β€˜π‘Š)
5 isassa.t . . 3 Γ— = (.rβ€˜π‘Š)
61, 2, 3, 4, 5assalem 21279 . 2 ((π‘Š ∈ AssAlg ∧ (𝐴 ∈ 𝐡 ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉)) β†’ (((𝐴 Β· 𝑋) Γ— π‘Œ) = (𝐴 Β· (𝑋 Γ— π‘Œ)) ∧ (𝑋 Γ— (𝐴 Β· π‘Œ)) = (𝐴 Β· (𝑋 Γ— π‘Œ))))
76simpld 496 1 ((π‘Š ∈ AssAlg ∧ (𝐴 ∈ 𝐡 ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉)) β†’ ((𝐴 Β· 𝑋) Γ— π‘Œ) = (𝐴 Β· (𝑋 Γ— π‘Œ)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  β€˜cfv 6497  (class class class)co 7358  Basecbs 17088  .rcmulr 17139  Scalarcsca 17141   ·𝑠 cvsca 17142  AssAlgcasa 21272
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-nul 5264
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-ral 3062  df-rab 3407  df-v 3446  df-sbc 3741  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-iota 6449  df-fv 6505  df-ov 7361  df-assa 21275
This theorem is referenced by:  assa2ass  21285  issubassa3  21287  asclmul1  21305  assamulgscmlem2  21319  mplmon2mul  21493  matinv  22042
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