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Theorem assaassr 21800
Description: Right-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
assaassr ((π‘Š ∈ AssAlg ∧ (𝐴 ∈ 𝐡 ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉)) β†’ (𝑋 Γ— (𝐴 Β· π‘Œ)) = (𝐴 Β· (𝑋 Γ— π‘Œ)))

Proof of Theorem assaassr
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 21798 . 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))))
76simprd 494 1 ((π‘Š ∈ AssAlg ∧ (𝐴 ∈ 𝐡 ∧ 𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉)) β†’ (𝑋 Γ— (𝐴 Β· π‘Œ)) = (𝐴 Β· (𝑋 Γ— π‘Œ)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  β€˜cfv 6553  (class class class)co 7426  Basecbs 17187  .rcmulr 17241  Scalarcsca 17243   ·𝑠 cvsca 17244  AssAlgcasa 21791
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699  ax-nul 5310
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rab 3431  df-v 3475  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-iota 6505  df-fv 6561  df-ov 7429  df-assa 21794
This theorem is referenced by:  assa2ass  21804  issubassa3  21806  sraassab  21808  asclmul2  21827  assamulgscmlem2  21840  mplmon2mul  22020  matinv  22599  cpmadugsumlemC  22797
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