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Theorem assaassr 17389
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  |-  .x.  =  ( .s `  W )
isassa.t  |-  .X.  =  ( .r `  W )
Assertion
Ref Expression
assaassr  |-  ( ( W  e. AssAlg  /\  ( A  e.  B  /\  X  e.  V  /\  Y  e.  V )
)  ->  ( X  .X.  ( A  .x.  Y
) )  =  ( A  .x.  ( X 
.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  |-  .x.  =  ( .s `  W )
5 isassa.t . . 3  |-  .X.  =  ( .r `  W )
61, 2, 3, 4, 5assalem 17387 . 2  |-  ( ( W  e. AssAlg  /\  ( A  e.  B  /\  X  e.  V  /\  Y  e.  V )
)  ->  ( (
( A  .x.  X
)  .X.  Y )  =  ( A  .x.  ( X  .X.  Y ) )  /\  ( X 
.X.  ( A  .x.  Y ) )  =  ( A  .x.  ( X  .X.  Y ) ) ) )
76simprd 463 1  |-  ( ( W  e. AssAlg  /\  ( A  e.  B  /\  X  e.  V  /\  Y  e.  V )
)  ->  ( X  .X.  ( A  .x.  Y
) )  =  ( A  .x.  ( X 
.X.  Y ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   ` cfv 5417  (class class class)co 6090   Basecbs 14173   .rcmulr 14238  Scalarcsca 14240   .scvsca 14241  AssAlgcasa 17380
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-nul 4420
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-rab 2723  df-v 2973  df-sbc 3186  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-br 4292  df-iota 5380  df-fv 5425  df-ov 6093  df-assa 17383
This theorem is referenced by:  issubassa  17394  asclmul2  17410  asclrhm  17411  mplmon2mul  17582  matinv  18482  assa2ass  30817  assamulgscmlem2  30819
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