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Theorem assaassr 18093
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 18091 . 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 973    = wceq 1395    e. wcel 1819   ` cfv 5594  (class class class)co 6296   Basecbs 14643   .rcmulr 14712  Scalarcsca 14714   .scvsca 14715  AssAlgcasa 18084
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-nul 4586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-iota 5557  df-fv 5602  df-ov 6299  df-assa 18087
This theorem is referenced by:  assa2ass  18097  issubassa  18099  asclmul2  18115  asclrhm  18117  assamulgscmlem2  18124  mplmon2mul  18292  matinv  19305  cpmadugsumlemC  19502
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