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Theorem assaass 18286
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  |-  .x.  =  ( .s `  W )
isassa.t  |-  .X.  =  ( .r `  W )
Assertion
Ref Expression
assaass  |-  ( ( W  e. AssAlg  /\  ( A  e.  B  /\  X  e.  V  /\  Y  e.  V )
)  ->  ( ( A  .x.  X )  .X.  Y )  =  ( A  .x.  ( X 
.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  |-  .x.  =  ( .s `  W )
5 isassa.t . . 3  |-  .X.  =  ( .r `  W )
61, 2, 3, 4, 5assalem 18285 . 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 ) ) ) )
76simpld 457 1  |-  ( ( W  e. AssAlg  /\  ( A  e.  B  /\  X  e.  V  /\  Y  e.  V )
)  ->  ( ( A  .x.  X )  .X.  Y )  =  ( A  .x.  ( X 
.X.  Y ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842   ` cfv 5569  (class class class)co 6278   Basecbs 14841   .rcmulr 14910  Scalarcsca 14912   .scvsca 14913  AssAlgcasa 18278
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-nul 4525
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-sbc 3278  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-iota 5533  df-fv 5577  df-ov 6281  df-assa 18281
This theorem is referenced by:  assa2ass  18291  issubassa  18293  asclmul1  18308  asclrhm  18311  assamulgscmlem2  18318  mplmon2mul  18486  matinv  19471
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