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Theorem assaass 16058
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 16057 . 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 445 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 set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   ` cfv 5255  (class class class)co 5858   Basecbs 13148   .rcmulr 13209  Scalarcsca 13211   .scvsca 13212  AssAlgcasa 16050
This theorem is referenced by:  issubassa  16064  asclmul1  16079  asclrhm  16081  mplmon2mul  16242
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-nul 4149
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ov 5861  df-assa 16053
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