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Theorem tendospdi1 31832
Description: Forward distributive law for endomorphism scalar product operation. (Contributed by NM, 10-Oct-2013.)
Hypotheses
Ref Expression
tendosp.h  |-  H  =  ( LHyp `  K
)
tendosp.t  |-  T  =  ( ( LTrn `  K
) `  W )
tendosp.e  |-  E  =  ( ( TEndo `  K
) `  W )
Assertion
Ref Expression
tendospdi1  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( U  e.  E  /\  F  e.  T  /\  G  e.  T ) )  -> 
( U `  ( F  o.  G )
)  =  ( ( U `  F )  o.  ( U `  G ) ) )

Proof of Theorem tendospdi1
StepHypRef Expression
1 simpll 730 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( U  e.  E  /\  F  e.  T  /\  G  e.  T ) )  ->  K  e.  V )
2 simplr 731 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( U  e.  E  /\  F  e.  T  /\  G  e.  T ) )  ->  W  e.  H )
3 simpr1 961 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( U  e.  E  /\  F  e.  T  /\  G  e.  T ) )  ->  U  e.  E )
4 simpr2 962 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( U  e.  E  /\  F  e.  T  /\  G  e.  T ) )  ->  F  e.  T )
5 simpr3 963 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( U  e.  E  /\  F  e.  T  /\  G  e.  T ) )  ->  G  e.  T )
6 tendosp.h . . 3  |-  H  =  ( LHyp `  K
)
7 tendosp.t . . 3  |-  T  =  ( ( LTrn `  K
) `  W )
8 tendosp.e . . 3  |-  E  =  ( ( TEndo `  K
) `  W )
96, 7, 8tendovalco 31576 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H  /\  U  e.  E
)  /\  ( F  e.  T  /\  G  e.  T ) )  -> 
( U `  ( F  o.  G )
)  =  ( ( U `  F )  o.  ( U `  G ) ) )
101, 2, 3, 4, 5, 9syl32anc 1190 1  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( U  e.  E  /\  F  e.  T  /\  G  e.  T ) )  -> 
( U `  ( F  o.  G )
)  =  ( ( U `  F )  o.  ( U `  G ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    o. ccom 4709   ` cfv 5271   LHypclh 30795   LTrncltrn 30912   TEndoctendo 31563
This theorem is referenced by:  tendocnv  31833  tendospcanN  31835  dvalveclem  31837  dvhlveclem  31920  dihjatcclem4  32233
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-map 6790  df-tendo 31566
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