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Theorem tendospdi1 34297
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 758 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( U  e.  E  /\  F  e.  T  /\  G  e.  T ) )  ->  K  e.  V )
2 simplr 760 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( U  e.  E  /\  F  e.  T  /\  G  e.  T ) )  ->  W  e.  H )
3 simpr1 1011 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( U  e.  E  /\  F  e.  T  /\  G  e.  T ) )  ->  U  e.  E )
4 simpr2 1012 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( U  e.  E  /\  F  e.  T  /\  G  e.  T ) )  ->  F  e.  T )
5 simpr3 1013 . 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 34041 . 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 1272 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 setvar class
Syntax hints:    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870    o. ccom 4858   ` cfv 5601   LHypclh 33258   LTrncltrn 33375   TEndoctendo 34028
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-map 7482  df-tendo 34031
This theorem is referenced by:  tendocnv  34298  tendospcanN  34300  dvalveclem  34302  dvhlveclem  34385  dihjatcclem4  34698
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