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Theorem tendospdi1 35835
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 753 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( U  e.  E  /\  F  e.  T  /\  G  e.  T ) )  ->  K  e.  V )
2 simplr 754 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( U  e.  E  /\  F  e.  T  /\  G  e.  T ) )  ->  W  e.  H )
3 simpr1 1002 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( U  e.  E  /\  F  e.  T  /\  G  e.  T ) )  ->  U  e.  E )
4 simpr2 1003 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( U  e.  E  /\  F  e.  T  /\  G  e.  T ) )  ->  F  e.  T )
5 simpr3 1004 . 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 35579 . 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 1236 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 369    /\ w3a 973    = wceq 1379    e. wcel 1767    o. ccom 5003   ` cfv 5588   LHypclh 34798   LTrncltrn 34915   TEndoctendo 35566
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-map 7422  df-tendo 35569
This theorem is referenced by:  tendocnv  35836  tendospcanN  35838  dvalveclem  35840  dvhlveclem  35923  dihjatcclem4  36236
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