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Theorem tendospdi1 31745
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 731 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( U  e.  E  /\  F  e.  T  /\  G  e.  T ) )  ->  K  e.  V )
2 simplr 732 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( U  e.  E  /\  F  e.  T  /\  G  e.  T ) )  ->  W  e.  H )
3 simpr1 963 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( U  e.  E  /\  F  e.  T  /\  G  e.  T ) )  ->  U  e.  E )
4 simpr2 964 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( U  e.  E  /\  F  e.  T  /\  G  e.  T ) )  ->  F  e.  T )
5 simpr3 965 . 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 31489 . 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 1192 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 359    /\ w3a 936    = wceq 1652    e. wcel 1725    o. ccom 4874   ` cfv 5446   LHypclh 30708   LTrncltrn 30825   TEndoctendo 31476
This theorem is referenced by:  tendocnv  31746  tendospcanN  31748  dvalveclem  31750  dvhlveclem  31833  dihjatcclem4  32146
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-map 7012  df-tendo 31479
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