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Theorem tendocoval 35562
Description: Value of composition of endomorphisms in a trace-preserving endomorphism. (Contributed by NM, 9-Jun-2013.)
Hypotheses
Ref Expression
tendof.h  |-  H  =  ( LHyp `  K
)
tendof.t  |-  T  =  ( ( LTrn `  K
) `  W )
tendof.e  |-  E  =  ( ( TEndo `  K
) `  W )
Assertion
Ref Expression
tendocoval  |-  ( ( ( K  e.  X  /\  W  e.  H
)  /\  ( U  e.  E  /\  V  e.  E )  /\  F  e.  T )  ->  (
( U  o.  V
) `  F )  =  ( U `  ( V `  F ) ) )

Proof of Theorem tendocoval
StepHypRef Expression
1 simp1 996 . . 3  |-  ( ( ( K  e.  X  /\  W  e.  H
)  /\  ( U  e.  E  /\  V  e.  E )  /\  F  e.  T )  ->  ( K  e.  X  /\  W  e.  H )
)
2 simp2r 1023 . . 3  |-  ( ( ( K  e.  X  /\  W  e.  H
)  /\  ( U  e.  E  /\  V  e.  E )  /\  F  e.  T )  ->  V  e.  E )
3 tendof.h . . . 4  |-  H  =  ( LHyp `  K
)
4 tendof.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
5 tendof.e . . . 4  |-  E  =  ( ( TEndo `  K
) `  W )
63, 4, 5tendof 35559 . . 3  |-  ( ( ( K  e.  X  /\  W  e.  H
)  /\  V  e.  E )  ->  V : T --> T )
71, 2, 6syl2anc 661 . 2  |-  ( ( ( K  e.  X  /\  W  e.  H
)  /\  ( U  e.  E  /\  V  e.  E )  /\  F  e.  T )  ->  V : T --> T )
8 simp3 998 . 2  |-  ( ( ( K  e.  X  /\  W  e.  H
)  /\  ( U  e.  E  /\  V  e.  E )  /\  F  e.  T )  ->  F  e.  T )
9 fvco3 5942 . 2  |-  ( ( V : T --> T  /\  F  e.  T )  ->  ( ( U  o.  V ) `  F
)  =  ( U `
 ( V `  F ) ) )
107, 8, 9syl2anc 661 1  |-  ( ( ( K  e.  X  /\  W  e.  H
)  /\  ( U  e.  E  /\  V  e.  E )  /\  F  e.  T )  ->  (
( U  o.  V
) `  F )  =  ( U `  ( V `  F ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    o. ccom 5003   -->wf 5582   ` cfv 5586   LHypclh 34780   LTrncltrn 34897   TEndoctendo 35548
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 6574
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 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-map 7419  df-tendo 35551
This theorem is referenced by:  tendococl  35568  tendodi1  35580  tendodi2  35581  tendo0mul  35622  tendo0mulr  35623  cdleml3N  35774  cdleml7  35778  dvhlveclem  35905  dih1dimatlem0  36125
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