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Theorem tendocoval 34719
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 988 . . 3  |-  ( ( ( K  e.  X  /\  W  e.  H
)  /\  ( U  e.  E  /\  V  e.  E )  /\  F  e.  T )  ->  ( K  e.  X  /\  W  e.  H )
)
2 simp2r 1015 . . 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 34716 . . 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 990 . 2  |-  ( ( ( K  e.  X  /\  W  e.  H
)  /\  ( U  e.  E  /\  V  e.  E )  /\  F  e.  T )  ->  F  e.  T )
9 fvco3 5870 . 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 965    = wceq 1370    e. wcel 1758    o. ccom 4945   -->wf 5515   ` cfv 5519   LHypclh 33937   LTrncltrn 34054   TEndoctendo 34705
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-map 7319  df-tendo 34708
This theorem is referenced by:  tendococl  34725  tendodi1  34737  tendodi2  34738  tendo0mul  34779  tendo0mulr  34780  cdleml3N  34931  cdleml7  34935  dvhlveclem  35062  dih1dimatlem0  35282
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