Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  tendocoval Structured version   Unicode version

Theorem tendocoval 33765
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 997 . . 3  |-  ( ( ( K  e.  X  /\  W  e.  H
)  /\  ( U  e.  E  /\  V  e.  E )  /\  F  e.  T )  ->  ( K  e.  X  /\  W  e.  H )
)
2 simp2r 1024 . . 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 33762 . . 3  |-  ( ( ( K  e.  X  /\  W  e.  H
)  /\  V  e.  E )  ->  V : T --> T )
71, 2, 6syl2anc 659 . 2  |-  ( ( ( K  e.  X  /\  W  e.  H
)  /\  ( U  e.  E  /\  V  e.  E )  /\  F  e.  T )  ->  V : T --> T )
8 simp3 999 . 2  |-  ( ( ( K  e.  X  /\  W  e.  H
)  /\  ( U  e.  E  /\  V  e.  E )  /\  F  e.  T )  ->  F  e.  T )
9 fvco3 5925 . 2  |-  ( ( V : T --> T  /\  F  e.  T )  ->  ( ( U  o.  V ) `  F
)  =  ( U `
 ( V `  F ) ) )
107, 8, 9syl2anc 659 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 367    /\ w3a 974    = wceq 1405    e. wcel 1842    o. ccom 4826   -->wf 5564   ` cfv 5568   LHypclh 32981   LTrncltrn 33098   TEndoctendo 33751
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-map 7458  df-tendo 33754
This theorem is referenced by:  tendococl  33771  tendodi1  33783  tendodi2  33784  tendo0mul  33825  tendo0mulr  33826  cdleml3N  33977  cdleml7  33981  dvhlveclem  34108  dih1dimatlem0  34328
  Copyright terms: Public domain W3C validator