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Theorem istendod 31560
Description: Deduce the predicate "is a trace-preserving endomorphism". (Contributed by NM, 9-Jun-2013.)
Hypotheses
Ref Expression
tendoset.l  |-  .<_  =  ( le `  K )
tendoset.h  |-  H  =  ( LHyp `  K
)
tendoset.t  |-  T  =  ( ( LTrn `  K
) `  W )
tendoset.r  |-  R  =  ( ( trL `  K
) `  W )
tendoset.e  |-  E  =  ( ( TEndo `  K
) `  W )
istendod.1  |-  ( ph  ->  ( K  e.  V  /\  W  e.  H
) )
istendod.2  |-  ( ph  ->  S : T --> T )
istendod.3  |-  ( (
ph  /\  f  e.  T  /\  g  e.  T
)  ->  ( S `  ( f  o.  g
) )  =  ( ( S `  f
)  o.  ( S `
 g ) ) )
istendod.4  |-  ( (
ph  /\  f  e.  T )  ->  ( R `  ( S `  f ) )  .<_  ( R `  f ) )
Assertion
Ref Expression
istendod  |-  ( ph  ->  S  e.  E )
Distinct variable groups:    f, g, K    T, f, g    f, W, g    S, f, g    .<_ , f    R, f    ph, f,
g
Allowed substitution hints:    R( g)    E( f, g)    H( f, g)    .<_ ( g)    V( f, g)

Proof of Theorem istendod
StepHypRef Expression
1 istendod.2 . 2  |-  ( ph  ->  S : T --> T )
2 istendod.3 . . . 4  |-  ( (
ph  /\  f  e.  T  /\  g  e.  T
)  ->  ( S `  ( f  o.  g
) )  =  ( ( S `  f
)  o.  ( S `
 g ) ) )
323expb 1155 . . 3  |-  ( (
ph  /\  ( f  e.  T  /\  g  e.  T ) )  -> 
( S `  (
f  o.  g ) )  =  ( ( S `  f )  o.  ( S `  g ) ) )
43ralrimivva 2799 . 2  |-  ( ph  ->  A. f  e.  T  A. g  e.  T  ( S `  ( f  o.  g ) )  =  ( ( S `
 f )  o.  ( S `  g
) ) )
5 istendod.4 . . 3  |-  ( (
ph  /\  f  e.  T )  ->  ( R `  ( S `  f ) )  .<_  ( R `  f ) )
65ralrimiva 2790 . 2  |-  ( ph  ->  A. f  e.  T  ( R `  ( S `
 f ) ) 
.<_  ( R `  f
) )
7 istendod.1 . . 3  |-  ( ph  ->  ( K  e.  V  /\  W  e.  H
) )
8 tendoset.l . . . 4  |-  .<_  =  ( le `  K )
9 tendoset.h . . . 4  |-  H  =  ( LHyp `  K
)
10 tendoset.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
11 tendoset.r . . . 4  |-  R  =  ( ( trL `  K
) `  W )
12 tendoset.e . . . 4  |-  E  =  ( ( TEndo `  K
) `  W )
138, 9, 10, 11, 12istendo 31558 . . 3  |-  ( ( K  e.  V  /\  W  e.  H )  ->  ( S  e.  E  <->  ( S : T --> T  /\  A. f  e.  T  A. g  e.  T  ( S `  ( f  o.  g ) )  =  ( ( S `  f )  o.  ( S `  g )
)  /\  A. f  e.  T  ( R `  ( S `  f
) )  .<_  ( R `
 f ) ) ) )
147, 13syl 16 . 2  |-  ( ph  ->  ( S  e.  E  <->  ( S : T --> T  /\  A. f  e.  T  A. g  e.  T  ( S `  ( f  o.  g ) )  =  ( ( S `  f )  o.  ( S `  g )
)  /\  A. f  e.  T  ( R `  ( S `  f
) )  .<_  ( R `
 f ) ) ) )
151, 4, 6, 14mpbir3and 1138 1  |-  ( ph  ->  S  e.  E )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   A.wral 2706   class class class wbr 4213    o. ccom 4883   -->wf 5451   ` cfv 5455   lecple 13537   LHypclh 30782   LTrncltrn 30899   trLctrl 30956   TEndoctendo 31550
This theorem is referenced by:  tendoidcl  31567  tendococl  31570  tendoplcl  31579  tendo0cl  31588  tendoicl  31594  cdlemk56  31769
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-map 7021  df-tendo 31553
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