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Theorem istendod 30951
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 1152 . . 3  |-  ( (
ph  /\  ( f  e.  T  /\  g  e.  T ) )  -> 
( S `  (
f  o.  g ) )  =  ( ( S `  f )  o.  ( S `  g ) ) )
43ralrimivva 2635 . 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 2626 . 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 30949 . . 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 15 . 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 1135 1  |-  ( ph  ->  S  e.  E )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   class class class wbr 4023    o. ccom 4693   -->wf 5251   ` cfv 5255   lecple 13215   LHypclh 30173   LTrncltrn 30290   trLctrl 30347   TEndoctendo 30941
This theorem is referenced by:  tendoidcl  30958  tendococl  30961  tendoplcl  30970  tendo0cl  30979  tendoicl  30985  cdlemk56  31160
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-map 6774  df-tendo 30944
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