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Theorem tendoidcl 31664
Description: The identity is a trace-preserving endomorphism. (Contributed by NM, 30-Jul-2013.)
Hypotheses
Ref Expression
tendof.h  |-  H  =  ( LHyp `  K
)
tendof.t  |-  T  =  ( ( LTrn `  K
) `  W )
tendof.e  |-  E  =  ( ( TEndo `  K
) `  W )
Assertion
Ref Expression
tendoidcl  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  _I  |`  T )  e.  E )

Proof of Theorem tendoidcl
Dummy variables  f 
g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2442 . 2  |-  ( le
`  K )  =  ( le `  K
)
2 tendof.h . 2  |-  H  =  ( LHyp `  K
)
3 tendof.t . 2  |-  T  =  ( ( LTrn `  K
) `  W )
4 eqid 2442 . 2  |-  ( ( trL `  K ) `
 W )  =  ( ( trL `  K
) `  W )
5 tendof.e . 2  |-  E  =  ( ( TEndo `  K
) `  W )
6 id 21 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( K  e.  HL  /\  W  e.  H ) )
7 f1oi 5742 . . 3  |-  (  _I  |`  T ) : T -1-1-onto-> T
8 f1of 5703 . . 3  |-  ( (  _I  |`  T ) : T -1-1-onto-> T  ->  (  _I  |`  T ) : T --> T )
97, 8mp1i 12 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  _I  |`  T ) : T --> T )
102, 3ltrnco 31614 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  T  /\  g  e.  T
)  ->  ( f  o.  g )  e.  T
)
11 fvresi 5953 . . . 4  |-  ( ( f  o.  g )  e.  T  ->  (
(  _I  |`  T ) `
 ( f  o.  g ) )  =  ( f  o.  g
) )
1210, 11syl 16 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  T  /\  g  e.  T
)  ->  ( (  _I  |`  T ) `  ( f  o.  g
) )  =  ( f  o.  g ) )
13 fvresi 5953 . . . . 5  |-  ( f  e.  T  ->  (
(  _I  |`  T ) `
 f )  =  f )
14133ad2ant2 980 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  T  /\  g  e.  T
)  ->  ( (  _I  |`  T ) `  f )  =  f )
15 fvresi 5953 . . . . 5  |-  ( g  e.  T  ->  (
(  _I  |`  T ) `
 g )  =  g )
16153ad2ant3 981 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  T  /\  g  e.  T
)  ->  ( (  _I  |`  T ) `  g )  =  g )
1714, 16coeq12d 5066 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  T  /\  g  e.  T
)  ->  ( (
(  _I  |`  T ) `
 f )  o.  ( (  _I  |`  T ) `
 g ) )  =  ( f  o.  g ) )
1812, 17eqtr4d 2477 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  T  /\  g  e.  T
)  ->  ( (  _I  |`  T ) `  ( f  o.  g
) )  =  ( ( (  _I  |`  T ) `
 f )  o.  ( (  _I  |`  T ) `
 g ) ) )
1913adantl 454 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  T
)  ->  ( (  _I  |`  T ) `  f )  =  f )
2019fveq2d 5761 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  T
)  ->  ( (
( trL `  K
) `  W ) `  ( (  _I  |`  T ) `
 f ) )  =  ( ( ( trL `  K ) `
 W ) `  f ) )
21 hllat 30259 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
2221ad2antrr 708 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  T
)  ->  K  e.  Lat )
23 eqid 2442 . . . . 5  |-  ( Base `  K )  =  (
Base `  K )
2423, 2, 3, 4trlcl 31059 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  T
)  ->  ( (
( trL `  K
) `  W ) `  f )  e.  (
Base `  K )
)
2523, 1latref 14513 . . . 4  |-  ( ( K  e.  Lat  /\  ( ( ( trL `  K ) `  W
) `  f )  e.  ( Base `  K
) )  ->  (
( ( trL `  K
) `  W ) `  f ) ( le
`  K ) ( ( ( trL `  K
) `  W ) `  f ) )
2622, 24, 25syl2anc 644 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  T
)  ->  ( (
( trL `  K
) `  W ) `  f ) ( le
`  K ) ( ( ( trL `  K
) `  W ) `  f ) )
2720, 26eqbrtrd 4257 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  T
)  ->  ( (
( trL `  K
) `  W ) `  ( (  _I  |`  T ) `
 f ) ) ( le `  K
) ( ( ( trL `  K ) `
 W ) `  f ) )
281, 2, 3, 4, 5, 6, 9, 18, 27istendod 31657 1  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  _I  |`  T )  e.  E )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1727   class class class wbr 4237    _I cid 4522    |` cres 4909    o. ccom 4911   -->wf 5479   -1-1-onto->wf1o 5482   ` cfv 5483   Basecbs 13500   lecple 13567   Latclat 14505   HLchlt 30246   LHypclh 30879   LTrncltrn 30996   trLctrl 31053   TEndoctendo 31647
This theorem is referenced by:  cdleml8  31878  erng1lem  31882  erngdvlem3  31885  erng1r  31890  erngdvlem3-rN  31893  erngdvlem4-rN  31894  dvalveclem  31921  dvhlveclem  32004  dvheveccl  32008  dvhopN  32012  diclspsn  32090  cdlemn4  32094  cdlemn4a  32095  cdlemn11a  32103  dihord6apre  32152  dihatlat  32230  dihatexv  32234
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2716  df-rex 2717  df-reu 2718  df-rmo 2719  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-iun 4119  df-iin 4120  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-1st 6378  df-2nd 6379  df-undef 6572  df-riota 6578  df-map 7049  df-poset 14434  df-plt 14446  df-lub 14462  df-glb 14463  df-join 14464  df-meet 14465  df-p0 14499  df-p1 14500  df-lat 14506  df-clat 14568  df-oposet 30072  df-ol 30074  df-oml 30075  df-covers 30162  df-ats 30163  df-atl 30194  df-cvlat 30218  df-hlat 30247  df-llines 30393  df-lplanes 30394  df-lvols 30395  df-lines 30396  df-psubsp 30398  df-pmap 30399  df-padd 30691  df-lhyp 30883  df-laut 30884  df-ldil 30999  df-ltrn 31000  df-trl 31054  df-tendo 31650
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