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Theorem tendoidcl 34716
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 2451 . 2  |-  ( le
`  K )  =  ( le `  K
)
2 tendof.h . 2  |-  H  =  ( LHyp `  K
)
3 tendof.t . 2  |-  T  =  ( ( LTrn `  K
) `  W )
4 eqid 2451 . 2  |-  ( ( trL `  K ) `
 W )  =  ( ( trL `  K
) `  W )
5 tendof.e . 2  |-  E  =  ( ( TEndo `  K
) `  W )
6 id 22 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( K  e.  HL  /\  W  e.  H ) )
7 f1oi 5771 . . 3  |-  (  _I  |`  T ) : T -1-1-onto-> T
8 f1of 5736 . . 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 34666 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  T  /\  g  e.  T
)  ->  ( f  o.  g )  e.  T
)
11 fvresi 6000 . . . 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 6000 . . . . 5  |-  ( f  e.  T  ->  (
(  _I  |`  T ) `
 f )  =  f )
14133ad2ant2 1010 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  T  /\  g  e.  T
)  ->  ( (  _I  |`  T ) `  f )  =  f )
15 fvresi 6000 . . . . 5  |-  ( g  e.  T  ->  (
(  _I  |`  T ) `
 g )  =  g )
16153ad2ant3 1011 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  T  /\  g  e.  T
)  ->  ( (  _I  |`  T ) `  g )  =  g )
1714, 16coeq12d 5099 . . 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 2494 . 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 466 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  T
)  ->  ( (  _I  |`  T ) `  f )  =  f )
2019fveq2d 5790 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  T
)  ->  ( (
( trL `  K
) `  W ) `  ( (  _I  |`  T ) `
 f ) )  =  ( ( ( trL `  K ) `
 W ) `  f ) )
21 hllat 33311 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
2221ad2antrr 725 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  T
)  ->  K  e.  Lat )
23 eqid 2451 . . . . 5  |-  ( Base `  K )  =  (
Base `  K )
2423, 2, 3, 4trlcl 34111 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  T
)  ->  ( (
( trL `  K
) `  W ) `  f )  e.  (
Base `  K )
)
2523, 1latref 15322 . . . 4  |-  ( ( K  e.  Lat  /\  ( ( ( trL `  K ) `  W
) `  f )  e.  ( Base `  K
) )  ->  (
( ( trL `  K
) `  W ) `  f ) ( le
`  K ) ( ( ( trL `  K
) `  W ) `  f ) )
2622, 24, 25syl2anc 661 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  T
)  ->  ( (
( trL `  K
) `  W ) `  f ) ( le
`  K ) ( ( ( trL `  K
) `  W ) `  f ) )
2720, 26eqbrtrd 4407 . 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 34709 1  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  _I  |`  T )  e.  E )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   class class class wbr 4387    _I cid 4726    |` cres 4937    o. ccom 4939   -->wf 5509   -1-1-onto->wf1o 5512   ` cfv 5513   Basecbs 14273   lecple 14344   Latclat 15314   HLchlt 33298   LHypclh 33931   LTrncltrn 34048   trLctrl 34105   TEndoctendo 34699
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 4498  ax-sep 4508  ax-nul 4516  ax-pow 4565  ax-pr 4626  ax-un 6469  ax-riotaBAD 32907
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  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 2599  df-ne 2644  df-nel 2645  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3067  df-sbc 3282  df-csb 3384  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-nul 3733  df-if 3887  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4187  df-iun 4268  df-iin 4269  df-br 4388  df-opab 4446  df-mpt 4447  df-id 4731  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-res 4947  df-ima 4948  df-iota 5476  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-riota 6148  df-ov 6190  df-oprab 6191  df-mpt2 6192  df-1st 6674  df-2nd 6675  df-undef 6889  df-map 7313  df-poset 15215  df-plt 15227  df-lub 15243  df-glb 15244  df-join 15245  df-meet 15246  df-p0 15308  df-p1 15309  df-lat 15315  df-clat 15377  df-oposet 33124  df-ol 33126  df-oml 33127  df-covers 33214  df-ats 33215  df-atl 33246  df-cvlat 33270  df-hlat 33299  df-llines 33445  df-lplanes 33446  df-lvols 33447  df-lines 33448  df-psubsp 33450  df-pmap 33451  df-padd 33743  df-lhyp 33935  df-laut 33936  df-ldil 34051  df-ltrn 34052  df-trl 34106  df-tendo 34702
This theorem is referenced by:  cdleml8  34930  erng1lem  34934  erngdvlem3  34937  erng1r  34942  erngdvlem3-rN  34945  erngdvlem4-rN  34946  dvalveclem  34973  dvhlveclem  35056  dvheveccl  35060  dvhopN  35064  diclspsn  35142  cdlemn4  35146  cdlemn4a  35147  cdlemn11a  35155  dihord6apre  35204  dihatlat  35282  dihatexv  35286
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