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Theorem tendoidcl 30958
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 2283 . 2  |-  ( le
`  K )  =  ( le `  K
)
2 tendof.h . 2  |-  H  =  ( LHyp `  K
)
3 tendof.t . 2  |-  T  =  ( ( LTrn `  K
) `  W )
4 eqid 2283 . 2  |-  ( ( trL `  K ) `
 W )  =  ( ( trL `  K
) `  W )
5 tendof.e . 2  |-  E  =  ( ( TEndo `  K
) `  W )
6 id 19 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( K  e.  HL  /\  W  e.  H ) )
7 f1oi 5511 . . 3  |-  (  _I  |`  T ) : T -1-1-onto-> T
8 f1of 5472 . . 3  |-  ( (  _I  |`  T ) : T -1-1-onto-> T  ->  (  _I  |`  T ) : T --> T )
97, 8mp1i 11 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  _I  |`  T ) : T --> T )
102, 3ltrnco 30908 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  T  /\  g  e.  T
)  ->  ( f  o.  g )  e.  T
)
11 fvresi 5711 . . . 4  |-  ( ( f  o.  g )  e.  T  ->  (
(  _I  |`  T ) `
 ( f  o.  g ) )  =  ( f  o.  g
) )
1210, 11syl 15 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  T  /\  g  e.  T
)  ->  ( (  _I  |`  T ) `  ( f  o.  g
) )  =  ( f  o.  g ) )
13 fvresi 5711 . . . . 5  |-  ( f  e.  T  ->  (
(  _I  |`  T ) `
 f )  =  f )
14133ad2ant2 977 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  T  /\  g  e.  T
)  ->  ( (  _I  |`  T ) `  f )  =  f )
15 fvresi 5711 . . . . 5  |-  ( g  e.  T  ->  (
(  _I  |`  T ) `
 g )  =  g )
16153ad2ant3 978 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  T  /\  g  e.  T
)  ->  ( (  _I  |`  T ) `  g )  =  g )
1714, 16coeq12d 4848 . . 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 2318 . 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 452 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  T
)  ->  ( (  _I  |`  T ) `  f )  =  f )
2019fveq2d 5529 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  T
)  ->  ( (
( trL `  K
) `  W ) `  ( (  _I  |`  T ) `
 f ) )  =  ( ( ( trL `  K ) `
 W ) `  f ) )
21 hllat 29553 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
2221ad2antrr 706 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  T
)  ->  K  e.  Lat )
23 eqid 2283 . . . . 5  |-  ( Base `  K )  =  (
Base `  K )
2423, 2, 3, 4trlcl 30353 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  T
)  ->  ( (
( trL `  K
) `  W ) `  f )  e.  (
Base `  K )
)
2523, 1latref 14159 . . . 4  |-  ( ( K  e.  Lat  /\  ( ( ( trL `  K ) `  W
) `  f )  e.  ( Base `  K
) )  ->  (
( ( trL `  K
) `  W ) `  f ) ( le
`  K ) ( ( ( trL `  K
) `  W ) `  f ) )
2622, 24, 25syl2anc 642 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  T
)  ->  ( (
( trL `  K
) `  W ) `  f ) ( le
`  K ) ( ( ( trL `  K
) `  W ) `  f ) )
2720, 26eqbrtrd 4043 . 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 30951 1  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  _I  |`  T )  e.  E )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   class class class wbr 4023    _I cid 4304    |` cres 4691    o. ccom 4693   -->wf 5251   -1-1-onto->wf1o 5254   ` cfv 5255   Basecbs 13148   lecple 13215   Latclat 14151   HLchlt 29540   LHypclh 30173   LTrncltrn 30290   trLctrl 30347   TEndoctendo 30941
This theorem is referenced by:  cdleml8  31172  erng1lem  31176  erngdvlem3  31179  erng1r  31184  erngdvlem3-rN  31187  erngdvlem4-rN  31188  dvalveclem  31215  dvhlveclem  31298  dvheveccl  31302  dvhopN  31306  diclspsn  31384  cdlemn4  31388  cdlemn4a  31389  cdlemn11a  31397  dihord6apre  31446  dihatlat  31524  dihatexv  31528
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-3or 935  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-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  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-iin 3908  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-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-map 6774  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-p1 14146  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-llines 29687  df-lplanes 29688  df-lvols 29689  df-lines 29690  df-psubsp 29692  df-pmap 29693  df-padd 29985  df-lhyp 30177  df-laut 30178  df-ldil 30293  df-ltrn 30294  df-trl 30348  df-tendo 30944
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