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Theorem tendoidcl 31029
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 2366 . 2  |-  ( le
`  K )  =  ( le `  K
)
2 tendof.h . 2  |-  H  =  ( LHyp `  K
)
3 tendof.t . 2  |-  T  =  ( ( LTrn `  K
) `  W )
4 eqid 2366 . 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 5617 . . 3  |-  (  _I  |`  T ) : T -1-1-onto-> T
8 f1of 5578 . . 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 30979 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  T  /\  g  e.  T
)  ->  ( f  o.  g )  e.  T
)
11 fvresi 5824 . . . 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 5824 . . . . 5  |-  ( f  e.  T  ->  (
(  _I  |`  T ) `
 f )  =  f )
14133ad2ant2 978 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  T  /\  g  e.  T
)  ->  ( (  _I  |`  T ) `  f )  =  f )
15 fvresi 5824 . . . . 5  |-  ( g  e.  T  ->  (
(  _I  |`  T ) `
 g )  =  g )
16153ad2ant3 979 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  T  /\  g  e.  T
)  ->  ( (  _I  |`  T ) `  g )  =  g )
1714, 16coeq12d 4951 . . 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 2401 . 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 5636 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  T
)  ->  ( (
( trL `  K
) `  W ) `  ( (  _I  |`  T ) `
 f ) )  =  ( ( ( trL `  K ) `
 W ) `  f ) )
21 hllat 29624 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
2221ad2antrr 706 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  T
)  ->  K  e.  Lat )
23 eqid 2366 . . . . 5  |-  ( Base `  K )  =  (
Base `  K )
2423, 2, 3, 4trlcl 30424 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  T
)  ->  ( (
( trL `  K
) `  W ) `  f )  e.  (
Base `  K )
)
2523, 1latref 14369 . . . 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 4145 . 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 31022 1  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  _I  |`  T )  e.  E )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 935    = wceq 1647    e. wcel 1715   class class class wbr 4125    _I cid 4407    |` cres 4794    o. ccom 4796   -->wf 5354   -1-1-onto->wf1o 5357   ` cfv 5358   Basecbs 13356   lecple 13423   Latclat 14361   HLchlt 29611   LHypclh 30244   LTrncltrn 30361   trLctrl 30418   TEndoctendo 31012
This theorem is referenced by:  cdleml8  31243  erng1lem  31247  erngdvlem3  31250  erng1r  31255  erngdvlem3-rN  31258  erngdvlem4-rN  31259  dvalveclem  31286  dvhlveclem  31369  dvheveccl  31373  dvhopN  31377  diclspsn  31455  cdlemn4  31459  cdlemn4a  31460  cdlemn11a  31468  dihord6apre  31517  dihatlat  31595  dihatexv  31599
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-iin 4010  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-undef 6440  df-riota 6446  df-map 6917  df-poset 14290  df-plt 14302  df-lub 14318  df-glb 14319  df-join 14320  df-meet 14321  df-p0 14355  df-p1 14356  df-lat 14362  df-clat 14424  df-oposet 29437  df-ol 29439  df-oml 29440  df-covers 29527  df-ats 29528  df-atl 29559  df-cvlat 29583  df-hlat 29612  df-llines 29758  df-lplanes 29759  df-lvols 29760  df-lines 29761  df-psubsp 29763  df-pmap 29764  df-padd 30056  df-lhyp 30248  df-laut 30249  df-ldil 30364  df-ltrn 30365  df-trl 30419  df-tendo 31015
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