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Theorem tendoidcl 31255
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 2408 . 2  |-  ( le
`  K )  =  ( le `  K
)
2 tendof.h . 2  |-  H  =  ( LHyp `  K
)
3 tendof.t . 2  |-  T  =  ( ( LTrn `  K
) `  W )
4 eqid 2408 . 2  |-  ( ( trL `  K ) `
 W )  =  ( ( trL `  K
) `  W )
5 tendof.e . 2  |-  E  =  ( ( TEndo `  K
) `  W )
6 id 20 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( K  e.  HL  /\  W  e.  H ) )
7 f1oi 5676 . . 3  |-  (  _I  |`  T ) : T -1-1-onto-> T
8 f1of 5637 . . 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 31205 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  T  /\  g  e.  T
)  ->  ( f  o.  g )  e.  T
)
11 fvresi 5887 . . . 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 5887 . . . . 5  |-  ( f  e.  T  ->  (
(  _I  |`  T ) `
 f )  =  f )
14133ad2ant2 979 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  T  /\  g  e.  T
)  ->  ( (  _I  |`  T ) `  f )  =  f )
15 fvresi 5887 . . . . 5  |-  ( g  e.  T  ->  (
(  _I  |`  T ) `
 g )  =  g )
16153ad2ant3 980 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  T  /\  g  e.  T
)  ->  ( (  _I  |`  T ) `  g )  =  g )
1714, 16coeq12d 5000 . . 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 2443 . 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 453 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  T
)  ->  ( (  _I  |`  T ) `  f )  =  f )
2019fveq2d 5695 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  T
)  ->  ( (
( trL `  K
) `  W ) `  ( (  _I  |`  T ) `
 f ) )  =  ( ( ( trL `  K ) `
 W ) `  f ) )
21 hllat 29850 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
2221ad2antrr 707 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  T
)  ->  K  e.  Lat )
23 eqid 2408 . . . . 5  |-  ( Base `  K )  =  (
Base `  K )
2423, 2, 3, 4trlcl 30650 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  T
)  ->  ( (
( trL `  K
) `  W ) `  f )  e.  (
Base `  K )
)
2523, 1latref 14441 . . . 4  |-  ( ( K  e.  Lat  /\  ( ( ( trL `  K ) `  W
) `  f )  e.  ( Base `  K
) )  ->  (
( ( trL `  K
) `  W ) `  f ) ( le
`  K ) ( ( ( trL `  K
) `  W ) `  f ) )
2622, 24, 25syl2anc 643 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  f  e.  T
)  ->  ( (
( trL `  K
) `  W ) `  f ) ( le
`  K ) ( ( ( trL `  K
) `  W ) `  f ) )
2720, 26eqbrtrd 4196 . 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 31248 1  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  _I  |`  T )  e.  E )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   class class class wbr 4176    _I cid 4457    |` cres 4843    o. ccom 4845   -->wf 5413   -1-1-onto->wf1o 5416   ` cfv 5417   Basecbs 13428   lecple 13495   Latclat 14433   HLchlt 29837   LHypclh 30470   LTrncltrn 30587   trLctrl 30644   TEndoctendo 31238
This theorem is referenced by:  cdleml8  31469  erng1lem  31473  erngdvlem3  31476  erng1r  31481  erngdvlem3-rN  31484  erngdvlem4-rN  31485  dvalveclem  31512  dvhlveclem  31595  dvheveccl  31599  dvhopN  31603  diclspsn  31681  cdlemn4  31685  cdlemn4a  31686  cdlemn11a  31694  dihord6apre  31743  dihatlat  31821  dihatexv  31825
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-iun 4059  df-iin 4060  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1st 6312  df-2nd 6313  df-undef 6506  df-riota 6512  df-map 6983  df-poset 14362  df-plt 14374  df-lub 14390  df-glb 14391  df-join 14392  df-meet 14393  df-p0 14427  df-p1 14428  df-lat 14434  df-clat 14496  df-oposet 29663  df-ol 29665  df-oml 29666  df-covers 29753  df-ats 29754  df-atl 29785  df-cvlat 29809  df-hlat 29838  df-llines 29984  df-lplanes 29985  df-lvols 29986  df-lines 29987  df-psubsp 29989  df-pmap 29990  df-padd 30282  df-lhyp 30474  df-laut 30475  df-ldil 30590  df-ltrn 30591  df-trl 30645  df-tendo 31241
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