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Theorem tendoid 35925
Description: The identity value of a trace-preserving endomorphism. (Contributed by NM, 21-Jun-2013.)
Hypotheses
Ref Expression
tendoid.b  |-  B  =  ( Base `  K
)
tendoid.h  |-  H  =  ( LHyp `  K
)
tendoid.e  |-  E  =  ( ( TEndo `  K
) `  W )
Assertion
Ref Expression
tendoid  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E
)  ->  ( S `  (  _I  |`  B ) )  =  (  _I  |`  B ) )

Proof of Theorem tendoid
StepHypRef Expression
1 tendoid.b . . . . . . 7  |-  B  =  ( Base `  K
)
2 tendoid.h . . . . . . 7  |-  H  =  ( LHyp `  K
)
3 eqid 2467 . . . . . . 7  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
41, 2, 3idltrn 35302 . . . . . 6  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  _I  |`  B )  e.  ( ( LTrn `  K ) `  W
) )
54adantr 465 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E
)  ->  (  _I  |`  B )  e.  ( ( LTrn `  K
) `  W )
)
6 eqid 2467 . . . . . 6  |-  ( le
`  K )  =  ( le `  K
)
7 eqid 2467 . . . . . 6  |-  ( ( trL `  K ) `
 W )  =  ( ( trL `  K
) `  W )
8 tendoid.e . . . . . 6  |-  E  =  ( ( TEndo `  K
) `  W )
96, 2, 3, 7, 8tendotp 35913 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E  /\  (  _I  |`  B )  e.  ( ( LTrn `  K ) `  W
) )  ->  (
( ( trL `  K
) `  W ) `  ( S `  (  _I  |`  B ) ) ) ( le `  K ) ( ( ( trL `  K
) `  W ) `  (  _I  |`  B ) ) )
105, 9mpd3an3 1325 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E
)  ->  ( (
( trL `  K
) `  W ) `  ( S `  (  _I  |`  B ) ) ) ( le `  K ) ( ( ( trL `  K
) `  W ) `  (  _I  |`  B ) ) )
11 eqid 2467 . . . . . 6  |-  ( 0.
`  K )  =  ( 0. `  K
)
121, 11, 2, 7trlid0 35328 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( ( trL `  K ) `  W
) `  (  _I  |`  B ) )  =  ( 0. `  K
) )
1312adantr 465 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E
)  ->  ( (
( trL `  K
) `  W ) `  (  _I  |`  B ) )  =  ( 0.
`  K ) )
1410, 13breqtrd 4477 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E
)  ->  ( (
( trL `  K
) `  W ) `  ( S `  (  _I  |`  B ) ) ) ( le `  K ) ( 0.
`  K ) )
15 hlop 34515 . . . . 5  |-  ( K  e.  HL  ->  K  e.  OP )
1615ad2antrr 725 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E
)  ->  K  e.  OP )
172, 3, 8tendocl 35919 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E  /\  (  _I  |`  B )  e.  ( ( LTrn `  K ) `  W
) )  ->  ( S `  (  _I  |`  B ) )  e.  ( ( LTrn `  K
) `  W )
)
185, 17mpd3an3 1325 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E
)  ->  ( S `  (  _I  |`  B ) )  e.  ( (
LTrn `  K ) `  W ) )
191, 2, 3, 7trlcl 35316 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S `  (  _I  |`  B ) )  e.  ( (
LTrn `  K ) `  W ) )  -> 
( ( ( trL `  K ) `  W
) `  ( S `  (  _I  |`  B ) ) )  e.  B
)
2018, 19syldan 470 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E
)  ->  ( (
( trL `  K
) `  W ) `  ( S `  (  _I  |`  B ) ) )  e.  B )
211, 6, 11ople0 34340 . . . 4  |-  ( ( K  e.  OP  /\  ( ( ( trL `  K ) `  W
) `  ( S `  (  _I  |`  B ) ) )  e.  B
)  ->  ( (
( ( trL `  K
) `  W ) `  ( S `  (  _I  |`  B ) ) ) ( le `  K ) ( 0.
`  K )  <->  ( (
( trL `  K
) `  W ) `  ( S `  (  _I  |`  B ) ) )  =  ( 0.
`  K ) ) )
2216, 20, 21syl2anc 661 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E
)  ->  ( (
( ( trL `  K
) `  W ) `  ( S `  (  _I  |`  B ) ) ) ( le `  K ) ( 0.
`  K )  <->  ( (
( trL `  K
) `  W ) `  ( S `  (  _I  |`  B ) ) )  =  ( 0.
`  K ) ) )
2314, 22mpbid 210 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E
)  ->  ( (
( trL `  K
) `  W ) `  ( S `  (  _I  |`  B ) ) )  =  ( 0.
`  K ) )
241, 11, 2, 3, 7trlid0b 35330 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S `  (  _I  |`  B ) )  e.  ( (
LTrn `  K ) `  W ) )  -> 
( ( S `  (  _I  |`  B ) )  =  (  _I  |`  B )  <->  ( (
( trL `  K
) `  W ) `  ( S `  (  _I  |`  B ) ) )  =  ( 0.
`  K ) ) )
2518, 24syldan 470 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E
)  ->  ( ( S `  (  _I  |`  B ) )  =  (  _I  |`  B )  <-> 
( ( ( trL `  K ) `  W
) `  ( S `  (  _I  |`  B ) ) )  =  ( 0. `  K ) ) )
2623, 25mpbird 232 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E
)  ->  ( S `  (  _I  |`  B ) )  =  (  _I  |`  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   class class class wbr 4453    _I cid 4796    |` cres 5007   ` cfv 5594   Basecbs 14507   lecple 14579   0.cp0 15541   OPcops 34325   HLchlt 34503   LHypclh 35136   LTrncltrn 35253   trLctrl 35310   TEndoctendo 35904
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-map 7434  df-poset 15450  df-plt 15462  df-lub 15478  df-glb 15479  df-join 15480  df-meet 15481  df-p0 15543  df-p1 15544  df-lat 15550  df-clat 15612  df-oposet 34329  df-ol 34331  df-oml 34332  df-covers 34419  df-ats 34420  df-atl 34451  df-cvlat 34475  df-hlat 34504  df-lhyp 35140  df-laut 35141  df-ldil 35256  df-ltrn 35257  df-trl 35311  df-tendo 35907
This theorem is referenced by:  tendoeq2  35926  tendo0mulr  35979  tendotr  35982  tendocnv  36174  dvhopN  36269  dihpN  36489
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