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Theorem tendoid 30887
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 2387 . . . . . . 7  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
41, 2, 3idltrn 30264 . . . . . 6  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  _I  |`  B )  e.  ( ( LTrn `  K ) `  W
) )
54adantr 452 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E
)  ->  (  _I  |`  B )  e.  ( ( LTrn `  K
) `  W )
)
6 eqid 2387 . . . . . 6  |-  ( le
`  K )  =  ( le `  K
)
7 eqid 2387 . . . . . 6  |-  ( ( trL `  K ) `
 W )  =  ( ( trL `  K
) `  W )
8 tendoid.e . . . . . 6  |-  E  =  ( ( TEndo `  K
) `  W )
96, 2, 3, 7, 8tendotp 30875 . . . . 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 1280 . . . 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 2387 . . . . . 6  |-  ( 0.
`  K )  =  ( 0. `  K
)
121, 11, 2, 7trlid0 30290 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( ( trL `  K ) `  W
) `  (  _I  |`  B ) )  =  ( 0. `  K
) )
1312adantr 452 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E
)  ->  ( (
( trL `  K
) `  W ) `  (  _I  |`  B ) )  =  ( 0.
`  K ) )
1410, 13breqtrd 4177 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E
)  ->  ( (
( trL `  K
) `  W ) `  ( S `  (  _I  |`  B ) ) ) ( le `  K ) ( 0.
`  K ) )
15 hlop 29477 . . . . 5  |-  ( K  e.  HL  ->  K  e.  OP )
1615ad2antrr 707 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E
)  ->  K  e.  OP )
172, 3, 8tendocl 30881 . . . . . 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 1280 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E
)  ->  ( S `  (  _I  |`  B ) )  e.  ( (
LTrn `  K ) `  W ) )
191, 2, 3, 7trlcl 30278 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S `  (  _I  |`  B ) )  e.  ( (
LTrn `  K ) `  W ) )  -> 
( ( ( trL `  K ) `  W
) `  ( S `  (  _I  |`  B ) ) )  e.  B
)
2018, 19syldan 457 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E
)  ->  ( (
( trL `  K
) `  W ) `  ( S `  (  _I  |`  B ) ) )  e.  B )
211, 6, 11ople0 29302 . . . 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 643 . . 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 202 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E
)  ->  ( (
( trL `  K
) `  W ) `  ( S `  (  _I  |`  B ) ) )  =  ( 0.
`  K ) )
241, 11, 2, 3, 7trlid0b 30292 . . 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 457 . 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 224 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E
)  ->  ( S `  (  _I  |`  B ) )  =  (  _I  |`  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   class class class wbr 4153    _I cid 4434    |` cres 4820   ` cfv 5394   Basecbs 13396   lecple 13463   0.cp0 14393   OPcops 29287   HLchlt 29465   LHypclh 30098   LTrncltrn 30215   trLctrl 30272   TEndoctendo 30866
This theorem is referenced by:  tendoeq2  30888  tendo0mulr  30941  tendotr  30944  tendocnv  31136  dvhopN  31231  dihpN  31451
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-undef 6479  df-riota 6485  df-map 6956  df-poset 14330  df-plt 14342  df-lub 14358  df-glb 14359  df-join 14360  df-meet 14361  df-p0 14395  df-p1 14396  df-lat 14402  df-clat 14464  df-oposet 29291  df-ol 29293  df-oml 29294  df-covers 29381  df-ats 29382  df-atl 29413  df-cvlat 29437  df-hlat 29466  df-lhyp 30102  df-laut 30103  df-ldil 30218  df-ltrn 30219  df-trl 30273  df-tendo 30869
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