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Theorem tendoid 31584
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 2296 . . . . . . 7  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
41, 2, 3idltrn 30961 . . . . . 6  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  _I  |`  B )  e.  ( ( LTrn `  K ) `  W
) )
54adantr 451 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E
)  ->  (  _I  |`  B )  e.  ( ( LTrn `  K
) `  W )
)
6 eqid 2296 . . . . . 6  |-  ( le
`  K )  =  ( le `  K
)
7 eqid 2296 . . . . . 6  |-  ( ( trL `  K ) `
 W )  =  ( ( trL `  K
) `  W )
8 tendoid.e . . . . . 6  |-  E  =  ( ( TEndo `  K
) `  W )
96, 2, 3, 7, 8tendotp 31572 . . . . 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 1278 . . . 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 2296 . . . . . 6  |-  ( 0.
`  K )  =  ( 0. `  K
)
121, 11, 2, 7trlid0 30987 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( ( trL `  K ) `  W
) `  (  _I  |`  B ) )  =  ( 0. `  K
) )
1312adantr 451 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E
)  ->  ( (
( trL `  K
) `  W ) `  (  _I  |`  B ) )  =  ( 0.
`  K ) )
1410, 13breqtrd 4063 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E
)  ->  ( (
( trL `  K
) `  W ) `  ( S `  (  _I  |`  B ) ) ) ( le `  K ) ( 0.
`  K ) )
15 hlop 30174 . . . . 5  |-  ( K  e.  HL  ->  K  e.  OP )
1615ad2antrr 706 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E
)  ->  K  e.  OP )
172, 3, 8tendocl 31578 . . . . . 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 1278 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E
)  ->  ( S `  (  _I  |`  B ) )  e.  ( (
LTrn `  K ) `  W ) )
191, 2, 3, 7trlcl 30975 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S `  (  _I  |`  B ) )  e.  ( (
LTrn `  K ) `  W ) )  -> 
( ( ( trL `  K ) `  W
) `  ( S `  (  _I  |`  B ) ) )  e.  B
)
2018, 19syldan 456 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E
)  ->  ( (
( trL `  K
) `  W ) `  ( S `  (  _I  |`  B ) ) )  e.  B )
211, 6, 11ople0 29999 . . . 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 642 . . 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 201 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E
)  ->  ( (
( trL `  K
) `  W ) `  ( S `  (  _I  |`  B ) ) )  =  ( 0.
`  K ) )
241, 11, 2, 3, 7trlid0b 30989 . . 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 456 . 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 223 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 176    /\ wa 358    = wceq 1632    e. wcel 1696   class class class wbr 4039    _I cid 4320    |` cres 4707   ` cfv 5271   Basecbs 13164   lecple 13231   0.cp0 14159   OPcops 29984   HLchlt 30162   LHypclh 30795   LTrncltrn 30912   trLctrl 30969   TEndoctendo 31563
This theorem is referenced by:  tendoeq2  31585  tendo0mulr  31638  tendotr  31641  tendocnv  31833  dvhopN  31928  dihpN  32148
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-map 6790  df-poset 14096  df-plt 14108  df-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-p0 14161  df-p1 14162  df-lat 14168  df-clat 14230  df-oposet 29988  df-ol 29990  df-oml 29991  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163  df-lhyp 30799  df-laut 30800  df-ldil 30915  df-ltrn 30916  df-trl 30970  df-tendo 31566
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