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Theorem tendoid 34049
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 2429 . . . . . . 7  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
41, 2, 3idltrn 33424 . . . . . 6  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  _I  |`  B )  e.  ( ( LTrn `  K ) `  W
) )
54adantr 466 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E
)  ->  (  _I  |`  B )  e.  ( ( LTrn `  K
) `  W )
)
6 eqid 2429 . . . . . 6  |-  ( le
`  K )  =  ( le `  K
)
7 eqid 2429 . . . . . 6  |-  ( ( trL `  K ) `
 W )  =  ( ( trL `  K
) `  W )
8 tendoid.e . . . . . 6  |-  E  =  ( ( TEndo `  K
) `  W )
96, 2, 3, 7, 8tendotp 34037 . . . . 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 1361 . . . 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 2429 . . . . . 6  |-  ( 0.
`  K )  =  ( 0. `  K
)
121, 11, 2, 7trlid0 33451 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( ( trL `  K ) `  W
) `  (  _I  |`  B ) )  =  ( 0. `  K
) )
1312adantr 466 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E
)  ->  ( (
( trL `  K
) `  W ) `  (  _I  |`  B ) )  =  ( 0.
`  K ) )
1410, 13breqtrd 4450 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E
)  ->  ( (
( trL `  K
) `  W ) `  ( S `  (  _I  |`  B ) ) ) ( le `  K ) ( 0.
`  K ) )
15 hlop 32637 . . . . 5  |-  ( K  e.  HL  ->  K  e.  OP )
1615ad2antrr 730 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E
)  ->  K  e.  OP )
172, 3, 8tendocl 34043 . . . . . 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 1361 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E
)  ->  ( S `  (  _I  |`  B ) )  e.  ( (
LTrn `  K ) `  W ) )
191, 2, 3, 7trlcl 33439 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S `  (  _I  |`  B ) )  e.  ( (
LTrn `  K ) `  W ) )  -> 
( ( ( trL `  K ) `  W
) `  ( S `  (  _I  |`  B ) ) )  e.  B
)
2018, 19syldan 472 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E
)  ->  ( (
( trL `  K
) `  W ) `  ( S `  (  _I  |`  B ) ) )  e.  B )
211, 6, 11ople0 32462 . . . 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 665 . . 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 213 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  S  e.  E
)  ->  ( (
( trL `  K
) `  W ) `  ( S `  (  _I  |`  B ) ) )  =  ( 0.
`  K ) )
241, 11, 2, 3, 7trlid0b 33453 . . 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 472 . 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 235 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 187    /\ wa 370    = wceq 1437    e. wcel 1870   class class class wbr 4426    _I cid 4764    |` cres 4856   ` cfv 5601   Basecbs 15084   lecple 15159   0.cp0 16234   OPcops 32447   HLchlt 32625   LHypclh 33258   LTrncltrn 33375   trLctrl 33433   TEndoctendo 34028
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-map 7482  df-preset 16124  df-poset 16142  df-plt 16155  df-lub 16171  df-glb 16172  df-join 16173  df-meet 16174  df-p0 16236  df-p1 16237  df-lat 16243  df-clat 16305  df-oposet 32451  df-ol 32453  df-oml 32454  df-covers 32541  df-ats 32542  df-atl 32573  df-cvlat 32597  df-hlat 32626  df-lhyp 33262  df-laut 33263  df-ldil 33378  df-ltrn 33379  df-trl 33434  df-tendo 34031
This theorem is referenced by:  tendoeq2  34050  tendo0mulr  34103  tendotr  34106  tendocnv  34298  dvhopN  34393  dihpN  34613
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