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Theorem tendotr 31564
Description: The trace of the value of a non-zero trace-preserving endomorphism equals the trace of the argument. (Contributed by NM, 11-Aug-2013.)
Hypotheses
Ref Expression
tendotr.b  |-  B  =  ( Base `  K
)
tendotr.h  |-  H  =  ( LHyp `  K
)
tendotr.t  |-  T  =  ( ( LTrn `  K
) `  W )
tendotr.r  |-  R  =  ( ( trL `  K
) `  W )
tendotr.e  |-  E  =  ( ( TEndo `  K
) `  W )
tendotr.o  |-  O  =  ( f  e.  T  |->  (  _I  |`  B ) )
Assertion
Ref Expression
tendotr  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  U  =/= 
O )  /\  F  e.  T )  ->  ( R `  ( U `  F ) )  =  ( R `  F
) )
Distinct variable groups:    B, f    T, f
Allowed substitution hints:    R( f)    U( f)    E( f)    F( f)    H( f)    K( f)    O( f)    W( f)

Proof of Theorem tendotr
StepHypRef Expression
1 simpl1 960 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  U  =/=  O )  /\  F  e.  T )  /\  F  =  (  _I  |`  B ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
2 simpl2l 1010 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  U  =/=  O )  /\  F  e.  T )  /\  F  =  (  _I  |`  B ) )  ->  U  e.  E
)
3 tendotr.b . . . . . 6  |-  B  =  ( Base `  K
)
4 tendotr.h . . . . . 6  |-  H  =  ( LHyp `  K
)
5 tendotr.e . . . . . 6  |-  E  =  ( ( TEndo `  K
) `  W )
63, 4, 5tendoid 31507 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E
)  ->  ( U `  (  _I  |`  B ) )  =  (  _I  |`  B ) )
71, 2, 6syl2anc 643 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  U  =/=  O )  /\  F  e.  T )  /\  F  =  (  _I  |`  B ) )  ->  ( U `  (  _I  |`  B ) )  =  (  _I  |`  B ) )
8 simpr 448 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  U  =/=  O )  /\  F  e.  T )  /\  F  =  (  _I  |`  B ) )  ->  F  =  (  _I  |`  B )
)
98fveq2d 5724 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  U  =/=  O )  /\  F  e.  T )  /\  F  =  (  _I  |`  B ) )  ->  ( U `  F )  =  ( U `  (  _I  |`  B ) ) )
107, 9, 83eqtr4d 2477 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  U  =/=  O )  /\  F  e.  T )  /\  F  =  (  _I  |`  B ) )  ->  ( U `  F )  =  F )
1110fveq2d 5724 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  U  =/=  O )  /\  F  e.  T )  /\  F  =  (  _I  |`  B ) )  ->  ( R `  ( U `  F ) )  =  ( R `
 F ) )
12 simpl1 960 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  U  =/=  O )  /\  F  e.  T )  /\  F  =/=  (  _I  |`  B ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
13 simpl2l 1010 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  U  =/=  O )  /\  F  e.  T )  /\  F  =/=  (  _I  |`  B ) )  ->  U  e.  E
)
14 simpl3 962 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  U  =/=  O )  /\  F  e.  T )  /\  F  =/=  (  _I  |`  B ) )  ->  F  e.  T
)
15 eqid 2435 . . . . 5  |-  ( le
`  K )  =  ( le `  K
)
16 tendotr.t . . . . 5  |-  T  =  ( ( LTrn `  K
) `  W )
17 tendotr.r . . . . 5  |-  R  =  ( ( trL `  K
) `  W )
1815, 4, 16, 17, 5tendotp 31495 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  F  e.  T
)  ->  ( R `  ( U `  F
) ) ( le
`  K ) ( R `  F ) )
1912, 13, 14, 18syl3anc 1184 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  U  =/=  O )  /\  F  e.  T )  /\  F  =/=  (  _I  |`  B ) )  ->  ( R `  ( U `  F ) ) ( le `  K ) ( R `
 F ) )
20 simpl1l 1008 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  U  =/=  O )  /\  F  e.  T )  /\  F  =/=  (  _I  |`  B ) )  ->  K  e.  HL )
21 hlatl 30095 . . . . 5  |-  ( K  e.  HL  ->  K  e.  AtLat )
2220, 21syl 16 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  U  =/=  O )  /\  F  e.  T )  /\  F  =/=  (  _I  |`  B ) )  ->  K  e.  AtLat )
234, 16, 5tendocl 31501 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  F  e.  T
)  ->  ( U `  F )  e.  T
)
2412, 13, 14, 23syl3anc 1184 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  U  =/=  O )  /\  F  e.  T )  /\  F  =/=  (  _I  |`  B ) )  ->  ( U `  F )  e.  T
)
25 simpl2r 1011 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  U  =/=  O )  /\  F  e.  T )  /\  F  =/=  (  _I  |`  B ) )  ->  U  =/=  O
)
26 simpr 448 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  U  =/=  O )  /\  F  e.  T )  /\  F  =/=  (  _I  |`  B ) )  ->  F  =/=  (  _I  |`  B ) )
27 tendotr.o . . . . . . . . 9  |-  O  =  ( f  e.  T  |->  (  _I  |`  B ) )
283, 4, 16, 5, 27tendoid0 31559 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  ( F  e.  T  /\  F  =/=  (  _I  |`  B ) ) )  ->  ( ( U `  F )  =  (  _I  |`  B )  <-> 
U  =  O ) )
2912, 13, 14, 26, 28syl112anc 1188 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  U  =/=  O )  /\  F  e.  T )  /\  F  =/=  (  _I  |`  B ) )  ->  ( ( U `
 F )  =  (  _I  |`  B )  <-> 
U  =  O ) )
3029necon3bid 2633 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  U  =/=  O )  /\  F  e.  T )  /\  F  =/=  (  _I  |`  B ) )  ->  ( ( U `
 F )  =/=  (  _I  |`  B )  <-> 
U  =/=  O ) )
3125, 30mpbird 224 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  U  =/=  O )  /\  F  e.  T )  /\  F  =/=  (  _I  |`  B ) )  ->  ( U `  F )  =/=  (  _I  |`  B ) )
32 eqid 2435 . . . . . 6  |-  ( Atoms `  K )  =  (
Atoms `  K )
333, 32, 4, 16, 17trlnidat 30907 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U `  F )  e.  T  /\  ( U `  F
)  =/=  (  _I  |`  B ) )  -> 
( R `  ( U `  F )
)  e.  ( Atoms `  K ) )
3412, 24, 31, 33syl3anc 1184 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  U  =/=  O )  /\  F  e.  T )  /\  F  =/=  (  _I  |`  B ) )  ->  ( R `  ( U `  F ) )  e.  ( Atoms `  K ) )
353, 32, 4, 16, 17trlnidat 30907 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  F  =/=  (  _I  |`  B ) )  ->  ( R `  F )  e.  (
Atoms `  K ) )
3612, 14, 26, 35syl3anc 1184 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  U  =/=  O )  /\  F  e.  T )  /\  F  =/=  (  _I  |`  B ) )  ->  ( R `  F )  e.  (
Atoms `  K ) )
3715, 32atcmp 30046 . . . 4  |-  ( ( K  e.  AtLat  /\  ( R `  ( U `  F ) )  e.  ( Atoms `  K )  /\  ( R `  F
)  e.  ( Atoms `  K ) )  -> 
( ( R `  ( U `  F ) ) ( le `  K ) ( R `
 F )  <->  ( R `  ( U `  F
) )  =  ( R `  F ) ) )
3822, 34, 36, 37syl3anc 1184 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  U  =/=  O )  /\  F  e.  T )  /\  F  =/=  (  _I  |`  B ) )  ->  ( ( R `
 ( U `  F ) ) ( le `  K ) ( R `  F
)  <->  ( R `  ( U `  F ) )  =  ( R `
 F ) ) )
3919, 38mpbid 202 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  U  =/=  O )  /\  F  e.  T )  /\  F  =/=  (  _I  |`  B ) )  ->  ( R `  ( U `  F ) )  =  ( R `
 F ) )
4011, 39pm2.61dane 2676 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  U  =/= 
O )  /\  F  e.  T )  ->  ( R `  ( U `  F ) )  =  ( R `  F
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   class class class wbr 4204    e. cmpt 4258    _I cid 4485    |` cres 4872   ` cfv 5446   Basecbs 13461   lecple 13528   Atomscatm 29998   AtLatcal 29999   HLchlt 30085   LHypclh 30718   LTrncltrn 30835   trLctrl 30892   TEndoctendo 31486
This theorem is referenced by:  cdleml6  31715
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-fal 1329  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-undef 6535  df-riota 6541  df-map 7012  df-poset 14395  df-plt 14407  df-lub 14423  df-glb 14424  df-join 14425  df-meet 14426  df-p0 14460  df-p1 14461  df-lat 14467  df-clat 14529  df-oposet 29911  df-ol 29913  df-oml 29914  df-covers 30001  df-ats 30002  df-atl 30033  df-cvlat 30057  df-hlat 30086  df-llines 30232  df-lplanes 30233  df-lvols 30234  df-lines 30235  df-psubsp 30237  df-pmap 30238  df-padd 30530  df-lhyp 30722  df-laut 30723  df-ldil 30838  df-ltrn 30839  df-trl 30893  df-tendo 31489
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