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Theorem tendotr 34314
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 991 . . . . 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 1041 . . . . 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 34257 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E
)  ->  ( U `  (  _I  |`  B ) )  =  (  _I  |`  B ) )
71, 2, 6syl2anc 661 . . . 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 461 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  U  =/=  O )  /\  F  e.  T )  /\  F  =  (  _I  |`  B ) )  ->  F  =  (  _I  |`  B )
)
98fveq2d 5690 . . . 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 2480 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  U  =/=  O )  /\  F  e.  T )  /\  F  =  (  _I  |`  B ) )  ->  ( U `  F )  =  F )
1110fveq2d 5690 . 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 991 . . . 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 1041 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  U  =/=  O )  /\  F  e.  T )  /\  F  =/=  (  _I  |`  B ) )  ->  U  e.  E
)
14 simpl3 993 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  U  =/=  O )  /\  F  e.  T )  /\  F  =/=  (  _I  |`  B ) )  ->  F  e.  T
)
15 eqid 2438 . . . . 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 34245 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  F  e.  T
)  ->  ( R `  ( U `  F
) ) ( le
`  K ) ( R `  F ) )
1912, 13, 14, 18syl3anc 1218 . . 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 1039 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  U  =/=  O )  /\  F  e.  T )  /\  F  =/=  (  _I  |`  B ) )  ->  K  e.  HL )
21 hlatl 32845 . . . . 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 34251 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  F  e.  T
)  ->  ( U `  F )  e.  T
)
2412, 13, 14, 23syl3anc 1218 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  U  =/=  O )  /\  F  e.  T )  /\  F  =/=  (  _I  |`  B ) )  ->  ( U `  F )  e.  T
)
25 simpl2r 1042 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  U  =/=  O )  /\  F  e.  T )  /\  F  =/=  (  _I  |`  B ) )  ->  U  =/=  O
)
26 simpr 461 . . . . . . . 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 34309 . . . . . . . 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 1222 . . . . . . 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 2638 . . . . . 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 232 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  U  =/=  O )  /\  F  e.  T )  /\  F  =/=  (  _I  |`  B ) )  ->  ( U `  F )  =/=  (  _I  |`  B ) )
32 eqid 2438 . . . . . 6  |-  ( Atoms `  K )  =  (
Atoms `  K )
333, 32, 4, 16, 17trlnidat 33657 . . . . 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 1218 . . . 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 33657 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  F  =/=  (  _I  |`  B ) )  ->  ( R `  F )  e.  (
Atoms `  K ) )
3612, 14, 26, 35syl3anc 1218 . . . 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 32796 . . . 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 1218 . . 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 210 . 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 2684 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  U  =/= 
O )  /\  F  e.  T )  ->  ( R `  ( U `  F ) )  =  ( R `  F
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2601   class class class wbr 4287    e. cmpt 4345    _I cid 4626    |` cres 4837   ` cfv 5413   Basecbs 14166   lecple 14237   Atomscatm 32748   AtLatcal 32749   HLchlt 32835   LHypclh 33468   LTrncltrn 33585   trLctrl 33642   TEndoctendo 34236
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-riotaBAD 32444
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-iun 4168  df-iin 4169  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-1st 6572  df-2nd 6573  df-undef 6784  df-map 7208  df-poset 15108  df-plt 15120  df-lub 15136  df-glb 15137  df-join 15138  df-meet 15139  df-p0 15201  df-p1 15202  df-lat 15208  df-clat 15270  df-oposet 32661  df-ol 32663  df-oml 32664  df-covers 32751  df-ats 32752  df-atl 32783  df-cvlat 32807  df-hlat 32836  df-llines 32982  df-lplanes 32983  df-lvols 32984  df-lines 32985  df-psubsp 32987  df-pmap 32988  df-padd 33280  df-lhyp 33472  df-laut 33473  df-ldil 33588  df-ltrn 33589  df-trl 33643  df-tendo 34239
This theorem is referenced by:  cdleml6  34465
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