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Theorem tendotr 30995
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 30938 . . . . 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 5665 . . . 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 2422 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  U  =/=  O )  /\  F  e.  T )  /\  F  =  (  _I  |`  B ) )  ->  ( U `  F )  =  F )
1110fveq2d 5665 . 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 2380 . . . . 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 30926 . . . 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 29526 . . . . 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 30932 . . . . . 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 30990 . . . . . . . 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 2578 . . . . . 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 2380 . . . . . 6  |-  ( Atoms `  K )  =  (
Atoms `  K )
333, 32, 4, 16, 17trlnidat 30338 . . . . 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 30338 . . . . 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 29477 . . . 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 2621 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 1649    e. wcel 1717    =/= wne 2543   class class class wbr 4146    e. cmpt 4200    _I cid 4427    |` cres 4813   ` cfv 5387   Basecbs 13389   lecple 13456   Atomscatm 29429   AtLatcal 29430   HLchlt 29516   LHypclh 30149   LTrncltrn 30266   trLctrl 30323   TEndoctendo 30917
This theorem is referenced by:  cdleml6  31146
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 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-fal 1326  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-iin 4031  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-undef 6472  df-riota 6478  df-map 6949  df-poset 14323  df-plt 14335  df-lub 14351  df-glb 14352  df-join 14353  df-meet 14354  df-p0 14388  df-p1 14389  df-lat 14395  df-clat 14457  df-oposet 29342  df-ol 29344  df-oml 29345  df-covers 29432  df-ats 29433  df-atl 29464  df-cvlat 29488  df-hlat 29517  df-llines 29663  df-lplanes 29664  df-lvols 29665  df-lines 29666  df-psubsp 29668  df-pmap 29669  df-padd 29961  df-lhyp 30153  df-laut 30154  df-ldil 30269  df-ltrn 30270  df-trl 30324  df-tendo 30920
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