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Theorem tendotr 36258
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 998 . . . . 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 1048 . . . . 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 36201 . . . . 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 5856 . . . 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 2492 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  U  =/=  O )  /\  F  e.  T )  /\  F  =  (  _I  |`  B ) )  ->  ( U `  F )  =  F )
1110fveq2d 5856 . 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 998 . . . 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 1048 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  U  =/=  O )  /\  F  e.  T )  /\  F  =/=  (  _I  |`  B ) )  ->  U  e.  E
)
14 simpl3 1000 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  U  =/=  O )  /\  F  e.  T )  /\  F  =/=  (  _I  |`  B ) )  ->  F  e.  T
)
15 eqid 2441 . . . . 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 36189 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  F  e.  T
)  ->  ( R `  ( U `  F
) ) ( le
`  K ) ( R `  F ) )
1912, 13, 14, 18syl3anc 1227 . . 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 1046 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  U  =/=  O )  /\  F  e.  T )  /\  F  =/=  (  _I  |`  B ) )  ->  K  e.  HL )
21 hlatl 34787 . . . . 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 36195 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  F  e.  T
)  ->  ( U `  F )  e.  T
)
2412, 13, 14, 23syl3anc 1227 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  U  =/=  O )  /\  F  e.  T )  /\  F  =/=  (  _I  |`  B ) )  ->  ( U `  F )  e.  T
)
25 simpl2r 1049 . . . . . 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 36253 . . . . . . . 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 1231 . . . . . . 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 2699 . . . . . 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 2441 . . . . . 6  |-  ( Atoms `  K )  =  (
Atoms `  K )
333, 32, 4, 16, 17trlnidat 35600 . . . . 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 1227 . . . 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 35600 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  F  =/=  (  _I  |`  B ) )  ->  ( R `  F )  e.  (
Atoms `  K ) )
3612, 14, 26, 35syl3anc 1227 . . . 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 34738 . . . 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 1227 . . 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 2759 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 972    = wceq 1381    e. wcel 1802    =/= wne 2636   class class class wbr 4433    |-> cmpt 4491    _I cid 4776    |` cres 4987   ` cfv 5574   Basecbs 14504   lecple 14576   Atomscatm 34690   AtLatcal 34691   HLchlt 34777   LHypclh 35410   LTrncltrn 35527   trLctrl 35585   TEndoctendo 36180
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-riotaBAD 34386
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-fal 1387  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-iun 4313  df-iin 4314  df-br 4434  df-opab 4492  df-mpt 4493  df-id 4781  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6781  df-2nd 6782  df-undef 7000  df-map 7420  df-preset 15426  df-poset 15444  df-plt 15457  df-lub 15473  df-glb 15474  df-join 15475  df-meet 15476  df-p0 15538  df-p1 15539  df-lat 15545  df-clat 15607  df-oposet 34603  df-ol 34605  df-oml 34606  df-covers 34693  df-ats 34694  df-atl 34725  df-cvlat 34749  df-hlat 34778  df-llines 34924  df-lplanes 34925  df-lvols 34926  df-lines 34927  df-psubsp 34929  df-pmap 34930  df-padd 35222  df-lhyp 35414  df-laut 35415  df-ldil 35530  df-ltrn 35531  df-trl 35586  df-tendo 36183
This theorem is referenced by:  cdleml6  36409
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