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Theorem tendotr 31641
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 958 . . . . 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 1008 . . . . 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 31584 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E
)  ->  ( U `  (  _I  |`  B ) )  =  (  _I  |`  B ) )
71, 2, 6syl2anc 642 . . . 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 447 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  U  =/=  O )  /\  F  e.  T )  /\  F  =  (  _I  |`  B ) )  ->  F  =  (  _I  |`  B )
)
98fveq2d 5545 . . . 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 2338 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  U  =/=  O )  /\  F  e.  T )  /\  F  =  (  _I  |`  B ) )  ->  ( U `  F )  =  F )
1110fveq2d 5545 . 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 958 . . . 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 1008 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  U  =/=  O )  /\  F  e.  T )  /\  F  =/=  (  _I  |`  B ) )  ->  U  e.  E
)
14 simpl3 960 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  U  =/=  O )  /\  F  e.  T )  /\  F  =/=  (  _I  |`  B ) )  ->  F  e.  T
)
15 eqid 2296 . . . . 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 31572 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  F  e.  T
)  ->  ( R `  ( U `  F
) ) ( le
`  K ) ( R `  F ) )
1912, 13, 14, 18syl3anc 1182 . . 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 1006 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  U  =/=  O )  /\  F  e.  T )  /\  F  =/=  (  _I  |`  B ) )  ->  K  e.  HL )
21 hlatl 30172 . . . . 5  |-  ( K  e.  HL  ->  K  e.  AtLat )
2220, 21syl 15 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  U  =/=  O )  /\  F  e.  T )  /\  F  =/=  (  _I  |`  B ) )  ->  K  e.  AtLat )
234, 16, 5tendocl 31578 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  U  e.  E  /\  F  e.  T
)  ->  ( U `  F )  e.  T
)
2412, 13, 14, 23syl3anc 1182 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  U  =/=  O )  /\  F  e.  T )  /\  F  =/=  (  _I  |`  B ) )  ->  ( U `  F )  e.  T
)
25 simpl2r 1009 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  U  =/=  O )  /\  F  e.  T )  /\  F  =/=  (  _I  |`  B ) )  ->  U  =/=  O
)
26 simpr 447 . . . . . . . 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 31636 . . . . . . . 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 1186 . . . . . . 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 2494 . . . . . 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 223 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  E  /\  U  =/=  O )  /\  F  e.  T )  /\  F  =/=  (  _I  |`  B ) )  ->  ( U `  F )  =/=  (  _I  |`  B ) )
32 eqid 2296 . . . . . 6  |-  ( Atoms `  K )  =  (
Atoms `  K )
333, 32, 4, 16, 17trlnidat 30984 . . . . 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 1182 . . . 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 30984 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  F  =/=  (  _I  |`  B ) )  ->  ( R `  F )  e.  (
Atoms `  K ) )
3612, 14, 26, 35syl3anc 1182 . . . 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 30123 . . . 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 1182 . . 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 201 . 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 2537 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 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   class class class wbr 4039    e. cmpt 4093    _I cid 4320    |` cres 4707   ` cfv 5271   Basecbs 13164   lecple 13231   Atomscatm 30075   AtLatcal 30076   HLchlt 30162   LHypclh 30795   LTrncltrn 30912   trLctrl 30969   TEndoctendo 31563
This theorem is referenced by:  cdleml6  31792
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-fal 1311  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-map 6790  df-poset 14096  df-plt 14108  df-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-p0 14161  df-p1 14162  df-lat 14168  df-clat 14230  df-oposet 29988  df-ol 29990  df-oml 29991  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163  df-llines 30309  df-lplanes 30310  df-lvols 30311  df-lines 30312  df-psubsp 30314  df-pmap 30315  df-padd 30607  df-lhyp 30799  df-laut 30800  df-ldil 30915  df-ltrn 30916  df-trl 30970  df-tendo 31566
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