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Theorem trl0 35597
Description: If an atom not under the fiducial co-atom  W equals its lattice translation, the trace of the translation is zero. (Contributed by NM, 24-May-2012.)
Hypotheses
Ref Expression
trl0.l  |-  .<_  =  ( le `  K )
trl0.z  |-  .0.  =  ( 0. `  K )
trl0.a  |-  A  =  ( Atoms `  K )
trl0.h  |-  H  =  ( LHyp `  K
)
trl0.t  |-  T  =  ( ( LTrn `  K
) `  W )
trl0.r  |-  R  =  ( ( trL `  K
) `  W )
Assertion
Ref Expression
trl0  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F  e.  T  /\  ( F `  P )  =  P ) )  ->  ( R `  F )  =  .0.  )

Proof of Theorem trl0
StepHypRef Expression
1 simp1 995 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F  e.  T  /\  ( F `  P )  =  P ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
2 simp3l 1023 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F  e.  T  /\  ( F `  P )  =  P ) )  ->  F  e.  T
)
3 simp2 996 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F  e.  T  /\  ( F `  P )  =  P ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
4 trl0.l . . . 4  |-  .<_  =  ( le `  K )
5 eqid 2441 . . . 4  |-  ( join `  K )  =  (
join `  K )
6 eqid 2441 . . . 4  |-  ( meet `  K )  =  (
meet `  K )
7 trl0.a . . . 4  |-  A  =  ( Atoms `  K )
8 trl0.h . . . 4  |-  H  =  ( LHyp `  K
)
9 trl0.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
10 trl0.r . . . 4  |-  R  =  ( ( trL `  K
) `  W )
114, 5, 6, 7, 8, 9, 10trlval2 35590 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( R `  F )  =  ( ( P ( join `  K ) ( F `
 P ) ) ( meet `  K
) W ) )
121, 2, 3, 11syl3anc 1227 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F  e.  T  /\  ( F `  P )  =  P ) )  ->  ( R `  F )  =  ( ( P ( join `  K ) ( F `
 P ) ) ( meet `  K
) W ) )
13 simp3r 1024 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F  e.  T  /\  ( F `  P )  =  P ) )  ->  ( F `  P )  =  P )
1413oveq2d 6293 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F  e.  T  /\  ( F `  P )  =  P ) )  ->  ( P (
join `  K )
( F `  P
) )  =  ( P ( join `  K
) P ) )
15 simp1l 1019 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F  e.  T  /\  ( F `  P )  =  P ) )  ->  K  e.  HL )
16 simp2l 1021 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F  e.  T  /\  ( F `  P )  =  P ) )  ->  P  e.  A
)
175, 7hlatjidm 34795 . . . . 5  |-  ( ( K  e.  HL  /\  P  e.  A )  ->  ( P ( join `  K ) P )  =  P )
1815, 16, 17syl2anc 661 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F  e.  T  /\  ( F `  P )  =  P ) )  ->  ( P (
join `  K ) P )  =  P )
1914, 18eqtrd 2482 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F  e.  T  /\  ( F `  P )  =  P ) )  ->  ( P (
join `  K )
( F `  P
) )  =  P )
2019oveq1d 6292 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F  e.  T  /\  ( F `  P )  =  P ) )  ->  ( ( P ( join `  K
) ( F `  P ) ) (
meet `  K ) W )  =  ( P ( meet `  K
) W ) )
21 trl0.z . . . 4  |-  .0.  =  ( 0. `  K )
224, 6, 21, 7, 8lhpmat 35456 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  -> 
( P ( meet `  K ) W )  =  .0.  )
231, 3, 22syl2anc 661 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F  e.  T  /\  ( F `  P )  =  P ) )  ->  ( P (
meet `  K ) W )  =  .0.  )
2412, 20, 233eqtrd 2486 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F  e.  T  /\  ( F `  P )  =  P ) )  ->  ( R `  F )  =  .0.  )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 972    = wceq 1381    e. wcel 1802   class class class wbr 4433   ` cfv 5574  (class class class)co 6277   lecple 14576   joincjn 15442   meetcmee 15443   0.cp0 15536   Atomscatm 34690   HLchlt 34777   LHypclh 35410   LTrncltrn 35527   trLctrl 35585
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
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  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-ral 2796  df-rex 2797  df-reu 2798  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-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-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-lat 15545  df-covers 34693  df-ats 34694  df-atl 34725  df-cvlat 34749  df-hlat 34778  df-lhyp 35414  df-laut 35415  df-ldil 35530  df-ltrn 35531  df-trl 35586
This theorem is referenced by:  trlator0  35598  ltrnnidn  35601  trlid0  35603  trlnidatb  35604  trlnle  35613  trlval3  35614  trlval4  35615  cdlemc6  35623  cdlemg31d  36128
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