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Theorem trl0 34841
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 991 . . 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 1019 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F  e.  T  /\  ( F `  P )  =  P ) )  ->  F  e.  T
)
3 simp2 992 . . 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 2460 . . . 4  |-  ( join `  K )  =  (
join `  K )
6 eqid 2460 . . . 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 34834 . . 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 1223 . 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 1020 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F  e.  T  /\  ( F `  P )  =  P ) )  ->  ( F `  P )  =  P )
1413oveq2d 6291 . . . 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 1015 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F  e.  T  /\  ( F `  P )  =  P ) )  ->  K  e.  HL )
16 simp2l 1017 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F  e.  T  /\  ( F `  P )  =  P ) )  ->  P  e.  A
)
175, 7hlatjidm 34040 . . . . 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 2501 . . 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 6290 . 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 34701 . . 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 2505 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 968    = wceq 1374    e. wcel 1762   class class class wbr 4440   ` cfv 5579  (class class class)co 6275   lecple 14551   joincjn 15420   meetcmee 15421   0.cp0 15513   Atomscatm 33935   HLchlt 34022   LHypclh 34655   LTrncltrn 34772   trLctrl 34829
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-map 7412  df-poset 15422  df-plt 15434  df-lub 15450  df-glb 15451  df-join 15452  df-meet 15453  df-p0 15515  df-lat 15522  df-covers 33938  df-ats 33939  df-atl 33970  df-cvlat 33994  df-hlat 34023  df-lhyp 34659  df-laut 34660  df-ldil 34775  df-ltrn 34776  df-trl 34830
This theorem is referenced by:  trlator0  34842  ltrnnidn  34845  trlid0  34847  trlnidatb  34848  trlnle  34857  trlval3  34858  trlval4  34859  cdlemc6  34867  cdlemg31d  35371
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