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Theorem ltrnmw 35348
Description: Property of lattice translation value. Remark below Lemma B in [Crawley] p. 112. TODO: Can this be used in more places? (Contributed by NM, 20-May-2012.)
Hypotheses
Ref Expression
ltrnmw.l  |-  .<_  =  ( le `  K )
ltrnmw.m  |-  ./\  =  ( meet `  K )
ltrnmw.z  |-  .0.  =  ( 0. `  K )
ltrnmw.a  |-  A  =  ( Atoms `  K )
ltrnmw.h  |-  H  =  ( LHyp `  K
)
ltrnmw.t  |-  T  =  ( ( LTrn `  K
) `  W )
Assertion
Ref Expression
ltrnmw  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( ( F `  P )  ./\  W )  =  .0.  )

Proof of Theorem ltrnmw
StepHypRef Expression
1 simp1 996 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
2 simp2 997 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  F  e.  T )
3 simp3l 1024 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  P  e.  A )
4 eqid 2467 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
5 ltrnmw.a . . . . . 6  |-  A  =  ( Atoms `  K )
64, 5atbase 34487 . . . . 5  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
73, 6syl 16 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  P  e.  ( Base `  K )
)
8 simp1r 1021 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  W  e.  H )
9 ltrnmw.h . . . . . 6  |-  H  =  ( LHyp `  K
)
104, 9lhpbase 35195 . . . . 5  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
118, 10syl 16 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  W  e.  ( Base `  K )
)
12 ltrnmw.m . . . . 5  |-  ./\  =  ( meet `  K )
13 ltrnmw.t . . . . 5  |-  T  =  ( ( LTrn `  K
) `  W )
144, 12, 9, 13ltrnm 35328 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  (
Base `  K )  /\  W  e.  ( Base `  K ) ) )  ->  ( F `  ( P  ./\  W
) )  =  ( ( F `  P
)  ./\  ( F `  W ) ) )
151, 2, 7, 11, 14syl112anc 1232 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( F `  ( P  ./\  W
) )  =  ( ( F `  P
)  ./\  ( F `  W ) ) )
16 simp3r 1025 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  -.  P  .<_  W )
17 simp1l 1020 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  K  e.  HL )
18 hlatl 34558 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  AtLat )
1917, 18syl 16 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  K  e.  AtLat
)
20 ltrnmw.l . . . . . . 7  |-  .<_  =  ( le `  K )
21 ltrnmw.z . . . . . . 7  |-  .0.  =  ( 0. `  K )
224, 20, 12, 21, 5atnle 34515 . . . . . 6  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  W  e.  ( Base `  K
) )  ->  ( -.  P  .<_  W  <->  ( P  ./\ 
W )  =  .0.  ) )
2319, 3, 11, 22syl3anc 1228 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( -.  P  .<_  W  <->  ( P  ./\ 
W )  =  .0.  ) )
2416, 23mpbid 210 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( P  ./\ 
W )  =  .0.  )
2524fveq2d 5876 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( F `  ( P  ./\  W
) )  =  ( F `  .0.  )
)
2615, 25eqtr3d 2510 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( ( F `  P )  ./\  ( F `  W
) )  =  ( F `  .0.  )
)
27 hllat 34561 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
2817, 27syl 16 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  K  e.  Lat )
294, 20latref 15557 . . . . 5  |-  ( ( K  e.  Lat  /\  W  e.  ( Base `  K ) )  ->  W  .<_  W )
3028, 11, 29syl2anc 661 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  W  .<_  W )
314, 20, 9, 13ltrnval1 35331 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( W  e.  (
Base `  K )  /\  W  .<_  W ) )  ->  ( F `  W )  =  W )
321, 2, 11, 30, 31syl112anc 1232 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( F `  W )  =  W )
3332oveq2d 6311 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( ( F `  P )  ./\  ( F `  W
) )  =  ( ( F `  P
)  ./\  W )
)
34 hlop 34560 . . . . 5  |-  ( K  e.  HL  ->  K  e.  OP )
3517, 34syl 16 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  K  e.  OP )
364, 21op0cl 34382 . . . 4  |-  ( K  e.  OP  ->  .0.  e.  ( Base `  K
) )
3735, 36syl 16 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  .0.  e.  ( Base `  K )
)
384, 20, 21op0le 34384 . . . 4  |-  ( ( K  e.  OP  /\  W  e.  ( Base `  K ) )  ->  .0.  .<_  W )
3935, 11, 38syl2anc 661 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  .0.  .<_  W )
404, 20, 9, 13ltrnval1 35331 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  (  .0.  e.  ( Base `  K )  /\  .0.  .<_  W ) )  ->  ( F `  .0.  )  =  .0.  )
411, 2, 37, 39, 40syl112anc 1232 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( F `  .0.  )  =  .0.  )
4226, 33, 413eqtr3d 2516 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( ( F `  P )  ./\  W )  =  .0.  )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   class class class wbr 4453   ` cfv 5594  (class class class)co 6295   Basecbs 14507   lecple 14579   meetcmee 15449   0.cp0 15541   Latclat 15549   OPcops 34370   Atomscatm 34461   AtLatcal 34462   HLchlt 34548   LHypclh 35181   LTrncltrn 35298
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-map 7434  df-poset 15450  df-plt 15462  df-lub 15478  df-glb 15479  df-join 15480  df-meet 15481  df-p0 15543  df-lat 15550  df-oposet 34374  df-ol 34376  df-oml 34377  df-covers 34464  df-ats 34465  df-atl 34496  df-cvlat 34520  df-hlat 34549  df-lhyp 35185  df-laut 35186  df-ldil 35301  df-ltrn 35302
This theorem is referenced by:  cdlemg2m  35801
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