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Theorem ltrnmw 31010
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 958 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
2 simp2 959 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  F  e.  T )
3 simp3l 986 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  P  e.  A )
4 eqid 2438 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
5 ltrnmw.a . . . . . 6  |-  A  =  ( Atoms `  K )
64, 5atbase 30149 . . . . 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 983 . . . . 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 30857 . . . . 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 30990 . . . 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 1189 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( F `  ( P  ./\  W
) )  =  ( ( F `  P
)  ./\  ( F `  W ) ) )
16 simp3r 987 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  -.  P  .<_  W )
17 simp1l 982 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  K  e.  HL )
18 hlatl 30220 . . . . . . 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 30177 . . . . . 6  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  W  e.  ( Base `  K
) )  ->  ( -.  P  .<_  W  <->  ( P  ./\ 
W )  =  .0.  ) )
2319, 3, 11, 22syl3anc 1185 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( -.  P  .<_  W  <->  ( P  ./\ 
W )  =  .0.  ) )
2416, 23mpbid 203 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( P  ./\ 
W )  =  .0.  )
2524fveq2d 5734 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( F `  ( P  ./\  W
) )  =  ( F `  .0.  )
)
2615, 25eqtr3d 2472 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( ( F `  P )  ./\  ( F `  W
) )  =  ( F `  .0.  )
)
27 hllat 30223 . . . . . 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 14484 . . . . 5  |-  ( ( K  e.  Lat  /\  W  e.  ( Base `  K ) )  ->  W  .<_  W )
3028, 11, 29syl2anc 644 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  W  .<_  W )
314, 20, 9, 13ltrnval1 30993 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( W  e.  (
Base `  K )  /\  W  .<_  W ) )  ->  ( F `  W )  =  W )
321, 2, 11, 30, 31syl112anc 1189 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( F `  W )  =  W )
3332oveq2d 6099 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( ( F `  P )  ./\  ( F `  W
) )  =  ( ( F `  P
)  ./\  W )
)
34 hlop 30222 . . . . 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 30044 . . . 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 30046 . . . 4  |-  ( ( K  e.  OP  /\  W  e.  ( Base `  K ) )  ->  .0.  .<_  W )
3935, 11, 38syl2anc 644 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  .0.  .<_  W )
404, 20, 9, 13ltrnval1 30993 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  (  .0.  e.  ( Base `  K )  /\  .0.  .<_  W ) )  ->  ( F `  .0.  )  =  .0.  )
411, 2, 37, 39, 40syl112anc 1189 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( F `  .0.  )  =  .0.  )
4226, 33, 413eqtr3d 2478 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( ( F `  P )  ./\  W )  =  .0.  )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   class class class wbr 4214   ` cfv 5456  (class class class)co 6083   Basecbs 13471   lecple 13538   meetcmee 14404   0.cp0 14468   Latclat 14476   OPcops 30032   Atomscatm 30123   AtLatcal 30124   HLchlt 30210   LHypclh 30843   LTrncltrn 30960
This theorem is referenced by:  cdlemg2m  31463
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-undef 6545  df-riota 6551  df-map 7022  df-poset 14405  df-plt 14417  df-glb 14434  df-meet 14436  df-p0 14470  df-lat 14477  df-oposet 30036  df-ol 30038  df-oml 30039  df-covers 30126  df-ats 30127  df-atl 30158  df-cvlat 30182  df-hlat 30211  df-lhyp 30847  df-laut 30848  df-ldil 30963  df-ltrn 30964
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