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Theorem ltrnmw 29619
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 955 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
2 simp2 956 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  F  e.  T )
3 simp3l 983 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  P  e.  A )
4 eqid 2284 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
5 ltrnmw.a . . . . . 6  |-  A  =  ( Atoms `  K )
64, 5atbase 28758 . . . . 5  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
73, 6syl 15 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  P  e.  ( Base `  K )
)
8 simp1r 980 . . . . 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 29466 . . . . 5  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
118, 10syl 15 . . . 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 29599 . . . 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 1186 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( F `  ( P  ./\  W
) )  =  ( ( F `  P
)  ./\  ( F `  W ) ) )
16 simp3r 984 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  -.  P  .<_  W )
17 simp1l 979 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  K  e.  HL )
18 hlatl 28829 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  AtLat )
1917, 18syl 15 . . . . . 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 28786 . . . . . 6  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  W  e.  ( Base `  K
) )  ->  ( -.  P  .<_  W  <->  ( P  ./\ 
W )  =  .0.  ) )
2319, 3, 11, 22syl3anc 1182 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( -.  P  .<_  W  <->  ( P  ./\ 
W )  =  .0.  ) )
2416, 23mpbid 201 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( P  ./\ 
W )  =  .0.  )
2524fveq2d 5490 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( F `  ( P  ./\  W
) )  =  ( F `  .0.  )
)
2615, 25eqtr3d 2318 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( ( F `  P )  ./\  ( F `  W
) )  =  ( F `  .0.  )
)
27 hllat 28832 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
2817, 27syl 15 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  K  e.  Lat )
294, 20latref 14155 . . . . 5  |-  ( ( K  e.  Lat  /\  W  e.  ( Base `  K ) )  ->  W  .<_  W )
3028, 11, 29syl2anc 642 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  W  .<_  W )
314, 20, 9, 13ltrnval1 29602 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( W  e.  (
Base `  K )  /\  W  .<_  W ) )  ->  ( F `  W )  =  W )
321, 2, 11, 30, 31syl112anc 1186 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( F `  W )  =  W )
3332oveq2d 5836 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( ( F `  P )  ./\  ( F `  W
) )  =  ( ( F `  P
)  ./\  W )
)
34 hlop 28831 . . . . 5  |-  ( K  e.  HL  ->  K  e.  OP )
3517, 34syl 15 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  K  e.  OP )
364, 21op0cl 28653 . . . 4  |-  ( K  e.  OP  ->  .0.  e.  ( Base `  K
) )
3735, 36syl 15 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  .0.  e.  ( Base `  K )
)
384, 20, 21op0le 28655 . . . 4  |-  ( ( K  e.  OP  /\  W  e.  ( Base `  K ) )  ->  .0.  .<_  W )
3935, 11, 38syl2anc 642 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  .0.  .<_  W )
404, 20, 9, 13ltrnval1 29602 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  (  .0.  e.  ( Base `  K )  /\  .0.  .<_  W ) )  ->  ( F `  .0.  )  =  .0.  )
411, 2, 37, 39, 40syl112anc 1186 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( F `  .0.  )  =  .0.  )
4226, 33, 413eqtr3d 2324 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1685   class class class wbr 4024   ` cfv 5221  (class class class)co 5820   Basecbs 13144   lecple 13211   meetcmee 14075   0.cp0 14139   Latclat 14147   OPcops 28641   Atomscatm 28732   AtLatcal 28733   HLchlt 28819   LHypclh 29452   LTrncltrn 29569
This theorem is referenced by:  cdlemg2m  30072
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-1st 6084  df-2nd 6085  df-iota 6253  df-undef 6292  df-riota 6300  df-map 6770  df-poset 14076  df-plt 14088  df-glb 14105  df-meet 14107  df-p0 14141  df-lat 14148  df-oposet 28645  df-ol 28647  df-oml 28648  df-covers 28735  df-ats 28736  df-atl 28767  df-cvlat 28791  df-hlat 28820  df-lhyp 29456  df-laut 29457  df-ldil 29572  df-ltrn 29573
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