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Theorem ltrnmw 29141
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 960 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
2 simp2 961 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  F  e.  T )
3 simp3l 988 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  P  e.  A )
4 eqid 2253 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
5 ltrnmw.a . . . . . 6  |-  A  =  ( Atoms `  K )
64, 5atbase 28280 . . . . 5  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
73, 6syl 17 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  P  e.  ( Base `  K )
)
8 simp1r 985 . . . . 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 28988 . . . . 5  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
118, 10syl 17 . . . 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 29121 . . . 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 1191 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( F `  ( P  ./\  W
) )  =  ( ( F `  P
)  ./\  ( F `  W ) ) )
16 simp3r 989 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  -.  P  .<_  W )
17 simp1l 984 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  K  e.  HL )
18 hlatl 28351 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  AtLat )
1917, 18syl 17 . . . . . 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 28308 . . . . . 6  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  W  e.  ( Base `  K
) )  ->  ( -.  P  .<_  W  <->  ( P  ./\ 
W )  =  .0.  ) )
2319, 3, 11, 22syl3anc 1187 . . . . 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 5381 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( F `  ( P  ./\  W
) )  =  ( F `  .0.  )
)
2615, 25eqtr3d 2287 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( ( F `  P )  ./\  ( F `  W
) )  =  ( F `  .0.  )
)
27 hllat 28354 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
2817, 27syl 17 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  K  e.  Lat )
294, 20latref 14003 . . . . 5  |-  ( ( K  e.  Lat  /\  W  e.  ( Base `  K ) )  ->  W  .<_  W )
3028, 11, 29syl2anc 645 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  W  .<_  W )
314, 20, 9, 13ltrnval1 29124 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( W  e.  (
Base `  K )  /\  W  .<_  W ) )  ->  ( F `  W )  =  W )
321, 2, 11, 30, 31syl112anc 1191 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( F `  W )  =  W )
3332oveq2d 5726 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( ( F `  P )  ./\  ( F `  W
) )  =  ( ( F `  P
)  ./\  W )
)
34 hlop 28353 . . . . 5  |-  ( K  e.  HL  ->  K  e.  OP )
3517, 34syl 17 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  K  e.  OP )
364, 21op0cl 28175 . . . 4  |-  ( K  e.  OP  ->  .0.  e.  ( Base `  K
) )
3735, 36syl 17 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  .0.  e.  ( Base `  K )
)
384, 20, 21op0le 28177 . . . 4  |-  ( ( K  e.  OP  /\  W  e.  ( Base `  K ) )  ->  .0.  .<_  W )
3935, 11, 38syl2anc 645 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  .0.  .<_  W )
404, 20, 9, 13ltrnval1 29124 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  (  .0.  e.  ( Base `  K )  /\  .0.  .<_  W ) )  ->  ( F `  .0.  )  =  .0.  )
411, 2, 37, 39, 40syl112anc 1191 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( F `  .0.  )  =  .0.  )
4226, 33, 413eqtr3d 2293 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 5    -> wi 6    <-> wb 178    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621   class class class wbr 3920   ` cfv 4592  (class class class)co 5710   Basecbs 13022   lecple 13089   meetcmee 13923   0.cp0 13987   Latclat 13995   OPcops 28163   Atomscatm 28254   AtLatcal 28255   HLchlt 28341   LHypclh 28974   LTrncltrn 29091
This theorem is referenced by:  cdlemg2m  29594
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-iota 6143  df-undef 6182  df-riota 6190  df-map 6660  df-poset 13924  df-plt 13936  df-glb 13953  df-meet 13955  df-p0 13989  df-lat 13996  df-oposet 28167  df-ol 28169  df-oml 28170  df-covers 28257  df-ats 28258  df-atl 28289  df-cvlat 28313  df-hlat 28342  df-lhyp 28978  df-laut 28979  df-ldil 29094  df-ltrn 29095
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