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Theorem ltrnmw 29590
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 2258 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
5 ltrnmw.a . . . . . 6  |-  A  =  ( Atoms `  K )
64, 5atbase 28729 . . . . 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 29437 . . . . 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 29570 . . . 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 28800 . . . . . . 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 28757 . . . . . 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 5462 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( F `  ( P  ./\  W
) )  =  ( F `  .0.  )
)
2615, 25eqtr3d 2292 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( ( F `  P )  ./\  ( F `  W
) )  =  ( F `  .0.  )
)
27 hllat 28803 . . . . . 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 14122 . . . . 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 29573 . . . 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 5808 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( ( F `  P )  ./\  ( F `  W
) )  =  ( ( F `  P
)  ./\  W )
)
34 hlop 28802 . . . . 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 28624 . . . 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 28626 . . . 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 29573 . . 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 2298 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 3997   ` cfv 4673  (class class class)co 5792   Basecbs 13111   lecple 13178   meetcmee 14042   0.cp0 14106   Latclat 14114   OPcops 28612   Atomscatm 28703   AtLatcal 28704   HLchlt 28790   LHypclh 29423   LTrncltrn 29540
This theorem is referenced by:  cdlemg2m  30043
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 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484
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 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-id 4281  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-1st 6056  df-2nd 6057  df-iota 6225  df-undef 6264  df-riota 6272  df-map 6742  df-poset 14043  df-plt 14055  df-glb 14072  df-meet 14074  df-p0 14108  df-lat 14115  df-oposet 28616  df-ol 28618  df-oml 28619  df-covers 28706  df-ats 28707  df-atl 28738  df-cvlat 28762  df-hlat 28791  df-lhyp 29427  df-laut 29428  df-ldil 29543  df-ltrn 29544
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