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Theorem dihmeetlem17N 31513
Description: Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
dihmeetlem14.b  |-  B  =  ( Base `  K
)
dihmeetlem14.l  |-  .<_  =  ( le `  K )
dihmeetlem14.h  |-  H  =  ( LHyp `  K
)
dihmeetlem14.j  |-  .\/  =  ( join `  K )
dihmeetlem14.m  |-  ./\  =  ( meet `  K )
dihmeetlem14.a  |-  A  =  ( Atoms `  K )
dihmeetlem14.u  |-  U  =  ( ( DVecH `  K
) `  W )
dihmeetlem14.s  |-  .(+)  =  (
LSSum `  U )
dihmeetlem14.i  |-  I  =  ( ( DIsoH `  K
) `  W )
dihmeetlem17.o  |-  .0.  =  ( 0. `  K )
Assertion
Ref Expression
dihmeetlem17N  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( p  e.  A  /\  -.  p  .<_  W ) )  /\  ( Y  e.  B  /\  ( X  ./\  Y
)  .<_  W  /\  p  .<_  X ) )  -> 
( Y  ./\  p
)  =  .0.  )

Proof of Theorem dihmeetlem17N
StepHypRef Expression
1 simpl1l 1006 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( p  e.  A  /\  -.  p  .<_  W ) )  /\  ( Y  e.  B  /\  ( X  ./\  Y
)  .<_  W  /\  p  .<_  X ) )  ->  K  e.  HL )
2 hllat 29553 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
31, 2syl 15 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( p  e.  A  /\  -.  p  .<_  W ) )  /\  ( Y  e.  B  /\  ( X  ./\  Y
)  .<_  W  /\  p  .<_  X ) )  ->  K  e.  Lat )
4 simpl3l 1010 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( p  e.  A  /\  -.  p  .<_  W ) )  /\  ( Y  e.  B  /\  ( X  ./\  Y
)  .<_  W  /\  p  .<_  X ) )  ->  p  e.  A )
5 dihmeetlem14.b . . . . 5  |-  B  =  ( Base `  K
)
6 dihmeetlem14.a . . . . 5  |-  A  =  ( Atoms `  K )
75, 6atbase 29479 . . . 4  |-  ( p  e.  A  ->  p  e.  B )
84, 7syl 15 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( p  e.  A  /\  -.  p  .<_  W ) )  /\  ( Y  e.  B  /\  ( X  ./\  Y
)  .<_  W  /\  p  .<_  X ) )  ->  p  e.  B )
9 simpr1 961 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( p  e.  A  /\  -.  p  .<_  W ) )  /\  ( Y  e.  B  /\  ( X  ./\  Y
)  .<_  W  /\  p  .<_  X ) )  ->  Y  e.  B )
10 dihmeetlem14.m . . . 4  |-  ./\  =  ( meet `  K )
115, 10latmcom 14181 . . 3  |-  ( ( K  e.  Lat  /\  p  e.  B  /\  Y  e.  B )  ->  ( p  ./\  Y
)  =  ( Y 
./\  p ) )
123, 8, 9, 11syl3anc 1182 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( p  e.  A  /\  -.  p  .<_  W ) )  /\  ( Y  e.  B  /\  ( X  ./\  Y
)  .<_  W  /\  p  .<_  X ) )  -> 
( p  ./\  Y
)  =  ( Y 
./\  p ) )
13 simpl1 958 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( p  e.  A  /\  -.  p  .<_  W ) )  /\  ( Y  e.  B  /\  ( X  ./\  Y
)  .<_  W  /\  p  .<_  X ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
14 simpl2 959 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( p  e.  A  /\  -.  p  .<_  W ) )  /\  ( Y  e.  B  /\  ( X  ./\  Y
)  .<_  W  /\  p  .<_  X ) )  -> 
( X  e.  B  /\  -.  X  .<_  W ) )
15 simpl3 960 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( p  e.  A  /\  -.  p  .<_  W ) )  /\  ( Y  e.  B  /\  ( X  ./\  Y
)  .<_  W  /\  p  .<_  X ) )  -> 
( p  e.  A  /\  -.  p  .<_  W ) )
16 simpr2 962 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( p  e.  A  /\  -.  p  .<_  W ) )  /\  ( Y  e.  B  /\  ( X  ./\  Y
)  .<_  W  /\  p  .<_  X ) )  -> 
( X  ./\  Y
)  .<_  W )
17 simpr3 963 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( p  e.  A  /\  -.  p  .<_  W ) )  /\  ( Y  e.  B  /\  ( X  ./\  Y
)  .<_  W  /\  p  .<_  X ) )  ->  p  .<_  X )
18 dihmeetlem14.l . . . . 5  |-  .<_  =  ( le `  K )
19 dihmeetlem14.j . . . . 5  |-  .\/  =  ( join `  K )
20 dihmeetlem14.h . . . . 5  |-  H  =  ( LHyp `  K
)
215, 18, 19, 10, 6, 20lhpmcvr4N 30215 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( X  e.  B  /\  -.  X  .<_  W )  /\  ( p  e.  A  /\  -.  p  .<_  W ) )  /\  ( Y  e.  B  /\  ( X  ./\  Y )  .<_  W  /\  p  .<_  X ) )  ->  -.  p  .<_  Y )
2213, 14, 15, 9, 16, 17, 21syl123anc 1199 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( p  e.  A  /\  -.  p  .<_  W ) )  /\  ( Y  e.  B  /\  ( X  ./\  Y
)  .<_  W  /\  p  .<_  X ) )  ->  -.  p  .<_  Y )
23 hlatl 29550 . . . . 5  |-  ( K  e.  HL  ->  K  e.  AtLat )
241, 23syl 15 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( p  e.  A  /\  -.  p  .<_  W ) )  /\  ( Y  e.  B  /\  ( X  ./\  Y
)  .<_  W  /\  p  .<_  X ) )  ->  K  e.  AtLat )
25 dihmeetlem17.o . . . . 5  |-  .0.  =  ( 0. `  K )
265, 18, 10, 25, 6atnle 29507 . . . 4  |-  ( ( K  e.  AtLat  /\  p  e.  A  /\  Y  e.  B )  ->  ( -.  p  .<_  Y  <->  ( p  ./\ 
Y )  =  .0.  ) )
2724, 4, 9, 26syl3anc 1182 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( p  e.  A  /\  -.  p  .<_  W ) )  /\  ( Y  e.  B  /\  ( X  ./\  Y
)  .<_  W  /\  p  .<_  X ) )  -> 
( -.  p  .<_  Y  <-> 
( p  ./\  Y
)  =  .0.  )
)
2822, 27mpbid 201 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( p  e.  A  /\  -.  p  .<_  W ) )  /\  ( Y  e.  B  /\  ( X  ./\  Y
)  .<_  W  /\  p  .<_  X ) )  -> 
( p  ./\  Y
)  =  .0.  )
2912, 28eqtr3d 2317 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( p  e.  A  /\  -.  p  .<_  W ) )  /\  ( Y  e.  B  /\  ( X  ./\  Y
)  .<_  W  /\  p  .<_  X ) )  -> 
( Y  ./\  p
)  =  .0.  )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215   joincjn 14078   meetcmee 14079   0.cp0 14143   Latclat 14151   LSSumclsm 14945   Atomscatm 29453   AtLatcal 29454   HLchlt 29540   LHypclh 30173   DVecHcdvh 31268   DIsoHcdih 31418
This theorem is referenced by:  dihmeetlem18N  31514
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 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-poset 14080  df-plt 14092  df-glb 14109  df-meet 14111  df-p0 14145  df-lat 14152  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-lhyp 30177
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