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Theorem dihmeetlem17N 32195
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 1009 . . . 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 30235 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
31, 2syl 16 . . 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 1013 . . . 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 30161 . . . 4  |-  ( p  e.  A  ->  p  e.  B )
84, 7syl 16 . . 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 964 . . 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 14509 . . 3  |-  ( ( K  e.  Lat  /\  p  e.  B  /\  Y  e.  B )  ->  ( p  ./\  Y
)  =  ( Y 
./\  p ) )
123, 8, 9, 11syl3anc 1185 . 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 961 . . . 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 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  e.  B  /\  -.  X  .<_  W ) )
15 simpl3 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  e.  A  /\  -.  p  .<_  W ) )
16 simpr2 965 . . . 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 966 . . . 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 30897 . . . 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 1202 . . 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 30232 . . . . 5  |-  ( K  e.  HL  ->  K  e.  AtLat )
241, 23syl 16 . . . 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 30189 . . . 4  |-  ( ( K  e.  AtLat  /\  p  e.  A  /\  Y  e.  B )  ->  ( -.  p  .<_  Y  <->  ( p  ./\ 
Y )  =  .0.  ) )
2724, 4, 9, 26syl3anc 1185 . . 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 203 . 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 2472 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 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   class class class wbr 4215   ` cfv 5457  (class class class)co 6084   Basecbs 13474   lecple 13541   joincjn 14406   meetcmee 14407   0.cp0 14471   Latclat 14479   LSSumclsm 15273   Atomscatm 30135   AtLatcal 30136   HLchlt 30222   LHypclh 30855   DVecHcdvh 31950   DIsoHcdih 32100
This theorem is referenced by:  dihmeetlem18N  32196
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
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 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-undef 6546  df-riota 6552  df-poset 14408  df-plt 14420  df-glb 14437  df-meet 14439  df-p0 14473  df-lat 14480  df-covers 30138  df-ats 30139  df-atl 30170  df-cvlat 30194  df-hlat 30223  df-lhyp 30859
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