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Theorem dihmeetlem17N 31572
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 1007 . . . 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 29612 . . . 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 1011 . . . 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 29538 . . . 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 962 . . 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 14391 . . 3  |-  ( ( K  e.  Lat  /\  p  e.  B  /\  Y  e.  B )  ->  ( p  ./\  Y
)  =  ( Y 
./\  p ) )
123, 8, 9, 11syl3anc 1183 . 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 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 ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
14 simpl2 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 ) )  -> 
( X  e.  B  /\  -.  X  .<_  W ) )
15 simpl3 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 ) )  -> 
( p  e.  A  /\  -.  p  .<_  W ) )
16 simpr2 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 ) )  -> 
( X  ./\  Y
)  .<_  W )
17 simpr3 964 . . . 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 30274 . . . 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 1200 . . 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 29609 . . . . 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 29566 . . . 4  |-  ( ( K  e.  AtLat  /\  p  e.  A  /\  Y  e.  B )  ->  ( -.  p  .<_  Y  <->  ( p  ./\ 
Y )  =  .0.  ) )
2724, 4, 9, 26syl3anc 1183 . . 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 2400 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 935    = wceq 1647    e. wcel 1715   class class class wbr 4125   ` cfv 5358  (class class class)co 5981   Basecbs 13356   lecple 13423   joincjn 14288   meetcmee 14289   0.cp0 14353   Latclat 14361   LSSumclsm 15155   Atomscatm 29512   AtLatcal 29513   HLchlt 29599   LHypclh 30232   DVecHcdvh 31327   DIsoHcdih 31477
This theorem is referenced by:  dihmeetlem18N  31573
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-undef 6440  df-riota 6446  df-poset 14290  df-plt 14302  df-glb 14319  df-meet 14321  df-p0 14355  df-lat 14362  df-covers 29515  df-ats 29516  df-atl 29547  df-cvlat 29571  df-hlat 29600  df-lhyp 30236
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