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Theorem lhpmcvr6N 30193
Description: Specialization of lhpmcvr2 30189. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
lhpmcvr2.b  |-  B  =  ( Base `  K
)
lhpmcvr2.l  |-  .<_  =  ( le `  K )
lhpmcvr2.j  |-  .\/  =  ( join `  K )
lhpmcvr2.m  |-  ./\  =  ( meet `  K )
lhpmcvr2.a  |-  A  =  ( Atoms `  K )
lhpmcvr2.h  |-  H  =  ( LHyp `  K
)
Assertion
Ref Expression
lhpmcvr6N  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Y  e.  B  /\  ( X  ./\  Y ) 
.<_  W ) )  ->  E. p  e.  A  ( -.  p  .<_  W  /\  -.  p  .<_  Y  /\  p  .<_  X ) )
Distinct variable groups:    A, p    B, p    K, p    .<_ , p    ./\ , p    X, p    W, p    H, p    Y, p
Allowed substitution hint:    .\/ ( p)

Proof of Theorem lhpmcvr6N
StepHypRef Expression
1 lhpmcvr2.b . . 3  |-  B  =  ( Base `  K
)
2 lhpmcvr2.l . . 3  |-  .<_  =  ( le `  K )
3 lhpmcvr2.j . . 3  |-  .\/  =  ( join `  K )
4 lhpmcvr2.m . . 3  |-  ./\  =  ( meet `  K )
5 lhpmcvr2.a . . 3  |-  A  =  ( Atoms `  K )
6 lhpmcvr2.h . . 3  |-  H  =  ( LHyp `  K
)
71, 2, 3, 4, 5, 6lhpmcvr5N 30192 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Y  e.  B  /\  ( X  ./\  Y ) 
.<_  W ) )  ->  E. p  e.  A  ( -.  p  .<_  W  /\  -.  p  .<_  Y  /\  ( p  .\/  ( X  ./\  W ) )  =  X ) )
8 simp31 993 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Y  e.  B  /\  ( X 
./\  Y )  .<_  W ) )  /\  p  e.  A  /\  ( -.  p  .<_  W  /\  -.  p  .<_  Y  /\  ( p  .\/  ( X  ./\  W ) )  =  X ) )  ->  -.  p  .<_  W )
9 simp32 994 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Y  e.  B  /\  ( X 
./\  Y )  .<_  W ) )  /\  p  e.  A  /\  ( -.  p  .<_  W  /\  -.  p  .<_  Y  /\  ( p  .\/  ( X  ./\  W ) )  =  X ) )  ->  -.  p  .<_  Y )
10 simp11l 1068 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Y  e.  B  /\  ( X 
./\  Y )  .<_  W ) )  /\  p  e.  A  /\  ( -.  p  .<_  W  /\  -.  p  .<_  Y  /\  ( p  .\/  ( X  ./\  W ) )  =  X ) )  ->  K  e.  HL )
11 hllat 29529 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  Lat )
1210, 11syl 16 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Y  e.  B  /\  ( X 
./\  Y )  .<_  W ) )  /\  p  e.  A  /\  ( -.  p  .<_  W  /\  -.  p  .<_  Y  /\  ( p  .\/  ( X  ./\  W ) )  =  X ) )  ->  K  e.  Lat )
131, 5atbase 29455 . . . . . . . 8  |-  ( p  e.  A  ->  p  e.  B )
14133ad2ant2 979 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Y  e.  B  /\  ( X 
./\  Y )  .<_  W ) )  /\  p  e.  A  /\  ( -.  p  .<_  W  /\  -.  p  .<_  Y  /\  ( p  .\/  ( X  ./\  W ) )  =  X ) )  ->  p  e.  B )
15 simp12l 1070 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Y  e.  B  /\  ( X 
./\  Y )  .<_  W ) )  /\  p  e.  A  /\  ( -.  p  .<_  W  /\  -.  p  .<_  Y  /\  ( p  .\/  ( X  ./\  W ) )  =  X ) )  ->  X  e.  B )
16 simp11r 1069 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Y  e.  B  /\  ( X 
./\  Y )  .<_  W ) )  /\  p  e.  A  /\  ( -.  p  .<_  W  /\  -.  p  .<_  Y  /\  ( p  .\/  ( X  ./\  W ) )  =  X ) )  ->  W  e.  H )
171, 6lhpbase 30163 . . . . . . . . 9  |-  ( W  e.  H  ->  W  e.  B )
1816, 17syl 16 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Y  e.  B  /\  ( X 
./\  Y )  .<_  W ) )  /\  p  e.  A  /\  ( -.  p  .<_  W  /\  -.  p  .<_  Y  /\  ( p  .\/  ( X  ./\  W ) )  =  X ) )  ->  W  e.  B )
191, 4latmcl 14400 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  W  e.  B )  ->  ( X  ./\  W
)  e.  B )
2012, 15, 18, 19syl3anc 1184 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Y  e.  B  /\  ( X 
./\  Y )  .<_  W ) )  /\  p  e.  A  /\  ( -.  p  .<_  W  /\  -.  p  .<_  Y  /\  ( p  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( X  ./\ 
W )  e.  B
)
211, 2, 3latlej1 14409 . . . . . . 7  |-  ( ( K  e.  Lat  /\  p  e.  B  /\  ( X  ./\  W )  e.  B )  ->  p  .<_  ( p  .\/  ( X  ./\  W ) ) )
2212, 14, 20, 21syl3anc 1184 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Y  e.  B  /\  ( X 
./\  Y )  .<_  W ) )  /\  p  e.  A  /\  ( -.  p  .<_  W  /\  -.  p  .<_  Y  /\  ( p  .\/  ( X  ./\  W ) )  =  X ) )  ->  p  .<_  ( p  .\/  ( X 
./\  W ) ) )
23 simp33 995 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Y  e.  B  /\  ( X 
./\  Y )  .<_  W ) )  /\  p  e.  A  /\  ( -.  p  .<_  W  /\  -.  p  .<_  Y  /\  ( p  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( p  .\/  ( X  ./\  W
) )  =  X )
2422, 23breqtrd 4170 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Y  e.  B  /\  ( X 
./\  Y )  .<_  W ) )  /\  p  e.  A  /\  ( -.  p  .<_  W  /\  -.  p  .<_  Y  /\  ( p  .\/  ( X  ./\  W ) )  =  X ) )  ->  p  .<_  X )
258, 9, 243jca 1134 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Y  e.  B  /\  ( X 
./\  Y )  .<_  W ) )  /\  p  e.  A  /\  ( -.  p  .<_  W  /\  -.  p  .<_  Y  /\  ( p  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( -.  p  .<_  W  /\  -.  p  .<_  Y  /\  p  .<_  X ) )
26253expia 1155 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Y  e.  B  /\  ( X 
./\  Y )  .<_  W ) )  /\  p  e.  A )  ->  ( ( -.  p  .<_  W  /\  -.  p  .<_  Y  /\  ( p 
.\/  ( X  ./\  W ) )  =  X )  ->  ( -.  p  .<_  W  /\  -.  p  .<_  Y  /\  p  .<_  X ) ) )
2726reximdva 2754 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Y  e.  B  /\  ( X  ./\  Y ) 
.<_  W ) )  -> 
( E. p  e.  A  ( -.  p  .<_  W  /\  -.  p  .<_  Y  /\  ( p 
.\/  ( X  ./\  W ) )  =  X )  ->  E. p  e.  A  ( -.  p  .<_  W  /\  -.  p  .<_  Y  /\  p  .<_  X ) ) )
287, 27mpd 15 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Y  e.  B  /\  ( X  ./\  Y ) 
.<_  W ) )  ->  E. p  e.  A  ( -.  p  .<_  W  /\  -.  p  .<_  Y  /\  p  .<_  X ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   E.wrex 2643   class class class wbr 4146   ` cfv 5387  (class class class)co 6013   Basecbs 13389   lecple 13456   joincjn 14321   meetcmee 14322   Latclat 14394   Atomscatm 29429   HLchlt 29516   LHypclh 30149
This theorem is referenced by:  dihmeetlem20N  31492
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-undef 6472  df-riota 6478  df-poset 14323  df-plt 14335  df-lub 14351  df-glb 14352  df-join 14353  df-meet 14354  df-p0 14388  df-p1 14389  df-lat 14395  df-clat 14457  df-oposet 29342  df-ol 29344  df-oml 29345  df-covers 29432  df-ats 29433  df-atl 29464  df-cvlat 29488  df-hlat 29517  df-lhyp 30153
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