Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lhpmcvr6N Structured version   Unicode version

Theorem lhpmcvr6N 30887
Description: Specialization of lhpmcvr2 30883. (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 30886 . 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 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  .<_  W )
9 simp32 995 . . . . 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 1069 . . . . . . . 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 30223 . . . . . . . 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 30149 . . . . . . . 8  |-  ( p  e.  A  ->  p  e.  B )
14133ad2ant2 980 . . . . . . 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 1071 . . . . . . . 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 1070 . . . . . . . . 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 30857 . . . . . . . . 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 14482 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  W  e.  B )  ->  ( X  ./\  W
)  e.  B )
2012, 15, 18, 19syl3anc 1185 . . . . . . 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 14491 . . . . . . 7  |-  ( ( K  e.  Lat  /\  p  e.  B  /\  ( X  ./\  W )  e.  B )  ->  p  .<_  ( p  .\/  ( X  ./\  W ) ) )
2212, 14, 20, 21syl3anc 1185 . . . . . 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 996 . . . . . 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 4238 . . . . 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 1135 . . . 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 1156 . . 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 2820 . 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 360    /\ w3a 937    = wceq 1653    e. wcel 1726   E.wrex 2708   class class class wbr 4214   ` cfv 5456  (class class class)co 6083   Basecbs 13471   lecple 13538   joincjn 14403   meetcmee 14404   Latclat 14476   Atomscatm 30123   HLchlt 30210   LHypclh 30843
This theorem is referenced by:  dihmeetlem20N  32186
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
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 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-undef 6545  df-riota 6551  df-poset 14405  df-plt 14417  df-lub 14433  df-glb 14434  df-join 14435  df-meet 14436  df-p0 14470  df-p1 14471  df-lat 14477  df-clat 14539  df-oposet 30036  df-ol 30038  df-oml 30039  df-covers 30126  df-ats 30127  df-atl 30158  df-cvlat 30182  df-hlat 30211  df-lhyp 30847
  Copyright terms: Public domain W3C validator