Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lhpmcvr5N Unicode version

Theorem lhpmcvr5N 30838
Description: Specialization of lhpmcvr2 30835. (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
lhpmcvr5N  |-  ( ( ( 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 ) )
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 lhpmcvr5N
StepHypRef Expression
1 lhpmcvr2.b . . . 4  |-  B  =  ( Base `  K
)
2 lhpmcvr2.l . . . 4  |-  .<_  =  ( le `  K )
3 lhpmcvr2.j . . . 4  |-  .\/  =  ( join `  K )
4 lhpmcvr2.m . . . 4  |-  ./\  =  ( meet `  K )
5 lhpmcvr2.a . . . 4  |-  A  =  ( Atoms `  K )
6 lhpmcvr2.h . . . 4  |-  H  =  ( LHyp `  K
)
71, 2, 3, 4, 5, 6lhpmcvr2 30835 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  ->  E. p  e.  A  ( -.  p  .<_  W  /\  ( p  .\/  ( X  ./\  W ) )  =  X ) )
873adant3 975 . 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  .\/  ( X  ./\  W ) )  =  X ) )
9 simp3l 983 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Y  e.  B  /\  ( X 
./\  Y )  .<_  W ) )  /\  p  e.  A  /\  ( -.  p  .<_  W  /\  ( p  .\/  ( X  ./\  W ) )  =  X ) )  ->  -.  p  .<_  W )
10 simp11 985 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Y  e.  B  /\  ( X 
./\  Y )  .<_  W ) )  /\  p  e.  A  /\  ( -.  p  .<_  W  /\  ( p  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
11 simp12 986 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Y  e.  B  /\  ( X 
./\  Y )  .<_  W ) )  /\  p  e.  A  /\  ( -.  p  .<_  W  /\  ( p  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( X  e.  B  /\  -.  X  .<_  W ) )
12 simp2 956 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Y  e.  B  /\  ( X 
./\  Y )  .<_  W ) )  /\  p  e.  A  /\  ( -.  p  .<_  W  /\  ( p  .\/  ( X  ./\  W ) )  =  X ) )  ->  p  e.  A )
1312, 9jca 518 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Y  e.  B  /\  ( X 
./\  Y )  .<_  W ) )  /\  p  e.  A  /\  ( -.  p  .<_  W  /\  ( p  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( p  e.  A  /\  -.  p  .<_  W ) )
14 simp13l 1070 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Y  e.  B  /\  ( X 
./\  Y )  .<_  W ) )  /\  p  e.  A  /\  ( -.  p  .<_  W  /\  ( p  .\/  ( X  ./\  W ) )  =  X ) )  ->  Y  e.  B )
15 simp13r 1071 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Y  e.  B  /\  ( X 
./\  Y )  .<_  W ) )  /\  p  e.  A  /\  ( -.  p  .<_  W  /\  ( p  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( X  ./\ 
Y )  .<_  W )
16 simp11l 1066 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Y  e.  B  /\  ( X 
./\  Y )  .<_  W ) )  /\  p  e.  A  /\  ( -.  p  .<_  W  /\  ( p  .\/  ( X  ./\  W ) )  =  X ) )  ->  K  e.  HL )
17 hllat 30175 . . . . . . . . 9  |-  ( K  e.  HL  ->  K  e.  Lat )
1816, 17syl 15 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Y  e.  B  /\  ( X 
./\  Y )  .<_  W ) )  /\  p  e.  A  /\  ( -.  p  .<_  W  /\  ( p  .\/  ( X  ./\  W ) )  =  X ) )  ->  K  e.  Lat )
191, 5atbase 30101 . . . . . . . . 9  |-  ( p  e.  A  ->  p  e.  B )
20193ad2ant2 977 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Y  e.  B  /\  ( X 
./\  Y )  .<_  W ) )  /\  p  e.  A  /\  ( -.  p  .<_  W  /\  ( p  .\/  ( X  ./\  W ) )  =  X ) )  ->  p  e.  B )
21 simp12l 1068 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Y  e.  B  /\  ( X 
./\  Y )  .<_  W ) )  /\  p  e.  A  /\  ( -.  p  .<_  W  /\  ( p  .\/  ( X  ./\  W ) )  =  X ) )  ->  X  e.  B )
22 simp11r 1067 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Y  e.  B  /\  ( X 
./\  Y )  .<_  W ) )  /\  p  e.  A  /\  ( -.  p  .<_  W  /\  ( p  .\/  ( X  ./\  W ) )  =  X ) )  ->  W  e.  H )
231, 6lhpbase 30809 . . . . . . . . . 10  |-  ( W  e.  H  ->  W  e.  B )
2422, 23syl 15 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Y  e.  B  /\  ( X 
./\  Y )  .<_  W ) )  /\  p  e.  A  /\  ( -.  p  .<_  W  /\  ( p  .\/  ( X  ./\  W ) )  =  X ) )  ->  W  e.  B )
251, 4latmcl 14173 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  W  e.  B )  ->  ( X  ./\  W
)  e.  B )
2618, 21, 24, 25syl3anc 1182 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Y  e.  B  /\  ( X 
./\  Y )  .<_  W ) )  /\  p  e.  A  /\  ( -.  p  .<_  W  /\  ( p  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( X  ./\ 
W )  e.  B
)
271, 2, 3latlej1 14182 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  p  e.  B  /\  ( X  ./\  W )  e.  B )  ->  p  .<_  ( p  .\/  ( X  ./\  W ) ) )
2818, 20, 26, 27syl3anc 1182 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Y  e.  B  /\  ( X 
./\  Y )  .<_  W ) )  /\  p  e.  A  /\  ( -.  p  .<_  W  /\  ( p  .\/  ( X  ./\  W ) )  =  X ) )  ->  p  .<_  ( p  .\/  ( X 
./\  W ) ) )
29 simp3r 984 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Y  e.  B  /\  ( X 
./\  Y )  .<_  W ) )  /\  p  e.  A  /\  ( -.  p  .<_  W  /\  ( p  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( p  .\/  ( X  ./\  W
) )  =  X )
3028, 29breqtrd 4063 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Y  e.  B  /\  ( X 
./\  Y )  .<_  W ) )  /\  p  e.  A  /\  ( -.  p  .<_  W  /\  ( p  .\/  ( X  ./\  W ) )  =  X ) )  ->  p  .<_  X )
311, 2, 3, 4, 5, 6lhpmcvr4N 30837 . . . . . 6  |-  ( ( ( 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 )
3210, 11, 13, 14, 15, 30, 31syl123anc 1199 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Y  e.  B  /\  ( X 
./\  Y )  .<_  W ) )  /\  p  e.  A  /\  ( -.  p  .<_  W  /\  ( p  .\/  ( X  ./\  W ) )  =  X ) )  ->  -.  p  .<_  Y )
339, 32, 293jca 1132 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Y  e.  B  /\  ( X 
./\  Y )  .<_  W ) )  /\  p  e.  A  /\  ( -.  p  .<_  W  /\  ( p  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( -.  p  .<_  W  /\  -.  p  .<_  Y  /\  (
p  .\/  ( X  ./\ 
W ) )  =  X ) )
34333expia 1153 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( Y  e.  B  /\  ( X 
./\  Y )  .<_  W ) )  /\  p  e.  A )  ->  ( ( -.  p  .<_  W  /\  ( p 
.\/  ( X  ./\  W ) )  =  X )  ->  ( -.  p  .<_  W  /\  -.  p  .<_  Y  /\  (
p  .\/  ( X  ./\ 
W ) )  =  X ) ) )
3534reximdva 2668 . 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 
.\/  ( X  ./\  W ) )  =  X )  ->  E. p  e.  A  ( -.  p  .<_  W  /\  -.  p  .<_  Y  /\  (
p  .\/  ( X  ./\ 
W ) )  =  X ) ) )
368, 35mpd 14 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  ./\  W ) )  =  X ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   E.wrex 2557   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Basecbs 13164   lecple 13231   joincjn 14094   meetcmee 14095   Latclat 14167   Atomscatm 30075   HLchlt 30162   LHypclh 30795
This theorem is referenced by:  lhpmcvr6N  30839
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-poset 14096  df-plt 14108  df-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-p0 14161  df-p1 14162  df-lat 14168  df-clat 14230  df-oposet 29988  df-ol 29990  df-oml 29991  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163  df-lhyp 30799
  Copyright terms: Public domain W3C validator