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Theorem lhpmcvr5N 34010
Description: Specialization of lhpmcvr2 34007. (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 34007 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  ->  E. p  e.  A  ( -.  p  .<_  W  /\  ( p  .\/  ( X  ./\  W ) )  =  X ) )
873adant3 1008 . 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 1016 . . . . 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 1018 . . . . . 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 1019 . . . . . 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 989 . . . . . . 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 532 . . . . . 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 1103 . . . . . 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 1104 . . . . . 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 1099 . . . . . . . . 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 33347 . . . . . . . . 9  |-  ( K  e.  HL  ->  K  e.  Lat )
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  .\/  ( X  ./\  W ) )  =  X ) )  ->  K  e.  Lat )
191, 5atbase 33273 . . . . . . . . 9  |-  ( p  e.  A  ->  p  e.  B )
20193ad2ant2 1010 . . . . . . . 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 1101 . . . . . . . . 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 1100 . . . . . . . . . 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 33981 . . . . . . . . . 10  |-  ( W  e.  H  ->  W  e.  B )
2422, 23syl 16 . . . . . . . . 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 15342 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  W  e.  B )  ->  ( X  ./\  W
)  e.  B )
2618, 21, 24, 25syl3anc 1219 . . . . . . . 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 15350 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  p  e.  B  /\  ( X  ./\  W )  e.  B )  ->  p  .<_  ( p  .\/  ( X  ./\  W ) ) )
2818, 20, 26, 27syl3anc 1219 . . . . . . 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 1017 . . . . . . 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 4425 . . . . . 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 34009 . . . . . 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 1236 . . . . 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 1168 . . . 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 1190 . . 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 2934 . 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 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  ./\  W ) )  =  X ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   E.wrex 2800   class class class wbr 4401   ` cfv 5527  (class class class)co 6201   Basecbs 14293   lecple 14365   joincjn 15234   meetcmee 15235   Latclat 15335   Atomscatm 33247   HLchlt 33334   LHypclh 33967
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-poset 15236  df-plt 15248  df-lub 15264  df-glb 15265  df-join 15266  df-meet 15267  df-p0 15329  df-p1 15330  df-lat 15336  df-clat 15398  df-oposet 33160  df-ol 33162  df-oml 33163  df-covers 33250  df-ats 33251  df-atl 33282  df-cvlat 33306  df-hlat 33335  df-lhyp 33971
This theorem is referenced by:  lhpmcvr6N  34011
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