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

Theorem lhpmcvr5N 36148
Description: Specialization of lhpmcvr2 36145. (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 36145 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  -.  X  .<_  W ) )  ->  E. p  e.  A  ( -.  p  .<_  W  /\  ( p  .\/  ( X  ./\  W ) )  =  X ) )
873adant3 1014 . 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 1022 . . . . 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 1024 . . . . . 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 1025 . . . . . 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 995 . . . . . . 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 530 . . . . . 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 1109 . . . . . 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 1110 . . . . . 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 1105 . . . . . . . . 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 35485 . . . . . . . . 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 35411 . . . . . . . . 9  |-  ( p  e.  A  ->  p  e.  B )
20193ad2ant2 1016 . . . . . . . 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 1107 . . . . . . . . 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 1106 . . . . . . . . . 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 36119 . . . . . . . . . 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 15881 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  W  e.  B )  ->  ( X  ./\  W
)  e.  B )
2618, 21, 24, 25syl3anc 1226 . . . . . . . 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 15889 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  p  e.  B  /\  ( X  ./\  W )  e.  B )  ->  p  .<_  ( p  .\/  ( X  ./\  W ) ) )
2818, 20, 26, 27syl3anc 1226 . . . . . . 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 1023 . . . . . . 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 4463 . . . . . 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 36147 . . . . . 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 1243 . . . . 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 1174 . . . 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 1196 . . 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 2929 . 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823   E.wrex 2805   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   Basecbs 14716   lecple 14791   joincjn 15772   meetcmee 15773   Latclat 15874   Atomscatm 35385   HLchlt 35472   LHypclh 36105
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-preset 15756  df-poset 15774  df-plt 15787  df-lub 15803  df-glb 15804  df-join 15805  df-meet 15806  df-p0 15868  df-p1 15869  df-lat 15875  df-clat 15937  df-oposet 35298  df-ol 35300  df-oml 35301  df-covers 35388  df-ats 35389  df-atl 35420  df-cvlat 35444  df-hlat 35473  df-lhyp 36109
This theorem is referenced by:  lhpmcvr6N  36149
  Copyright terms: Public domain W3C validator