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Theorem lhpexle2lem 33493
Description: Lemma for lhpexle2 33494. (Contributed by NM, 19-Jun-2013.)
Hypotheses
Ref Expression
lhpex1.l  |-  .<_  =  ( le `  K )
lhpex1.a  |-  A  =  ( Atoms `  K )
lhpex1.h  |-  H  =  ( LHyp `  K
)
Assertion
Ref Expression
lhpexle2lem  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  A  /\  X  .<_  W )  /\  ( Y  e.  A  /\  Y  .<_  W ) )  ->  E. p  e.  A  ( p  .<_  W  /\  p  =/=  X  /\  p  =/=  Y ) )
Distinct variable groups:    .<_ , p    A, p    H, p    K, p    W, p    X, p    Y, p

Proof of Theorem lhpexle2lem
StepHypRef Expression
1 simpl1 991 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  A  /\  X  .<_  W )  /\  ( Y  e.  A  /\  Y  .<_  W ) )  /\  X  =  Y )  ->  ( K  e.  HL  /\  W  e.  H ) )
2 lhpex1.l . . . . 5  |-  .<_  =  ( le `  K )
3 lhpex1.a . . . . 5  |-  A  =  ( Atoms `  K )
4 lhpex1.h . . . . 5  |-  H  =  ( LHyp `  K
)
52, 3, 4lhpexle1 33492 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  A  ( p  .<_  W  /\  p  =/=  X ) )
61, 5syl 16 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  A  /\  X  .<_  W )  /\  ( Y  e.  A  /\  Y  .<_  W ) )  /\  X  =  Y )  ->  E. p  e.  A  ( p  .<_  W  /\  p  =/= 
X ) )
7 simp3l 1016 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  A  /\  X  .<_  W )  /\  ( Y  e.  A  /\  Y  .<_  W ) )  /\  X  =  Y  /\  ( p 
.<_  W  /\  p  =/= 
X ) )  ->  p  .<_  W )
8 simp3r 1017 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  A  /\  X  .<_  W )  /\  ( Y  e.  A  /\  Y  .<_  W ) )  /\  X  =  Y  /\  ( p 
.<_  W  /\  p  =/= 
X ) )  ->  p  =/=  X )
9 simp2 989 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  A  /\  X  .<_  W )  /\  ( Y  e.  A  /\  Y  .<_  W ) )  /\  X  =  Y  /\  ( p 
.<_  W  /\  p  =/= 
X ) )  ->  X  =  Y )
108, 9neeqtrd 2625 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  A  /\  X  .<_  W )  /\  ( Y  e.  A  /\  Y  .<_  W ) )  /\  X  =  Y  /\  ( p 
.<_  W  /\  p  =/= 
X ) )  ->  p  =/=  Y )
117, 8, 103jca 1168 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  A  /\  X  .<_  W )  /\  ( Y  e.  A  /\  Y  .<_  W ) )  /\  X  =  Y  /\  ( p 
.<_  W  /\  p  =/= 
X ) )  -> 
( p  .<_  W  /\  p  =/=  X  /\  p  =/=  Y ) )
12113expia 1189 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  A  /\  X  .<_  W )  /\  ( Y  e.  A  /\  Y  .<_  W ) )  /\  X  =  Y )  ->  (
( p  .<_  W  /\  p  =/=  X )  -> 
( p  .<_  W  /\  p  =/=  X  /\  p  =/=  Y ) ) )
1312reximdv 2822 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  A  /\  X  .<_  W )  /\  ( Y  e.  A  /\  Y  .<_  W ) )  /\  X  =  Y )  ->  ( E. p  e.  A  ( p  .<_  W  /\  p  =/=  X )  ->  E. p  e.  A  ( p  .<_  W  /\  p  =/=  X  /\  p  =/=  Y ) ) )
146, 13mpd 15 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  A  /\  X  .<_  W )  /\  ( Y  e.  A  /\  Y  .<_  W ) )  /\  X  =  Y )  ->  E. p  e.  A  ( p  .<_  W  /\  p  =/= 
X  /\  p  =/=  Y ) )
15 simpl1l 1039 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  A  /\  X  .<_  W )  /\  ( Y  e.  A  /\  Y  .<_  W ) )  /\  X  =/= 
Y )  ->  K  e.  HL )
16 simpl2l 1041 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  A  /\  X  .<_  W )  /\  ( Y  e.  A  /\  Y  .<_  W ) )  /\  X  =/= 
Y )  ->  X  e.  A )
17 simpl3l 1043 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  A  /\  X  .<_  W )  /\  ( Y  e.  A  /\  Y  .<_  W ) )  /\  X  =/= 
Y )  ->  Y  e.  A )
18 simpr 461 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  A  /\  X  .<_  W )  /\  ( Y  e.  A  /\  Y  .<_  W ) )  /\  X  =/= 
Y )  ->  X  =/=  Y )
19 eqid 2438 . . . . 5  |-  ( join `  K )  =  (
join `  K )
202, 19, 3hlsupr 32870 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  A  /\  Y  e.  A )  /\  X  =/=  Y
)  ->  E. p  e.  A  ( p  =/=  X  /\  p  =/= 
Y  /\  p  .<_  ( X ( join `  K
) Y ) ) )
2115, 16, 17, 18, 20syl31anc 1221 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  A  /\  X  .<_  W )  /\  ( Y  e.  A  /\  Y  .<_  W ) )  /\  X  =/= 
Y )  ->  E. p  e.  A  ( p  =/=  X  /\  p  =/= 
Y  /\  p  .<_  ( X ( join `  K
) Y ) ) )
22 eqid 2438 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  K )
23 simpl1l 1039 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  A  /\  X  .<_  W )  /\  ( Y  e.  A  /\  Y  .<_  W ) )  /\  ( ( X  =/=  Y  /\  p  e.  A )  /\  ( p  =/=  X  /\  p  =/=  Y  /\  p  .<_  ( X ( join `  K
) Y ) ) ) )  ->  K  e.  HL )
24 hllat 32848 . . . . . . . . 9  |-  ( K  e.  HL  ->  K  e.  Lat )
2523, 24syl 16 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  A  /\  X  .<_  W )  /\  ( Y  e.  A  /\  Y  .<_  W ) )  /\  ( ( X  =/=  Y  /\  p  e.  A )  /\  ( p  =/=  X  /\  p  =/=  Y  /\  p  .<_  ( X ( join `  K
) Y ) ) ) )  ->  K  e.  Lat )
26 simprlr 762 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  A  /\  X  .<_  W )  /\  ( Y  e.  A  /\  Y  .<_  W ) )  /\  ( ( X  =/=  Y  /\  p  e.  A )  /\  ( p  =/=  X  /\  p  =/=  Y  /\  p  .<_  ( X ( join `  K
) Y ) ) ) )  ->  p  e.  A )
2722, 3atbase 32774 . . . . . . . . 9  |-  ( p  e.  A  ->  p  e.  ( Base `  K
) )
2826, 27syl 16 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  A  /\  X  .<_  W )  /\  ( Y  e.  A  /\  Y  .<_  W ) )  /\  ( ( X  =/=  Y  /\  p  e.  A )  /\  ( p  =/=  X  /\  p  =/=  Y  /\  p  .<_  ( X ( join `  K
) Y ) ) ) )  ->  p  e.  ( Base `  K
) )
29 simpl2l 1041 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  A  /\  X  .<_  W )  /\  ( Y  e.  A  /\  Y  .<_  W ) )  /\  ( ( X  =/=  Y  /\  p  e.  A )  /\  ( p  =/=  X  /\  p  =/=  Y  /\  p  .<_  ( X ( join `  K
) Y ) ) ) )  ->  X  e.  A )
30 simpl3l 1043 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  A  /\  X  .<_  W )  /\  ( Y  e.  A  /\  Y  .<_  W ) )  /\  ( ( X  =/=  Y  /\  p  e.  A )  /\  ( p  =/=  X  /\  p  =/=  Y  /\  p  .<_  ( X ( join `  K
) Y ) ) ) )  ->  Y  e.  A )
3122, 19, 3hlatjcl 32851 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  X  e.  A  /\  Y  e.  A )  ->  ( X ( join `  K ) Y )  e.  ( Base `  K
) )
3223, 29, 30, 31syl3anc 1218 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  A  /\  X  .<_  W )  /\  ( Y  e.  A  /\  Y  .<_  W ) )  /\  ( ( X  =/=  Y  /\  p  e.  A )  /\  ( p  =/=  X  /\  p  =/=  Y  /\  p  .<_  ( X ( join `  K
) Y ) ) ) )  ->  ( X ( join `  K
) Y )  e.  ( Base `  K
) )
33 simpl1r 1040 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  A  /\  X  .<_  W )  /\  ( Y  e.  A  /\  Y  .<_  W ) )  /\  ( ( X  =/=  Y  /\  p  e.  A )  /\  ( p  =/=  X  /\  p  =/=  Y  /\  p  .<_  ( X ( join `  K
) Y ) ) ) )  ->  W  e.  H )
3422, 4lhpbase 33482 . . . . . . . . 9  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
3533, 34syl 16 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  A  /\  X  .<_  W )  /\  ( Y  e.  A  /\  Y  .<_  W ) )  /\  ( ( X  =/=  Y  /\  p  e.  A )  /\  ( p  =/=  X  /\  p  =/=  Y  /\  p  .<_  ( X ( join `  K
) Y ) ) ) )  ->  W  e.  ( Base `  K
) )
36 simprr3 1038 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  A  /\  X  .<_  W )  /\  ( Y  e.  A  /\  Y  .<_  W ) )  /\  ( ( X  =/=  Y  /\  p  e.  A )  /\  ( p  =/=  X  /\  p  =/=  Y  /\  p  .<_  ( X ( join `  K
) Y ) ) ) )  ->  p  .<_  ( X ( join `  K ) Y ) )
37 simpl2r 1042 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  A  /\  X  .<_  W )  /\  ( Y  e.  A  /\  Y  .<_  W ) )  /\  ( ( X  =/=  Y  /\  p  e.  A )  /\  ( p  =/=  X  /\  p  =/=  Y  /\  p  .<_  ( X ( join `  K
) Y ) ) ) )  ->  X  .<_  W )
38 simpl3r 1044 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  A  /\  X  .<_  W )  /\  ( Y  e.  A  /\  Y  .<_  W ) )  /\  ( ( X  =/=  Y  /\  p  e.  A )  /\  ( p  =/=  X  /\  p  =/=  Y  /\  p  .<_  ( X ( join `  K
) Y ) ) ) )  ->  Y  .<_  W )
3922, 3atbase 32774 . . . . . . . . . . 11  |-  ( X  e.  A  ->  X  e.  ( Base `  K
) )
4029, 39syl 16 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  A  /\  X  .<_  W )  /\  ( Y  e.  A  /\  Y  .<_  W ) )  /\  ( ( X  =/=  Y  /\  p  e.  A )  /\  ( p  =/=  X  /\  p  =/=  Y  /\  p  .<_  ( X ( join `  K
) Y ) ) ) )  ->  X  e.  ( Base `  K
) )
4122, 3atbase 32774 . . . . . . . . . . 11  |-  ( Y  e.  A  ->  Y  e.  ( Base `  K
) )
4230, 41syl 16 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  A  /\  X  .<_  W )  /\  ( Y  e.  A  /\  Y  .<_  W ) )  /\  ( ( X  =/=  Y  /\  p  e.  A )  /\  ( p  =/=  X  /\  p  =/=  Y  /\  p  .<_  ( X ( join `  K
) Y ) ) ) )  ->  Y  e.  ( Base `  K
) )
4322, 2, 19latjle12 15224 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  ( X  e.  ( Base `  K )  /\  Y  e.  ( Base `  K )  /\  W  e.  ( Base `  K
) ) )  -> 
( ( X  .<_  W  /\  Y  .<_  W )  <-> 
( X ( join `  K ) Y ) 
.<_  W ) )
4425, 40, 42, 35, 43syl13anc 1220 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  A  /\  X  .<_  W )  /\  ( Y  e.  A  /\  Y  .<_  W ) )  /\  ( ( X  =/=  Y  /\  p  e.  A )  /\  ( p  =/=  X  /\  p  =/=  Y  /\  p  .<_  ( X ( join `  K
) Y ) ) ) )  ->  (
( X  .<_  W  /\  Y  .<_  W )  <->  ( X
( join `  K ) Y )  .<_  W ) )
4537, 38, 44mpbi2and 912 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  A  /\  X  .<_  W )  /\  ( Y  e.  A  /\  Y  .<_  W ) )  /\  ( ( X  =/=  Y  /\  p  e.  A )  /\  ( p  =/=  X  /\  p  =/=  Y  /\  p  .<_  ( X ( join `  K
) Y ) ) ) )  ->  ( X ( join `  K
) Y )  .<_  W )
4622, 2, 25, 28, 32, 35, 36, 45lattrd 15220 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  A  /\  X  .<_  W )  /\  ( Y  e.  A  /\  Y  .<_  W ) )  /\  ( ( X  =/=  Y  /\  p  e.  A )  /\  ( p  =/=  X  /\  p  =/=  Y  /\  p  .<_  ( X ( join `  K
) Y ) ) ) )  ->  p  .<_  W )
47 simprr1 1036 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  A  /\  X  .<_  W )  /\  ( Y  e.  A  /\  Y  .<_  W ) )  /\  ( ( X  =/=  Y  /\  p  e.  A )  /\  ( p  =/=  X  /\  p  =/=  Y  /\  p  .<_  ( X ( join `  K
) Y ) ) ) )  ->  p  =/=  X )
48 simprr2 1037 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  A  /\  X  .<_  W )  /\  ( Y  e.  A  /\  Y  .<_  W ) )  /\  ( ( X  =/=  Y  /\  p  e.  A )  /\  ( p  =/=  X  /\  p  =/=  Y  /\  p  .<_  ( X ( join `  K
) Y ) ) ) )  ->  p  =/=  Y )
4946, 47, 483jca 1168 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  A  /\  X  .<_  W )  /\  ( Y  e.  A  /\  Y  .<_  W ) )  /\  ( ( X  =/=  Y  /\  p  e.  A )  /\  ( p  =/=  X  /\  p  =/=  Y  /\  p  .<_  ( X ( join `  K
) Y ) ) ) )  ->  (
p  .<_  W  /\  p  =/=  X  /\  p  =/= 
Y ) )
5049exp44 613 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  A  /\  X  .<_  W )  /\  ( Y  e.  A  /\  Y  .<_  W ) )  -> 
( X  =/=  Y  ->  ( p  e.  A  ->  ( ( p  =/= 
X  /\  p  =/=  Y  /\  p  .<_  ( X ( join `  K
) Y ) )  ->  ( p  .<_  W  /\  p  =/=  X  /\  p  =/=  Y
) ) ) ) )
5150imp31 432 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  A  /\  X  .<_  W )  /\  ( Y  e.  A  /\  Y  .<_  W ) )  /\  X  =/= 
Y )  /\  p  e.  A )  ->  (
( p  =/=  X  /\  p  =/=  Y  /\  p  .<_  ( X ( join `  K
) Y ) )  ->  ( p  .<_  W  /\  p  =/=  X  /\  p  =/=  Y
) ) )
5251reximdva 2823 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  A  /\  X  .<_  W )  /\  ( Y  e.  A  /\  Y  .<_  W ) )  /\  X  =/= 
Y )  ->  ( E. p  e.  A  ( p  =/=  X  /\  p  =/=  Y  /\  p  .<_  ( X ( join `  K
) Y ) )  ->  E. p  e.  A  ( p  .<_  W  /\  p  =/=  X  /\  p  =/=  Y ) ) )
5321, 52mpd 15 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  A  /\  X  .<_  W )  /\  ( Y  e.  A  /\  Y  .<_  W ) )  /\  X  =/= 
Y )  ->  E. p  e.  A  ( p  .<_  W  /\  p  =/= 
X  /\  p  =/=  Y ) )
5414, 53pm2.61dane 2684 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  A  /\  X  .<_  W )  /\  ( Y  e.  A  /\  Y  .<_  W ) )  ->  E. p  e.  A  ( p  .<_  W  /\  p  =/=  X  /\  p  =/=  Y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2601   E.wrex 2711   class class class wbr 4287   ` cfv 5413  (class class class)co 6086   Basecbs 14166   lecple 14237   joincjn 15106   Latclat 15207   Atomscatm 32748   HLchlt 32835   LHypclh 33468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-poset 15108  df-plt 15120  df-lub 15136  df-glb 15137  df-join 15138  df-meet 15139  df-p0 15201  df-p1 15202  df-lat 15208  df-clat 15270  df-oposet 32661  df-ol 32663  df-oml 32664  df-covers 32751  df-ats 32752  df-atl 32783  df-cvlat 32807  df-hlat 32836  df-lhyp 33472
This theorem is referenced by:  lhpexle2  33494
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