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Theorem lhpexle2lem 30491
Description: Lemma for lhpexle2 30492. (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 960 . . . 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 30490 . . . 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 985 . . . . . 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 986 . . . . . 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 958 . . . . . . 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 2589 . . . . . 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 1134 . . . . 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 1155 . . . 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 2777 . . 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 1008 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  A  /\  X  .<_  W )  /\  ( Y  e.  A  /\  Y  .<_  W ) )  /\  X  =/= 
Y )  ->  K  e.  HL )
16 simpl2l 1010 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  A  /\  X  .<_  W )  /\  ( Y  e.  A  /\  Y  .<_  W ) )  /\  X  =/= 
Y )  ->  X  e.  A )
17 simpl3l 1012 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  A  /\  X  .<_  W )  /\  ( Y  e.  A  /\  Y  .<_  W ) )  /\  X  =/= 
Y )  ->  Y  e.  A )
18 simpr 448 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  A  /\  X  .<_  W )  /\  ( Y  e.  A  /\  Y  .<_  W ) )  /\  X  =/= 
Y )  ->  X  =/=  Y )
19 eqid 2404 . . . . 5  |-  ( join `  K )  =  (
join `  K )
202, 19, 3hlsupr 29868 . . . 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 1187 . . 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 2404 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  K )
23 simpl1l 1008 . . . . . . . . 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 29846 . . . . . . . . 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 740 . . . . . . . . 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 29772 . . . . . . . . 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 1010 . . . . . . . . 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 1012 . . . . . . . . 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 29849 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  X  e.  A  /\  Y  e.  A )  ->  ( X ( join `  K ) Y )  e.  ( Base `  K
) )
3223, 29, 30, 31syl3anc 1184 . . . . . . . 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 1009 . . . . . . . . 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 30480 . . . . . . . . 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 1007 . . . . . . . 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 1011 . . . . . . . . 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 1013 . . . . . . . . 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 29772 . . . . . . . . . . 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 29772 . . . . . . . . . . 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 14446 . . . . . . . . . 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 1186 . . . . . . . . 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 888 . . . . . . . 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 14442 . . . . . . 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 1005 . . . . . . 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 1006 . . . . . . 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 1134 . . . . . 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 597 . . . . 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 422 . . . 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 2778 . . 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 2645 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 set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567   E.wrex 2667   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   Basecbs 13424   lecple 13491   joincjn 14356   Latclat 14429   Atomscatm 29746   HLchlt 29833   LHypclh 30466
This theorem is referenced by:  lhpexle2  30492
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-undef 6502  df-riota 6508  df-poset 14358  df-plt 14370  df-lub 14386  df-glb 14387  df-join 14388  df-meet 14389  df-p0 14423  df-p1 14424  df-lat 14430  df-clat 14492  df-oposet 29659  df-ol 29661  df-oml 29662  df-covers 29749  df-ats 29750  df-atl 29781  df-cvlat 29805  df-hlat 29834  df-lhyp 30470
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