Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lhpexle3lem Structured version   Visualization version   Unicode version
Theorem lhpexle3lem 35297
Description: There exists atom under a co-atom different from any 3 other atoms. TODO: study if adant*,simp* usage can be improved. (Contributed by NM, 9-Jul-2013.)
Hypotheses
Ref Expression
lhpex1.l |- .<_ = ( le ` K )
lhpex1.a |- A = ( Atoms ` K )
lhpex1.h |- H = ( LHyp ` K )
Assertion
Ref Expression
lhpexle3lem |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) -> E. p e. A ( p .<_ W /\ ( p =/= X /\ p =/= Y /\ p =/= Z ) ) )
Distinct variable groups:   .<_ , p   A, p   H, p   K, p   W, p   X, p   Y, p   Z, p
Proof of Theorem lhpexle3lem
StepHypRef Expression
1 simpl1 1064 . . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ 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, 4lhpexle2 35296 . . . 4 |- ( ( K e. HL /\ W e. H ) -> E. p e. A ( p .<_ W /\ p =/= X /\ p =/= Z ) )
61, 5syl 17 . . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X = Y ) -> E. p e. A ( p .<_ W /\ p =/= X /\ p =/= Z ) )
7 simp31 1097 . . . . . 6 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X = Y ) /\ p e. A /\ ( p .<_ W /\ p =/= X /\ p =/= Z ) ) -> p .<_ W )
8 simp32 1098 . . . . . . 7 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X = Y ) /\ p e. A /\ ( p .<_ W /\ p =/= X /\ p =/= Z ) ) -> p =/= X )
9 simp1r 1086 . . . . . . . 8 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X = Y ) /\ p e. A /\ ( p .<_ W /\ p =/= X /\ p =/= Z ) ) -> X = Y )
108, 9neeqtrd 2863 . . . . . . 7 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X = Y ) /\ p e. A /\ ( p .<_ W /\ p =/= X /\ p =/= Z ) ) -> p =/= Y )
11 simp33 1099 . . . . . . 7 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X = Y ) /\ p e. A /\ ( p .<_ W /\ p =/= X /\ p =/= Z ) ) -> p =/= Z )
128, 10, 113jca 1242 . . . . . 6 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X = Y ) /\ p e. A /\ ( p .<_ W /\ p =/= X /\ p =/= Z ) ) -> ( p =/= X /\ p =/= Y /\ p =/= Z ) )
137, 12jca 554 . . . . 5 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X = Y ) /\ p e. A /\ ( p .<_ W /\ p =/= X /\ p =/= Z ) ) -> ( p .<_ W /\ ( p =/= X /\ p =/= Y /\ p =/= Z ) ) )
14133exp 1264 . . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X = Y ) -> ( p e. A -> ( ( p .<_ W /\ p =/= X /\ p =/= Z ) -> ( p .<_ W /\ ( p =/= X /\ p =/= Y /\ p =/= Z ) ) ) ) )
1514reximdvai 3015 . . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X = Y ) -> ( E. p e. A ( p .<_ W /\ p =/= X /\ p =/= Z ) -> E. p e. A ( p .<_ W /\ ( p =/= X /\ p =/= Y /\ p =/= Z ) ) ) )
166, 15mpd 15 . 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X = Y ) -> E. p e. A ( p .<_ W /\ ( p =/= X /\ p =/= Y /\ p =/= Z ) ) )
17 simprrr 805 . . . . . 6 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ Z .<_ ( X ( join ` K ) Y ) ) /\ ( p e. A /\ ( -. p .<_ ( X ( join ` K ) Y ) /\ p .<_ W ) ) ) -> p .<_ W )
18 simp11l 1172 . . . . . . . . . 10 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ Z .<_ ( X ( join ` K ) Y ) ) -> K e. HL )
1918adantr 481 . . . . . . . . 9 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ Z .<_ ( X ( join ` K ) Y ) ) /\ ( p e. A /\ ( -. p .<_ ( X ( join ` K ) Y ) /\ p .<_ W ) ) ) -> K e. HL )
20 hllat 34650 . . . . . . . . 9 |- ( K e. HL -> K e. Lat )
2119, 20syl 17 . . . . . . . 8 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ Z .<_ ( X ( join ` K ) Y ) ) /\ ( p e. A /\ ( -. p .<_ ( X ( join ` K ) Y ) /\ p .<_ W ) ) ) -> K e. Lat )
22 eqid 2622 . . . . . . . . . 10 |- ( Base ` K ) = ( Base ` K )
2322, 3atbase 34576 . . . . . . . . 9 |- ( p e. A -> p e. ( Base ` K ) )
2423ad2antrl 764 . . . . . . . 8 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ Z .<_ ( X ( join ` K ) Y ) ) /\ ( p e. A /\ ( -. p .<_ ( X ( join ` K ) Y ) /\ p .<_ W ) ) ) -> p e. ( Base ` K ) )
25 simp121 1193 . . . . . . . . . 10 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ Z .<_ ( X ( join ` K ) Y ) ) -> X e. A )
2625adantr 481 . . . . . . . . 9 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ Z .<_ ( X ( join ` K ) Y ) ) /\ ( p e. A /\ ( -. p .<_ ( X ( join ` K ) Y ) /\ p .<_ W ) ) ) -> X e. A )
2722, 3atbase 34576 . . . . . . . . 9 |- ( X e. A -> X e. ( Base ` K ) )
2826, 27syl 17 . . . . . . . 8 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ Z .<_ ( X ( join ` K ) Y ) ) /\ ( p e. A /\ ( -. p .<_ ( X ( join ` K ) Y ) /\ p .<_ W ) ) ) -> X e. ( Base ` K ) )
29 simp122 1194 . . . . . . . . . 10 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ Z .<_ ( X ( join ` K ) Y ) ) -> Y e. A )
3029adantr 481 . . . . . . . . 9 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ Z .<_ ( X ( join ` K ) Y ) ) /\ ( p e. A /\ ( -. p .<_ ( X ( join ` K ) Y ) /\ p .<_ W ) ) ) -> Y e. A )
3122, 3atbase 34576 . . . . . . . . 9 |- ( Y e. A -> Y e. ( Base ` K ) )
3230, 31syl 17 . . . . . . . 8 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ Z .<_ ( X ( join ` K ) Y ) ) /\ ( p e. A /\ ( -. p .<_ ( X ( join ` K ) Y ) /\ p .<_ W ) ) ) -> Y e. ( Base ` K ) )
33 simprrl 804 . . . . . . . 8 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ Z .<_ ( X ( join ` K ) Y ) ) /\ ( p e. A /\ ( -. p .<_ ( X ( join ` K ) Y ) /\ p .<_ W ) ) ) -> -. p .<_ ( X ( join ` K ) Y ) )
34 eqid 2622 . . . . . . . . 9 |- ( join ` K ) = ( join ` K )
3522, 2, 34latnlej1l 17069 . . . . . . . 8 |- ( ( K e. Lat /\ ( p e. ( Base ` K ) /\ X e. ( Base ` K ) /\ Y e. ( Base ` K ) ) /\ -. p .<_ ( X ( join ` K ) Y ) ) -> p =/= X )
3621, 24, 28, 32, 33, 35syl131anc 1339 . . . . . . 7 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ Z .<_ ( X ( join ` K ) Y ) ) /\ ( p e. A /\ ( -. p .<_ ( X ( join ` K ) Y ) /\ p .<_ W ) ) ) -> p =/= X )
3722, 2, 34latnlej1r 17070 . . . . . . . 8 |- ( ( K e. Lat /\ ( p e. ( Base ` K ) /\ X e. ( Base ` K ) /\ Y e. ( Base ` K ) ) /\ -. p .<_ ( X ( join ` K ) Y ) ) -> p =/= Y )
3821, 24, 28, 32, 33, 37syl131anc 1339 . . . . . . 7 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ Z .<_ ( X ( join ` K ) Y ) ) /\ ( p e. A /\ ( -. p .<_ ( X ( join ` K ) Y ) /\ p .<_ W ) ) ) -> p =/= Y )
39 simpl3 1066 . . . . . . . 8 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ Z .<_ ( X ( join ` K ) Y ) ) /\ ( p e. A /\ ( -. p .<_ ( X ( join ` K ) Y ) /\ p .<_ W ) ) ) -> Z .<_ ( X ( join ` K ) Y ) )
40 nbrne2 4673 . . . . . . . . 9 |- ( ( Z .<_ ( X ( join ` K ) Y ) /\ -. p .<_ ( X ( join ` K ) Y ) ) -> Z =/= p )
4140necomd 2849 . . . . . . . 8 |- ( ( Z .<_ ( X ( join ` K ) Y ) /\ -. p .<_ ( X ( join ` K ) Y ) ) -> p =/= Z )
4239, 33, 41syl2anc 693 . . . . . . 7 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ Z .<_ ( X ( join ` K ) Y ) ) /\ ( p e. A /\ ( -. p .<_ ( X ( join ` K ) Y ) /\ p .<_ W ) ) ) -> p =/= Z )
4336, 38, 423jca 1242 . . . . . 6 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ Z .<_ ( X ( join ` K ) Y ) ) /\ ( p e. A /\ ( -. p .<_ ( X ( join ` K ) Y ) /\ p .<_ W ) ) ) -> ( p =/= X /\ p =/= Y /\ p =/= Z ) )
4417, 43jca 554 . . . . 5 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ Z .<_ ( X ( join ` K ) Y ) ) /\ ( p e. A /\ ( -. p .<_ ( X ( join ` K ) Y ) /\ p .<_ W ) ) ) -> ( p .<_ W /\ ( p =/= X /\ p =/= Y /\ p =/= Z ) ) )
45 simp11 1091 . . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ Z .<_ ( X ( join ` K ) Y ) ) -> ( K e. HL /\ W e. H ) )
46 simp131 1196 . . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ Z .<_ ( X ( join ` K ) Y ) ) -> X .<_ W )
47 simp132 1197 . . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ Z .<_ ( X ( join ` K ) Y ) ) -> Y .<_ W )
48 eqid 2622 . . . . . . . 8 |- ( lt ` K ) = ( lt ` K )
492, 48, 34, 3, 4lhp2lt 35287 . . . . . . 7 |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ X .<_ W ) /\ ( Y e. A /\ Y .<_ W ) ) -> ( X ( join ` K ) Y ) ( lt ` K ) W )
5045, 25, 46, 29, 47, 49syl122anc 1335 . . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ Z .<_ ( X ( join ` K ) Y ) ) -> ( X ( join ` K ) Y ) ( lt ` K ) W )
5122, 34, 3hlatjcl 34653 . . . . . . . 8 |- ( ( K e. HL /\ X e. A /\ Y e. A ) -> ( X ( join ` K ) Y ) e. ( Base ` K ) )
5218, 25, 29, 51syl3anc 1326 . . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ Z .<_ ( X ( join ` K ) Y ) ) -> ( X ( join ` K ) Y ) e. ( Base ` K ) )
53 simp11r 1173 . . . . . . . 8 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ Z .<_ ( X ( join ` K ) Y ) ) -> W e. H )
5422, 4lhpbase 35284 . . . . . . . 8 |- ( W e. H -> W e. ( Base ` K ) )
5553, 54syl 17 . . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ Z .<_ ( X ( join ` K ) Y ) ) -> W e. ( Base ` K ) )
5622, 2, 48, 3hlrelat1 34686 . . . . . . 7 |- ( ( K e. HL /\ ( X ( join ` K ) Y ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( ( X ( join ` K ) Y ) ( lt ` K ) W -> E. p e. A ( -. p .<_ ( X ( join ` K ) Y ) /\ p .<_ W ) ) )
5718, 52, 55, 56syl3anc 1326 . . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ Z .<_ ( X ( join ` K ) Y ) ) -> ( ( X ( join ` K ) Y ) ( lt ` K ) W -> E. p e. A ( -. p .<_ ( X ( join ` K ) Y ) /\ p .<_ W ) ) )
5850, 57mpd 15 . . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ Z .<_ ( X ( join ` K ) Y ) ) -> E. p e. A ( -. p .<_ ( X ( join ` K ) Y ) /\ p .<_ W ) )
5944, 58reximddv 3018 . . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ Z .<_ ( X ( join ` K ) Y ) ) -> E. p e. A ( p .<_ W /\ ( p =/= X /\ p =/= Y /\ p =/= Z ) ) )
60593expa 1265 . . 3 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y ) /\ Z .<_ ( X ( join ` K ) Y ) ) -> E. p e. A ( p .<_ W /\ ( p =/= X /\ p =/= Y /\ p =/= Z ) ) )
61 simp11l 1172 . . . . . . . . 9 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ -. Z .<_ ( X ( join ` K ) Y ) ) -> K e. HL )
6261adantr 481 . . . . . . . 8 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ -. Z .<_ ( X ( join ` K ) Y ) ) /\ ( p e. A /\ ( p =/= X /\ p =/= Y /\ p .<_ ( X ( join ` K ) Y ) ) ) ) -> K e. HL )
6362, 20syl 17 . . . . . . 7 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ -. Z .<_ ( X ( join ` K ) Y ) ) /\ ( p e. A /\ ( p =/= X /\ p =/= Y /\ p .<_ ( X ( join ` K ) Y ) ) ) ) -> K e. Lat )
6423ad2antrl 764 . . . . . . 7 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ -. Z .<_ ( X ( join ` K ) Y ) ) /\ ( p e. A /\ ( p =/= X /\ p =/= Y /\ p .<_ ( X ( join ` K ) Y ) ) ) ) -> p e. ( Base ` K ) )
65 simp121 1193 . . . . . . . . 9 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ -. Z .<_ ( X ( join ` K ) Y ) ) -> X e. A )
6665adantr 481 . . . . . . . 8 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ -. Z .<_ ( X ( join ` K ) Y ) ) /\ ( p e. A /\ ( p =/= X /\ p =/= Y /\ p .<_ ( X ( join ` K ) Y ) ) ) ) -> X e. A )
67 simp122 1194 . . . . . . . . 9 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ -. Z .<_ ( X ( join ` K ) Y ) ) -> Y e. A )
6867adantr 481 . . . . . . . 8 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ -. Z .<_ ( X ( join ` K ) Y ) ) /\ ( p e. A /\ ( p =/= X /\ p =/= Y /\ p .<_ ( X ( join ` K ) Y ) ) ) ) -> Y e. A )
6962, 66, 68, 51syl3anc 1326 . . . . . . 7 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ -. Z .<_ ( X ( join ` K ) Y ) ) /\ ( p e. A /\ ( p =/= X /\ p =/= Y /\ p .<_ ( X ( join ` K ) Y ) ) ) ) -> ( X ( join ` K ) Y ) e. ( Base ` K ) )
70 simp11r 1173 . . . . . . . . 9 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ -. Z .<_ ( X ( join ` K ) Y ) ) -> W e. H )
7170adantr 481 . . . . . . . 8 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ -. Z .<_ ( X ( join ` K ) Y ) ) /\ ( p e. A /\ ( p =/= X /\ p =/= Y /\ p .<_ ( X ( join ` K ) Y ) ) ) ) -> W e. H )
7271, 54syl 17 . . . . . . 7 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ -. Z .<_ ( X ( join ` K ) Y ) ) /\ ( p e. A /\ ( p =/= X /\ p =/= Y /\ p .<_ ( X ( join ` K ) Y ) ) ) ) -> W e. ( Base ` K ) )
73 simprr3 1111 . . . . . . 7 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ -. Z .<_ ( X ( join ` K ) Y ) ) /\ ( p e. A /\ ( p =/= X /\ p =/= Y /\ p .<_ ( X ( join ` K ) Y ) ) ) ) -> p .<_ ( X ( join ` K ) Y ) )
74 simp131 1196 . . . . . . . . 9 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ -. Z .<_ ( X ( join ` K ) Y ) ) -> X .<_ W )
7574adantr 481 . . . . . . . 8 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ -. Z .<_ ( X ( join ` K ) Y ) ) /\ ( p e. A /\ ( p =/= X /\ p =/= Y /\ p .<_ ( X ( join ` K ) Y ) ) ) ) -> X .<_ W )
76 simp132 1197 . . . . . . . . 9 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ -. Z .<_ ( X ( join ` K ) Y ) ) -> Y .<_ W )
7776adantr 481 . . . . . . . 8 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ -. Z .<_ ( X ( join ` K ) Y ) ) /\ ( p e. A /\ ( p =/= X /\ p =/= Y /\ p .<_ ( X ( join ` K ) Y ) ) ) ) -> Y .<_ W )
7866, 27syl 17 . . . . . . . . 9 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ -. Z .<_ ( X ( join ` K ) Y ) ) /\ ( p e. A /\ ( p =/= X /\ p =/= Y /\ p .<_ ( X ( join ` K ) Y ) ) ) ) -> X e. ( Base ` K ) )
7968, 31syl 17 . . . . . . . . 9 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ -. Z .<_ ( X ( join ` K ) Y ) ) /\ ( p e. A /\ ( p =/= X /\ p =/= Y /\ p .<_ ( X ( join ` K ) Y ) ) ) ) -> Y e. ( Base ` K ) )
8022, 2, 34latjle12 17062 . . . . . . . . 9 |- ( ( K e. Lat /\ ( X e. ( Base ` K ) /\ Y e. ( Base ` K ) /\ W e. ( Base ` K ) ) ) -> ( ( X .<_ W /\ Y .<_ W ) <-> ( X ( join ` K ) Y ) .<_ W ) )
8163, 78, 79, 72, 80syl13anc 1328 . . . . . . . 8 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ -. Z .<_ ( X ( join ` K ) Y ) ) /\ ( p e. A /\ ( p =/= X /\ p =/= Y /\ p .<_ ( X ( join ` K ) Y ) ) ) ) -> ( ( X .<_ W /\ Y .<_ W ) <-> ( X ( join ` K ) Y ) .<_ W ) )
8275, 77, 81mpbi2and 956 . . . . . . 7 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ -. Z .<_ ( X ( join ` K ) Y ) ) /\ ( p e. A /\ ( p =/= X /\ p =/= Y /\ p .<_ ( X ( join ` K ) Y ) ) ) ) -> ( X ( join ` K ) Y ) .<_ W )
8322, 2, 63, 64, 69, 72, 73, 82lattrd 17058 . . . . . 6 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ -. Z .<_ ( X ( join ` K ) Y ) ) /\ ( p e. A /\ ( p =/= X /\ p =/= Y /\ p .<_ ( X ( join ` K ) Y ) ) ) ) -> p .<_ W )
84 simprr1 1109 . . . . . . 7 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ -. Z .<_ ( X ( join ` K ) Y ) ) /\ ( p e. A /\ ( p =/= X /\ p =/= Y /\ p .<_ ( X ( join ` K ) Y ) ) ) ) -> p =/= X )
85 simprr2 1110 . . . . . . 7 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ -. Z .<_ ( X ( join ` K ) Y ) ) /\ ( p e. A /\ ( p =/= X /\ p =/= Y /\ p .<_ ( X ( join ` K ) Y ) ) ) ) -> p =/= Y )
86 simpl3 1066 . . . . . . . 8 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ -. Z .<_ ( X ( join ` K ) Y ) ) /\ ( p e. A /\ ( p =/= X /\ p =/= Y /\ p .<_ ( X ( join ` K ) Y ) ) ) ) -> -. Z .<_ ( X ( join ` K ) Y ) )
87 nbrne2 4673 . . . . . . . 8 |- ( ( p .<_ ( X ( join ` K ) Y ) /\ -. Z .<_ ( X ( join ` K ) Y ) ) -> p =/= Z )
8873, 86, 87syl2anc 693 . . . . . . 7 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ -. Z .<_ ( X ( join ` K ) Y ) ) /\ ( p e. A /\ ( p =/= X /\ p =/= Y /\ p .<_ ( X ( join ` K ) Y ) ) ) ) -> p =/= Z )
8984, 85, 883jca 1242 . . . . . 6 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ -. Z .<_ ( X ( join ` K ) Y ) ) /\ ( p e. A /\ ( p =/= X /\ p =/= Y /\ p .<_ ( X ( join ` K ) Y ) ) ) ) -> ( p =/= X /\ p =/= Y /\ p =/= Z ) )
9083, 89jca 554 . . . . 5 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ -. Z .<_ ( X ( join ` K ) Y ) ) /\ ( p e. A /\ ( p =/= X /\ p =/= Y /\ p .<_ ( X ( join ` K ) Y ) ) ) ) -> ( p .<_ W /\ ( p =/= X /\ p =/= Y /\ p =/= Z ) ) )
91 simp2 1062 . . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ -. Z .<_ ( X ( join ` K ) Y ) ) -> X =/= Y )
922, 34, 3hlsupr 34672 . . . . . 6 |- ( ( ( K e. HL /\ X e. A /\ Y e. A ) /\ X =/= Y ) -> E. p e. A ( p =/= X /\ p =/= Y /\ p .<_ ( X ( join ` K ) Y ) ) )
9361, 65, 67, 91, 92syl31anc 1329 . . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ -. Z .<_ ( X ( join ` K ) Y ) ) -> E. p e. A ( p =/= X /\ p =/= Y /\ p .<_ ( X ( join ` K ) Y ) ) )
9490, 93reximddv 3018 . . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y /\ -. Z .<_ ( X ( join ` K ) Y ) ) -> E. p e. A ( p .<_ W /\ ( p =/= X /\ p =/= Y /\ p =/= Z ) ) )
95943expa 1265 . . 3 |- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y ) /\ -. Z .<_ ( X ( join ` K ) Y ) ) -> E. p e. A ( p .<_ W /\ ( p =/= X /\ p =/= Y /\ p =/= Z ) ) )
9660, 95pm2.61dan 832 . 2 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) /\ X =/= Y ) -> E. p e. A ( p .<_ W /\ ( p =/= X /\ p =/= Y /\ p =/= Z ) ) )
9716, 96pm2.61dane 2881 1 |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ Y e. A /\ Z e. A ) /\ ( X .<_ W /\ Y .<_ W /\ Z .<_ W ) ) -> E. p e. A ( p .<_ W /\ ( p =/= X /\ p =/= Y /\ p =/= Z ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Basecbs 15857   lecple 15948   ltcplt 16941   joincjn 16944   Latclat 17045   Atomscatm 34550   HLchlt 34637   LHypclh 35270
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-preset 16928  df-poset 16946  df-plt 16958  df-lub 16974  df-glb 16975  df-join 16976  df-meet 16977  df-p0 17039  df-p1 17040  df-lat 17046  df-clat 17108  df-oposet 34463  df-ol 34465  df-oml 34466  df-covers 34553  df-ats 34554  df-atl 34585  df-cvlat 34609  df-hlat 34638  df-lhyp 35274
This theorem is referenced by:  lhpexle3  35298
  Copyright terms: Public domain W3C validator