Metamath Proof Explorer

Theorem lhpexle1

Description: There exists an atom under a co-atom different from any given element. (Contributed by NM, 24-Jul-2013)

Ref Expression
Hypotheses lhpex1.l 
|- .<_ = ( le ` K )
lhpex1.a 
|- A = ( Atoms ` K )
lhpex1.h 
|- H = ( LHyp ` K )
Assertion lhpexle1 
|- ( ( K e. HL /\ W e. H ) -> E. p e. A ( p .<_ W /\ p =/= X ) )

Proof

Step Hyp Ref Expression
1 lhpex1.l 
 |-  .<_ = ( le ` K )
2 lhpex1.a 
 |-  A = ( Atoms ` K )
3 lhpex1.h 
 |-  H = ( LHyp ` K )
4 1 2 3 lhpexle 
 |-  ( ( K e. HL /\ W e. H ) -> E. p e. A p .<_ W )
5 tru 
 |-  T.
6 5 jctr 
 |-  ( p .<_ W -> ( p .<_ W /\ T. ) )
7 6 reximi 
 |-  ( E. p e. A p .<_ W -> E. p e. A ( p .<_ W /\ T. ) )
8 4 7 syl 
 |-  ( ( K e. HL /\ W e. H ) -> E. p e. A ( p .<_ W /\ T. ) )
9 simpll 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ X .<_ W ) ) -> K e. HL )
10 simprl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ X .<_ W ) ) -> X e. A )
11 eqid 
 |-  ( Base ` K ) = ( Base ` K )
12 11 3 lhpbase 
 |-  ( W e. H -> W e. ( Base ` K ) )
13 12 ad2antlr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ X .<_ W ) ) -> W e. ( Base ` K ) )
14 eqid 
 |-  ( lt ` K ) = ( lt ` K )
15 1 14 2 3 lhplt 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ X .<_ W ) ) -> X ( lt ` K ) W )
16 11 14 2 2atlt 
 |-  ( ( ( K e. HL /\ X e. A /\ W e. ( Base ` K ) ) /\ X ( lt ` K ) W ) -> E. p e. A ( p =/= X /\ p ( lt ` K ) W ) )
17 9 10 13 15 16 syl31anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ X .<_ W ) ) -> E. p e. A ( p =/= X /\ p ( lt ` K ) W ) )
18 simp3r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ X .<_ W ) ) /\ p e. A /\ ( p =/= X /\ p ( lt ` K ) W ) ) -> p ( lt ` K ) W )
19 simp1ll 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ X .<_ W ) ) /\ p e. A /\ ( p =/= X /\ p ( lt ` K ) W ) ) -> K e. HL )
20 simp2 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ X .<_ W ) ) /\ p e. A /\ ( p =/= X /\ p ( lt ` K ) W ) ) -> p e. A )
21 simp1lr 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ X .<_ W ) ) /\ p e. A /\ ( p =/= X /\ p ( lt ` K ) W ) ) -> W e. H )
22 1 14 pltle 
 |-  ( ( K e. HL /\ p e. A /\ W e. H ) -> ( p ( lt ` K ) W -> p .<_ W ) )
23 19 20 21 22 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ X .<_ W ) ) /\ p e. A /\ ( p =/= X /\ p ( lt ` K ) W ) ) -> ( p ( lt ` K ) W -> p .<_ W ) )
24 18 23 mpd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ X .<_ W ) ) /\ p e. A /\ ( p =/= X /\ p ( lt ` K ) W ) ) -> p .<_ W )
25 trud 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ X .<_ W ) ) /\ p e. A /\ ( p =/= X /\ p ( lt ` K ) W ) ) -> T. )
26 simp3l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ X .<_ W ) ) /\ p e. A /\ ( p =/= X /\ p ( lt ` K ) W ) ) -> p =/= X )
27 24 25 26 3jca 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ X .<_ W ) ) /\ p e. A /\ ( p =/= X /\ p ( lt ` K ) W ) ) -> ( p .<_ W /\ T. /\ p =/= X ) )
28 27 3expia 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ X .<_ W ) ) /\ p e. A ) -> ( ( p =/= X /\ p ( lt ` K ) W ) -> ( p .<_ W /\ T. /\ p =/= X ) ) )
29 28 reximdva 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ X .<_ W ) ) -> ( E. p e. A ( p =/= X /\ p ( lt ` K ) W ) -> E. p e. A ( p .<_ W /\ T. /\ p =/= X ) ) )
30 17 29 mpd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. A /\ X .<_ W ) ) -> E. p e. A ( p .<_ W /\ T. /\ p =/= X ) )
31 8 30 lhpexle1lem 
 |-  ( ( K e. HL /\ W e. H ) -> E. p e. A ( p .<_ W /\ T. /\ p =/= X ) )
32 3simpb 
 |-  ( ( p .<_ W /\ T. /\ p =/= X ) -> ( p .<_ W /\ p =/= X ) )
33 32 reximi 
 |-  ( E. p e. A ( p .<_ W /\ T. /\ p =/= X ) -> E. p e. A ( p .<_ W /\ p =/= X ) )
34 31 33 syl 
 |-  ( ( K e. HL /\ W e. H ) -> E. p e. A ( p .<_ W /\ p =/= X ) )