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Metamath Proof Explorer

Theorem cdlemg35

Description: TODO: Fix comment. TODO: should we have a more general version of hlsupr to avoid the =/= conditions? (Contributed by NM, 31-May-2013)

Ref Expression
Hypotheses cdlemg35.l 
|- .<_ = ( le ` K )
cdlemg35.j 
|- .\/ = ( join ` K )
cdlemg35.m 
|- ./\ = ( meet ` K )
cdlemg35.a 
|- A = ( Atoms ` K )
cdlemg35.h 
|- H = ( LHyp ` K )
cdlemg35.t 
|- T = ( ( LTrn ` K ) ` W )
cdlemg35.r 
|- R = ( ( trL ` K ) ` W )
Assertion cdlemg35 
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) -> E. v e. A ( v .<_ W /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) ) )

Proof

Step Hyp Ref Expression
1 cdlemg35.l 
 |-  .<_ = ( le ` K )
2 cdlemg35.j 
 |-  .\/ = ( join ` K )
3 cdlemg35.m 
 |-  ./\ = ( meet ` K )
4 cdlemg35.a 
 |-  A = ( Atoms ` K )
5 cdlemg35.h 
 |-  H = ( LHyp ` K )
6 cdlemg35.t 
 |-  T = ( ( LTrn ` K ) ` W )
7 cdlemg35.r 
 |-  R = ( ( trL ` K ) ` W )
8 simp1l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) -> K e. HL )
9 simp1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( K e. HL /\ W e. H ) )
10 simp21 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( P e. A /\ -. P .<_ W ) )
11 simp22 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) -> F e. T )
12 simp31 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( F ` P ) =/= P )
13 1 4 5 6 7 trlat 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( F e. T /\ ( F ` P ) =/= P ) ) -> ( R ` F ) e. A )
14 9 10 11 12 13 syl112anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( R ` F ) e. A )
15 simp23 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) -> G e. T )
16 simp32 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( G ` P ) =/= P )
17 1 4 5 6 7 trlat 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( G e. T /\ ( G ` P ) =/= P ) ) -> ( R ` G ) e. A )
18 9 10 15 16 17 syl112anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( R ` G ) e. A )
19 simp33 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( R ` F ) =/= ( R ` G ) )
20 1 2 4 hlsupr 
 |-  ( ( ( K e. HL /\ ( R ` F ) e. A /\ ( R ` G ) e. A ) /\ ( R ` F ) =/= ( R ` G ) ) -> E. v e. A ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ v .<_ ( ( R ` F ) .\/ ( R ` G ) ) ) )
21 8 14 18 19 20 syl31anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) -> E. v e. A ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ v .<_ ( ( R ` F ) .\/ ( R ` G ) ) ) )
22 eqid 
 |-  ( Base ` K ) = ( Base ` K )
23 simp11l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) /\ v e. A /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ v .<_ ( ( R ` F ) .\/ ( R ` G ) ) ) ) -> K e. HL )
24 23 hllatd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) /\ v e. A /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ v .<_ ( ( R ` F ) .\/ ( R ` G ) ) ) ) -> K e. Lat )
25 22 4 atbase 
 |-  ( v e. A -> v e. ( Base ` K ) )
26 25 3ad2ant2 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) /\ v e. A /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ v .<_ ( ( R ` F ) .\/ ( R ` G ) ) ) ) -> v e. ( Base ` K ) )
27 simp11 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) /\ v e. A /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ v .<_ ( ( R ` F ) .\/ ( R ` G ) ) ) ) -> ( K e. HL /\ W e. H ) )
28 simp122 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) /\ v e. A /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ v .<_ ( ( R ` F ) .\/ ( R ` G ) ) ) ) -> F e. T )
29 22 5 6 7 trlcl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( R ` F ) e. ( Base ` K ) )
30 27 28 29 syl2anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) /\ v e. A /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ v .<_ ( ( R ` F ) .\/ ( R ` G ) ) ) ) -> ( R ` F ) e. ( Base ` K ) )
31 simp123 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) /\ v e. A /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ v .<_ ( ( R ` F ) .\/ ( R ` G ) ) ) ) -> G e. T )
32 22 5 6 7 trlcl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ G e. T ) -> ( R ` G ) e. ( Base ` K ) )
33 27 31 32 syl2anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) /\ v e. A /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ v .<_ ( ( R ` F ) .\/ ( R ` G ) ) ) ) -> ( R ` G ) e. ( Base ` K ) )
34 22 2 latjcl 
 |-  ( ( K e. Lat /\ ( R ` F ) e. ( Base ` K ) /\ ( R ` G ) e. ( Base ` K ) ) -> ( ( R ` F ) .\/ ( R ` G ) ) e. ( Base ` K ) )
35 24 30 33 34 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) /\ v e. A /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ v .<_ ( ( R ` F ) .\/ ( R ` G ) ) ) ) -> ( ( R ` F ) .\/ ( R ` G ) ) e. ( Base ` K ) )
36 simp11r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) /\ v e. A /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ v .<_ ( ( R ` F ) .\/ ( R ` G ) ) ) ) -> W e. H )
37 22 5 lhpbase 
 |-  ( W e. H -> W e. ( Base ` K ) )
38 36 37 syl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) /\ v e. A /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ v .<_ ( ( R ` F ) .\/ ( R ` G ) ) ) ) -> W e. ( Base ` K ) )
39 simp33 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) /\ v e. A /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ v .<_ ( ( R ` F ) .\/ ( R ` G ) ) ) ) -> v .<_ ( ( R ` F ) .\/ ( R ` G ) ) )
40 1 5 6 7 trlle 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( R ` F ) .<_ W )
41 27 28 40 syl2anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) /\ v e. A /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ v .<_ ( ( R ` F ) .\/ ( R ` G ) ) ) ) -> ( R ` F ) .<_ W )
42 1 5 6 7 trlle 
 |-  ( ( ( K e. HL /\ W e. H ) /\ G e. T ) -> ( R ` G ) .<_ W )
43 27 31 42 syl2anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) /\ v e. A /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ v .<_ ( ( R ` F ) .\/ ( R ` G ) ) ) ) -> ( R ` G ) .<_ W )
44 22 1 2 latjle12 
 |-  ( ( K e. Lat /\ ( ( R ` F ) e. ( Base ` K ) /\ ( R ` G ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) ) -> ( ( ( R ` F ) .<_ W /\ ( R ` G ) .<_ W ) <-> ( ( R ` F ) .\/ ( R ` G ) ) .<_ W ) )
45 24 30 33 38 44 syl13anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) /\ v e. A /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ v .<_ ( ( R ` F ) .\/ ( R ` G ) ) ) ) -> ( ( ( R ` F ) .<_ W /\ ( R ` G ) .<_ W ) <-> ( ( R ` F ) .\/ ( R ` G ) ) .<_ W ) )
46 41 43 45 mpbi2and 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) /\ v e. A /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ v .<_ ( ( R ` F ) .\/ ( R ` G ) ) ) ) -> ( ( R ` F ) .\/ ( R ` G ) ) .<_ W )
47 22 1 24 26 35 38 39 46 lattrd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) /\ v e. A /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ v .<_ ( ( R ` F ) .\/ ( R ` G ) ) ) ) -> v .<_ W )
48 simp31 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) /\ v e. A /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ v .<_ ( ( R ` F ) .\/ ( R ` G ) ) ) ) -> v =/= ( R ` F ) )
49 simp32 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) /\ v e. A /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ v .<_ ( ( R ` F ) .\/ ( R ` G ) ) ) ) -> v =/= ( R ` G ) )
50 47 48 49 jca32 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) /\ v e. A /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ v .<_ ( ( R ` F ) .\/ ( R ` G ) ) ) ) -> ( v .<_ W /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) ) )
51 50 3expia 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) /\ v e. A ) -> ( ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ v .<_ ( ( R ` F ) .\/ ( R ` G ) ) ) -> ( v .<_ W /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) ) ) )
52 51 reximdva 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( E. v e. A ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ v .<_ ( ( R ` F ) .\/ ( R ` G ) ) ) -> E. v e. A ( v .<_ W /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) ) ) )
53 21 52 mpd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) -> E. v e. A ( v .<_ W /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) ) )