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Metamath Proof Explorer

Theorem cdlemg11b

Description: TODO: FIX COMMENT. (Contributed by NM, 5-May-2013)

Ref Expression
Hypotheses cdlemg8.l 
|- .<_ = ( le ` K )
cdlemg8.j 
|- .\/ = ( join ` K )
cdlemg8.m 
|- ./\ = ( meet ` K )
cdlemg8.a 
|- A = ( Atoms ` K )
cdlemg8.h 
|- H = ( LHyp ` K )
cdlemg8.t 
|- T = ( ( LTrn ` K ) ` W )
cdlemg10.r 
|- R = ( ( trL ` K ) ` W )
Assertion cdlemg11b 
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( G e. T /\ P =/= Q /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) -> ( P .\/ Q ) =/= ( ( G ` P ) .\/ ( G ` Q ) ) )

Proof

Step Hyp Ref Expression
1 cdlemg8.l 
 |-  .<_ = ( le ` K )
2 cdlemg8.j 
 |-  .\/ = ( join ` K )
3 cdlemg8.m 
 |-  ./\ = ( meet ` K )
4 cdlemg8.a 
 |-  A = ( Atoms ` K )
5 cdlemg8.h 
 |-  H = ( LHyp ` K )
6 cdlemg8.t 
 |-  T = ( ( LTrn ` K ) ` W )
7 cdlemg10.r 
 |-  R = ( ( trL ` K ) ` W )
8 simp33 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( G e. T /\ P =/= Q /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) -> -. ( R ` G ) .<_ ( P .\/ Q ) )
9 simpl1 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( G e. T /\ P =/= Q /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ Q ) = ( ( G ` P ) .\/ ( G ` Q ) ) ) -> ( K e. HL /\ W e. H ) )
10 simpl31 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( G e. T /\ P =/= Q /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ Q ) = ( ( G ` P ) .\/ ( G ` Q ) ) ) -> G e. T )
11 simpl2l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( G e. T /\ P =/= Q /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ Q ) = ( ( G ` P ) .\/ ( G ` Q ) ) ) -> ( P e. A /\ -. P .<_ W ) )
12 1 2 3 4 5 6 7 trlval2 
 |-  ( ( ( K e. HL /\ W e. H ) /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( R ` G ) = ( ( P .\/ ( G ` P ) ) ./\ W ) )
13 9 10 11 12 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( G e. T /\ P =/= Q /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ Q ) = ( ( G ` P ) .\/ ( G ` Q ) ) ) -> ( R ` G ) = ( ( P .\/ ( G ` P ) ) ./\ W ) )
14 eqid 
 |-  ( Base ` K ) = ( Base ` K )
15 simpl1l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( G e. T /\ P =/= Q /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ Q ) = ( ( G ` P ) .\/ ( G ` Q ) ) ) -> K e. HL )
16 15 hllatd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( G e. T /\ P =/= Q /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ Q ) = ( ( G ` P ) .\/ ( G ` Q ) ) ) -> K e. Lat )
17 simp2ll 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( G e. T /\ P =/= Q /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) -> P e. A )
18 17 adantr 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( G e. T /\ P =/= Q /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ Q ) = ( ( G ` P ) .\/ ( G ` Q ) ) ) -> P e. A )
19 14 4 atbase 
 |-  ( P e. A -> P e. ( Base ` K ) )
20 18 19 syl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( G e. T /\ P =/= Q /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ Q ) = ( ( G ` P ) .\/ ( G ` Q ) ) ) -> P e. ( Base ` K ) )
21 14 5 6 ltrncl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ G e. T /\ P e. ( Base ` K ) ) -> ( G ` P ) e. ( Base ` K ) )
22 9 10 20 21 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( G e. T /\ P =/= Q /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ Q ) = ( ( G ` P ) .\/ ( G ` Q ) ) ) -> ( G ` P ) e. ( Base ` K ) )
23 14 2 latjcl 
 |-  ( ( K e. Lat /\ P e. ( Base ` K ) /\ ( G ` P ) e. ( Base ` K ) ) -> ( P .\/ ( G ` P ) ) e. ( Base ` K ) )
24 16 20 22 23 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( G e. T /\ P =/= Q /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ Q ) = ( ( G ` P ) .\/ ( G ` Q ) ) ) -> ( P .\/ ( G ` P ) ) e. ( Base ` K ) )
25 simpl1r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( G e. T /\ P =/= Q /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ Q ) = ( ( G ` P ) .\/ ( G ` Q ) ) ) -> W e. H )
26 14 5 lhpbase 
 |-  ( W e. H -> W e. ( Base ` K ) )
27 25 26 syl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( G e. T /\ P =/= Q /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ Q ) = ( ( G ` P ) .\/ ( G ` Q ) ) ) -> W e. ( Base ` K ) )
28 14 3 latmcl 
 |-  ( ( K e. Lat /\ ( P .\/ ( G ` P ) ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( ( P .\/ ( G ` P ) ) ./\ W ) e. ( Base ` K ) )
29 16 24 27 28 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( G e. T /\ P =/= Q /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ Q ) = ( ( G ` P ) .\/ ( G ` Q ) ) ) -> ( ( P .\/ ( G ` P ) ) ./\ W ) e. ( Base ` K ) )
30 simpl2r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( G e. T /\ P =/= Q /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ Q ) = ( ( G ` P ) .\/ ( G ` Q ) ) ) -> Q e. A )
31 14 4 atbase 
 |-  ( Q e. A -> Q e. ( Base ` K ) )
32 30 31 syl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( G e. T /\ P =/= Q /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ Q ) = ( ( G ` P ) .\/ ( G ` Q ) ) ) -> Q e. ( Base ` K ) )
33 14 2 latjcl 
 |-  ( ( K e. Lat /\ P e. ( Base ` K ) /\ Q e. ( Base ` K ) ) -> ( P .\/ Q ) e. ( Base ` K ) )
34 16 20 32 33 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( G e. T /\ P =/= Q /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ Q ) = ( ( G ` P ) .\/ ( G ` Q ) ) ) -> ( P .\/ Q ) e. ( Base ` K ) )
35 14 1 3 latmle1 
 |-  ( ( K e. Lat /\ ( P .\/ ( G ` P ) ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( ( P .\/ ( G ` P ) ) ./\ W ) .<_ ( P .\/ ( G ` P ) ) )
36 16 24 27 35 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( G e. T /\ P =/= Q /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ Q ) = ( ( G ` P ) .\/ ( G ` Q ) ) ) -> ( ( P .\/ ( G ` P ) ) ./\ W ) .<_ ( P .\/ ( G ` P ) ) )
37 14 1 2 latlej1 
 |-  ( ( K e. Lat /\ P e. ( Base ` K ) /\ Q e. ( Base ` K ) ) -> P .<_ ( P .\/ Q ) )
38 16 20 32 37 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( G e. T /\ P =/= Q /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ Q ) = ( ( G ` P ) .\/ ( G ` Q ) ) ) -> P .<_ ( P .\/ Q ) )
39 14 5 6 ltrncl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ G e. T /\ Q e. ( Base ` K ) ) -> ( G ` Q ) e. ( Base ` K ) )
40 9 10 32 39 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( G e. T /\ P =/= Q /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ Q ) = ( ( G ` P ) .\/ ( G ` Q ) ) ) -> ( G ` Q ) e. ( Base ` K ) )
41 14 1 2 latlej1 
 |-  ( ( K e. Lat /\ ( G ` P ) e. ( Base ` K ) /\ ( G ` Q ) e. ( Base ` K ) ) -> ( G ` P ) .<_ ( ( G ` P ) .\/ ( G ` Q ) ) )
42 16 22 40 41 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( G e. T /\ P =/= Q /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ Q ) = ( ( G ` P ) .\/ ( G ` Q ) ) ) -> ( G ` P ) .<_ ( ( G ` P ) .\/ ( G ` Q ) ) )
43 simpr 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( G e. T /\ P =/= Q /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ Q ) = ( ( G ` P ) .\/ ( G ` Q ) ) ) -> ( P .\/ Q ) = ( ( G ` P ) .\/ ( G ` Q ) ) )
44 42 43 breqtrrd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( G e. T /\ P =/= Q /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ Q ) = ( ( G ` P ) .\/ ( G ` Q ) ) ) -> ( G ` P ) .<_ ( P .\/ Q ) )
45 14 1 2 latjle12 
 |-  ( ( K e. Lat /\ ( P e. ( Base ` K ) /\ ( G ` P ) e. ( Base ` K ) /\ ( P .\/ Q ) e. ( Base ` K ) ) ) -> ( ( P .<_ ( P .\/ Q ) /\ ( G ` P ) .<_ ( P .\/ Q ) ) <-> ( P .\/ ( G ` P ) ) .<_ ( P .\/ Q ) ) )
46 16 20 22 34 45 syl13anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( G e. T /\ P =/= Q /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ Q ) = ( ( G ` P ) .\/ ( G ` Q ) ) ) -> ( ( P .<_ ( P .\/ Q ) /\ ( G ` P ) .<_ ( P .\/ Q ) ) <-> ( P .\/ ( G ` P ) ) .<_ ( P .\/ Q ) ) )
47 38 44 46 mpbi2and 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( G e. T /\ P =/= Q /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ Q ) = ( ( G ` P ) .\/ ( G ` Q ) ) ) -> ( P .\/ ( G ` P ) ) .<_ ( P .\/ Q ) )
48 14 1 16 29 24 34 36 47 lattrd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( G e. T /\ P =/= Q /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ Q ) = ( ( G ` P ) .\/ ( G ` Q ) ) ) -> ( ( P .\/ ( G ` P ) ) ./\ W ) .<_ ( P .\/ Q ) )
49 13 48 eqbrtrd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( G e. T /\ P =/= Q /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) /\ ( P .\/ Q ) = ( ( G ` P ) .\/ ( G ` Q ) ) ) -> ( R ` G ) .<_ ( P .\/ Q ) )
50 49 ex 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( G e. T /\ P =/= Q /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) -> ( ( P .\/ Q ) = ( ( G ` P ) .\/ ( G ` Q ) ) -> ( R ` G ) .<_ ( P .\/ Q ) ) )
51 50 necon3bd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( G e. T /\ P =/= Q /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) -> ( -. ( R ` G ) .<_ ( P .\/ Q ) -> ( P .\/ Q ) =/= ( ( G ` P ) .\/ ( G ` Q ) ) ) )
52 8 51 mpd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( G e. T /\ P =/= Q /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) -> ( P .\/ Q ) =/= ( ( G ` P ) .\/ ( G ` Q ) ) )