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Metamath Proof Explorer

Theorem cdlemg17dALTN

Description: Same as cdlemg17dN with fewer antecedents but longer proof TODO: fix comment. (Contributed by NM, 9-May-2013) (New usage is discouraged.)

Ref Expression
Hypotheses cdlemg12.l 
|- .<_ = ( le ` K )
cdlemg12.j 
|- .\/ = ( join ` K )
cdlemg12.m 
|- ./\ = ( meet ` K )
cdlemg12.a 
|- A = ( Atoms ` K )
cdlemg12.h 
|- H = ( LHyp ` K )
cdlemg12.t 
|- T = ( ( LTrn ` K ) ` W )
cdlemg12b.r 
|- R = ( ( trL ` K ) ` W )
Assertion cdlemg17dALTN 
|- ( ( ( K e. HL /\ W e. H /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ P =/= Q ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ ( G ` P ) =/= P ) ) -> ( R ` G ) = ( ( P .\/ Q ) ./\ W ) )

Proof

Step Hyp Ref Expression
1 cdlemg12.l 
 |-  .<_ = ( le ` K )
2 cdlemg12.j 
 |-  .\/ = ( join ` K )
3 cdlemg12.m 
 |-  ./\ = ( meet ` K )
4 cdlemg12.a 
 |-  A = ( Atoms ` K )
5 cdlemg12.h 
 |-  H = ( LHyp ` K )
6 cdlemg12.t 
 |-  T = ( ( LTrn ` K ) ` W )
7 cdlemg12b.r 
 |-  R = ( ( trL ` K ) ` W )
8 simp3l 
 |-  ( ( ( K e. HL /\ W e. H /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ P =/= Q ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ ( G ` P ) =/= P ) ) -> ( R ` G ) .<_ ( P .\/ Q ) )
9 simp11 
 |-  ( ( ( K e. HL /\ W e. H /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ P =/= Q ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ ( G ` P ) =/= P ) ) -> K e. HL )
10 simp12 
 |-  ( ( ( K e. HL /\ W e. H /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ P =/= Q ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ ( G ` P ) =/= P ) ) -> W e. H )
11 simp13 
 |-  ( ( ( K e. HL /\ W e. H /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ P =/= Q ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ ( G ` P ) =/= P ) ) -> G e. T )
12 1 5 6 7 trlle 
 |-  ( ( ( K e. HL /\ W e. H ) /\ G e. T ) -> ( R ` G ) .<_ W )
13 9 10 11 12 syl21anc 
 |-  ( ( ( K e. HL /\ W e. H /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ P =/= Q ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ ( G ` P ) =/= P ) ) -> ( R ` G ) .<_ W )
14 9 hllatd 
 |-  ( ( ( K e. HL /\ W e. H /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ P =/= Q ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ ( G ` P ) =/= P ) ) -> K e. Lat )
15 eqid 
 |-  ( Base ` K ) = ( Base ` K )
16 15 5 6 7 trlcl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ G e. T ) -> ( R ` G ) e. ( Base ` K ) )
17 9 10 11 16 syl21anc 
 |-  ( ( ( K e. HL /\ W e. H /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ P =/= Q ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ ( G ` P ) =/= P ) ) -> ( R ` G ) e. ( Base ` K ) )
18 simp21l 
 |-  ( ( ( K e. HL /\ W e. H /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ P =/= Q ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ ( G ` P ) =/= P ) ) -> P e. A )
19 simp22 
 |-  ( ( ( K e. HL /\ W e. H /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ P =/= Q ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ ( G ` P ) =/= P ) ) -> Q e. A )
20 15 2 4 hlatjcl 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .\/ Q ) e. ( Base ` K ) )
21 9 18 19 20 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ P =/= Q ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ ( G ` P ) =/= P ) ) -> ( P .\/ Q ) e. ( Base ` K ) )
22 15 5 lhpbase 
 |-  ( W e. H -> W e. ( Base ` K ) )
23 10 22 syl 
 |-  ( ( ( K e. HL /\ W e. H /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ P =/= Q ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ ( G ` P ) =/= P ) ) -> W e. ( Base ` K ) )
24 15 1 3 latlem12 
 |-  ( ( K e. Lat /\ ( ( R ` G ) e. ( Base ` K ) /\ ( P .\/ Q ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) ) -> ( ( ( R ` G ) .<_ ( P .\/ Q ) /\ ( R ` G ) .<_ W ) <-> ( R ` G ) .<_ ( ( P .\/ Q ) ./\ W ) ) )
25 14 17 21 23 24 syl13anc 
 |-  ( ( ( K e. HL /\ W e. H /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ P =/= Q ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ ( G ` P ) =/= P ) ) -> ( ( ( R ` G ) .<_ ( P .\/ Q ) /\ ( R ` G ) .<_ W ) <-> ( R ` G ) .<_ ( ( P .\/ Q ) ./\ W ) ) )
26 8 13 25 mpbi2and 
 |-  ( ( ( K e. HL /\ W e. H /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ P =/= Q ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ ( G ` P ) =/= P ) ) -> ( R ` G ) .<_ ( ( P .\/ Q ) ./\ W ) )
27 hlatl 
 |-  ( K e. HL -> K e. AtLat )
28 9 27 syl 
 |-  ( ( ( K e. HL /\ W e. H /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ P =/= Q ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ ( G ` P ) =/= P ) ) -> K e. AtLat )
29 simp21 
 |-  ( ( ( K e. HL /\ W e. H /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ P =/= Q ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ ( G ` P ) =/= P ) ) -> ( P e. A /\ -. P .<_ W ) )
30 simp3r 
 |-  ( ( ( K e. HL /\ W e. H /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ P =/= Q ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ ( G ` P ) =/= P ) ) -> ( G ` P ) =/= P )
31 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 )
32 9 10 29 11 30 31 syl212anc 
 |-  ( ( ( K e. HL /\ W e. H /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ P =/= Q ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ ( G ` P ) =/= P ) ) -> ( R ` G ) e. A )
33 simp23 
 |-  ( ( ( K e. HL /\ W e. H /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ P =/= Q ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ ( G ` P ) =/= P ) ) -> P =/= Q )
34 1 2 3 4 5 lhpat 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) ) -> ( ( P .\/ Q ) ./\ W ) e. A )
35 9 10 29 19 33 34 syl212anc 
 |-  ( ( ( K e. HL /\ W e. H /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ P =/= Q ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ ( G ` P ) =/= P ) ) -> ( ( P .\/ Q ) ./\ W ) e. A )
36 1 4 atcmp 
 |-  ( ( K e. AtLat /\ ( R ` G ) e. A /\ ( ( P .\/ Q ) ./\ W ) e. A ) -> ( ( R ` G ) .<_ ( ( P .\/ Q ) ./\ W ) <-> ( R ` G ) = ( ( P .\/ Q ) ./\ W ) ) )
37 28 32 35 36 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ P =/= Q ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ ( G ` P ) =/= P ) ) -> ( ( R ` G ) .<_ ( ( P .\/ Q ) ./\ W ) <-> ( R ` G ) = ( ( P .\/ Q ) ./\ W ) ) )
38 26 37 mpbid 
 |-  ( ( ( K e. HL /\ W e. H /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ P =/= Q ) /\ ( ( R ` G ) .<_ ( P .\/ Q ) /\ ( G ` P ) =/= P ) ) -> ( R ` G ) = ( ( P .\/ Q ) ./\ W ) )