This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem cdlemg10bALTN

Description: TODO: FIX COMMENT. TODO: Can this be moved up as a stand-alone theorem in ltrn* area? TODO: Compare this proof to cdlemg2m and pick best, if moved to ltrn* area. (Contributed by NM, 4-May-2013) (New usage is discouraged.)

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 )
Assertion cdlemg10bALTN 
|- ( ( ( K e. HL /\ W e. H /\ F e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( ( ( F ` P ) .\/ ( F ` Q ) ) ./\ W ) = ( ( P .\/ Q ) ./\ W ) )

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 simp11 
 |-  ( ( ( K e. HL /\ W e. H /\ F e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> K e. HL )
8 simp12 
 |-  ( ( ( K e. HL /\ W e. H /\ F e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> W e. H )
9 7 8 jca 
 |-  ( ( ( K e. HL /\ W e. H /\ F e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( K e. HL /\ W e. H ) )
10 3simpc 
 |-  ( ( ( K e. HL /\ W e. H /\ F e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) )
11 simp13 
 |-  ( ( ( K e. HL /\ W e. H /\ F e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> F e. T )
12 eqid 
 |-  ( ( P .\/ Q ) ./\ W ) = ( ( P .\/ Q ) ./\ W )
13 5 6 1 2 4 3 12 cdlemg2k 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ F e. T ) -> ( ( F ` P ) .\/ ( F ` Q ) ) = ( ( F ` P ) .\/ ( ( P .\/ Q ) ./\ W ) ) )
14 9 10 11 13 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H /\ F e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( ( F ` P ) .\/ ( F ` Q ) ) = ( ( F ` P ) .\/ ( ( P .\/ Q ) ./\ W ) ) )
15 14 oveq1d 
 |-  ( ( ( K e. HL /\ W e. H /\ F e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( ( ( F ` P ) .\/ ( F ` Q ) ) ./\ W ) = ( ( ( F ` P ) .\/ ( ( P .\/ Q ) ./\ W ) ) ./\ W ) )
16 simp2 
 |-  ( ( ( K e. HL /\ W e. H /\ F e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( P e. A /\ -. P .<_ W ) )
17 1 4 5 6 ltrnel 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( ( F ` P ) e. A /\ -. ( F ` P ) .<_ W ) )
18 9 11 16 17 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H /\ F e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( ( F ` P ) e. A /\ -. ( F ` P ) .<_ W ) )
19 eqid 
 |-  ( 0. ` K ) = ( 0. ` K )
20 1 3 19 4 5 lhpmat 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( F ` P ) e. A /\ -. ( F ` P ) .<_ W ) ) -> ( ( F ` P ) ./\ W ) = ( 0. ` K ) )
21 9 18 20 syl2anc 
 |-  ( ( ( K e. HL /\ W e. H /\ F e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( ( F ` P ) ./\ W ) = ( 0. ` K ) )
22 21 oveq1d 
 |-  ( ( ( K e. HL /\ W e. H /\ F e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( ( ( F ` P ) ./\ W ) .\/ ( ( P .\/ Q ) ./\ W ) ) = ( ( 0. ` K ) .\/ ( ( P .\/ Q ) ./\ W ) ) )
23 simp2l 
 |-  ( ( ( K e. HL /\ W e. H /\ F e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> P e. A )
24 1 4 5 6 ltrnat 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ P e. A ) -> ( F ` P ) e. A )
25 9 11 23 24 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H /\ F e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( F ` P ) e. A )
26 7 hllatd 
 |-  ( ( ( K e. HL /\ W e. H /\ F e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> K e. Lat )
27 simp3l 
 |-  ( ( ( K e. HL /\ W e. H /\ F e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> Q e. A )
28 eqid 
 |-  ( Base ` K ) = ( Base ` K )
29 28 2 4 hlatjcl 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .\/ Q ) e. ( Base ` K ) )
30 7 23 27 29 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H /\ F e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( P .\/ Q ) e. ( Base ` K ) )
31 28 5 lhpbase 
 |-  ( W e. H -> W e. ( Base ` K ) )
32 8 31 syl 
 |-  ( ( ( K e. HL /\ W e. H /\ F e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> W e. ( Base ` K ) )
33 28 3 latmcl 
 |-  ( ( K e. Lat /\ ( P .\/ Q ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( ( P .\/ Q ) ./\ W ) e. ( Base ` K ) )
34 26 30 32 33 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H /\ F e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( ( P .\/ Q ) ./\ W ) e. ( Base ` K ) )
35 28 1 3 latmle2 
 |-  ( ( K e. Lat /\ ( P .\/ Q ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( ( P .\/ Q ) ./\ W ) .<_ W )
36 26 30 32 35 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H /\ F e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( ( P .\/ Q ) ./\ W ) .<_ W )
37 28 1 2 3 4 atmod4i2 
 |-  ( ( K e. HL /\ ( ( F ` P ) e. A /\ ( ( P .\/ Q ) ./\ W ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) /\ ( ( P .\/ Q ) ./\ W ) .<_ W ) -> ( ( ( F ` P ) ./\ W ) .\/ ( ( P .\/ Q ) ./\ W ) ) = ( ( ( F ` P ) .\/ ( ( P .\/ Q ) ./\ W ) ) ./\ W ) )
38 7 25 34 32 36 37 syl131anc 
 |-  ( ( ( K e. HL /\ W e. H /\ F e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( ( ( F ` P ) ./\ W ) .\/ ( ( P .\/ Q ) ./\ W ) ) = ( ( ( F ` P ) .\/ ( ( P .\/ Q ) ./\ W ) ) ./\ W ) )
39 hlol 
 |-  ( K e. HL -> K e. OL )
40 7 39 syl 
 |-  ( ( ( K e. HL /\ W e. H /\ F e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> K e. OL )
41 28 2 19 olj02 
 |-  ( ( K e. OL /\ ( ( P .\/ Q ) ./\ W ) e. ( Base ` K ) ) -> ( ( 0. ` K ) .\/ ( ( P .\/ Q ) ./\ W ) ) = ( ( P .\/ Q ) ./\ W ) )
42 40 34 41 syl2anc 
 |-  ( ( ( K e. HL /\ W e. H /\ F e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( ( 0. ` K ) .\/ ( ( P .\/ Q ) ./\ W ) ) = ( ( P .\/ Q ) ./\ W ) )
43 22 38 42 3eqtr3d 
 |-  ( ( ( K e. HL /\ W e. H /\ F e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( ( ( F ` P ) .\/ ( ( P .\/ Q ) ./\ W ) ) ./\ W ) = ( ( P .\/ Q ) ./\ W ) )
44 15 43 eqtrd 
 |-  ( ( ( K e. HL /\ W e. H /\ F e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( ( ( F ` P ) .\/ ( F ` Q ) ) ./\ W ) = ( ( P .\/ Q ) ./\ W ) )