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Metamath Proof Explorer

Theorem cdlemg18b

Description: Lemma for cdlemg18c . TODO: fix comment. (Contributed by NM, 15-May-2013)

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 )
cdlemg18b.u 
|- U = ( ( P .\/ Q ) ./\ W )
Assertion cdlemg18b 
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) ) -> -. P .<_ ( U .\/ ( F ` Q ) ) )

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 cdlemg18b.u 
 |-  U = ( ( P .\/ Q ) ./\ W )
9 simp33 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) ) -> ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) )
10 simp3r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) /\ P .<_ ( U .\/ ( F ` Q ) ) ) ) -> P .<_ ( U .\/ ( F ` Q ) ) )
11 simp1l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) /\ P .<_ ( U .\/ ( F ` Q ) ) ) ) -> K e. HL )
12 simp1r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) /\ P .<_ ( U .\/ ( F ` Q ) ) ) ) -> W e. H )
13 simp21 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) /\ P .<_ ( U .\/ ( F ` Q ) ) ) ) -> ( P e. A /\ -. P .<_ W ) )
14 simp22l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) /\ P .<_ ( U .\/ ( F ` Q ) ) ) ) -> Q e. A )
15 simp3l1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) /\ P .<_ ( U .\/ ( F ` Q ) ) ) ) -> P =/= Q )
16 1 2 3 4 5 8 cdleme0a 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) ) -> U e. A )
17 11 12 13 14 15 16 syl212anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) /\ P .<_ ( U .\/ ( F ` Q ) ) ) ) -> U e. A )
18 simp1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) /\ P .<_ ( U .\/ ( F ` Q ) ) ) ) -> ( K e. HL /\ W e. H ) )
19 simp23 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) /\ P .<_ ( U .\/ ( F ` Q ) ) ) ) -> F e. T )
20 1 4 5 6 ltrnat 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ Q e. A ) -> ( F ` Q ) e. A )
21 18 19 14 20 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) /\ P .<_ ( U .\/ ( F ` Q ) ) ) ) -> ( F ` Q ) e. A )
22 1 2 4 hlatlej1 
 |-  ( ( K e. HL /\ U e. A /\ ( F ` Q ) e. A ) -> U .<_ ( U .\/ ( F ` Q ) ) )
23 11 17 21 22 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) /\ P .<_ ( U .\/ ( F ` Q ) ) ) ) -> U .<_ ( U .\/ ( F ` Q ) ) )
24 11 hllatd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) /\ P .<_ ( U .\/ ( F ` Q ) ) ) ) -> K e. Lat )
25 simp21l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) /\ P .<_ ( U .\/ ( F ` Q ) ) ) ) -> P e. A )
26 eqid 
 |-  ( Base ` K ) = ( Base ` K )
27 26 4 atbase 
 |-  ( P e. A -> P e. ( Base ` K ) )
28 25 27 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) /\ P .<_ ( U .\/ ( F ` Q ) ) ) ) -> P e. ( Base ` K ) )
29 26 4 atbase 
 |-  ( U e. A -> U e. ( Base ` K ) )
30 17 29 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) /\ P .<_ ( U .\/ ( F ` Q ) ) ) ) -> U e. ( Base ` K ) )
31 26 2 4 hlatjcl 
 |-  ( ( K e. HL /\ U e. A /\ ( F ` Q ) e. A ) -> ( U .\/ ( F ` Q ) ) e. ( Base ` K ) )
32 11 17 21 31 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) /\ P .<_ ( U .\/ ( F ` Q ) ) ) ) -> ( U .\/ ( F ` Q ) ) e. ( Base ` K ) )
33 26 1 2 latjle12 
 |-  ( ( K e. Lat /\ ( P e. ( Base ` K ) /\ U e. ( Base ` K ) /\ ( U .\/ ( F ` Q ) ) e. ( Base ` K ) ) ) -> ( ( P .<_ ( U .\/ ( F ` Q ) ) /\ U .<_ ( U .\/ ( F ` Q ) ) ) <-> ( P .\/ U ) .<_ ( U .\/ ( F ` Q ) ) ) )
34 24 28 30 32 33 syl13anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) /\ P .<_ ( U .\/ ( F ` Q ) ) ) ) -> ( ( P .<_ ( U .\/ ( F ` Q ) ) /\ U .<_ ( U .\/ ( F ` Q ) ) ) <-> ( P .\/ U ) .<_ ( U .\/ ( F ` Q ) ) ) )
35 10 23 34 mpbi2and 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) /\ P .<_ ( U .\/ ( F ` Q ) ) ) ) -> ( P .\/ U ) .<_ ( U .\/ ( F ` Q ) ) )
36 1 2 3 4 5 8 cdleme0cp 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) ) -> ( P .\/ U ) = ( P .\/ Q ) )
37 11 12 13 14 36 syl22anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) /\ P .<_ ( U .\/ ( F ` Q ) ) ) ) -> ( P .\/ U ) = ( P .\/ Q ) )
38 simp22 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) /\ P .<_ ( U .\/ ( F ` Q ) ) ) ) -> ( Q e. A /\ -. Q .<_ W ) )
39 5 6 1 2 4 3 8 cdlemg2kq 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ F e. T ) -> ( ( F ` P ) .\/ ( F ` Q ) ) = ( ( F ` Q ) .\/ U ) )
40 18 13 38 19 39 syl121anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) /\ P .<_ ( U .\/ ( F ` Q ) ) ) ) -> ( ( F ` P ) .\/ ( F ` Q ) ) = ( ( F ` Q ) .\/ U ) )
41 2 4 hlatjcom 
 |-  ( ( K e. HL /\ ( F ` Q ) e. A /\ U e. A ) -> ( ( F ` Q ) .\/ U ) = ( U .\/ ( F ` Q ) ) )
42 11 21 17 41 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) /\ P .<_ ( U .\/ ( F ` Q ) ) ) ) -> ( ( F ` Q ) .\/ U ) = ( U .\/ ( F ` Q ) ) )
43 40 42 eqtr2d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) /\ P .<_ ( U .\/ ( F ` Q ) ) ) ) -> ( U .\/ ( F ` Q ) ) = ( ( F ` P ) .\/ ( F ` Q ) ) )
44 35 37 43 3brtr3d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) /\ P .<_ ( U .\/ ( F ` Q ) ) ) ) -> ( P .\/ Q ) .<_ ( ( F ` P ) .\/ ( F ` Q ) ) )
45 1 4 5 6 ltrnat 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ P e. A ) -> ( F ` P ) e. A )
46 18 19 25 45 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) /\ P .<_ ( U .\/ ( F ` Q ) ) ) ) -> ( F ` P ) e. A )
47 1 2 4 ps-1 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( ( F ` P ) e. A /\ ( F ` Q ) e. A ) ) -> ( ( P .\/ Q ) .<_ ( ( F ` P ) .\/ ( F ` Q ) ) <-> ( P .\/ Q ) = ( ( F ` P ) .\/ ( F ` Q ) ) ) )
48 11 25 14 15 46 21 47 syl132anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) /\ P .<_ ( U .\/ ( F ` Q ) ) ) ) -> ( ( P .\/ Q ) .<_ ( ( F ` P ) .\/ ( F ` Q ) ) <-> ( P .\/ Q ) = ( ( F ` P ) .\/ ( F ` Q ) ) ) )
49 44 48 mpbid 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) /\ P .<_ ( U .\/ ( F ` Q ) ) ) ) -> ( P .\/ Q ) = ( ( F ` P ) .\/ ( F ` Q ) ) )
50 2 4 hlatjcom 
 |-  ( ( K e. HL /\ ( F ` P ) e. A /\ ( F ` Q ) e. A ) -> ( ( F ` P ) .\/ ( F ` Q ) ) = ( ( F ` Q ) .\/ ( F ` P ) ) )
51 11 46 21 50 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) /\ P .<_ ( U .\/ ( F ` Q ) ) ) ) -> ( ( F ` P ) .\/ ( F ` Q ) ) = ( ( F ` Q ) .\/ ( F ` P ) ) )
52 49 51 eqtr2d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) /\ P .<_ ( U .\/ ( F ` Q ) ) ) ) -> ( ( F ` Q ) .\/ ( F ` P ) ) = ( P .\/ Q ) )
53 52 3exp 
 |-  ( ( K e. HL /\ W e. H ) -> ( ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) -> ( ( ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) /\ P .<_ ( U .\/ ( F ` Q ) ) ) -> ( ( F ` Q ) .\/ ( F ` P ) ) = ( P .\/ Q ) ) ) )
54 53 exp4a 
 |-  ( ( K e. HL /\ W e. H ) -> ( ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) -> ( ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) -> ( P .<_ ( U .\/ ( F ` Q ) ) -> ( ( F ` Q ) .\/ ( F ` P ) ) = ( P .\/ Q ) ) ) ) )
55 54 3imp 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) ) -> ( P .<_ ( U .\/ ( F ` Q ) ) -> ( ( F ` Q ) .\/ ( F ` P ) ) = ( P .\/ Q ) ) )
56 55 necon3ad 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) ) -> ( ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) -> -. P .<_ ( U .\/ ( F ` Q ) ) ) )
57 9 56 mpd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) ) -> -. P .<_ ( U .\/ ( F ` Q ) ) )