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Metamath Proof Explorer

Theorem dalawlem2

Description: Lemma for dalaw . Utility lemma that breaks ( ( P .\/ Q ) ./\ ( S .\/ T ) ) into a join of two pieces. (Contributed by NM, 6-Oct-2012)

Ref Expression
Hypotheses dalawlem.l 
|- .<_ = ( le ` K )
dalawlem.j 
|- .\/ = ( join ` K )
dalawlem.m 
|- ./\ = ( meet ` K )
dalawlem.a 
|- A = ( Atoms ` K )
Assertion dalawlem2 
|- ( ( K e. HL /\ ( P e. A /\ Q e. A ) /\ ( S e. A /\ T e. A ) ) -> ( ( P .\/ Q ) ./\ ( S .\/ T ) ) .<_ ( ( ( ( P .\/ Q ) .\/ T ) ./\ S ) .\/ ( ( ( P .\/ Q ) .\/ S ) ./\ T ) ) )

Proof

Step Hyp Ref Expression
1 dalawlem.l 
 |-  .<_ = ( le ` K )
2 dalawlem.j 
 |-  .\/ = ( join ` K )
3 dalawlem.m 
 |-  ./\ = ( meet ` K )
4 dalawlem.a 
 |-  A = ( Atoms ` K )
5 simp1 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A ) /\ ( S e. A /\ T e. A ) ) -> K e. HL )
6 5 hllatd 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A ) /\ ( S e. A /\ T e. A ) ) -> K e. Lat )
7 simp2l 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A ) /\ ( S e. A /\ T e. A ) ) -> P e. A )
8 simp2r 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A ) /\ ( S e. A /\ T e. A ) ) -> Q e. A )
9 eqid 
 |-  ( Base ` K ) = ( Base ` K )
10 9 2 4 hlatjcl 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .\/ Q ) e. ( Base ` K ) )
11 5 7 8 10 syl3anc 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A ) /\ ( S e. A /\ T e. A ) ) -> ( P .\/ Q ) e. ( Base ` K ) )
12 simp3r 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A ) /\ ( S e. A /\ T e. A ) ) -> T e. A )
13 9 4 atbase 
 |-  ( T e. A -> T e. ( Base ` K ) )
14 12 13 syl 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A ) /\ ( S e. A /\ T e. A ) ) -> T e. ( Base ` K ) )
15 9 1 2 latlej1 
 |-  ( ( K e. Lat /\ ( P .\/ Q ) e. ( Base ` K ) /\ T e. ( Base ` K ) ) -> ( P .\/ Q ) .<_ ( ( P .\/ Q ) .\/ T ) )
16 6 11 14 15 syl3anc 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A ) /\ ( S e. A /\ T e. A ) ) -> ( P .\/ Q ) .<_ ( ( P .\/ Q ) .\/ T ) )
17 simp3l 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A ) /\ ( S e. A /\ T e. A ) ) -> S e. A )
18 9 4 atbase 
 |-  ( S e. A -> S e. ( Base ` K ) )
19 17 18 syl 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A ) /\ ( S e. A /\ T e. A ) ) -> S e. ( Base ` K ) )
20 9 1 2 latlej1 
 |-  ( ( K e. Lat /\ ( P .\/ Q ) e. ( Base ` K ) /\ S e. ( Base ` K ) ) -> ( P .\/ Q ) .<_ ( ( P .\/ Q ) .\/ S ) )
21 6 11 19 20 syl3anc 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A ) /\ ( S e. A /\ T e. A ) ) -> ( P .\/ Q ) .<_ ( ( P .\/ Q ) .\/ S ) )
22 9 2 latjcl 
 |-  ( ( K e. Lat /\ ( P .\/ Q ) e. ( Base ` K ) /\ T e. ( Base ` K ) ) -> ( ( P .\/ Q ) .\/ T ) e. ( Base ` K ) )
23 6 11 14 22 syl3anc 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A ) /\ ( S e. A /\ T e. A ) ) -> ( ( P .\/ Q ) .\/ T ) e. ( Base ` K ) )
24 9 2 latjcl 
 |-  ( ( K e. Lat /\ ( P .\/ Q ) e. ( Base ` K ) /\ S e. ( Base ` K ) ) -> ( ( P .\/ Q ) .\/ S ) e. ( Base ` K ) )
25 6 11 19 24 syl3anc 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A ) /\ ( S e. A /\ T e. A ) ) -> ( ( P .\/ Q ) .\/ S ) e. ( Base ` K ) )
26 9 1 3 latlem12 
 |-  ( ( K e. Lat /\ ( ( P .\/ Q ) e. ( Base ` K ) /\ ( ( P .\/ Q ) .\/ T ) e. ( Base ` K ) /\ ( ( P .\/ Q ) .\/ S ) e. ( Base ` K ) ) ) -> ( ( ( P .\/ Q ) .<_ ( ( P .\/ Q ) .\/ T ) /\ ( P .\/ Q ) .<_ ( ( P .\/ Q ) .\/ S ) ) <-> ( P .\/ Q ) .<_ ( ( ( P .\/ Q ) .\/ T ) ./\ ( ( P .\/ Q ) .\/ S ) ) ) )
27 6 11 23 25 26 syl13anc 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A ) /\ ( S e. A /\ T e. A ) ) -> ( ( ( P .\/ Q ) .<_ ( ( P .\/ Q ) .\/ T ) /\ ( P .\/ Q ) .<_ ( ( P .\/ Q ) .\/ S ) ) <-> ( P .\/ Q ) .<_ ( ( ( P .\/ Q ) .\/ T ) ./\ ( ( P .\/ Q ) .\/ S ) ) ) )
28 16 21 27 mpbi2and 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A ) /\ ( S e. A /\ T e. A ) ) -> ( P .\/ Q ) .<_ ( ( ( P .\/ Q ) .\/ T ) ./\ ( ( P .\/ Q ) .\/ S ) ) )
29 9 3 latmcl 
 |-  ( ( K e. Lat /\ ( ( P .\/ Q ) .\/ T ) e. ( Base ` K ) /\ ( ( P .\/ Q ) .\/ S ) e. ( Base ` K ) ) -> ( ( ( P .\/ Q ) .\/ T ) ./\ ( ( P .\/ Q ) .\/ S ) ) e. ( Base ` K ) )
30 6 23 25 29 syl3anc 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A ) /\ ( S e. A /\ T e. A ) ) -> ( ( ( P .\/ Q ) .\/ T ) ./\ ( ( P .\/ Q ) .\/ S ) ) e. ( Base ` K ) )
31 9 2 4 hlatjcl 
 |-  ( ( K e. HL /\ S e. A /\ T e. A ) -> ( S .\/ T ) e. ( Base ` K ) )
32 5 17 12 31 syl3anc 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A ) /\ ( S e. A /\ T e. A ) ) -> ( S .\/ T ) e. ( Base ` K ) )
33 9 1 3 latmlem1 
 |-  ( ( K e. Lat /\ ( ( P .\/ Q ) e. ( Base ` K ) /\ ( ( ( P .\/ Q ) .\/ T ) ./\ ( ( P .\/ Q ) .\/ S ) ) e. ( Base ` K ) /\ ( S .\/ T ) e. ( Base ` K ) ) ) -> ( ( P .\/ Q ) .<_ ( ( ( P .\/ Q ) .\/ T ) ./\ ( ( P .\/ Q ) .\/ S ) ) -> ( ( P .\/ Q ) ./\ ( S .\/ T ) ) .<_ ( ( ( ( P .\/ Q ) .\/ T ) ./\ ( ( P .\/ Q ) .\/ S ) ) ./\ ( S .\/ T ) ) ) )
34 6 11 30 32 33 syl13anc 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A ) /\ ( S e. A /\ T e. A ) ) -> ( ( P .\/ Q ) .<_ ( ( ( P .\/ Q ) .\/ T ) ./\ ( ( P .\/ Q ) .\/ S ) ) -> ( ( P .\/ Q ) ./\ ( S .\/ T ) ) .<_ ( ( ( ( P .\/ Q ) .\/ T ) ./\ ( ( P .\/ Q ) .\/ S ) ) ./\ ( S .\/ T ) ) ) )
35 28 34 mpd 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A ) /\ ( S e. A /\ T e. A ) ) -> ( ( P .\/ Q ) ./\ ( S .\/ T ) ) .<_ ( ( ( ( P .\/ Q ) .\/ T ) ./\ ( ( P .\/ Q ) .\/ S ) ) ./\ ( S .\/ T ) ) )
36 9 1 2 latlej2 
 |-  ( ( K e. Lat /\ ( P .\/ Q ) e. ( Base ` K ) /\ S e. ( Base ` K ) ) -> S .<_ ( ( P .\/ Q ) .\/ S ) )
37 6 11 19 36 syl3anc 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A ) /\ ( S e. A /\ T e. A ) ) -> S .<_ ( ( P .\/ Q ) .\/ S ) )
38 9 1 2 3 4 atmod3i1 
 |-  ( ( K e. HL /\ ( S e. A /\ ( ( P .\/ Q ) .\/ S ) e. ( Base ` K ) /\ T e. ( Base ` K ) ) /\ S .<_ ( ( P .\/ Q ) .\/ S ) ) -> ( S .\/ ( ( ( P .\/ Q ) .\/ S ) ./\ T ) ) = ( ( ( P .\/ Q ) .\/ S ) ./\ ( S .\/ T ) ) )
39 5 17 25 14 37 38 syl131anc 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A ) /\ ( S e. A /\ T e. A ) ) -> ( S .\/ ( ( ( P .\/ Q ) .\/ S ) ./\ T ) ) = ( ( ( P .\/ Q ) .\/ S ) ./\ ( S .\/ T ) ) )
40 39 oveq2d 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A ) /\ ( S e. A /\ T e. A ) ) -> ( ( ( P .\/ Q ) .\/ T ) ./\ ( S .\/ ( ( ( P .\/ Q ) .\/ S ) ./\ T ) ) ) = ( ( ( P .\/ Q ) .\/ T ) ./\ ( ( ( P .\/ Q ) .\/ S ) ./\ ( S .\/ T ) ) ) )
41 9 3 latmcl 
 |-  ( ( K e. Lat /\ ( ( P .\/ Q ) .\/ S ) e. ( Base ` K ) /\ T e. ( Base ` K ) ) -> ( ( ( P .\/ Q ) .\/ S ) ./\ T ) e. ( Base ` K ) )
42 6 25 14 41 syl3anc 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A ) /\ ( S e. A /\ T e. A ) ) -> ( ( ( P .\/ Q ) .\/ S ) ./\ T ) e. ( Base ` K ) )
43 9 1 2 3 latmlej22 
 |-  ( ( K e. Lat /\ ( T e. ( Base ` K ) /\ ( ( P .\/ Q ) .\/ S ) e. ( Base ` K ) /\ ( P .\/ Q ) e. ( Base ` K ) ) ) -> ( ( ( P .\/ Q ) .\/ S ) ./\ T ) .<_ ( ( P .\/ Q ) .\/ T ) )
44 6 14 25 11 43 syl13anc 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A ) /\ ( S e. A /\ T e. A ) ) -> ( ( ( P .\/ Q ) .\/ S ) ./\ T ) .<_ ( ( P .\/ Q ) .\/ T ) )
45 9 1 2 3 4 atmod2i2 
 |-  ( ( K e. HL /\ ( S e. A /\ ( ( P .\/ Q ) .\/ T ) e. ( Base ` K ) /\ ( ( ( P .\/ Q ) .\/ S ) ./\ T ) e. ( Base ` K ) ) /\ ( ( ( P .\/ Q ) .\/ S ) ./\ T ) .<_ ( ( P .\/ Q ) .\/ T ) ) -> ( ( ( ( P .\/ Q ) .\/ T ) ./\ S ) .\/ ( ( ( P .\/ Q ) .\/ S ) ./\ T ) ) = ( ( ( P .\/ Q ) .\/ T ) ./\ ( S .\/ ( ( ( P .\/ Q ) .\/ S ) ./\ T ) ) ) )
46 5 17 23 42 44 45 syl131anc 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A ) /\ ( S e. A /\ T e. A ) ) -> ( ( ( ( P .\/ Q ) .\/ T ) ./\ S ) .\/ ( ( ( P .\/ Q ) .\/ S ) ./\ T ) ) = ( ( ( P .\/ Q ) .\/ T ) ./\ ( S .\/ ( ( ( P .\/ Q ) .\/ S ) ./\ T ) ) ) )
47 hlol 
 |-  ( K e. HL -> K e. OL )
48 5 47 syl 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A ) /\ ( S e. A /\ T e. A ) ) -> K e. OL )
49 9 3 latmassOLD 
 |-  ( ( K e. OL /\ ( ( ( P .\/ Q ) .\/ T ) e. ( Base ` K ) /\ ( ( P .\/ Q ) .\/ S ) e. ( Base ` K ) /\ ( S .\/ T ) e. ( Base ` K ) ) ) -> ( ( ( ( P .\/ Q ) .\/ T ) ./\ ( ( P .\/ Q ) .\/ S ) ) ./\ ( S .\/ T ) ) = ( ( ( P .\/ Q ) .\/ T ) ./\ ( ( ( P .\/ Q ) .\/ S ) ./\ ( S .\/ T ) ) ) )
50 48 23 25 32 49 syl13anc 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A ) /\ ( S e. A /\ T e. A ) ) -> ( ( ( ( P .\/ Q ) .\/ T ) ./\ ( ( P .\/ Q ) .\/ S ) ) ./\ ( S .\/ T ) ) = ( ( ( P .\/ Q ) .\/ T ) ./\ ( ( ( P .\/ Q ) .\/ S ) ./\ ( S .\/ T ) ) ) )
51 40 46 50 3eqtr4rd 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A ) /\ ( S e. A /\ T e. A ) ) -> ( ( ( ( P .\/ Q ) .\/ T ) ./\ ( ( P .\/ Q ) .\/ S ) ) ./\ ( S .\/ T ) ) = ( ( ( ( P .\/ Q ) .\/ T ) ./\ S ) .\/ ( ( ( P .\/ Q ) .\/ S ) ./\ T ) ) )
52 35 51 breqtrd 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A ) /\ ( S e. A /\ T e. A ) ) -> ( ( P .\/ Q ) ./\ ( S .\/ T ) ) .<_ ( ( ( ( P .\/ Q ) .\/ T ) ./\ S ) .\/ ( ( ( P .\/ Q ) .\/ S ) ./\ T ) ) )