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Metamath Proof Explorer

Theorem dalawlem8

Description: Lemma for dalaw . Special case to eliminate the requirement -. ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) in dalawlem1 . (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 dalawlem8 
|- ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> ( ( P .\/ Q ) ./\ ( S .\/ T ) ) .<_ ( ( ( Q .\/ R ) ./\ ( T .\/ U ) ) .\/ ( ( R .\/ P ) ./\ ( U .\/ S ) ) ) )

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 eqid 
 |-  ( Base ` K ) = ( Base ` K )
6 simp11 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> K e. HL )
7 6 hllatd 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> K e. Lat )
8 simp21 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> P e. A )
9 simp22 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> Q e. A )
10 5 2 4 hlatjcl 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .\/ Q ) e. ( Base ` K ) )
11 6 8 9 10 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> ( P .\/ Q ) e. ( Base ` K ) )
12 simp31 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> S e. A )
13 simp32 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> T e. A )
14 5 2 4 hlatjcl 
 |-  ( ( K e. HL /\ S e. A /\ T e. A ) -> ( S .\/ T ) e. ( Base ` K ) )
15 6 12 13 14 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> ( S .\/ T ) e. ( Base ` K ) )
16 5 3 latmcl 
 |-  ( ( K e. Lat /\ ( P .\/ Q ) e. ( Base ` K ) /\ ( S .\/ T ) e. ( Base ` K ) ) -> ( ( P .\/ Q ) ./\ ( S .\/ T ) ) e. ( Base ` K ) )
17 7 11 15 16 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> ( ( P .\/ Q ) ./\ ( S .\/ T ) ) e. ( Base ` K ) )
18 5 4 atbase 
 |-  ( T e. A -> T e. ( Base ` K ) )
19 13 18 syl 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> T e. ( Base ` K ) )
20 5 2 latjcl 
 |-  ( ( K e. Lat /\ ( P .\/ Q ) e. ( Base ` K ) /\ T e. ( Base ` K ) ) -> ( ( P .\/ Q ) .\/ T ) e. ( Base ` K ) )
21 7 11 19 20 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> ( ( P .\/ Q ) .\/ T ) e. ( Base ` K ) )
22 5 4 atbase 
 |-  ( S e. A -> S e. ( Base ` K ) )
23 12 22 syl 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> S e. ( Base ` K ) )
24 5 3 latmcl 
 |-  ( ( K e. Lat /\ ( ( P .\/ Q ) .\/ T ) e. ( Base ` K ) /\ S e. ( Base ` K ) ) -> ( ( ( P .\/ Q ) .\/ T ) ./\ S ) e. ( Base ` K ) )
25 7 21 23 24 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> ( ( ( P .\/ Q ) .\/ T ) ./\ S ) e. ( Base ` K ) )
26 5 2 latjcl 
 |-  ( ( K e. Lat /\ ( P .\/ Q ) e. ( Base ` K ) /\ S e. ( Base ` K ) ) -> ( ( P .\/ Q ) .\/ S ) e. ( Base ` K ) )
27 7 11 23 26 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> ( ( P .\/ Q ) .\/ S ) e. ( Base ` K ) )
28 5 3 latmcl 
 |-  ( ( K e. Lat /\ ( ( P .\/ Q ) .\/ S ) e. ( Base ` K ) /\ T e. ( Base ` K ) ) -> ( ( ( P .\/ Q ) .\/ S ) ./\ T ) e. ( Base ` K ) )
29 7 27 19 28 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> ( ( ( P .\/ Q ) .\/ S ) ./\ T ) e. ( Base ` K ) )
30 5 2 latjcl 
 |-  ( ( K e. Lat /\ ( ( ( P .\/ Q ) .\/ T ) ./\ S ) e. ( Base ` K ) /\ ( ( ( P .\/ Q ) .\/ S ) ./\ T ) e. ( Base ` K ) ) -> ( ( ( ( P .\/ Q ) .\/ T ) ./\ S ) .\/ ( ( ( P .\/ Q ) .\/ S ) ./\ T ) ) e. ( Base ` K ) )
31 7 25 29 30 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> ( ( ( ( P .\/ Q ) .\/ T ) ./\ S ) .\/ ( ( ( P .\/ Q ) .\/ S ) ./\ T ) ) e. ( Base ` K ) )
32 simp23 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> R e. A )
33 5 2 4 hlatjcl 
 |-  ( ( K e. HL /\ Q e. A /\ R e. A ) -> ( Q .\/ R ) e. ( Base ` K ) )
34 6 9 32 33 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> ( Q .\/ R ) e. ( Base ` K ) )
35 simp33 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> U e. A )
36 5 2 4 hlatjcl 
 |-  ( ( K e. HL /\ T e. A /\ U e. A ) -> ( T .\/ U ) e. ( Base ` K ) )
37 6 13 35 36 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> ( T .\/ U ) e. ( Base ` K ) )
38 5 3 latmcl 
 |-  ( ( K e. Lat /\ ( Q .\/ R ) e. ( Base ` K ) /\ ( T .\/ U ) e. ( Base ` K ) ) -> ( ( Q .\/ R ) ./\ ( T .\/ U ) ) e. ( Base ` K ) )
39 7 34 37 38 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> ( ( Q .\/ R ) ./\ ( T .\/ U ) ) e. ( Base ` K ) )
40 5 2 4 hlatjcl 
 |-  ( ( K e. HL /\ R e. A /\ P e. A ) -> ( R .\/ P ) e. ( Base ` K ) )
41 6 32 8 40 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> ( R .\/ P ) e. ( Base ` K ) )
42 5 2 4 hlatjcl 
 |-  ( ( K e. HL /\ U e. A /\ S e. A ) -> ( U .\/ S ) e. ( Base ` K ) )
43 6 35 12 42 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> ( U .\/ S ) e. ( Base ` K ) )
44 5 3 latmcl 
 |-  ( ( K e. Lat /\ ( R .\/ P ) e. ( Base ` K ) /\ ( U .\/ S ) e. ( Base ` K ) ) -> ( ( R .\/ P ) ./\ ( U .\/ S ) ) e. ( Base ` K ) )
45 7 41 43 44 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> ( ( R .\/ P ) ./\ ( U .\/ S ) ) e. ( Base ` K ) )
46 5 2 latjcl 
 |-  ( ( K e. Lat /\ ( ( Q .\/ R ) ./\ ( T .\/ U ) ) e. ( Base ` K ) /\ ( ( R .\/ P ) ./\ ( U .\/ S ) ) e. ( Base ` K ) ) -> ( ( ( Q .\/ R ) ./\ ( T .\/ U ) ) .\/ ( ( R .\/ P ) ./\ ( U .\/ S ) ) ) e. ( Base ` K ) )
47 7 39 45 46 syl3anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> ( ( ( Q .\/ R ) ./\ ( T .\/ U ) ) .\/ ( ( R .\/ P ) ./\ ( U .\/ S ) ) ) e. ( Base ` K ) )
48 1 2 3 4 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 ) ) )
49 6 8 9 12 13 48 syl122anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> ( ( P .\/ Q ) ./\ ( S .\/ T ) ) .<_ ( ( ( ( P .\/ Q ) .\/ T ) ./\ S ) .\/ ( ( ( P .\/ Q ) .\/ S ) ./\ T ) ) )
50 1 2 3 4 dalawlem6 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> ( ( ( P .\/ Q ) .\/ T ) ./\ S ) .<_ ( ( ( Q .\/ R ) ./\ ( T .\/ U ) ) .\/ ( ( R .\/ P ) ./\ ( U .\/ S ) ) ) )
51 1 2 3 4 dalawlem7 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> ( ( ( P .\/ Q ) .\/ S ) ./\ T ) .<_ ( ( ( Q .\/ R ) ./\ ( T .\/ U ) ) .\/ ( ( R .\/ P ) ./\ ( U .\/ S ) ) ) )
52 5 1 2 latjle12 
 |-  ( ( K e. Lat /\ ( ( ( ( P .\/ Q ) .\/ T ) ./\ S ) e. ( Base ` K ) /\ ( ( ( P .\/ Q ) .\/ S ) ./\ T ) e. ( Base ` K ) /\ ( ( ( Q .\/ R ) ./\ ( T .\/ U ) ) .\/ ( ( R .\/ P ) ./\ ( U .\/ S ) ) ) e. ( Base ` K ) ) ) -> ( ( ( ( ( P .\/ Q ) .\/ T ) ./\ S ) .<_ ( ( ( Q .\/ R ) ./\ ( T .\/ U ) ) .\/ ( ( R .\/ P ) ./\ ( U .\/ S ) ) ) /\ ( ( ( P .\/ Q ) .\/ S ) ./\ T ) .<_ ( ( ( Q .\/ R ) ./\ ( T .\/ U ) ) .\/ ( ( R .\/ P ) ./\ ( U .\/ S ) ) ) ) <-> ( ( ( ( P .\/ Q ) .\/ T ) ./\ S ) .\/ ( ( ( P .\/ Q ) .\/ S ) ./\ T ) ) .<_ ( ( ( Q .\/ R ) ./\ ( T .\/ U ) ) .\/ ( ( R .\/ P ) ./\ ( U .\/ S ) ) ) ) )
53 7 25 29 47 52 syl13anc 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> ( ( ( ( ( P .\/ Q ) .\/ T ) ./\ S ) .<_ ( ( ( Q .\/ R ) ./\ ( T .\/ U ) ) .\/ ( ( R .\/ P ) ./\ ( U .\/ S ) ) ) /\ ( ( ( P .\/ Q ) .\/ S ) ./\ T ) .<_ ( ( ( Q .\/ R ) ./\ ( T .\/ U ) ) .\/ ( ( R .\/ P ) ./\ ( U .\/ S ) ) ) ) <-> ( ( ( ( P .\/ Q ) .\/ T ) ./\ S ) .\/ ( ( ( P .\/ Q ) .\/ S ) ./\ T ) ) .<_ ( ( ( Q .\/ R ) ./\ ( T .\/ U ) ) .\/ ( ( R .\/ P ) ./\ ( U .\/ S ) ) ) ) )
54 50 51 53 mpbi2and 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> ( ( ( ( P .\/ Q ) .\/ T ) ./\ S ) .\/ ( ( ( P .\/ Q ) .\/ S ) ./\ T ) ) .<_ ( ( ( Q .\/ R ) ./\ ( T .\/ U ) ) .\/ ( ( R .\/ P ) ./\ ( U .\/ S ) ) ) )
55 5 1 7 17 31 47 49 54 lattrd 
 |-  ( ( ( K e. HL /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( Q .\/ R ) /\ ( ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( R .\/ U ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) -> ( ( P .\/ Q ) ./\ ( S .\/ T ) ) .<_ ( ( ( Q .\/ R ) ./\ ( T .\/ U ) ) .\/ ( ( R .\/ P ) ./\ ( U .\/ S ) ) ) )