Metamath Proof Explorer

Theorem 3dimlem4OLDN

Description: Lemma for 3dim1 . (Contributed by NM, 25-Jul-2012) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Hypotheses 3dim0.j 
|- .\/ = ( join ` K )
3dim0.l 
|- .<_ = ( le ` K )
3dim0.a 
|- A = ( Atoms ` K )
Assertion 3dimlem4OLDN 
|- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( Q =/= R /\ -. S .<_ ( Q .\/ R ) ) ) /\ ( P =/= Q /\ -. P .<_ ( Q .\/ R ) ) /\ -. P .<_ ( ( Q .\/ R ) .\/ S ) ) -> ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) )

Proof

Step Hyp Ref Expression
1 3dim0.j 
 |-  .\/ = ( join ` K )
2 3dim0.l 
 |-  .<_ = ( le ` K )
3 3dim0.a 
 |-  A = ( Atoms ` K )
4 simp2l 
 |-  ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( Q =/= R /\ -. S .<_ ( Q .\/ R ) ) ) /\ ( P =/= Q /\ -. P .<_ ( Q .\/ R ) ) /\ -. P .<_ ( ( Q .\/ R ) .\/ S ) ) -> P =/= Q )
5 simp2r 
 |-  ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( Q =/= R /\ -. S .<_ ( Q .\/ R ) ) ) /\ ( P =/= Q /\ -. P .<_ ( Q .\/ R ) ) /\ -. P .<_ ( ( Q .\/ R ) .\/ S ) ) -> -. P .<_ ( Q .\/ R ) )
6 simp11 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( Q =/= R /\ -. S .<_ ( Q .\/ R ) ) ) -> K e. HL )
7 simp2l 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( Q =/= R /\ -. S .<_ ( Q .\/ R ) ) ) -> R e. A )
8 simp12 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( Q =/= R /\ -. S .<_ ( Q .\/ R ) ) ) -> P e. A )
9 simp13 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( Q =/= R /\ -. S .<_ ( Q .\/ R ) ) ) -> Q e. A )
10 simp3l 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( Q =/= R /\ -. S .<_ ( Q .\/ R ) ) ) -> Q =/= R )
11 10 necomd 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( Q =/= R /\ -. S .<_ ( Q .\/ R ) ) ) -> R =/= Q )
12 2 1 3 hlatexch2 
 |-  ( ( K e. HL /\ ( R e. A /\ P e. A /\ Q e. A ) /\ R =/= Q ) -> ( R .<_ ( P .\/ Q ) -> P .<_ ( R .\/ Q ) ) )
13 6 7 8 9 11 12 syl131anc 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( Q =/= R /\ -. S .<_ ( Q .\/ R ) ) ) -> ( R .<_ ( P .\/ Q ) -> P .<_ ( R .\/ Q ) ) )
14 1 3 hlatjcom 
 |-  ( ( K e. HL /\ Q e. A /\ R e. A ) -> ( Q .\/ R ) = ( R .\/ Q ) )
15 6 9 7 14 syl3anc 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( Q =/= R /\ -. S .<_ ( Q .\/ R ) ) ) -> ( Q .\/ R ) = ( R .\/ Q ) )
16 15 breq2d 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( Q =/= R /\ -. S .<_ ( Q .\/ R ) ) ) -> ( P .<_ ( Q .\/ R ) <-> P .<_ ( R .\/ Q ) ) )
17 13 16 sylibrd 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( Q =/= R /\ -. S .<_ ( Q .\/ R ) ) ) -> ( R .<_ ( P .\/ Q ) -> P .<_ ( Q .\/ R ) ) )
18 17 3ad2ant1 
 |-  ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( Q =/= R /\ -. S .<_ ( Q .\/ R ) ) ) /\ ( P =/= Q /\ -. P .<_ ( Q .\/ R ) ) /\ -. P .<_ ( ( Q .\/ R ) .\/ S ) ) -> ( R .<_ ( P .\/ Q ) -> P .<_ ( Q .\/ R ) ) )
19 5 18 mtod 
 |-  ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( Q =/= R /\ -. S .<_ ( Q .\/ R ) ) ) /\ ( P =/= Q /\ -. P .<_ ( Q .\/ R ) ) /\ -. P .<_ ( ( Q .\/ R ) .\/ S ) ) -> -. R .<_ ( P .\/ Q ) )
20 simp3 
 |-  ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( Q =/= R /\ -. S .<_ ( Q .\/ R ) ) ) /\ ( P =/= Q /\ -. P .<_ ( Q .\/ R ) ) /\ -. P .<_ ( ( Q .\/ R ) .\/ S ) ) -> -. P .<_ ( ( Q .\/ R ) .\/ S ) )
21 hllat 
 |-  ( K e. HL -> K e. Lat )
22 6 21 syl 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( Q =/= R /\ -. S .<_ ( Q .\/ R ) ) ) -> K e. Lat )
23 eqid 
 |-  ( Base ` K ) = ( Base ` K )
24 23 3 atbase 
 |-  ( Q e. A -> Q e. ( Base ` K ) )
25 9 24 syl 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( Q =/= R /\ -. S .<_ ( Q .\/ R ) ) ) -> Q e. ( Base ` K ) )
26 23 3 atbase 
 |-  ( R e. A -> R e. ( Base ` K ) )
27 7 26 syl 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( Q =/= R /\ -. S .<_ ( Q .\/ R ) ) ) -> R e. ( Base ` K ) )
28 23 3 atbase 
 |-  ( P e. A -> P e. ( Base ` K ) )
29 8 28 syl 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( Q =/= R /\ -. S .<_ ( Q .\/ R ) ) ) -> P e. ( Base ` K ) )
30 23 1 latjrot 
 |-  ( ( K e. Lat /\ ( Q e. ( Base ` K ) /\ R e. ( Base ` K ) /\ P e. ( Base ` K ) ) ) -> ( ( Q .\/ R ) .\/ P ) = ( ( P .\/ Q ) .\/ R ) )
31 22 25 27 29 30 syl13anc 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( Q =/= R /\ -. S .<_ ( Q .\/ R ) ) ) -> ( ( Q .\/ R ) .\/ P ) = ( ( P .\/ Q ) .\/ R ) )
32 31 breq2d 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( Q =/= R /\ -. S .<_ ( Q .\/ R ) ) ) -> ( S .<_ ( ( Q .\/ R ) .\/ P ) <-> S .<_ ( ( P .\/ Q ) .\/ R ) ) )
33 simp2r 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( Q =/= R /\ -. S .<_ ( Q .\/ R ) ) ) -> S e. A )
34 23 1 3 hlatjcl 
 |-  ( ( K e. HL /\ Q e. A /\ R e. A ) -> ( Q .\/ R ) e. ( Base ` K ) )
35 6 9 7 34 syl3anc 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( Q =/= R /\ -. S .<_ ( Q .\/ R ) ) ) -> ( Q .\/ R ) e. ( Base ` K ) )
36 simp3r 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( Q =/= R /\ -. S .<_ ( Q .\/ R ) ) ) -> -. S .<_ ( Q .\/ R ) )
37 23 2 1 3 hlexch1 
 |-  ( ( K e. HL /\ ( S e. A /\ P e. A /\ ( Q .\/ R ) e. ( Base ` K ) ) /\ -. S .<_ ( Q .\/ R ) ) -> ( S .<_ ( ( Q .\/ R ) .\/ P ) -> P .<_ ( ( Q .\/ R ) .\/ S ) ) )
38 6 33 8 35 36 37 syl131anc 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( Q =/= R /\ -. S .<_ ( Q .\/ R ) ) ) -> ( S .<_ ( ( Q .\/ R ) .\/ P ) -> P .<_ ( ( Q .\/ R ) .\/ S ) ) )
39 32 38 sylbird 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( Q =/= R /\ -. S .<_ ( Q .\/ R ) ) ) -> ( S .<_ ( ( P .\/ Q ) .\/ R ) -> P .<_ ( ( Q .\/ R ) .\/ S ) ) )
40 39 3ad2ant1 
 |-  ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( Q =/= R /\ -. S .<_ ( Q .\/ R ) ) ) /\ ( P =/= Q /\ -. P .<_ ( Q .\/ R ) ) /\ -. P .<_ ( ( Q .\/ R ) .\/ S ) ) -> ( S .<_ ( ( P .\/ Q ) .\/ R ) -> P .<_ ( ( Q .\/ R ) .\/ S ) ) )
41 20 40 mtod 
 |-  ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( Q =/= R /\ -. S .<_ ( Q .\/ R ) ) ) /\ ( P =/= Q /\ -. P .<_ ( Q .\/ R ) ) /\ -. P .<_ ( ( Q .\/ R ) .\/ S ) ) -> -. S .<_ ( ( P .\/ Q ) .\/ R ) )
42 4 19 41 3jca 
 |-  ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( Q =/= R /\ -. S .<_ ( Q .\/ R ) ) ) /\ ( P =/= Q /\ -. P .<_ ( Q .\/ R ) ) /\ -. P .<_ ( ( Q .\/ R ) .\/ S ) ) -> ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) )