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Metamath Proof Explorer

Theorem 4noncolr3

Description: A way to express 4 non-colinear atoms (rotated right 3 places). (Contributed by NM, 11-Jul-2012)

Ref Expression
Hypotheses 3noncol.l 
|- .<_ = ( le ` K )
3noncol.j 
|- .\/ = ( join ` K )
3noncol.a 
|- A = ( Atoms ` K )
Assertion 4noncolr3 
|- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) -> ( Q =/= R /\ -. S .<_ ( Q .\/ R ) /\ -. P .<_ ( ( Q .\/ R ) .\/ S ) ) )

Proof

Step Hyp Ref Expression
1 3noncol.l 
 |-  .<_ = ( le ` K )
2 3noncol.j 
 |-  .\/ = ( join ` K )
3 3noncol.a 
 |-  A = ( Atoms ` K )
4 simp11 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) -> K e. HL )
5 4 hllatd 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) -> K e. Lat )
6 simp2l 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) -> R e. A )
7 eqid 
 |-  ( Base ` K ) = ( Base ` K )
8 7 3 atbase 
 |-  ( R e. A -> R e. ( Base ` K ) )
9 6 8 syl 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) -> R e. ( Base ` K ) )
10 simp12 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) -> P e. A )
11 7 3 atbase 
 |-  ( P e. A -> P e. ( Base ` K ) )
12 10 11 syl 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) -> P e. ( Base ` K ) )
13 simp13 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) -> Q e. A )
14 7 3 atbase 
 |-  ( Q e. A -> Q e. ( Base ` K ) )
15 13 14 syl 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) -> Q e. ( Base ` K ) )
16 simp32 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) -> -. R .<_ ( P .\/ Q ) )
17 7 1 2 latnlej1r 
 |-  ( ( K e. Lat /\ ( R e. ( Base ` K ) /\ P e. ( Base ` K ) /\ Q e. ( Base ` K ) ) /\ -. R .<_ ( P .\/ Q ) ) -> R =/= Q )
18 5 9 12 15 16 17 syl131anc 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) -> R =/= Q )
19 18 necomd 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) -> Q =/= R )
20 simp2r 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) -> S e. A )
21 7 3 atbase 
 |-  ( S e. A -> S e. ( Base ` K ) )
22 20 21 syl 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) -> S e. ( Base ` K ) )
23 7 2 latjcl 
 |-  ( ( K e. Lat /\ Q e. ( Base ` K ) /\ R e. ( Base ` K ) ) -> ( Q .\/ R ) e. ( Base ` K ) )
24 5 15 9 23 syl3anc 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) -> ( Q .\/ R ) e. ( Base ` K ) )
25 simp33 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) -> -. S .<_ ( ( P .\/ Q ) .\/ R ) )
26 2 3 hlatjass 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> ( ( P .\/ Q ) .\/ R ) = ( P .\/ ( Q .\/ R ) ) )
27 4 10 13 6 26 syl13anc 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) -> ( ( P .\/ Q ) .\/ R ) = ( P .\/ ( Q .\/ R ) ) )
28 27 breq2d 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) -> ( S .<_ ( ( P .\/ Q ) .\/ R ) <-> S .<_ ( P .\/ ( Q .\/ R ) ) ) )
29 25 28 mtbid 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) -> -. S .<_ ( P .\/ ( Q .\/ R ) ) )
30 7 1 2 latnlej2r 
 |-  ( ( K e. Lat /\ ( S e. ( Base ` K ) /\ P e. ( Base ` K ) /\ ( Q .\/ R ) e. ( Base ` K ) ) /\ -. S .<_ ( P .\/ ( Q .\/ R ) ) ) -> -. S .<_ ( Q .\/ R ) )
31 5 22 12 24 29 30 syl131anc 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) -> -. S .<_ ( Q .\/ R ) )
32 simp31 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) -> P =/= Q )
33 1 2 3 hlatexch1 
 |-  ( ( K e. HL /\ ( P e. A /\ R e. A /\ Q e. A ) /\ P =/= Q ) -> ( P .<_ ( Q .\/ R ) -> R .<_ ( Q .\/ P ) ) )
34 4 10 6 13 32 33 syl131anc 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) -> ( P .<_ ( Q .\/ R ) -> R .<_ ( Q .\/ P ) ) )
35 7 2 latjcom 
 |-  ( ( K e. Lat /\ P e. ( Base ` K ) /\ Q e. ( Base ` K ) ) -> ( P .\/ Q ) = ( Q .\/ P ) )
36 5 12 15 35 syl3anc 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) -> ( P .\/ Q ) = ( Q .\/ P ) )
37 36 breq2d 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) -> ( R .<_ ( P .\/ Q ) <-> R .<_ ( Q .\/ P ) ) )
38 34 37 sylibrd 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) -> ( P .<_ ( Q .\/ R ) -> R .<_ ( P .\/ Q ) ) )
39 16 38 mtod 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) -> -. P .<_ ( Q .\/ R ) )
40 7 1 2 3 hlexch1 
 |-  ( ( K e. HL /\ ( P e. A /\ S e. A /\ ( Q .\/ R ) e. ( Base ` K ) ) /\ -. P .<_ ( Q .\/ R ) ) -> ( P .<_ ( ( Q .\/ R ) .\/ S ) -> S .<_ ( ( Q .\/ R ) .\/ P ) ) )
41 4 10 20 24 39 40 syl131anc 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) -> ( P .<_ ( ( Q .\/ R ) .\/ S ) -> S .<_ ( ( Q .\/ R ) .\/ P ) ) )
42 7 2 latjcom 
 |-  ( ( K e. Lat /\ Q e. ( Base ` K ) /\ R e. ( Base ` K ) ) -> ( Q .\/ R ) = ( R .\/ Q ) )
43 5 15 9 42 syl3anc 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) -> ( Q .\/ R ) = ( R .\/ Q ) )
44 43 oveq1d 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) -> ( ( Q .\/ R ) .\/ P ) = ( ( R .\/ Q ) .\/ P ) )
45 7 2 latj31 
 |-  ( ( K e. Lat /\ ( R e. ( Base ` K ) /\ Q e. ( Base ` K ) /\ P e. ( Base ` K ) ) ) -> ( ( R .\/ Q ) .\/ P ) = ( ( P .\/ Q ) .\/ R ) )
46 5 9 15 12 45 syl13anc 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) -> ( ( R .\/ Q ) .\/ P ) = ( ( P .\/ Q ) .\/ R ) )
47 44 46 eqtrd 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) -> ( ( Q .\/ R ) .\/ P ) = ( ( P .\/ Q ) .\/ R ) )
48 47 breq2d 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) -> ( S .<_ ( ( Q .\/ R ) .\/ P ) <-> S .<_ ( ( P .\/ Q ) .\/ R ) ) )
49 41 48 sylibd 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) -> ( P .<_ ( ( Q .\/ R ) .\/ S ) -> S .<_ ( ( P .\/ Q ) .\/ R ) ) )
50 25 49 mtod 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) -> -. P .<_ ( ( Q .\/ R ) .\/ S ) )
51 19 31 50 3jca 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) -> ( Q =/= R /\ -. S .<_ ( Q .\/ R ) /\ -. P .<_ ( ( Q .\/ R ) .\/ S ) ) )