Metamath Proof Explorer

Theorem 3noncolr2

Description: Two ways to express 3 non-colinear atoms (rotated right 2 places). (Contributed by NM, 12-Jul-2012)

Ref Expression
Hypotheses 3noncol.l 
|- .<_ = ( le ` K )
3noncol.j 
|- .\/ = ( join ` K )
3noncol.a 
|- A = ( Atoms ` K )
Assertion 3noncolr2 
|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> ( Q =/= R /\ -. P .<_ ( Q .\/ R ) ) )

Proof

Step Hyp Ref Expression
1 3noncol.l 
 |-  .<_ = ( le ` K )
2 3noncol.j 
 |-  .\/ = ( join ` K )
3 3noncol.a 
 |-  A = ( Atoms ` K )
4 hllat 
 |-  ( K e. HL -> K e. Lat )
5 4 3ad2ant1 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> K e. Lat )
6 simp23 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> 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 ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> R e. ( Base ` K ) )
10 simp21 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> 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 ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> P e. ( Base ` K ) )
13 simp22 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> 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 ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> Q e. ( Base ` K ) )
16 simp3r 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> -. 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 ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> R =/= Q )
19 18 necomd 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> Q =/= R )
20 simp1 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> K e. HL )
21 simp3l 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> P =/= Q )
22 1 2 3 hlatexch1 
 |-  ( ( K e. HL /\ ( P e. A /\ R e. A /\ Q e. A ) /\ P =/= Q ) -> ( P .<_ ( Q .\/ R ) -> R .<_ ( Q .\/ P ) ) )
23 20 10 6 13 21 22 syl131anc 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> ( P .<_ ( Q .\/ R ) -> R .<_ ( Q .\/ P ) ) )
24 2 3 hlatjcom 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .\/ Q ) = ( Q .\/ P ) )
25 20 10 13 24 syl3anc 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> ( P .\/ Q ) = ( Q .\/ P ) )
26 25 breq2d 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> ( R .<_ ( P .\/ Q ) <-> R .<_ ( Q .\/ P ) ) )
27 23 26 sylibrd 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> ( P .<_ ( Q .\/ R ) -> R .<_ ( P .\/ Q ) ) )
28 16 27 mtod 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> -. P .<_ ( Q .\/ R ) )
29 19 28 jca 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> ( Q =/= R /\ -. P .<_ ( Q .\/ R ) ) )