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Metamath Proof Explorer

Theorem 3noncolr1N

Description: Two ways to express 3 non-colinear atoms (rotated right 1 place). (Contributed by NM, 12-Jul-2012) (New usage is discouraged.)

Ref Expression
Hypotheses 3noncol.l 
|- .<_ = ( le ` K )
3noncol.j 
|- .\/ = ( join ` K )
3noncol.a 
|- A = ( Atoms ` K )
Assertion 3noncolr1N 
|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> ( R =/= P /\ -. Q .<_ ( R .\/ P ) ) )

Proof

Step Hyp Ref Expression
1 3noncol.l 
 |-  .<_ = ( le ` K )
2 3noncol.j 
 |-  .\/ = ( join ` K )
3 3noncol.a 
 |-  A = ( Atoms ` K )
4 simp1 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> K e. HL )
5 simp22 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> Q e. A )
6 simp23 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> R e. A )
7 simp21 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> P e. A )
8 1 2 3 3noncolr2 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> ( Q =/= R /\ -. P .<_ ( Q .\/ R ) ) )
9 1 2 3 3noncolr2 
 |-  ( ( K e. HL /\ ( Q e. A /\ R e. A /\ P e. A ) /\ ( Q =/= R /\ -. P .<_ ( Q .\/ R ) ) ) -> ( R =/= P /\ -. Q .<_ ( R .\/ P ) ) )
10 4 5 6 7 8 9 syl131anc 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> ( R =/= P /\ -. Q .<_ ( R .\/ P ) ) )