Metamath Proof Explorer

Theorem 4noncolr1

Description: A way to express 4 non-colinear atoms (rotated right 1 places). (Contributed by NM, 11-Jul-2012)

Ref Expression
Hypotheses 3noncol.l 
|- .<_ = ( le ` K )
3noncol.j 
|- .\/ = ( join ` K )
3noncol.a 
|- A = ( Atoms ` K )
Assertion 4noncolr1 
|- ( ( ( 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 .<_ ( S .\/ P ) /\ -. R .<_ ( ( S .\/ P ) .\/ Q ) ) )

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 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 )
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 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 )
8 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 )
9 1 2 3 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 ) ) )
10 1 2 3 4noncolr2 
 |-  ( ( ( K e. HL /\ Q e. A /\ R e. A ) /\ ( S e. A /\ P e. A ) /\ ( Q =/= R /\ -. S .<_ ( Q .\/ R ) /\ -. P .<_ ( ( Q .\/ R ) .\/ S ) ) ) -> ( S =/= P /\ -. Q .<_ ( S .\/ P ) /\ -. R .<_ ( ( S .\/ P ) .\/ Q ) ) )
11 4 5 6 7 8 9 10 syl321anc 
 |-  ( ( ( 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 .<_ ( S .\/ P ) /\ -. R .<_ ( ( S .\/ P ) .\/ Q ) ) )