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Metamath Proof Explorer

Theorem 2llnneN

Description: Condition implying that two intersecting lines are different. (Contributed by NM, 29-May-2012) (New usage is discouraged.)

Ref Expression
Hypotheses 2lnne.l 
|- .<_ = ( le ` K )
2lnne.j 
|- .\/ = ( join ` K )
2lnne.a 
|- A = ( Atoms ` K )
Assertion 2llnneN 
|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> ( R .\/ P ) =/= ( R .\/ Q ) )

Proof

Step Hyp Ref Expression
1 2lnne.l 
 |-  .<_ = ( le ` K )
2 2lnne.j 
 |-  .\/ = ( join ` K )
3 2lnne.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 simp21 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> P 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 ) -> P e. A )
8 simp23 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ P =/= Q ) -> R e. A )
9 simp22 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ P =/= Q ) -> Q e. A )
10 7 8 9 3jca 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ P =/= Q ) -> ( P e. A /\ R e. A /\ Q e. A ) )
11 1 2 3 hlatexch2 
 |-  ( ( K e. HL /\ ( P e. A /\ R e. A /\ Q e. A ) /\ P =/= Q ) -> ( P .<_ ( R .\/ Q ) -> R .<_ ( P .\/ Q ) ) )
12 10 11 syld3an2 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ P =/= Q ) -> ( P .<_ ( R .\/ Q ) -> R .<_ ( P .\/ Q ) ) )
13 12 con3d 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ P =/= Q ) -> ( -. R .<_ ( P .\/ Q ) -> -. P .<_ ( R .\/ Q ) ) )
14 13 3exp 
 |-  ( K e. HL -> ( ( P e. A /\ Q e. A /\ R e. A ) -> ( P =/= Q -> ( -. R .<_ ( P .\/ Q ) -> -. P .<_ ( R .\/ Q ) ) ) ) )
15 14 imp4a 
 |-  ( K e. HL -> ( ( P e. A /\ Q e. A /\ R e. A ) -> ( ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) -> -. P .<_ ( R .\/ Q ) ) ) )
16 15 3imp 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> -. P .<_ ( R .\/ Q ) )
17 1 2 3 2llnne2N 
 |-  ( ( K e. HL /\ ( P e. A /\ R e. A ) /\ -. P .<_ ( R .\/ Q ) ) -> ( R .\/ P ) =/= ( R .\/ Q ) )
18 4 5 6 16 17 syl121anc 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> ( R .\/ P ) =/= ( R .\/ Q ) )