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Metamath Proof Explorer

Theorem hlatcon2

Description: Atom exchange combined with contraposition. (Contributed by NM, 13-Jun-2012)

Ref Expression
Hypotheses 3noncol.l 
|- .<_ = ( le ` K )
3noncol.j 
|- .\/ = ( join ` K )
3noncol.a 
|- A = ( Atoms ` K )
Assertion hlatcon2 
|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> -. P .<_ ( R .\/ Q ) )

Proof

Step Hyp Ref Expression
1 3noncol.l 
 |-  .<_ = ( le ` K )
2 3noncol.j 
 |-  .\/ = ( join ` K )
3 3noncol.a 
 |-  A = ( Atoms ` K )
4 1 2 3 hlatcon3 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> -. P .<_ ( Q .\/ R ) )
5 simp1 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> K e. HL )
6 simp22 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> Q e. A )
7 simp23 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> R e. A )
8 2 3 hlatjcom 
 |-  ( ( K e. HL /\ Q e. A /\ R e. A ) -> ( Q .\/ R ) = ( R .\/ Q ) )
9 5 6 7 8 syl3anc 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> ( Q .\/ R ) = ( R .\/ Q ) )
10 9 breq2d 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> ( P .<_ ( Q .\/ R ) <-> P .<_ ( R .\/ Q ) ) )
11 4 10 mtbid 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> -. P .<_ ( R .\/ Q ) )