Metamath Proof Explorer

Theorem cvlsupr4

Description: Consequence of superposition condition ( P .\/ R ) = ( Q .\/ R ) . (Contributed by NM, 9-Nov-2012)

Ref Expression
Hypotheses cvlsupr2.a 
|- A = ( Atoms ` K )
cvlsupr2.l 
|- .<_ = ( le ` K )
cvlsupr2.j 
|- .\/ = ( join ` K )
Assertion cvlsupr4 
|- ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ ( P .\/ R ) = ( Q .\/ R ) ) ) -> R .<_ ( P .\/ Q ) )

Proof

Step Hyp Ref Expression
1 cvlsupr2.a 
 |-  A = ( Atoms ` K )
2 cvlsupr2.l 
 |-  .<_ = ( le ` K )
3 cvlsupr2.j 
 |-  .\/ = ( join ` K )
4 1 2 3 cvlsupr2 
 |-  ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ R e. A ) /\ P =/= Q ) -> ( ( P .\/ R ) = ( Q .\/ R ) <-> ( R =/= P /\ R =/= Q /\ R .<_ ( P .\/ Q ) ) ) )
5 simp3 
 |-  ( ( R =/= P /\ R =/= Q /\ R .<_ ( P .\/ Q ) ) -> R .<_ ( P .\/ Q ) )
6 4 5 syl6bi 
 |-  ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ R e. A ) /\ P =/= Q ) -> ( ( P .\/ R ) = ( Q .\/ R ) -> R .<_ ( P .\/ Q ) ) )
7 6 3exp 
 |-  ( K e. CvLat -> ( ( P e. A /\ Q e. A /\ R e. A ) -> ( P =/= Q -> ( ( P .\/ R ) = ( Q .\/ R ) -> R .<_ ( P .\/ Q ) ) ) ) )
8 7 imp4a 
 |-  ( K e. CvLat -> ( ( P e. A /\ Q e. A /\ R e. A ) -> ( ( P =/= Q /\ ( P .\/ R ) = ( Q .\/ R ) ) -> R .<_ ( P .\/ Q ) ) ) )
9 8 3imp 
 |-  ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ ( P .\/ R ) = ( Q .\/ R ) ) ) -> R .<_ ( P .\/ Q ) )