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Metamath Proof Explorer

Theorem latjlej2

Description: Add join to both sides of a lattice ordering. ( chlej2i analog.) (Contributed by NM, 8-Nov-2011)

Ref Expression
Hypotheses latlej.b 
|- B = ( Base ` K )
latlej.l 
|- .<_ = ( le ` K )
latlej.j 
|- .\/ = ( join ` K )
Assertion latjlej2 
|- ( ( K e. Lat /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( X .<_ Y -> ( Z .\/ X ) .<_ ( Z .\/ Y ) ) )

Proof

Step Hyp Ref Expression
1 latlej.b 
 |-  B = ( Base ` K )
2 latlej.l 
 |-  .<_ = ( le ` K )
3 latlej.j 
 |-  .\/ = ( join ` K )
4 1 2 3 latjlej1 
 |-  ( ( K e. Lat /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( X .<_ Y -> ( X .\/ Z ) .<_ ( Y .\/ Z ) ) )
5 1 3 latjcom 
 |-  ( ( K e. Lat /\ X e. B /\ Z e. B ) -> ( X .\/ Z ) = ( Z .\/ X ) )
6 5 3adant3r2 
 |-  ( ( K e. Lat /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( X .\/ Z ) = ( Z .\/ X ) )
7 1 3 latjcom 
 |-  ( ( K e. Lat /\ Y e. B /\ Z e. B ) -> ( Y .\/ Z ) = ( Z .\/ Y ) )
8 7 3adant3r1 
 |-  ( ( K e. Lat /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( Y .\/ Z ) = ( Z .\/ Y ) )
9 6 8 breq12d 
 |-  ( ( K e. Lat /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .\/ Z ) .<_ ( Y .\/ Z ) <-> ( Z .\/ X ) .<_ ( Z .\/ Y ) ) )
10 4 9 sylibd 
 |-  ( ( K e. Lat /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( X .<_ Y -> ( Z .\/ X ) .<_ ( Z .\/ Y ) ) )