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 ) ) )