Theorem latmlem2

Description: Add meet to both sides of a lattice ordering. ( sslin analog.) (Contributed by NM, 10-Nov-2011)

	Ref	Expression
Hypotheses	latmle.b	|- B = ( Base ` K )
	latmle.l	|- .<_ = ( le ` K )
	latmle.m	|- ./\ = ( meet ` K )
Assertion	latmlem2	|- ( ( K e. Lat /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( X .<_ Y -> ( Z ./\ X ) .<_ ( Z ./\ Y ) ) )

Proof

Step	Hyp	Ref	Expression
1		latmle.b	|- B = ( Base ` K )
2		latmle.l	|- .<_ = ( le ` K )
3		latmle.m	|- ./\ = ( meet ` K )
4	1 2 3	latmlem1	|- ( ( K e. Lat /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( X .<_ Y -> ( X ./\ Z ) .<_ ( Y ./\ Z ) ) )
5	1 3	latmcom	|- ( ( 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	latmcom	|- ( ( 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 ) ) )