Theorem lattr

Description: A lattice ordering is transitive. ( sstr analog.) (Contributed by NM, 17-Nov-2011)

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

Proof

Step	Hyp	Ref	Expression
1		latref.b	|- B = ( Base ` K )
2		latref.l	|- .<_ = ( le ` K )
3		latpos	|- ( K e. Lat -> K e. Poset )
4	1 2	postr	|- ( ( K e. Poset /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .<_ Y /\ Y .<_ Z ) -> X .<_ Z ) )
5	3 4	sylan	|- ( ( K e. Lat /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .<_ Y /\ Y .<_ Z ) -> X .<_ Z ) )