Theorem clatglble

Description: The greatest lower bound is the least element. (Contributed by NM, 5-Dec-2011)

	Ref	Expression
Hypotheses	clatglb.b	|- B = ( Base ` K )
	clatglb.l	|- .<_ = ( le ` K )
	clatglb.g	|- G = ( glb ` K )
Assertion	clatglble	|- ( ( K e. CLat /\ S C_ B /\ X e. S ) -> ( G ` S ) .<_ X )

Proof

Step	Hyp	Ref	Expression
1		clatglb.b	|- B = ( Base ` K )
2		clatglb.l	|- .<_ = ( le ` K )
3		clatglb.g	|- G = ( glb ` K )
4		simp1	|- ( ( K e. CLat /\ S C_ B /\ X e. S ) -> K e. CLat )
5	1 3	clatglbcl2	|- ( ( K e. CLat /\ S C_ B ) -> S e. dom G )
6	5	3adant3	|- ( ( K e. CLat /\ S C_ B /\ X e. S ) -> S e. dom G )
7		simp3	|- ( ( K e. CLat /\ S C_ B /\ X e. S ) -> X e. S )
8	1 2 3 4 6 7	glble	|- ( ( K e. CLat /\ S C_ B /\ X e. S ) -> ( G ` S ) .<_ X )