Theorem clatp0cl

Description: The poset zero of a complete lattice belongs to its base. (Contributed by Thierry Arnoux, 17-Feb-2018)

	Ref	Expression
Hypotheses	clatp0cl.b	|- B = ( Base ` W )
	clatp0cl.0	|- .0. = ( 0. ` W )
Assertion	clatp0cl	|- ( W e. CLat -> .0. e. B )

Proof

Step	Hyp	Ref	Expression
1		clatp0cl.b	|- B = ( Base ` W )
2		clatp0cl.0	|- .0. = ( 0. ` W )
3		eqid	|- ( glb ` W ) = ( glb ` W )
4	1 3 2	p0val	|- ( W e. CLat -> .0. = ( ( glb ` W ) ` B ) )
5		ssid	|- B C_ B
6	1 3	clatglbcl	|- ( ( W e. CLat /\ B C_ B ) -> ( ( glb ` W ) ` B ) e. B )
7	5 6	mpan2	|- ( W e. CLat -> ( ( glb ` W ) ` B ) e. B )
8	4 7	eqeltrd	|- ( W e. CLat -> .0. e. B )