Theorem atlex

Description: Every nonzero element of an atomic lattice is greater than or equal to an atom. ( hatomic analog.) (Contributed by NM, 21-Oct-2011)

	Ref	Expression
Hypotheses	atlex.b	|- B = ( Base ` K )
	atlex.l	|- .<_ = ( le ` K )
	atlex.z	|- .0. = ( 0. ` K )
	atlex.a	|- A = ( Atoms ` K )
Assertion	atlex	|- ( ( K e. AtLat /\ X e. B /\ X =/= .0. ) -> E. y e. A y .<_ X )

Proof

Step	Hyp	Ref	Expression
1		atlex.b	|- B = ( Base ` K )
2		atlex.l	|- .<_ = ( le ` K )
3		atlex.z	|- .0. = ( 0. ` K )
4		atlex.a	|- A = ( Atoms ` K )
5		eqid	|- ( glb ` K ) = ( glb ` K )
6	1 5 2 3 4	isatl	|- ( K e. AtLat <-> ( K e. Lat /\ B e. dom ( glb ` K ) /\ A. x e. B ( x =/= .0. -> E. y e. A y .<_ x ) ) )
7	6	simp3bi	|- ( K e. AtLat -> A. x e. B ( x =/= .0. -> E. y e. A y .<_ x ) )
8		neeq1	|- ( x = X -> ( x =/= .0. <-> X =/= .0. ) )
9		breq2	|- ( x = X -> ( y .<_ x <-> y .<_ X ) )
10	9	rexbidv	|- ( x = X -> ( E. y e. A y .<_ x <-> E. y e. A y .<_ X ) )
11	8 10	imbi12d	|- ( x = X -> ( ( x =/= .0. -> E. y e. A y .<_ x ) <-> ( X =/= .0. -> E. y e. A y .<_ X ) ) )
12	11	rspccv	|- ( A. x e. B ( x =/= .0. -> E. y e. A y .<_ x ) -> ( X e. B -> ( X =/= .0. -> E. y e. A y .<_ X ) ) )
13	7 12	syl	|- ( K e. AtLat -> ( X e. B -> ( X =/= .0. -> E. y e. A y .<_ X ) ) )
14	13	3imp	|- ( ( K e. AtLat /\ X e. B /\ X =/= .0. ) -> E. y e. A y .<_ X )