Theorem atn0

Description: An atom is not zero. ( atne0 analog.) (Contributed by NM, 5-Nov-2012)

	Ref	Expression
Hypotheses	atne0.z	|- .0. = ( 0. ` K )
	atne0.a	|- A = ( Atoms ` K )
Assertion	atn0	|- ( ( K e. AtLat /\ P e. A ) -> P =/= .0. )

Proof

Step	Hyp	Ref	Expression
1		atne0.z	|- .0. = ( 0. ` K )
2		atne0.a	|- A = ( Atoms ` K )
3		eqid	|- ( Base ` K ) = ( Base ` K )
4		eqid	|- ( le ` K ) = ( le ` K )
5	3 4 1 2	isat3	|- ( K e. AtLat -> ( P e. A <-> ( P e. ( Base ` K ) /\ P =/= .0. /\ A. x e. ( Base ` K ) ( x ( le ` K ) P -> ( x = P \/ x = .0. ) ) ) ) )
6		simp2	|- ( ( P e. ( Base ` K ) /\ P =/= .0. /\ A. x e. ( Base ` K ) ( x ( le ` K ) P -> ( x = P \/ x = .0. ) ) ) -> P =/= .0. )
7	5 6	biimtrdi	|- ( K e. AtLat -> ( P e. A -> P =/= .0. ) )
8	7	imp	|- ( ( K e. AtLat /\ P e. A ) -> P =/= .0. )