Theorem meetat

Description: The meet of any element with an atom is either the atom or zero. (Contributed by NM, 28-Aug-2012)

	Ref	Expression
Hypotheses	m.b	|- B = ( Base ` K )
	m.m	|- ./\ = ( meet ` K )
	m.z	|- .0. = ( 0. ` K )
	m.a	|- A = ( Atoms ` K )
Assertion	meetat	|- ( ( K e. OL /\ X e. B /\ P e. A ) -> ( ( X ./\ P ) = P \/ ( X ./\ P ) = .0. ) )

Proof

Step	Hyp	Ref	Expression
1		m.b	|- B = ( Base ` K )
2		m.m	|- ./\ = ( meet ` K )
3		m.z	|- .0. = ( 0. ` K )
4		m.a	|- A = ( Atoms ` K )
5		ollat	|- ( K e. OL -> K e. Lat )
6	5	3ad2ant1	|- ( ( K e. OL /\ X e. B /\ P e. A ) -> K e. Lat )
7		simp2	|- ( ( K e. OL /\ X e. B /\ P e. A ) -> X e. B )
8		simp3	|- ( ( K e. OL /\ X e. B /\ P e. A ) -> P e. A )
9	1 4	atbase	|- ( P e. A -> P e. B )
10	8 9	syl	|- ( ( K e. OL /\ X e. B /\ P e. A ) -> P e. B )
11		eqid	|- ( le ` K ) = ( le ` K )
12	1 11 2	latmle2	|- ( ( K e. Lat /\ X e. B /\ P e. B ) -> ( X ./\ P ) ( le ` K ) P )
13	6 7 10 12	syl3anc	|- ( ( K e. OL /\ X e. B /\ P e. A ) -> ( X ./\ P ) ( le ` K ) P )
14		olop	|- ( K e. OL -> K e. OP )
15	14	3ad2ant1	|- ( ( K e. OL /\ X e. B /\ P e. A ) -> K e. OP )
16	1 2	latmcl	|- ( ( K e. Lat /\ X e. B /\ P e. B ) -> ( X ./\ P ) e. B )
17	6 7 10 16	syl3anc	|- ( ( K e. OL /\ X e. B /\ P e. A ) -> ( X ./\ P ) e. B )
18	1 11 3 4	leatb	|- ( ( K e. OP /\ ( X ./\ P ) e. B /\ P e. A ) -> ( ( X ./\ P ) ( le ` K ) P <-> ( ( X ./\ P ) = P \/ ( X ./\ P ) = .0. ) ) )
19	15 17 8 18	syl3anc	|- ( ( K e. OL /\ X e. B /\ P e. A ) -> ( ( X ./\ P ) ( le ` K ) P <-> ( ( X ./\ P ) = P \/ ( X ./\ P ) = .0. ) ) )
20	13 19	mpbid	|- ( ( K e. OL /\ X e. B /\ P e. A ) -> ( ( X ./\ P ) = P \/ ( X ./\ P ) = .0. ) )