Theorem lhpmcvr2

Description: Alternate way to express that the meet of a lattice hyperplane with an element not under it is covered by the element. (Contributed by NM, 9-Apr-2013)

	Ref	Expression
Hypotheses	lhpmcvr2.b	|- B = ( Base ` K )
	lhpmcvr2.l	|- .<_ = ( le ` K )
	lhpmcvr2.j	|- .\/ = ( join ` K )
	lhpmcvr2.m	|- ./\ = ( meet ` K )
	lhpmcvr2.a	|- A = ( Atoms ` K )
	lhpmcvr2.h	|- H = ( LHyp ` K )
Assertion	lhpmcvr2	|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) ) -> E. p e. A ( -. p .<_ W /\ ( p .\/ ( X ./\ W ) ) = X ) )

Proof

Step	Hyp	Ref	Expression
1		lhpmcvr2.b	|- B = ( Base ` K )
2		lhpmcvr2.l	|- .<_ = ( le ` K )
3		lhpmcvr2.j	|- .\/ = ( join ` K )
4		lhpmcvr2.m	|- ./\ = ( meet ` K )
5		lhpmcvr2.a	|- A = ( Atoms ` K )
6		lhpmcvr2.h	|- H = ( LHyp ` K )
7		eqid	|- (
8	1 2 4 7 6	lhpmcvr	|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) ) -> ( X ./\ W ) (
9		simpll	|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) ) -> K e. HL )
10		simprl	|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) ) -> X e. B )
11	1 6	lhpbase	|- ( W e. H -> W e. B )
12	11	ad2antlr	|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) ) -> W e. B )
13	1 2 3 4 7 5	cvrval5	|- ( ( K e. HL /\ X e. B /\ W e. B ) -> ( ( X ./\ W ) ( E. p e. A ( -. p .<_ W /\ ( p .\/ ( X ./\ W ) ) = X ) ) )
14	9 10 12 13	syl3anc	|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) ) -> ( ( X ./\ W ) ( E. p e. A ( -. p .<_ W /\ ( p .\/ ( X ./\ W ) ) = X ) ) )
15	8 14	mpbid	|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) ) -> E. p e. A ( -. p .<_ W /\ ( p .\/ ( X ./\ W ) ) = X ) )