Theorem dihglbcN

Description: Isomorphism H of a lattice glb when the glb is not under the fiducial hyperplane W . (Contributed by NM, 26-Mar-2014) (New usage is discouraged.)

	Ref	Expression
Hypotheses	dihglbc.b	|- B = ( Base ` K )
	dihglbc.g	|- G = ( glb ` K )
	dihglbc.h	|- H = ( LHyp ` K )
	dihglbc.i	|- I = ( ( DIsoH ` K ) ` W )
	dihglbc.l	|- .<_ = ( le ` K )
Assertion	dihglbcN	|- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) /\ -. ( G ` S ) .<_ W ) -> ( I ` ( G ` S ) ) = |^|_ x e. S ( I ` x ) )

Proof

Step	Hyp	Ref	Expression
1		dihglbc.b	|- B = ( Base ` K )
2		dihglbc.g	|- G = ( glb ` K )
3		dihglbc.h	|- H = ( LHyp ` K )
4		dihglbc.i	|- I = ( ( DIsoH ` K ) ` W )
5		dihglbc.l	|- .<_ = ( le ` K )
6		eqid	|- ( join ` K ) = ( join ` K )
7		eqid	|- ( meet ` K ) = ( meet ` K )
8		eqid	|- ( Atoms ` K ) = ( Atoms ` K )
9		eqid	|- ( ( oc ` K ) ` W ) = ( ( oc ` K ) ` W )
10		eqid	|- ( ( LTrn ` K ) ` W ) = ( ( LTrn ` K ) ` W )
11		eqid	|- ( ( trL ` K ) ` W ) = ( ( trL ` K ) ` W )
12		eqid	|- ( ( TEndo ` K ) ` W ) = ( ( TEndo ` K ) ` W )
13		eqid	|- ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = q ) = ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = q )
14	1 2 3 4 5 6 7 8 9 10 11 12 13	dihglbcpreN	|- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) /\ -. ( G ` S ) .<_ W ) -> ( I ` ( G ` S ) ) = |^|_ x e. S ( I ` x ) )