Theorem dihmeetlem4N

Description: Lemma for isomorphism H of a lattice meet. (Contributed by NM, 30-Mar-2014) (New usage is discouraged.)

	Ref	Expression
Hypotheses	dihmeetlem4.b	|- B = ( Base ` K )
	dihmeetlem4.l	|- .<_ = ( le ` K )
	dihmeetlem4.m	|- ./\ = ( meet ` K )
	dihmeetlem4.a	|- A = ( Atoms ` K )
	dihmeetlem4.h	|- H = ( LHyp ` K )
	dihmeetlem4.i	|- I = ( ( DIsoH ` K ) ` W )
	dihmeetlem4.u	|- U = ( ( DVecH ` K ) ` W )
	dihmeetlem4.z	|- .0. = ( 0g ` U )
Assertion	dihmeetlem4N	|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( ( I ` Q ) i^i ( I ` ( X ./\ W ) ) ) = { .0. } )

Proof

Step	Hyp	Ref	Expression
1		dihmeetlem4.b	|- B = ( Base ` K )
2		dihmeetlem4.l	|- .<_ = ( le ` K )
3		dihmeetlem4.m	|- ./\ = ( meet ` K )
4		dihmeetlem4.a	|- A = ( Atoms ` K )
5		dihmeetlem4.h	|- H = ( LHyp ` K )
6		dihmeetlem4.i	|- I = ( ( DIsoH ` K ) ` W )
7		dihmeetlem4.u	|- U = ( ( DVecH ` K ) ` W )
8		dihmeetlem4.z	|- .0. = ( 0g ` U )
9		eqid	|- ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = Q ) = ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = Q )
10		eqid	|- ( ( oc ` K ) ` W ) = ( ( oc ` K ) ` W )
11		eqid	|- ( ( LTrn ` K ) ` W ) = ( ( LTrn ` K ) ` W )
12		eqid	|- ( ( trL ` K ) ` W ) = ( ( trL ` K ) ` W )
13		eqid	|- ( ( TEndo ` K ) ` W ) = ( ( TEndo ` K ) ` W )
14		eqid	|- ( h e. ( ( LTrn ` K ) ` W ) |-> ( _I |` B ) ) = ( h e. ( ( LTrn ` K ) ` W ) |-> ( _I |` B ) )
15	1 2 3 4 5 6 7 8 9 10 11 12 13 14	dihmeetlem4preN	|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( ( I ` Q ) i^i ( I ` ( X ./\ W ) ) ) = { .0. } )