Theorem dia0eldmN

Description: The lattice zero belongs to the domain of partial isomorphism A. (Contributed by NM, 5-Dec-2013) (New usage is discouraged.)

	Ref	Expression
Hypotheses	dia0eldm.z	|- .0. = ( 0. ` K )
	dia0eldm.h	|- H = ( LHyp ` K )
	dia0eldm.i	|- I = ( ( DIsoA ` K ) ` W )
Assertion	dia0eldmN	|- ( ( K e. HL /\ W e. H ) -> .0. e. dom I )

Proof

Step	Hyp	Ref	Expression
1		dia0eldm.z	|- .0. = ( 0. ` K )
2		dia0eldm.h	|- H = ( LHyp ` K )
3		dia0eldm.i	|- I = ( ( DIsoA ` K ) ` W )
4		hlop	|- ( K e. HL -> K e. OP )
5	4	adantr	|- ( ( K e. HL /\ W e. H ) -> K e. OP )
6		eqid	|- ( Base ` K ) = ( Base ` K )
7	6 1	op0cl	|- ( K e. OP -> .0. e. ( Base ` K ) )
8	5 7	syl	|- ( ( K e. HL /\ W e. H ) -> .0. e. ( Base ` K ) )
9	6 2	lhpbase	|- ( W e. H -> W e. ( Base ` K ) )
10		eqid	|- ( le ` K ) = ( le ` K )
11	6 10 1	op0le	|- ( ( K e. OP /\ W e. ( Base ` K ) ) -> .0. ( le ` K ) W )
12	4 9 11	syl2an	|- ( ( K e. HL /\ W e. H ) -> .0. ( le ` K ) W )
13	6 10 2 3	diaeldm	|- ( ( K e. HL /\ W e. H ) -> ( .0. e. dom I <-> ( .0. e. ( Base ` K ) /\ .0. ( le ` K ) W ) ) )
14	8 12 13	mpbir2and	|- ( ( K e. HL /\ W e. H ) -> .0. e. dom I )