Theorem dih0cnv

Description: The isomorphism H converse value of the zero subspace is the lattice zero. (Contributed by NM, 19-Jun-2014)

	Ref	Expression
Hypotheses	dih0cnv.h	|- H = ( LHyp ` K )
	dih0cnv.o	|- .0. = ( 0. ` K )
	dih0cnv.i	|- I = ( ( DIsoH ` K ) ` W )
	dih0cnv.u	|- U = ( ( DVecH ` K ) ` W )
	dih0cnv.z	|- Z = ( 0g ` U )
Assertion	dih0cnv	|- ( ( K e. HL /\ W e. H ) -> ( `' I ` { Z } ) = .0. )

Proof

Step	Hyp	Ref	Expression
1		dih0cnv.h	|- H = ( LHyp ` K )
2		dih0cnv.o	|- .0. = ( 0. ` K )
3		dih0cnv.i	|- I = ( ( DIsoH ` K ) ` W )
4		dih0cnv.u	|- U = ( ( DVecH ` K ) ` W )
5		dih0cnv.z	|- Z = ( 0g ` U )
6	2 1 3 4 5	dih0	|- ( ( K e. HL /\ W e. H ) -> ( I ` .0. ) = { Z } )
7	6	fveq2d	|- ( ( K e. HL /\ W e. H ) -> ( `' I ` ( I ` .0. ) ) = ( `' I ` { Z } ) )
8		hlatl	|- ( K e. HL -> K e. AtLat )
9	8	adantr	|- ( ( K e. HL /\ W e. H ) -> K e. AtLat )
10		eqid	|- ( Base ` K ) = ( Base ` K )
11	10 2	atl0cl	|- ( K e. AtLat -> .0. e. ( Base ` K ) )
12	9 11	syl	|- ( ( K e. HL /\ W e. H ) -> .0. e. ( Base ` K ) )
13	10 1 3	dihcnvid1	|- ( ( ( K e. HL /\ W e. H ) /\ .0. e. ( Base ` K ) ) -> ( `' I ` ( I ` .0. ) ) = .0. )
14	12 13	mpdan	|- ( ( K e. HL /\ W e. H ) -> ( `' I ` ( I ` .0. ) ) = .0. )
15	7 14	eqtr3d	|- ( ( K e. HL /\ W e. H ) -> ( `' I ` { Z } ) = .0. )