Theorem dih0vbN

Description: A vector is zero iff its span is the isomorphism of lattice zero. (Contributed by NM, 16-Aug-2014) (New usage is discouraged.)

	Ref	Expression
Hypotheses	dih0vb.h	|- H = ( LHyp ` K )
	dih0vb.o	|- .0. = ( 0. ` K )
	dih0vb.i	|- I = ( ( DIsoH ` K ) ` W )
	dih0vb.u	|- U = ( ( DVecH ` K ) ` W )
	dih0vb.v	|- V = ( Base ` U )
	dih0vb.z	|- Z = ( 0g ` U )
	dih0vb.n	|- N = ( LSpan ` U )
	dih0vb.k	|- ( ph -> ( K e. HL /\ W e. H ) )
	dih0vb.x	|- ( ph -> X e. V )
Assertion	dih0vbN	|- ( ph -> ( X = Z <-> ( N ` { X } ) = ( I ` .0. ) ) )

Proof

Step	Hyp	Ref	Expression
1		dih0vb.h	|- H = ( LHyp ` K )
2		dih0vb.o	|- .0. = ( 0. ` K )
3		dih0vb.i	|- I = ( ( DIsoH ` K ) ` W )
4		dih0vb.u	|- U = ( ( DVecH ` K ) ` W )
5		dih0vb.v	|- V = ( Base ` U )
6		dih0vb.z	|- Z = ( 0g ` U )
7		dih0vb.n	|- N = ( LSpan ` U )
8		dih0vb.k	|- ( ph -> ( K e. HL /\ W e. H ) )
9		dih0vb.x	|- ( ph -> X e. V )
10	2 1 3 4 6	dih0	|- ( ( K e. HL /\ W e. H ) -> ( I ` .0. ) = { Z } )
11	8 10	syl	|- ( ph -> ( I ` .0. ) = { Z } )
12	11	eqeq2d	|- ( ph -> ( ( N ` { X } ) = ( I ` .0. ) <-> ( N ` { X } ) = { Z } ) )
13	1 4 8	dvhlmod	|- ( ph -> U e. LMod )
14	5 6 7	lspsneq0	|- ( ( U e. LMod /\ X e. V ) -> ( ( N ` { X } ) = { Z } <-> X = Z ) )
15	13 9 14	syl2anc	|- ( ph -> ( ( N ` { X } ) = { Z } <-> X = Z ) )
16	12 15	bitr2d	|- ( ph -> ( X = Z <-> ( N ` { X } ) = ( I ` .0. ) ) )