Theorem dih0bN

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

	Ref	Expression
Hypotheses	dih0b.b	|- B = ( Base ` K )
	dih0b.h	|- H = ( LHyp ` K )
	dih0b.o	|- .0. = ( 0. ` K )
	dih0b.i	|- I = ( ( DIsoH ` K ) ` W )
	dih0b.u	|- U = ( ( DVecH ` K ) ` W )
	dih0b.z	|- Z = ( 0g ` U )
	dih0b.k	|- ( ph -> ( K e. HL /\ W e. H ) )
	dih0b.x	|- ( ph -> X e. B )
Assertion	dih0bN	|- ( ph -> ( X = .0. <-> ( I ` X ) = { Z } ) )

Proof

Step	Hyp	Ref	Expression
1		dih0b.b	|- B = ( Base ` K )
2		dih0b.h	|- H = ( LHyp ` K )
3		dih0b.o	|- .0. = ( 0. ` K )
4		dih0b.i	|- I = ( ( DIsoH ` K ) ` W )
5		dih0b.u	|- U = ( ( DVecH ` K ) ` W )
6		dih0b.z	|- Z = ( 0g ` U )
7		dih0b.k	|- ( ph -> ( K e. HL /\ W e. H ) )
8		dih0b.x	|- ( ph -> X e. B )
9	7	simpld	|- ( ph -> K e. HL )
10		hlop	|- ( K e. HL -> K e. OP )
11	1 3	op0cl	|- ( K e. OP -> .0. e. B )
12	9 10 11	3syl	|- ( ph -> .0. e. B )
13	1 2 4	dih11	|- ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ .0. e. B ) -> ( ( I ` X ) = ( I ` .0. ) <-> X = .0. ) )
14	7 8 12 13	syl3anc	|- ( ph -> ( ( I ` X ) = ( I ` .0. ) <-> X = .0. ) )
15	3 2 4 5 6	dih0	|- ( ( K e. HL /\ W e. H ) -> ( I ` .0. ) = { Z } )
16	7 15	syl	|- ( ph -> ( I ` .0. ) = { Z } )
17	16	eqeq2d	|- ( ph -> ( ( I ` X ) = ( I ` .0. ) <-> ( I ` X ) = { Z } ) )
18	14 17	bitr3d	|- ( ph -> ( X = .0. <-> ( I ` X ) = { Z } ) )