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Metamath Proof Explorer

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 } ) )