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Metamath Proof Explorer

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. ) ) )