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Metamath Proof Explorer

Theorem dochoc1

Description: The unit subspace (all vectors) is closed. (Contributed by NM, 16-Feb-2015)

Ref Expression
Hypotheses dochoc1.h 
|- H = ( LHyp ` K )
dochoc1.u 
|- U = ( ( DVecH ` K ) ` W )
dochoc1.o 
|- ._|_ = ( ( ocH ` K ) ` W )
dochoc1.v 
|- V = ( Base ` U )
dochoc1.k 
|- ( ph -> ( K e. HL /\ W e. H ) )
Assertion dochoc1 
|- ( ph -> ( ._|_ ` ( ._|_ ` V ) ) = V )

Proof

Step Hyp Ref Expression
1 dochoc1.h 
 |-  H = ( LHyp ` K )
2 dochoc1.u 
 |-  U = ( ( DVecH ` K ) ` W )
3 dochoc1.o 
 |-  ._|_ = ( ( ocH ` K ) ` W )
4 dochoc1.v 
 |-  V = ( Base ` U )
5 dochoc1.k 
 |-  ( ph -> ( K e. HL /\ W e. H ) )
6 eqid 
 |-  ( 0g ` U ) = ( 0g ` U )
7 1 2 3 4 6 doch1 
 |-  ( ( K e. HL /\ W e. H ) -> ( ._|_ ` V ) = { ( 0g ` U ) } )
8 7 fveq2d 
 |-  ( ( K e. HL /\ W e. H ) -> ( ._|_ ` ( ._|_ ` V ) ) = ( ._|_ ` { ( 0g ` U ) } ) )
9 1 2 3 4 6 doch0 
 |-  ( ( K e. HL /\ W e. H ) -> ( ._|_ ` { ( 0g ` U ) } ) = V )
10 8 9 eqtrd 
 |-  ( ( K e. HL /\ W e. H ) -> ( ._|_ ` ( ._|_ ` V ) ) = V )
11 5 10 syl 
 |-  ( ph -> ( ._|_ ` ( ._|_ ` V ) ) = V )