Metamath Proof Explorer

Theorem dochspss

Description: The span of a set of vectors is included in their double orthocomplement. (Contributed by NM, 26-Jul-2014)

Ref Expression
Hypotheses dochsp.h 
|- H = ( LHyp ` K )
dochsp.u 
|- U = ( ( DVecH ` K ) ` W )
dochsp.o 
|- ._|_ = ( ( ocH ` K ) ` W )
dochsp.v 
|- V = ( Base ` U )
dochsp.n 
|- N = ( LSpan ` U )
dochsp.k 
|- ( ph -> ( K e. HL /\ W e. H ) )
dochsp.x 
|- ( ph -> X C_ V )
Assertion dochspss 
|- ( ph -> ( N ` X ) C_ ( ._|_ ` ( ._|_ ` X ) ) )

Proof

Step Hyp Ref Expression
1 dochsp.h 
 |-  H = ( LHyp ` K )
2 dochsp.u 
 |-  U = ( ( DVecH ` K ) ` W )
3 dochsp.o 
 |-  ._|_ = ( ( ocH ` K ) ` W )
4 dochsp.v 
 |-  V = ( Base ` U )
5 dochsp.n 
 |-  N = ( LSpan ` U )
6 dochsp.k 
 |-  ( ph -> ( K e. HL /\ W e. H ) )
7 dochsp.x 
 |-  ( ph -> X C_ V )
8 eqid 
 |-  ( ( DIsoH ` K ) ` W ) = ( ( DIsoH ` K ) ` W )
9 eqid 
 |-  ( LSubSp ` U ) = ( LSubSp ` U )
10 1 2 8 9 dihsslss 
 |-  ( ( K e. HL /\ W e. H ) -> ran ( ( DIsoH ` K ) ` W ) C_ ( LSubSp ` U ) )
11 rabss2 
 |-  ( ran ( ( DIsoH ` K ) ` W ) C_ ( LSubSp ` U ) -> { z e. ran ( ( DIsoH ` K ) ` W ) | X C_ z } C_ { z e. ( LSubSp ` U ) | X C_ z } )
12 intss 
 |-  ( { z e. ran ( ( DIsoH ` K ) ` W ) | X C_ z } C_ { z e. ( LSubSp ` U ) | X C_ z } -> |^| { z e. ( LSubSp ` U ) | X C_ z } C_ |^| { z e. ran ( ( DIsoH ` K ) ` W ) | X C_ z } )
13 6 10 11 12 4syl 
 |-  ( ph -> |^| { z e. ( LSubSp ` U ) | X C_ z } C_ |^| { z e. ran ( ( DIsoH ` K ) ` W ) | X C_ z } )
14 1 2 6 dvhlmod 
 |-  ( ph -> U e. LMod )
15 4 9 5 lspval 
 |-  ( ( U e. LMod /\ X C_ V ) -> ( N ` X ) = |^| { z e. ( LSubSp ` U ) | X C_ z } )
16 14 7 15 syl2anc 
 |-  ( ph -> ( N ` X ) = |^| { z e. ( LSubSp ` U ) | X C_ z } )
17 1 8 2 4 3 6 7 doch2val2 
 |-  ( ph -> ( ._|_ ` ( ._|_ ` X ) ) = |^| { z e. ran ( ( DIsoH ` K ) ` W ) | X C_ z } )
18 13 16 17 3sstr4d 
 |-  ( ph -> ( N ` X ) C_ ( ._|_ ` ( ._|_ ` X ) ) )