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Metamath Proof Explorer

Theorem dochspocN

Description: The span of an orthocomplement equals the orthocomplement of the span. (Contributed by NM, 7-Aug-2014) (New usage is discouraged.)

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 dochspocN 
|- ( ph -> ( N ` ( ._|_ ` X ) ) = ( ._|_ ` ( N ` 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 1 2 6 dvhlmod 
 |-  ( ph -> U e. LMod )
9 eqid 
 |-  ( LSubSp ` U ) = ( LSubSp ` U )
10 1 2 4 9 3 dochlss 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X C_ V ) -> ( ._|_ ` X ) e. ( LSubSp ` U ) )
11 6 7 10 syl2anc 
 |-  ( ph -> ( ._|_ ` X ) e. ( LSubSp ` U ) )
12 9 5 lspid 
 |-  ( ( U e. LMod /\ ( ._|_ ` X ) e. ( LSubSp ` U ) ) -> ( N ` ( ._|_ ` X ) ) = ( ._|_ ` X ) )
13 8 11 12 syl2anc 
 |-  ( ph -> ( N ` ( ._|_ ` X ) ) = ( ._|_ ` X ) )
14 1 2 3 4 5 6 7 dochocsp 
 |-  ( ph -> ( ._|_ ` ( N ` X ) ) = ( ._|_ ` X ) )
15 13 14 eqtr4d 
 |-  ( ph -> ( N ` ( ._|_ ` X ) ) = ( ._|_ ` ( N ` X ) ) )