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Metamath Proof Explorer

Theorem dochocsn

Description: The double orthocomplement of a singleton is its span. (Contributed by NM, 13-Jan-2015)

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

Proof

Step Hyp Ref Expression
1 dochocsn.h 
 |-  H = ( LHyp ` K )
2 dochocsn.u 
 |-  U = ( ( DVecH ` K ) ` W )
3 dochocsn.o 
 |-  ._|_ = ( ( ocH ` K ) ` W )
4 dochocsn.v 
 |-  V = ( Base ` U )
5 dochocsn.n 
 |-  N = ( LSpan ` U )
6 dochocsn.k 
 |-  ( ph -> ( K e. HL /\ W e. H ) )
7 dochocsn.x 
 |-  ( ph -> X e. V )
8 7 snssd 
 |-  ( ph -> { X } C_ V )
9 1 2 3 4 5 6 8 dochocsp 
 |-  ( ph -> ( ._|_ ` ( N ` { X } ) ) = ( ._|_ ` { X } ) )
10 9 fveq2d 
 |-  ( ph -> ( ._|_ ` ( ._|_ ` ( N ` { X } ) ) ) = ( ._|_ ` ( ._|_ ` { X } ) ) )
11 eqid 
 |-  ( ( DIsoH ` K ) ` W ) = ( ( DIsoH ` K ) ` W )
12 1 2 4 5 11 dihlsprn 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. V ) -> ( N ` { X } ) e. ran ( ( DIsoH ` K ) ` W ) )
13 6 7 12 syl2anc 
 |-  ( ph -> ( N ` { X } ) e. ran ( ( DIsoH ` K ) ` W ) )
14 1 11 3 dochoc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( N ` { X } ) e. ran ( ( DIsoH ` K ) ` W ) ) -> ( ._|_ ` ( ._|_ ` ( N ` { X } ) ) ) = ( N ` { X } ) )
15 6 13 14 syl2anc 
 |-  ( ph -> ( ._|_ ` ( ._|_ ` ( N ` { X } ) ) ) = ( N ` { X } ) )
16 10 15 eqtr3d 
 |-  ( ph -> ( ._|_ ` ( ._|_ ` { X } ) ) = ( N ` { X } ) )