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Metamath Proof Explorer

Theorem dochvalr2

Description: Orthocomplement of a closed subspace. (Contributed by NM, 21-Jul-2014)

Ref Expression
Hypotheses dochvalr2.b 
|- B = ( Base ` K )
dochvalr2.o 
|- ._|_ = ( oc ` K )
dochvalr2.h 
|- H = ( LHyp ` K )
dochvalr2.i 
|- I = ( ( DIsoH ` K ) ` W )
dochvalr2.n 
|- N = ( ( ocH ` K ) ` W )
Assertion dochvalr2 
|- ( ( ( K e. HL /\ W e. H ) /\ X e. B ) -> ( N ` ( I ` X ) ) = ( I ` ( ._|_ ` X ) ) )

Proof

Step Hyp Ref Expression
1 dochvalr2.b 
 |-  B = ( Base ` K )
2 dochvalr2.o 
 |-  ._|_ = ( oc ` K )
3 dochvalr2.h 
 |-  H = ( LHyp ` K )
4 dochvalr2.i 
 |-  I = ( ( DIsoH ` K ) ` W )
5 dochvalr2.n 
 |-  N = ( ( ocH ` K ) ` W )
6 1 3 4 dihcl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. B ) -> ( I ` X ) e. ran I )
7 2 3 4 5 dochvalr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( I ` X ) e. ran I ) -> ( N ` ( I ` X ) ) = ( I ` ( ._|_ ` ( `' I ` ( I ` X ) ) ) ) )
8 6 7 syldan 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. B ) -> ( N ` ( I ` X ) ) = ( I ` ( ._|_ ` ( `' I ` ( I ` X ) ) ) ) )
9 1 3 4 dihcnvid1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. B ) -> ( `' I ` ( I ` X ) ) = X )
10 9 fveq2d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. B ) -> ( ._|_ ` ( `' I ` ( I ` X ) ) ) = ( ._|_ ` X ) )
11 10 fveq2d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. B ) -> ( I ` ( ._|_ ` ( `' I ` ( I ` X ) ) ) ) = ( I ` ( ._|_ ` X ) ) )
12 8 11 eqtrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. B ) -> ( N ` ( I ` X ) ) = ( I ` ( ._|_ ` X ) ) )