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Metamath Proof Explorer

Theorem dochoc

Description: Double negative law for orthocomplement of a closed subspace. (Contributed by NM, 14-Mar-2014)

Ref Expression
Hypotheses dochoc.h 
|- H = ( LHyp ` K )
dochoc.i 
|- I = ( ( DIsoH ` K ) ` W )
dochoc.o 
|- ._|_ = ( ( ocH ` K ) ` W )
Assertion dochoc 
|- ( ( ( K e. HL /\ W e. H ) /\ X e. ran I ) -> ( ._|_ ` ( ._|_ ` X ) ) = X )

Proof

Step Hyp Ref Expression
1 dochoc.h 
 |-  H = ( LHyp ` K )
2 dochoc.i 
 |-  I = ( ( DIsoH ` K ) ` W )
3 dochoc.o 
 |-  ._|_ = ( ( ocH ` K ) ` W )
4 eqid 
 |-  ( oc ` K ) = ( oc ` K )
5 4 1 2 3 dochvalr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. ran I ) -> ( ._|_ ` X ) = ( I ` ( ( oc ` K ) ` ( `' I ` X ) ) ) )
6 5 fveq2d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. ran I ) -> ( ._|_ ` ( ._|_ ` X ) ) = ( ._|_ ` ( I ` ( ( oc ` K ) ` ( `' I ` X ) ) ) ) )
7 hlop 
 |-  ( K e. HL -> K e. OP )
8 7 ad2antrr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. ran I ) -> K e. OP )
9 eqid 
 |-  ( Base ` K ) = ( Base ` K )
10 9 1 2 dihcnvcl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. ran I ) -> ( `' I ` X ) e. ( Base ` K ) )
11 9 4 opoccl 
 |-  ( ( K e. OP /\ ( `' I ` X ) e. ( Base ` K ) ) -> ( ( oc ` K ) ` ( `' I ` X ) ) e. ( Base ` K ) )
12 8 10 11 syl2anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. ran I ) -> ( ( oc ` K ) ` ( `' I ` X ) ) e. ( Base ` K ) )
13 9 1 2 dihcl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( oc ` K ) ` ( `' I ` X ) ) e. ( Base ` K ) ) -> ( I ` ( ( oc ` K ) ` ( `' I ` X ) ) ) e. ran I )
14 12 13 syldan 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. ran I ) -> ( I ` ( ( oc ` K ) ` ( `' I ` X ) ) ) e. ran I )
15 4 1 2 3 dochvalr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( I ` ( ( oc ` K ) ` ( `' I ` X ) ) ) e. ran I ) -> ( ._|_ ` ( I ` ( ( oc ` K ) ` ( `' I ` X ) ) ) ) = ( I ` ( ( oc ` K ) ` ( `' I ` ( I ` ( ( oc ` K ) ` ( `' I ` X ) ) ) ) ) ) )
16 14 15 syldan 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. ran I ) -> ( ._|_ ` ( I ` ( ( oc ` K ) ` ( `' I ` X ) ) ) ) = ( I ` ( ( oc ` K ) ` ( `' I ` ( I ` ( ( oc ` K ) ` ( `' I ` X ) ) ) ) ) ) )
17 9 1 2 dihcnvid1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( oc ` K ) ` ( `' I ` X ) ) e. ( Base ` K ) ) -> ( `' I ` ( I ` ( ( oc ` K ) ` ( `' I ` X ) ) ) ) = ( ( oc ` K ) ` ( `' I ` X ) ) )
18 12 17 syldan 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. ran I ) -> ( `' I ` ( I ` ( ( oc ` K ) ` ( `' I ` X ) ) ) ) = ( ( oc ` K ) ` ( `' I ` X ) ) )
19 18 fveq2d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. ran I ) -> ( ( oc ` K ) ` ( `' I ` ( I ` ( ( oc ` K ) ` ( `' I ` X ) ) ) ) ) = ( ( oc ` K ) ` ( ( oc ` K ) ` ( `' I ` X ) ) ) )
20 9 4 opococ 
 |-  ( ( K e. OP /\ ( `' I ` X ) e. ( Base ` K ) ) -> ( ( oc ` K ) ` ( ( oc ` K ) ` ( `' I ` X ) ) ) = ( `' I ` X ) )
21 8 10 20 syl2anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. ran I ) -> ( ( oc ` K ) ` ( ( oc ` K ) ` ( `' I ` X ) ) ) = ( `' I ` X ) )
22 19 21 eqtrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. ran I ) -> ( ( oc ` K ) ` ( `' I ` ( I ` ( ( oc ` K ) ` ( `' I ` X ) ) ) ) ) = ( `' I ` X ) )
23 22 fveq2d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. ran I ) -> ( I ` ( ( oc ` K ) ` ( `' I ` ( I ` ( ( oc ` K ) ` ( `' I ` X ) ) ) ) ) ) = ( I ` ( `' I ` X ) ) )
24 1 2 dihcnvid2 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. ran I ) -> ( I ` ( `' I ` X ) ) = X )
25 23 24 eqtrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. ran I ) -> ( I ` ( ( oc ` K ) ` ( `' I ` ( I ` ( ( oc ` K ) ` ( `' I ` X ) ) ) ) ) ) = X )
26 16 25 eqtrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. ran I ) -> ( ._|_ ` ( I ` ( ( oc ` K ) ` ( `' I ` X ) ) ) ) = X )
27 6 26 eqtrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. ran I ) -> ( ._|_ ` ( ._|_ ` X ) ) = X )