Metamath Proof Explorer

Theorem dochvalr3

Description: Orthocomplement of a closed subspace. (Contributed by NM, 15-Jan-2015)

Ref Expression
Hypotheses dochvalr3.o 
|- ._|_ = ( oc ` K )
dochvalr3.h 
|- H = ( LHyp ` K )
dochvalr3.i 
|- I = ( ( DIsoH ` K ) ` W )
dochvalr3.n 
|- N = ( ( ocH ` K ) ` W )
dochvalr3.k 
|- ( ph -> ( K e. HL /\ W e. H ) )
dochvalr3.x 
|- ( ph -> X e. ran I )
Assertion dochvalr3 
|- ( ph -> ( ._|_ ` ( `' I ` X ) ) = ( `' I ` ( N ` X ) ) )

Proof

Step Hyp Ref Expression
1 dochvalr3.o 
 |-  ._|_ = ( oc ` K )
2 dochvalr3.h 
 |-  H = ( LHyp ` K )
3 dochvalr3.i 
 |-  I = ( ( DIsoH ` K ) ` W )
4 dochvalr3.n 
 |-  N = ( ( ocH ` K ) ` W )
5 dochvalr3.k 
 |-  ( ph -> ( K e. HL /\ W e. H ) )
6 dochvalr3.x 
 |-  ( ph -> X e. ran I )
7 eqid 
 |-  ( ( DVecH ` K ) ` W ) = ( ( DVecH ` K ) ` W )
8 eqid 
 |-  ( Base ` ( ( DVecH ` K ) ` W ) ) = ( Base ` ( ( DVecH ` K ) ` W ) )
9 2 7 3 8 dihrnss 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. ran I ) -> X C_ ( Base ` ( ( DVecH ` K ) ` W ) ) )
10 5 6 9 syl2anc 
 |-  ( ph -> X C_ ( Base ` ( ( DVecH ` K ) ` W ) ) )
11 2 3 7 8 4 dochcl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X C_ ( Base ` ( ( DVecH ` K ) ` W ) ) ) -> ( N ` X ) e. ran I )
12 5 10 11 syl2anc 
 |-  ( ph -> ( N ` X ) e. ran I )
13 2 3 dihcnvid2 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( N ` X ) e. ran I ) -> ( I ` ( `' I ` ( N ` X ) ) ) = ( N ` X ) )
14 5 12 13 syl2anc 
 |-  ( ph -> ( I ` ( `' I ` ( N ` X ) ) ) = ( N ` X ) )
15 1 2 3 4 dochvalr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. ran I ) -> ( N ` X ) = ( I ` ( ._|_ ` ( `' I ` X ) ) ) )
16 5 6 15 syl2anc 
 |-  ( ph -> ( N ` X ) = ( I ` ( ._|_ ` ( `' I ` X ) ) ) )
17 14 16 eqtr2d 
 |-  ( ph -> ( I ` ( ._|_ ` ( `' I ` X ) ) ) = ( I ` ( `' I ` ( N ` X ) ) ) )
18 5 simpld 
 |-  ( ph -> K e. HL )
19 hlop 
 |-  ( K e. HL -> K e. OP )
20 18 19 syl 
 |-  ( ph -> K e. OP )
21 eqid 
 |-  ( Base ` K ) = ( Base ` K )
22 21 2 3 dihcnvcl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. ran I ) -> ( `' I ` X ) e. ( Base ` K ) )
23 5 6 22 syl2anc 
 |-  ( ph -> ( `' I ` X ) e. ( Base ` K ) )
24 21 1 opoccl 
 |-  ( ( K e. OP /\ ( `' I ` X ) e. ( Base ` K ) ) -> ( ._|_ ` ( `' I ` X ) ) e. ( Base ` K ) )
25 20 23 24 syl2anc 
 |-  ( ph -> ( ._|_ ` ( `' I ` X ) ) e. ( Base ` K ) )
26 21 2 3 dihcnvcl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( N ` X ) e. ran I ) -> ( `' I ` ( N ` X ) ) e. ( Base ` K ) )
27 5 12 26 syl2anc 
 |-  ( ph -> ( `' I ` ( N ` X ) ) e. ( Base ` K ) )
28 21 2 3 dih11 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ._|_ ` ( `' I ` X ) ) e. ( Base ` K ) /\ ( `' I ` ( N ` X ) ) e. ( Base ` K ) ) -> ( ( I ` ( ._|_ ` ( `' I ` X ) ) ) = ( I ` ( `' I ` ( N ` X ) ) ) <-> ( ._|_ ` ( `' I ` X ) ) = ( `' I ` ( N ` X ) ) ) )
29 5 25 27 28 syl3anc 
 |-  ( ph -> ( ( I ` ( ._|_ ` ( `' I ` X ) ) ) = ( I ` ( `' I ` ( N ` X ) ) ) <-> ( ._|_ ` ( `' I ` X ) ) = ( `' I ` ( N ` X ) ) ) )
30 17 29 mpbid 
 |-  ( ph -> ( ._|_ ` ( `' I ` X ) ) = ( `' I ` ( N ` X ) ) )