This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem dochoccl

Description: A set of vectors is closed iff it equals its double orthocomplent. (Contributed by NM, 1-Jan-2015)

Ref Expression
Hypotheses dochoccl.h 
|- H = ( LHyp ` K )
dochoccl.i 
|- I = ( ( DIsoH ` K ) ` W )
dochoccl.u 
|- U = ( ( DVecH ` K ) ` W )
dochoccl.v 
|- V = ( Base ` U )
dochoccl.o 
|- ._|_ = ( ( ocH ` K ) ` W )
dochoccl.k 
|- ( ph -> ( K e. HL /\ W e. H ) )
dochoccl.g 
|- ( ph -> X C_ V )
Assertion dochoccl 
|- ( ph -> ( X e. ran I <-> ( ._|_ ` ( ._|_ ` X ) ) = X ) )

Proof

Step Hyp Ref Expression
1 dochoccl.h 
 |-  H = ( LHyp ` K )
2 dochoccl.i 
 |-  I = ( ( DIsoH ` K ) ` W )
3 dochoccl.u 
 |-  U = ( ( DVecH ` K ) ` W )
4 dochoccl.v 
 |-  V = ( Base ` U )
5 dochoccl.o 
 |-  ._|_ = ( ( ocH ` K ) ` W )
6 dochoccl.k 
 |-  ( ph -> ( K e. HL /\ W e. H ) )
7 dochoccl.g 
 |-  ( ph -> X C_ V )
8 1 2 5 dochoc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. ran I ) -> ( ._|_ ` ( ._|_ ` X ) ) = X )
9 6 8 sylan 
 |-  ( ( ph /\ X e. ran I ) -> ( ._|_ ` ( ._|_ ` X ) ) = X )
10 simpr 
 |-  ( ( ph /\ ( ._|_ ` ( ._|_ ` X ) ) = X ) -> ( ._|_ ` ( ._|_ ` X ) ) = X )
11 1 3 4 5 dochssv 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X C_ V ) -> ( ._|_ ` X ) C_ V )
12 6 7 11 syl2anc 
 |-  ( ph -> ( ._|_ ` X ) C_ V )
13 1 2 3 4 5 dochcl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ._|_ ` X ) C_ V ) -> ( ._|_ ` ( ._|_ ` X ) ) e. ran I )
14 6 12 13 syl2anc 
 |-  ( ph -> ( ._|_ ` ( ._|_ ` X ) ) e. ran I )
15 14 adantr 
 |-  ( ( ph /\ ( ._|_ ` ( ._|_ ` X ) ) = X ) -> ( ._|_ ` ( ._|_ ` X ) ) e. ran I )
16 10 15 eqeltrrd 
 |-  ( ( ph /\ ( ._|_ ` ( ._|_ ` X ) ) = X ) -> X e. ran I )
17 9 16 impbida 
 |-  ( ph -> ( X e. ran I <-> ( ._|_ ` ( ._|_ ` X ) ) = X ) )