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Metamath Proof Explorer

Theorem dochfN

Description: Domain and codomain of the subspace orthocomplement for the DVecH vector space. (Contributed by NM, 27-Dec-2014) (New usage is discouraged.)

Ref Expression
Hypotheses dochf.h 
|- H = ( LHyp ` K )
dochf.i 
|- I = ( ( DIsoH ` K ) ` W )
dochf.u 
|- U = ( ( DVecH ` K ) ` W )
dochf.v 
|- V = ( Base ` U )
dochf.o 
|- ._|_ = ( ( ocH ` K ) ` W )
dochf.k 
|- ( ph -> ( K e. HL /\ W e. H ) )
Assertion dochfN 
|- ( ph -> ._|_ : ~P V --> ran I )

Proof

Step Hyp Ref Expression
1 dochf.h 
 |-  H = ( LHyp ` K )
2 dochf.i 
 |-  I = ( ( DIsoH ` K ) ` W )
3 dochf.u 
 |-  U = ( ( DVecH ` K ) ` W )
4 dochf.v 
 |-  V = ( Base ` U )
5 dochf.o 
 |-  ._|_ = ( ( ocH ` K ) ` W )
6 dochf.k 
 |-  ( ph -> ( K e. HL /\ W e. H ) )
7 fvexd 
 |-  ( ( ph /\ x e. ~P V ) -> ( I ` ( ( oc ` K ) ` ( ( glb ` K ) ` { y e. ( Base ` K ) | x C_ ( I ` y ) } ) ) ) e. _V )
8 eqid 
 |-  ( Base ` K ) = ( Base ` K )
9 eqid 
 |-  ( glb ` K ) = ( glb ` K )
10 eqid 
 |-  ( oc ` K ) = ( oc ` K )
11 8 9 10 1 2 3 4 5 dochfval 
 |-  ( ( K e. HL /\ W e. H ) -> ._|_ = ( x e. ~P V |-> ( I ` ( ( oc ` K ) ` ( ( glb ` K ) ` { y e. ( Base ` K ) | x C_ ( I ` y ) } ) ) ) ) )
12 6 11 syl 
 |-  ( ph -> ._|_ = ( x e. ~P V |-> ( I ` ( ( oc ` K ) ` ( ( glb ` K ) ` { y e. ( Base ` K ) | x C_ ( I ` y ) } ) ) ) ) )
13 elpwi 
 |-  ( y e. ~P V -> y C_ V )
14 1 2 3 4 5 dochcl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ y C_ V ) -> ( ._|_ ` y ) e. ran I )
15 6 13 14 syl2an 
 |-  ( ( ph /\ y e. ~P V ) -> ( ._|_ ` y ) e. ran I )
16 7 12 15 fmpt2d 
 |-  ( ph -> ._|_ : ~P V --> ran I )