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Metamath Proof Explorer

Theorem dochffval

Description: Subspace orthocomplement for DVecH vector space. (Contributed by NM, 14-Mar-2014)

Ref Expression
Hypotheses dochval.b 
|- B = ( Base ` K )
dochval.g 
|- G = ( glb ` K )
dochval.o 
|- ._|_ = ( oc ` K )
dochval.h 
|- H = ( LHyp ` K )
Assertion dochffval 
|- ( K e. V -> ( ocH ` K ) = ( w e. H |-> ( x e. ~P ( Base ` ( ( DVecH ` K ) ` w ) ) |-> ( ( ( DIsoH ` K ) ` w ) ` ( ._|_ ` ( G ` { y e. B | x C_ ( ( ( DIsoH ` K ) ` w ) ` y ) } ) ) ) ) ) )

Proof

Step Hyp Ref Expression
1 dochval.b 
 |-  B = ( Base ` K )
2 dochval.g 
 |-  G = ( glb ` K )
3 dochval.o 
 |-  ._|_ = ( oc ` K )
4 dochval.h 
 |-  H = ( LHyp ` K )
5 elex 
 |-  ( K e. V -> K e. _V )
6 fveq2 
 |-  ( k = K -> ( LHyp ` k ) = ( LHyp ` K ) )
7 6 4 eqtr4di 
 |-  ( k = K -> ( LHyp ` k ) = H )
8 fveq2 
 |-  ( k = K -> ( DVecH ` k ) = ( DVecH ` K ) )
9 8 fveq1d 
 |-  ( k = K -> ( ( DVecH ` k ) ` w ) = ( ( DVecH ` K ) ` w ) )
10 9 fveq2d 
 |-  ( k = K -> ( Base ` ( ( DVecH ` k ) ` w ) ) = ( Base ` ( ( DVecH ` K ) ` w ) ) )
11 10 pweqd 
 |-  ( k = K -> ~P ( Base ` ( ( DVecH ` k ) ` w ) ) = ~P ( Base ` ( ( DVecH ` K ) ` w ) ) )
12 fveq2 
 |-  ( k = K -> ( DIsoH ` k ) = ( DIsoH ` K ) )
13 12 fveq1d 
 |-  ( k = K -> ( ( DIsoH ` k ) ` w ) = ( ( DIsoH ` K ) ` w ) )
14 fveq2 
 |-  ( k = K -> ( oc ` k ) = ( oc ` K ) )
15 14 3 eqtr4di 
 |-  ( k = K -> ( oc ` k ) = ._|_ )
16 fveq2 
 |-  ( k = K -> ( glb ` k ) = ( glb ` K ) )
17 16 2 eqtr4di 
 |-  ( k = K -> ( glb ` k ) = G )
18 fveq2 
 |-  ( k = K -> ( Base ` k ) = ( Base ` K ) )
19 18 1 eqtr4di 
 |-  ( k = K -> ( Base ` k ) = B )
20 13 fveq1d 
 |-  ( k = K -> ( ( ( DIsoH ` k ) ` w ) ` y ) = ( ( ( DIsoH ` K ) ` w ) ` y ) )
21 20 sseq2d 
 |-  ( k = K -> ( x C_ ( ( ( DIsoH ` k ) ` w ) ` y ) <-> x C_ ( ( ( DIsoH ` K ) ` w ) ` y ) ) )
22 19 21 rabeqbidv 
 |-  ( k = K -> { y e. ( Base ` k ) | x C_ ( ( ( DIsoH ` k ) ` w ) ` y ) } = { y e. B | x C_ ( ( ( DIsoH ` K ) ` w ) ` y ) } )
23 17 22 fveq12d 
 |-  ( k = K -> ( ( glb ` k ) ` { y e. ( Base ` k ) | x C_ ( ( ( DIsoH ` k ) ` w ) ` y ) } ) = ( G ` { y e. B | x C_ ( ( ( DIsoH ` K ) ` w ) ` y ) } ) )
24 15 23 fveq12d 
 |-  ( k = K -> ( ( oc ` k ) ` ( ( glb ` k ) ` { y e. ( Base ` k ) | x C_ ( ( ( DIsoH ` k ) ` w ) ` y ) } ) ) = ( ._|_ ` ( G ` { y e. B | x C_ ( ( ( DIsoH ` K ) ` w ) ` y ) } ) ) )
25 13 24 fveq12d 
 |-  ( k = K -> ( ( ( DIsoH ` k ) ` w ) ` ( ( oc ` k ) ` ( ( glb ` k ) ` { y e. ( Base ` k ) | x C_ ( ( ( DIsoH ` k ) ` w ) ` y ) } ) ) ) = ( ( ( DIsoH ` K ) ` w ) ` ( ._|_ ` ( G ` { y e. B | x C_ ( ( ( DIsoH ` K ) ` w ) ` y ) } ) ) ) )
26 11 25 mpteq12dv 
 |-  ( k = K -> ( x e. ~P ( Base ` ( ( DVecH ` k ) ` w ) ) |-> ( ( ( DIsoH ` k ) ` w ) ` ( ( oc ` k ) ` ( ( glb ` k ) ` { y e. ( Base ` k ) | x C_ ( ( ( DIsoH ` k ) ` w ) ` y ) } ) ) ) ) = ( x e. ~P ( Base ` ( ( DVecH ` K ) ` w ) ) |-> ( ( ( DIsoH ` K ) ` w ) ` ( ._|_ ` ( G ` { y e. B | x C_ ( ( ( DIsoH ` K ) ` w ) ` y ) } ) ) ) ) )
27 7 26 mpteq12dv 
 |-  ( k = K -> ( w e. ( LHyp ` k ) |-> ( x e. ~P ( Base ` ( ( DVecH ` k ) ` w ) ) |-> ( ( ( DIsoH ` k ) ` w ) ` ( ( oc ` k ) ` ( ( glb ` k ) ` { y e. ( Base ` k ) | x C_ ( ( ( DIsoH ` k ) ` w ) ` y ) } ) ) ) ) ) = ( w e. H |-> ( x e. ~P ( Base ` ( ( DVecH ` K ) ` w ) ) |-> ( ( ( DIsoH ` K ) ` w ) ` ( ._|_ ` ( G ` { y e. B | x C_ ( ( ( DIsoH ` K ) ` w ) ` y ) } ) ) ) ) ) )
28 df-doch 
 |-  ocH = ( k e. _V |-> ( w e. ( LHyp ` k ) |-> ( x e. ~P ( Base ` ( ( DVecH ` k ) ` w ) ) |-> ( ( ( DIsoH ` k ) ` w ) ` ( ( oc ` k ) ` ( ( glb ` k ) ` { y e. ( Base ` k ) | x C_ ( ( ( DIsoH ` k ) ` w ) ` y ) } ) ) ) ) ) )
29 27 28 4 mptfvmpt 
 |-  ( K e. _V -> ( ocH ` K ) = ( w e. H |-> ( x e. ~P ( Base ` ( ( DVecH ` K ) ` w ) ) |-> ( ( ( DIsoH ` K ) ` w ) ` ( ._|_ ` ( G ` { y e. B | x C_ ( ( ( DIsoH ` K ) ` w ) ` y ) } ) ) ) ) ) )
30 5 29 syl 
 |-  ( K e. V -> ( ocH ` K ) = ( w e. H |-> ( x e. ~P ( Base ` ( ( DVecH ` K ) ` w ) ) |-> ( ( ( DIsoH ` K ) ` w ) ` ( ._|_ ` ( G ` { y e. B | x C_ ( ( ( DIsoH ` K ) ` w ) ` y ) } ) ) ) ) ) )