Theorem dochffval 36638

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

Hypotheses

Ref	Expression
dochval.b	|- B = ( Base ` K )
dochval.g	|- G = ( glb ` K )
dochval.o	|- ._|_ = ( oc ` K )
dochval.h	|- H = ( LHyp ` K )

Assertion

Ref	Expression
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 ) } ) ) ) ) ) )

Distinct variable groups:   y, B   w, H   x, w, y, K

Allowed substitution hints:   B( x, w)   G( x, y, w)   H( x, y)   ._|_ ( x, y, w)   V( x, y, w)

Proof of Theorem dochffval

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3212	. 2 |- ( K e. V -> K e. _V )
2		fveq2 6191	. . . . 5 |- ( k = K -> ( LHyp ` k ) = ( LHyp ` K ) )
3		dochval.h	. . . . 5 |- H = ( LHyp ` K )
4	2, 3	syl6eqr 2674	. . . 4 |- ( k = K -> ( LHyp ` k ) = H )
5		fveq2 6191	. . . . . . . 8 |- ( k = K -> ( DVecH ` k ) = ( DVecH ` K ) )
6	5	fveq1d 6193	. . . . . . 7 |- ( k = K -> ( ( DVecH ` k ) ` w ) = ( ( DVecH ` K ) ` w ) )
7	6	fveq2d 6195	. . . . . 6 |- ( k = K -> ( Base ` ( ( DVecH ` k ) ` w ) ) = ( Base ` ( ( DVecH ` K ) ` w ) ) )
8	7	pweqd 4163	. . . . 5 |- ( k = K -> ~P ( Base ` ( ( DVecH ` k ) ` w ) ) = ~P ( Base ` ( ( DVecH ` K ) ` w ) ) )
9		fveq2 6191	. . . . . . 7 |- ( k = K -> ( DIsoH ` k ) = ( DIsoH ` K ) )
10	9	fveq1d 6193	. . . . . 6 |- ( k = K -> ( ( DIsoH ` k ) ` w ) = ( ( DIsoH ` K ) ` w ) )
11		fveq2 6191	. . . . . . . 8 |- ( k = K -> ( oc ` k ) = ( oc ` K ) )
12		dochval.o	. . . . . . . 8 |- ._|_ = ( oc ` K )
13	11, 12	syl6eqr 2674	. . . . . . 7 |- ( k = K -> ( oc ` k ) = ._|_ )
14		fveq2 6191	. . . . . . . . 9 |- ( k = K -> ( glb ` k ) = ( glb ` K ) )
15		dochval.g	. . . . . . . . 9 |- G = ( glb ` K )
16	14, 15	syl6eqr 2674	. . . . . . . 8 |- ( k = K -> ( glb ` k ) = G )
17		fveq2 6191	. . . . . . . . . 10 |- ( k = K -> ( Base ` k ) = ( Base ` K ) )
18		dochval.b	. . . . . . . . . 10 |- B = ( Base ` K )
19	17, 18	syl6eqr 2674	. . . . . . . . 9 |- ( k = K -> ( Base ` k ) = B )
20	10	fveq1d 6193	. . . . . . . . . 10 |- ( k = K -> ( ( ( DIsoH ` k ) ` w ) ` y ) = ( ( ( DIsoH ` K ) ` w ) ` y ) )
21	20	sseq2d 3633	. . . . . . . . 9 |- ( k = K -> ( x C_ ( ( ( DIsoH ` k ) ` w ) ` y ) <-> x C_ ( ( ( DIsoH ` K ) ` w ) ` y ) ) )
22	19, 21	rabeqbidv 3195	. . . . . . . 8 |- ( k = K -> { y e. ( Base ` k ) | x C_ ( ( ( DIsoH ` k ) ` w ) ` y ) } = { y e. B | x C_ ( ( ( DIsoH ` K ) ` w ) ` y ) } )
23	16, 22	fveq12d 6197	. . . . . . 7 |- ( 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	13, 23	fveq12d 6197	. . . . . 6 |- ( 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	10, 24	fveq12d 6197	. . . . 5 |- ( 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	8, 25	mpteq12dv 4733	. . . 4 |- ( 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	4, 26	mpteq12dv 4733	. . 3 |- ( 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 36637	. . 3 |- 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		fvex 6201	. . . . 5 |- ( LHyp ` K ) e. _V
30	3, 29	eqeltri 2697	. . . 4 |- H e. _V
31	30	mptex 6486	. . 3 |- ( w e. H |-> ( x e. ~P ( Base ` ( ( DVecH ` K ) ` w ) ) |-> ( ( ( DIsoH ` K ) ` w ) ` ( ._|_ ` ( G ` { y e. B | x C_ ( ( ( DIsoH ` K ) ` w ) ` y ) } ) ) ) ) ) e. _V
32	27, 28, 31	fvmpt 6282	. 2 |- ( 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 ) } ) ) ) ) ) )
33	1, 32	syl 17	1 |- ( 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 ) } ) ) ) ) ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   C_ wss 3574   ~Pcpw 4158   |-> cmpt 4729   ` cfv 5888   Basecbs 15857   occoc 15949   glbcglb 16943   LHypclh 35270   DVecHcdvh 36367   DIsoHcdih 36517   ocHcoch 36636

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-doch 36637

This theorem is referenced by:  dochfval  36639