Theorem dochval 36640

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 )
dochval.i	|- I = ( ( DIsoH ` K ) ` W )
dochval.u	|- U = ( ( DVecH ` K ) ` W )
dochval.v	|- V = ( Base ` U )
dochval.n	|- N = ( ( ocH ` K ) ` W )

Assertion

Ref	Expression
dochval	|- ( ( ( K e. Y /\ W e. H ) /\ X C_ V ) -> ( N ` X ) = ( I ` ( ._|_ ` ( G ` { y e. B | X C_ ( I ` y ) } ) ) ) )

Distinct variable groups:   y, B   y, K   y, W   y, X

Allowed substitution hints:   U( y)   G( y)   H( y)   I( y)   N( y)   ._|_ ( y)   V( y)   Y( y)

Proof of Theorem dochval

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dochval.b	. . . . 5 |- B = ( Base ` K )
2		dochval.g	. . . . 5 |- G = ( glb ` K )
3		dochval.o	. . . . 5 |- ._|_ = ( oc ` K )
4		dochval.h	. . . . 5 |- H = ( LHyp ` K )
5		dochval.i	. . . . 5 |- I = ( ( DIsoH ` K ) ` W )
6		dochval.u	. . . . 5 |- U = ( ( DVecH ` K ) ` W )
7		dochval.v	. . . . 5 |- V = ( Base ` U )
8		dochval.n	. . . . 5 |- N = ( ( ocH ` K ) ` W )
9	1, 2, 3, 4, 5, 6, 7, 8	dochfval 36639	. . . 4 |- ( ( K e. Y /\ W e. H ) -> N = ( x e. ~P V |-> ( I ` ( ._|_ ` ( G ` { y e. B | x C_ ( I ` y ) } ) ) ) ) )
10	9	adantr 481	. . 3 |- ( ( ( K e. Y /\ W e. H ) /\ X C_ V ) -> N = ( x e. ~P V |-> ( I ` ( ._|_ ` ( G ` { y e. B | x C_ ( I ` y ) } ) ) ) ) )
11	10	fveq1d 6193	. 2 |- ( ( ( K e. Y /\ W e. H ) /\ X C_ V ) -> ( N ` X ) = ( ( x e. ~P V |-> ( I ` ( ._|_ ` ( G ` { y e. B | x C_ ( I ` y ) } ) ) ) ) ` X ) )
12		fvex 6201	. . . . . . 7 |- ( Base ` U ) e. _V
13	7, 12	eqeltri 2697	. . . . . 6 |- V e. _V
14	13	elpw2 4828	. . . . 5 |- ( X e. ~P V <-> X C_ V )
15	14	biimpri 218	. . . 4 |- ( X C_ V -> X e. ~P V )
16	15	adantl 482	. . 3 |- ( ( ( K e. Y /\ W e. H ) /\ X C_ V ) -> X e. ~P V )
17		fvex 6201	. . 3 |- ( I ` ( ._|_ ` ( G ` { y e. B | X C_ ( I ` y ) } ) ) ) e. _V
18		sseq1 3626	. . . . . . . 8 |- ( x = X -> ( x C_ ( I ` y ) <-> X C_ ( I ` y ) ) )
19	18	rabbidv 3189	. . . . . . 7 |- ( x = X -> { y e. B | x C_ ( I ` y ) } = { y e. B | X C_ ( I ` y ) } )
20	19	fveq2d 6195	. . . . . 6 |- ( x = X -> ( G ` { y e. B | x C_ ( I ` y ) } ) = ( G ` { y e. B | X C_ ( I ` y ) } ) )
21	20	fveq2d 6195	. . . . 5 |- ( x = X -> ( ._|_ ` ( G ` { y e. B | x C_ ( I ` y ) } ) ) = ( ._|_ ` ( G ` { y e. B | X C_ ( I ` y ) } ) ) )
22	21	fveq2d 6195	. . . 4 |- ( x = X -> ( I ` ( ._|_ ` ( G ` { y e. B | x C_ ( I ` y ) } ) ) ) = ( I ` ( ._|_ ` ( G ` { y e. B | X C_ ( I ` y ) } ) ) ) )
23		eqid 2622	. . . 4 |- ( x e. ~P V |-> ( I ` ( ._|_ ` ( G ` { y e. B | x C_ ( I ` y ) } ) ) ) ) = ( x e. ~P V |-> ( I ` ( ._|_ ` ( G ` { y e. B | x C_ ( I ` y ) } ) ) ) )
24	22, 23	fvmptg 6280	. . 3 |- ( ( X e. ~P V /\ ( I ` ( ._|_ ` ( G ` { y e. B | X C_ ( I ` y ) } ) ) ) e. _V ) -> ( ( x e. ~P V |-> ( I ` ( ._|_ ` ( G ` { y e. B | x C_ ( I ` y ) } ) ) ) ) ` X ) = ( I ` ( ._|_ ` ( G ` { y e. B | X C_ ( I ` y ) } ) ) ) )
25	16, 17, 24	sylancl 694	. 2 |- ( ( ( K e. Y /\ W e. H ) /\ X C_ V ) -> ( ( x e. ~P V |-> ( I ` ( ._|_ ` ( G ` { y e. B | x C_ ( I ` y ) } ) ) ) ) ` X ) = ( I ` ( ._|_ ` ( G ` { y e. B | X C_ ( I ` y ) } ) ) ) )
26	11, 25	eqtrd 2656	1 |- ( ( ( K e. Y /\ W e. H ) /\ X C_ V ) -> ( N ` X ) = ( I ` ( ._|_ ` ( G ` { y e. B | X C_ ( I ` y ) } ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = 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-pow 4843  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:  dochval2  36641  dochcl  36642  dochvalr  36646  dochss  36654