Theorem elocv 20012

Description: Elementhood in the orthocomplement of a subset (normally a subspace) of a pre-Hilbert space. (Contributed by Mario Carneiro, 13-Oct-2015.)

Hypotheses

Ref	Expression
ocvfval.v	|- V = ( Base ` W )
ocvfval.i	|- ., = ( .i ` W )
ocvfval.f	|- F = (Scalar ` W )
ocvfval.z	|- .0. = ( 0g ` F )
ocvfval.o	|- ._|_ = ( ocv ` W )

Assertion

Ref	Expression
elocv	|- ( A e. ( ._|_ ` S ) <-> ( S C_ V /\ A e. V /\ A. x e. S ( A ., x ) = .0. ) )

Distinct variable groups:   x, .0.   x, A   x, V   x, W   x, .,   x, S

Allowed substitution hints:   F( x)   ._|_ ( x)

Proof of Theorem elocv

Dummy variables s y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elfvdm 6220	. . . . 5 |- ( A e. ( ._|_ ` S ) -> S e. dom ._|_ )
2		n0i 3920	. . . . . . . . 9 |- ( A e. ( ._|_ ` S ) -> -. ( ._|_ ` S ) = (/) )
3		ocvfval.o	. . . . . . . . . . . 12 |- ._|_ = ( ocv ` W )
4		fvprc 6185	. . . . . . . . . . . 12 |- ( -. W e. _V -> ( ocv ` W ) = (/) )
5	3, 4	syl5eq 2668	. . . . . . . . . . 11 |- ( -. W e. _V -> ._|_ = (/) )
6	5	fveq1d 6193	. . . . . . . . . 10 |- ( -. W e. _V -> ( ._|_ ` S ) = ( (/) ` S ) )
7		0fv 6227	. . . . . . . . . 10 |- ( (/) ` S ) = (/)
8	6, 7	syl6eq 2672	. . . . . . . . 9 |- ( -. W e. _V -> ( ._|_ ` S ) = (/) )
9	2, 8	nsyl2 142	. . . . . . . 8 |- ( A e. ( ._|_ ` S ) -> W e. _V )
10		ocvfval.v	. . . . . . . . 9 |- V = ( Base ` W )
11		ocvfval.i	. . . . . . . . 9 |- ., = ( .i ` W )
12		ocvfval.f	. . . . . . . . 9 |- F = (Scalar ` W )
13		ocvfval.z	. . . . . . . . 9 |- .0. = ( 0g ` F )
14	10, 11, 12, 13, 3	ocvfval 20010	. . . . . . . 8 |- ( W e. _V -> ._|_ = ( s e. ~P V |-> { y e. V | A. x e. s ( y ., x ) = .0. } ) )
15	9, 14	syl 17	. . . . . . 7 |- ( A e. ( ._|_ ` S ) -> ._|_ = ( s e. ~P V |-> { y e. V | A. x e. s ( y ., x ) = .0. } ) )
16	15	dmeqd 5326	. . . . . 6 |- ( A e. ( ._|_ ` S ) -> dom ._|_ = dom ( s e. ~P V |-> { y e. V | A. x e. s ( y ., x ) = .0. } ) )
17		fvex 6201	. . . . . . . . 9 |- ( Base ` W ) e. _V
18	10, 17	eqeltri 2697	. . . . . . . 8 |- V e. _V
19	18	rabex 4813	. . . . . . 7 |- { y e. V | A. x e. s ( y ., x ) = .0. } e. _V
20		eqid 2622	. . . . . . 7 |- ( s e. ~P V |-> { y e. V | A. x e. s ( y ., x ) = .0. } ) = ( s e. ~P V |-> { y e. V | A. x e. s ( y ., x ) = .0. } )
21	19, 20	dmmpti 6023	. . . . . 6 |- dom ( s e. ~P V |-> { y e. V | A. x e. s ( y ., x ) = .0. } ) = ~P V
22	16, 21	syl6eq 2672	. . . . 5 |- ( A e. ( ._|_ ` S ) -> dom ._|_ = ~P V )
23	1, 22	eleqtrd 2703	. . . 4 |- ( A e. ( ._|_ ` S ) -> S e. ~P V )
24	23	elpwid 4170	. . 3 |- ( A e. ( ._|_ ` S ) -> S C_ V )
25	10, 11, 12, 13, 3	ocvval 20011	. . . . 5 |- ( S C_ V -> ( ._|_ ` S ) = { y e. V | A. x e. S ( y ., x ) = .0. } )
26	25	eleq2d 2687	. . . 4 |- ( S C_ V -> ( A e. ( ._|_ ` S ) <-> A e. { y e. V | A. x e. S ( y ., x ) = .0. } ) )
27		oveq1 6657	. . . . . . 7 |- ( y = A -> ( y ., x ) = ( A ., x ) )
28	27	eqeq1d 2624	. . . . . 6 |- ( y = A -> ( ( y ., x ) = .0. <-> ( A ., x ) = .0. ) )
29	28	ralbidv 2986	. . . . 5 |- ( y = A -> ( A. x e. S ( y ., x ) = .0. <-> A. x e. S ( A ., x ) = .0. ) )
30	29	elrab 3363	. . . 4 |- ( A e. { y e. V | A. x e. S ( y ., x ) = .0. } <-> ( A e. V /\ A. x e. S ( A ., x ) = .0. ) )
31	26, 30	syl6bb 276	. . 3 |- ( S C_ V -> ( A e. ( ._|_ ` S ) <-> ( A e. V /\ A. x e. S ( A ., x ) = .0. ) ) )
32	24, 31	biadan2 674	. 2 |- ( A e. ( ._|_ ` S ) <-> ( S C_ V /\ ( A e. V /\ A. x e. S ( A ., x ) = .0. ) ) )
33		3anass 1042	. 2 |- ( ( S C_ V /\ A e. V /\ A. x e. S ( A ., x ) = .0. ) <-> ( S C_ V /\ ( A e. V /\ A. x e. S ( A ., x ) = .0. ) ) )
34	32, 33	bitr4i 267	1 |- ( A e. ( ._|_ ` S ) <-> ( S C_ V /\ A e. V /\ A. x e. S ( A ., x ) = .0. ) )

Syntax hints:   -. wn 3   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   _Vcvv 3200   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   |-> cmpt 4729   dom cdm 5114   ` cfv 5888  (class class class)co 6650   Basecbs 15857  Scalarcsca 15944   .icip 15946   0gc0g 16100   ocvcocv 20004

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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

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-rab 2921  df-v 3202  df-sbc 3436  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-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-fv 5896  df-ov 6653  df-ocv 20007

This theorem is referenced by:  ocvi  20013  ocvss  20014  ocvocv  20015  ocvlss  20016  ocv2ss  20017  unocv  20024  iunocv  20025  obselocv  20072  clsocv  23049  pjthlem2  23209