Theorem elocv 18506

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 5892	. . . . 5 |- ( A e. ( ._|_ ` S ) -> S e. dom ._|_ )
2		n0i 3790	. . . . . . . . 9 |- ( A e. ( ._|_ ` S ) -> -. ( ._|_ ` S ) = (/) )
3		ocvfval.o	. . . . . . . . . . . 12 |- ._|_ = ( ocv ` W )
4		fvprc 5860	. . . . . . . . . . . 12 |- ( -. W e. _V -> ( ocv ` W ) = (/) )
5	3, 4	syl5eq 2520	. . . . . . . . . . 11 |- ( -. W e. _V -> ._|_ = (/) )
6	5	fveq1d 5868	. . . . . . . . . 10 |- ( -. W e. _V -> ( ._|_ ` S ) = ( (/) ` S ) )
7		0fv 5899	. . . . . . . . . 10 |- ( (/) ` S ) = (/)
8	6, 7	syl6eq 2524	. . . . . . . . 9 |- ( -. W e. _V -> ( ._|_ ` S ) = (/) )
9	2, 8	nsyl2 127	. . . . . . . 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 18504	. . . . . . . 8 |- ( W e. _V -> ._|_ = ( s e. ~P V |-> { y e. V | A. x e. s ( y ., x ) = .0. } ) )
15	9, 14	syl 16	. . . . . . 7 |- ( A e. ( ._|_ ` S ) -> ._|_ = ( s e. ~P V |-> { y e. V | A. x e. s ( y ., x ) = .0. } ) )
16	15	dmeqd 5205	. . . . . 6 |- ( A e. ( ._|_ ` S ) -> dom ._|_ = dom ( s e. ~P V |-> { y e. V | A. x e. s ( y ., x ) = .0. } ) )
17		fvex 5876	. . . . . . . . 9 |- ( Base ` W ) e. _V
18	10, 17	eqeltri 2551	. . . . . . . 8 |- V e. _V
19	18	rabex 4598	. . . . . . 7 |- { y e. V | A. x e. s ( y ., x ) = .0. } e. _V
20		eqid 2467	. . . . . . 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 5710	. . . . . 6 |- dom ( s e. ~P V |-> { y e. V | A. x e. s ( y ., x ) = .0. } ) = ~P V
22	16, 21	syl6eq 2524	. . . . 5 |- ( A e. ( ._|_ ` S ) -> dom ._|_ = ~P V )
23	1, 22	eleqtrd 2557	. . . 4 |- ( A e. ( ._|_ ` S ) -> S e. ~P V )
24	23	elpwid 4020	. . 3 |- ( A e. ( ._|_ ` S ) -> S C_ V )
25	10, 11, 12, 13, 3	ocvval 18505	. . . . 5 |- ( S C_ V -> ( ._|_ ` S ) = { y e. V | A. x e. S ( y ., x ) = .0. } )
26	25	eleq2d 2537	. . . 4 |- ( S C_ V -> ( A e. ( ._|_ ` S ) <-> A e. { y e. V | A. x e. S ( y ., x ) = .0. } ) )
27		oveq1 6292	. . . . . . 7 |- ( y = A -> ( y ., x ) = ( A ., x ) )
28	27	eqeq1d 2469	. . . . . 6 |- ( y = A -> ( ( y ., x ) = .0. <-> ( A ., x ) = .0. ) )
29	28	ralbidv 2903	. . . . 5 |- ( y = A -> ( A. x e. S ( y ., x ) = .0. <-> A. x e. S ( A ., x ) = .0. ) )
30	29	elrab 3261	. . . 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 261	. . 3 |- ( S C_ V -> ( A e. ( ._|_ ` S ) <-> ( A e. V /\ A. x e. S ( A ., x ) = .0. ) ) )
32	24, 31	biadan2 642	. 2 |- ( A e. ( ._|_ ` S ) <-> ( S C_ V /\ ( A e. V /\ A. x e. S ( A ., x ) = .0. ) ) )
33		3anass 977	. 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 252	1 |- ( A e. ( ._|_ ` S ) <-> ( S C_ V /\ A e. V /\ A. x e. S ( A ., x ) = .0. ) )

Syntax hints:   -. wn 3   <-> wb 184   /\ wa 369   /\ w3a 973   = wceq 1379   e. wcel 1767   A.wral 2814   {crab 2818   _Vcvv 3113   C_ wss 3476   (/)c0 3785   ~Pcpw 4010   |-> cmpt 4505   dom cdm 4999   ` cfv 5588  (class class class)co 6285   Basecbs 14493  Scalarcsca 14561   .icip 14563   0gc0g 14698   ocvcocv 18498

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-fv 5596  df-ov 6288  df-ocv 18501

This theorem is referenced by:  ocvi  18507  ocvss  18508  ocvocv  18509  ocvlss  18510  ocv2ss  18511  unocv  18518  iunocv  18519  obselocv  18566  clsocv  21517  pjthlem2  21680