Theorem elocv 19168

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 5898	. . . . 5 |- ( A e. ( ._|_ ` S ) -> S e. dom ._|_ )
2		n0i 3763	. . . . . . . . 9 |- ( A e. ( ._|_ ` S ) -> -. ( ._|_ ` S ) = (/) )
3		ocvfval.o	. . . . . . . . . . . 12 |- ._|_ = ( ocv ` W )
4		fvprc 5866	. . . . . . . . . . . 12 |- ( -. W e. _V -> ( ocv ` W ) = (/) )
5	3, 4	syl5eq 2473	. . . . . . . . . . 11 |- ( -. W e. _V -> ._|_ = (/) )
6	5	fveq1d 5874	. . . . . . . . . 10 |- ( -. W e. _V -> ( ._|_ ` S ) = ( (/) ` S ) )
7		0fv 5905	. . . . . . . . . 10 |- ( (/) ` S ) = (/)
8	6, 7	syl6eq 2477	. . . . . . . . 9 |- ( -. W e. _V -> ( ._|_ ` S ) = (/) )
9	2, 8	nsyl2 130	. . . . . . . 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 19166	. . . . . . . 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 5048	. . . . . 6 |- ( A e. ( ._|_ ` S ) -> dom ._|_ = dom ( s e. ~P V |-> { y e. V | A. x e. s ( y ., x ) = .0. } ) )
17		fvex 5882	. . . . . . . . 9 |- ( Base ` W ) e. _V
18	10, 17	eqeltri 2504	. . . . . . . 8 |- V e. _V
19	18	rabex 4567	. . . . . . 7 |- { y e. V | A. x e. s ( y ., x ) = .0. } e. _V
20		eqid 2420	. . . . . . 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 5716	. . . . . 6 |- dom ( s e. ~P V |-> { y e. V | A. x e. s ( y ., x ) = .0. } ) = ~P V
22	16, 21	syl6eq 2477	. . . . 5 |- ( A e. ( ._|_ ` S ) -> dom ._|_ = ~P V )
23	1, 22	eleqtrd 2510	. . . 4 |- ( A e. ( ._|_ ` S ) -> S e. ~P V )
24	23	elpwid 3986	. . 3 |- ( A e. ( ._|_ ` S ) -> S C_ V )
25	10, 11, 12, 13, 3	ocvval 19167	. . . . 5 |- ( S C_ V -> ( ._|_ ` S ) = { y e. V | A. x e. S ( y ., x ) = .0. } )
26	25	eleq2d 2490	. . . 4 |- ( S C_ V -> ( A e. ( ._|_ ` S ) <-> A e. { y e. V | A. x e. S ( y ., x ) = .0. } ) )
27		oveq1 6303	. . . . . . 7 |- ( y = A -> ( y ., x ) = ( A ., x ) )
28	27	eqeq1d 2422	. . . . . 6 |- ( y = A -> ( ( y ., x ) = .0. <-> ( A ., x ) = .0. ) )
29	28	ralbidv 2862	. . . . 5 |- ( y = A -> ( A. x e. S ( y ., x ) = .0. <-> A. x e. S ( A ., x ) = .0. ) )
30	29	elrab 3226	. . . 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 264	. . 3 |- ( S C_ V -> ( A e. ( ._|_ ` S ) <-> ( A e. V /\ A. x e. S ( A ., x ) = .0. ) ) )
32	24, 31	biadan2 646	. 2 |- ( A e. ( ._|_ ` S ) <-> ( S C_ V /\ ( A e. V /\ A. x e. S ( A ., x ) = .0. ) ) )
33		3anass 986	. 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 255	1 |- ( A e. ( ._|_ ` S ) <-> ( S C_ V /\ A e. V /\ A. x e. S ( A ., x ) = .0. ) )

Syntax hints:   -. wn 3   <-> wb 187   /\ wa 370   /\ w3a 982   = wceq 1437   e. wcel 1867   A.wral 2773   {crab 2777   _Vcvv 3078   C_ wss 3433   (/)c0 3758   ~Pcpw 3976   |-> cmpt 4475   dom cdm 4845   ` cfv 5592  (class class class)co 6296   Basecbs 15081  Scalarcsca 15153   .icip 15155   0gc0g 15298   ocvcocv 19160

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-sbc 3297  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-fv 5600  df-ov 6299  df-ocv 19163

This theorem is referenced by:  ocvi  19169  ocvss  19170  ocvocv  19171  ocvlss  19172  ocv2ss  19173  unocv  19180  iunocv  19181  obselocv  19228  clsocv  22140  pjthlem2  22299