Theorem elocv 18219

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 5826	. . . . 5 |- ( A e. ( ._|_ ` S ) -> S e. dom ._|_ )
2		n0i 3751	. . . . . . . . 9 |- ( A e. ( ._|_ ` S ) -> -. ( ._|_ ` S ) = (/) )
3		ocvfval.o	. . . . . . . . . . . 12 |- ._|_ = ( ocv ` W )
4		fvprc 5794	. . . . . . . . . . . 12 |- ( -. W e. _V -> ( ocv ` W ) = (/) )
5	3, 4	syl5eq 2507	. . . . . . . . . . 11 |- ( -. W e. _V -> ._|_ = (/) )
6	5	fveq1d 5802	. . . . . . . . . 10 |- ( -. W e. _V -> ( ._|_ ` S ) = ( (/) ` S ) )
7		0fv 5833	. . . . . . . . . 10 |- ( (/) ` S ) = (/)
8	6, 7	syl6eq 2511	. . . . . . . . 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 18217	. . . . . . . 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 5151	. . . . . 6 |- ( A e. ( ._|_ ` S ) -> dom ._|_ = dom ( s e. ~P V |-> { y e. V | A. x e. s ( y ., x ) = .0. } ) )
17		fvex 5810	. . . . . . . . 9 |- ( Base ` W ) e. _V
18	10, 17	eqeltri 2538	. . . . . . . 8 |- V e. _V
19	18	rabex 4552	. . . . . . 7 |- { y e. V | A. x e. s ( y ., x ) = .0. } e. _V
20		eqid 2454	. . . . . . 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 5649	. . . . . 6 |- dom ( s e. ~P V |-> { y e. V | A. x e. s ( y ., x ) = .0. } ) = ~P V
22	16, 21	syl6eq 2511	. . . . 5 |- ( A e. ( ._|_ ` S ) -> dom ._|_ = ~P V )
23	1, 22	eleqtrd 2544	. . . 4 |- ( A e. ( ._|_ ` S ) -> S e. ~P V )
24	23	elpwid 3979	. . 3 |- ( A e. ( ._|_ ` S ) -> S C_ V )
25	10, 11, 12, 13, 3	ocvval 18218	. . . . 5 |- ( S C_ V -> ( ._|_ ` S ) = { y e. V | A. x e. S ( y ., x ) = .0. } )
26	25	eleq2d 2524	. . . 4 |- ( S C_ V -> ( A e. ( ._|_ ` S ) <-> A e. { y e. V | A. x e. S ( y ., x ) = .0. } ) )
27		oveq1 6208	. . . . . . 7 |- ( y = A -> ( y ., x ) = ( A ., x ) )
28	27	eqeq1d 2456	. . . . . 6 |- ( y = A -> ( ( y ., x ) = .0. <-> ( A ., x ) = .0. ) )
29	28	ralbidv 2846	. . . . 5 |- ( y = A -> ( A. x e. S ( y ., x ) = .0. <-> A. x e. S ( A ., x ) = .0. ) )
30	29	elrab 3224	. . . 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 969	. 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 965   = wceq 1370   e. wcel 1758   A.wral 2799   {crab 2803   _Vcvv 3078   C_ wss 3437   (/)c0 3746   ~Pcpw 3969   |-> cmpt 4459   dom cdm 4949   ` cfv 5527  (class class class)co 6201   Basecbs 14293  Scalarcsca 14361   .icip 14363   0gc0g 14498   ocvcocv 18211

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-fv 5535  df-ov 6204  df-ocv 18214

This theorem is referenced by:  ocvi  18220  ocvss  18221  ocvocv  18222  ocvlss  18223  ocv2ss  18224  unocv  18231  iunocv  18232  obselocv  18279  clsocv  20895  pjthlem2  21058