Theorem elocv 19224

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 5889	. . . . 5 |- ( A e. ( ._|_ ` S ) -> S e. dom ._|_ )
2		n0i 3735	. . . . . . . . 9 |- ( A e. ( ._|_ ` S ) -> -. ( ._|_ ` S ) = (/) )
3		ocvfval.o	. . . . . . . . . . . 12 |- ._|_ = ( ocv ` W )
4		fvprc 5857	. . . . . . . . . . . 12 |- ( -. W e. _V -> ( ocv ` W ) = (/) )
5	3, 4	syl5eq 2496	. . . . . . . . . . 11 |- ( -. W e. _V -> ._|_ = (/) )
6	5	fveq1d 5865	. . . . . . . . . 10 |- ( -. W e. _V -> ( ._|_ ` S ) = ( (/) ` S ) )
7		0fv 5896	. . . . . . . . . 10 |- ( (/) ` S ) = (/)
8	6, 7	syl6eq 2500	. . . . . . . . 9 |- ( -. W e. _V -> ( ._|_ ` S ) = (/) )
9	2, 8	nsyl2 131	. . . . . . . 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 19222	. . . . . . . 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 5036	. . . . . 6 |- ( A e. ( ._|_ ` S ) -> dom ._|_ = dom ( s e. ~P V |-> { y e. V | A. x e. s ( y ., x ) = .0. } ) )
17		fvex 5873	. . . . . . . . 9 |- ( Base ` W ) e. _V
18	10, 17	eqeltri 2524	. . . . . . . 8 |- V e. _V
19	18	rabex 4553	. . . . . . 7 |- { y e. V | A. x e. s ( y ., x ) = .0. } e. _V
20		eqid 2450	. . . . . . 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 5705	. . . . . 6 |- dom ( s e. ~P V |-> { y e. V | A. x e. s ( y ., x ) = .0. } ) = ~P V
22	16, 21	syl6eq 2500	. . . . 5 |- ( A e. ( ._|_ ` S ) -> dom ._|_ = ~P V )
23	1, 22	eleqtrd 2530	. . . 4 |- ( A e. ( ._|_ ` S ) -> S e. ~P V )
24	23	elpwid 3960	. . 3 |- ( A e. ( ._|_ ` S ) -> S C_ V )
25	10, 11, 12, 13, 3	ocvval 19223	. . . . 5 |- ( S C_ V -> ( ._|_ ` S ) = { y e. V | A. x e. S ( y ., x ) = .0. } )
26	25	eleq2d 2513	. . . 4 |- ( S C_ V -> ( A e. ( ._|_ ` S ) <-> A e. { y e. V | A. x e. S ( y ., x ) = .0. } ) )
27		oveq1 6295	. . . . . . 7 |- ( y = A -> ( y ., x ) = ( A ., x ) )
28	27	eqeq1d 2452	. . . . . 6 |- ( y = A -> ( ( y ., x ) = .0. <-> ( A ., x ) = .0. ) )
29	28	ralbidv 2826	. . . . 5 |- ( y = A -> ( A. x e. S ( y ., x ) = .0. <-> A. x e. S ( A ., x ) = .0. ) )
30	29	elrab 3195	. . . 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 265	. . 3 |- ( S C_ V -> ( A e. ( ._|_ ` S ) <-> ( A e. V /\ A. x e. S ( A ., x ) = .0. ) ) )
32	24, 31	biadan2 647	. 2 |- ( A e. ( ._|_ ` S ) <-> ( S C_ V /\ ( A e. V /\ A. x e. S ( A ., x ) = .0. ) ) )
33		3anass 988	. 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 256	1 |- ( A e. ( ._|_ ` S ) <-> ( S C_ V /\ A e. V /\ A. x e. S ( A ., x ) = .0. ) )

Syntax hints:   -. wn 3   <-> wb 188   /\ wa 371   /\ w3a 984   = wceq 1443   e. wcel 1886   A.wral 2736   {crab 2740   _Vcvv 3044   C_ wss 3403   (/)c0 3730   ~Pcpw 3950   |-> cmpt 4460   dom cdm 4833   ` cfv 5581  (class class class)co 6288   Basecbs 15114  Scalarcsca 15186   .icip 15188   0gc0g 15331   ocvcocv 19216

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-rab 2745  df-v 3046  df-sbc 3267  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-op 3974  df-uni 4198  df-br 4402  df-opab 4461  df-mpt 4462  df-id 4748  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-fv 5589  df-ov 6291  df-ocv 19219

This theorem is referenced by:  ocvi  19225  ocvss  19226  ocvocv  19227  ocvlss  19228  ocv2ss  19229  unocv  19236  iunocv  19237  obselocv  19284  clsocv  22214  pjthlem2  22385