Theorem ocvval 20011

Description: Value of the orthocomplement of a subset (normally a subspace) of a pre-Hilbert space. (Contributed by NM, 7-Oct-2011.) (Revised 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
ocvval	|- ( S C_ V -> ( ._|_ ` S ) = { x e. V | A. y e. S ( x ., y ) = .0. } )

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

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

Proof of Theorem ocvval

Dummy variable s is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ocvfval.v	. . . 4 |- V = ( Base ` W )
2		fvex 6201	. . . 4 |- ( Base ` W ) e. _V
3	1, 2	eqeltri 2697	. . 3 |- V e. _V
4	3	elpw2 4828	. 2 |- ( S e. ~P V <-> S C_ V )
5		ocvfval.i	. . . . . 6 |- ., = ( .i ` W )
6		ocvfval.f	. . . . . 6 |- F = (Scalar ` W )
7		ocvfval.z	. . . . . 6 |- .0. = ( 0g ` F )
8		ocvfval.o	. . . . . 6 |- ._|_ = ( ocv ` W )
9	1, 5, 6, 7, 8	ocvfval 20010	. . . . 5 |- ( W e. _V -> ._|_ = ( s e. ~P V |-> { x e. V | A. y e. s ( x ., y ) = .0. } ) )
10	9	fveq1d 6193	. . . 4 |- ( W e. _V -> ( ._|_ ` S ) = ( ( s e. ~P V |-> { x e. V | A. y e. s ( x ., y ) = .0. } ) ` S ) )
11		raleq 3138	. . . . . 6 |- ( s = S -> ( A. y e. s ( x ., y ) = .0. <-> A. y e. S ( x ., y ) = .0. ) )
12	11	rabbidv 3189	. . . . 5 |- ( s = S -> { x e. V | A. y e. s ( x ., y ) = .0. } = { x e. V | A. y e. S ( x ., y ) = .0. } )
13		eqid 2622	. . . . 5 |- ( s e. ~P V |-> { x e. V | A. y e. s ( x ., y ) = .0. } ) = ( s e. ~P V |-> { x e. V | A. y e. s ( x ., y ) = .0. } )
14	3	rabex 4813	. . . . 5 |- { x e. V | A. y e. S ( x ., y ) = .0. } e. _V
15	12, 13, 14	fvmpt 6282	. . . 4 |- ( S e. ~P V -> ( ( s e. ~P V |-> { x e. V | A. y e. s ( x ., y ) = .0. } ) ` S ) = { x e. V | A. y e. S ( x ., y ) = .0. } )
16	10, 15	sylan9eq 2676	. . 3 |- ( ( W e. _V /\ S e. ~P V ) -> ( ._|_ ` S ) = { x e. V | A. y e. S ( x ., y ) = .0. } )
17		0fv 6227	. . . . 5 |- ( (/) ` S ) = (/)
18		fvprc 6185	. . . . . . 7 |- ( -. W e. _V -> ( ocv ` W ) = (/) )
19	8, 18	syl5eq 2668	. . . . . 6 |- ( -. W e. _V -> ._|_ = (/) )
20	19	fveq1d 6193	. . . . 5 |- ( -. W e. _V -> ( ._|_ ` S ) = ( (/) ` S ) )
21		ssrab2 3687	. . . . . 6 |- { x e. V | A. y e. S ( x ., y ) = .0. } C_ V
22		fvprc 6185	. . . . . . 7 |- ( -. W e. _V -> ( Base ` W ) = (/) )
23	1, 22	syl5eq 2668	. . . . . 6 |- ( -. W e. _V -> V = (/) )
24		sseq0 3975	. . . . . 6 |- ( ( { x e. V | A. y e. S ( x ., y ) = .0. } C_ V /\ V = (/) ) -> { x e. V | A. y e. S ( x ., y ) = .0. } = (/) )
25	21, 23, 24	sylancr 695	. . . . 5 |- ( -. W e. _V -> { x e. V | A. y e. S ( x ., y ) = .0. } = (/) )
26	17, 20, 25	3eqtr4a 2682	. . . 4 |- ( -. W e. _V -> ( ._|_ ` S ) = { x e. V | A. y e. S ( x ., y ) = .0. } )
27	26	adantr 481	. . 3 |- ( ( -. W e. _V /\ S e. ~P V ) -> ( ._|_ ` S ) = { x e. V | A. y e. S ( x ., y ) = .0. } )
28	16, 27	pm2.61ian 831	. 2 |- ( S e. ~P V -> ( ._|_ ` S ) = { x e. V | A. y e. S ( x ., y ) = .0. } )
29	4, 28	sylbir 225	1 |- ( S C_ V -> ( ._|_ ` S ) = { x e. V | A. y e. S ( x ., y ) = .0. } )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   _Vcvv 3200   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   |-> cmpt 4729   ` 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:  elocv  20012  ocv0  20021  csscld  23048  hlhilocv  37249