Theorem isobs 20064

Description: The predicate "is an orthonormal basis" (over a pre-Hilbert space). (Contributed by Mario Carneiro, 23-Oct-2015.)

Hypotheses

Ref	Expression
isobs.v	|- V = ( Base ` W )
isobs.h	|- ., = ( .i ` W )
isobs.f	|- F = (Scalar ` W )
isobs.u	|- .1. = ( 1r ` F )
isobs.z	|- .0. = ( 0g ` F )
isobs.o	|- ._|_ = ( ocv ` W )
isobs.y	|- Y = ( 0g ` W )

Assertion

Ref	Expression
isobs	|- ( B e. (OBasis ` W ) <-> ( W e. PreHil /\ B C_ V /\ ( A. x e. B A. y e. B ( x ., y ) = if ( x = y , .1. , .0. ) /\ ( ._|_ ` B ) = { Y } ) ) )

Distinct variable groups:   x, y, .,   x, .0. , y   x, .1. , y   x, B, y   x, W, y

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

Proof of Theorem isobs

Dummy variables h b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-obs 20049	. . . . 5 |- OBasis = ( h e. PreHil |-> { b e. ~P ( Base ` h ) | ( A. x e. b A. y e. b ( x ( .i ` h ) y ) = if ( x = y , ( 1r ` (Scalar ` h ) ) , ( 0g ` (Scalar ` h ) ) ) /\ ( ( ocv ` h ) ` b ) = { ( 0g ` h ) } ) } )
2	1	dmmptss 5631	. . . 4 |- dom OBasis C_ PreHil
3		elfvdm 6220	. . . 4 |- ( B e. (OBasis ` W ) -> W e. dom OBasis )
4	2, 3	sseldi 3601	. . 3 |- ( B e. (OBasis ` W ) -> W e. PreHil )
5		fveq2 6191	. . . . . . . . 9 |- ( h = W -> ( Base ` h ) = ( Base ` W ) )
6		isobs.v	. . . . . . . . 9 |- V = ( Base ` W )
7	5, 6	syl6eqr 2674	. . . . . . . 8 |- ( h = W -> ( Base ` h ) = V )
8	7	pweqd 4163	. . . . . . 7 |- ( h = W -> ~P ( Base ` h ) = ~P V )
9		fveq2 6191	. . . . . . . . . . . 12 |- ( h = W -> ( .i ` h ) = ( .i ` W ) )
10		isobs.h	. . . . . . . . . . . 12 |- ., = ( .i ` W )
11	9, 10	syl6eqr 2674	. . . . . . . . . . 11 |- ( h = W -> ( .i ` h ) = ., )
12	11	oveqd 6667	. . . . . . . . . 10 |- ( h = W -> ( x ( .i ` h ) y ) = ( x ., y ) )
13		fveq2 6191	. . . . . . . . . . . . . 14 |- ( h = W -> (Scalar ` h ) = (Scalar ` W ) )
14		isobs.f	. . . . . . . . . . . . . 14 |- F = (Scalar ` W )
15	13, 14	syl6eqr 2674	. . . . . . . . . . . . 13 |- ( h = W -> (Scalar ` h ) = F )
16	15	fveq2d 6195	. . . . . . . . . . . 12 |- ( h = W -> ( 1r ` (Scalar ` h ) ) = ( 1r ` F ) )
17		isobs.u	. . . . . . . . . . . 12 |- .1. = ( 1r ` F )
18	16, 17	syl6eqr 2674	. . . . . . . . . . 11 |- ( h = W -> ( 1r ` (Scalar ` h ) ) = .1. )
19	15	fveq2d 6195	. . . . . . . . . . . 12 |- ( h = W -> ( 0g ` (Scalar ` h ) ) = ( 0g ` F ) )
20		isobs.z	. . . . . . . . . . . 12 |- .0. = ( 0g ` F )
21	19, 20	syl6eqr 2674	. . . . . . . . . . 11 |- ( h = W -> ( 0g ` (Scalar ` h ) ) = .0. )
22	18, 21	ifeq12d 4106	. . . . . . . . . 10 |- ( h = W -> if ( x = y , ( 1r ` (Scalar ` h ) ) , ( 0g ` (Scalar ` h ) ) ) = if ( x = y , .1. , .0. ) )
23	12, 22	eqeq12d 2637	. . . . . . . . 9 |- ( h = W -> ( ( x ( .i ` h ) y ) = if ( x = y , ( 1r ` (Scalar ` h ) ) , ( 0g ` (Scalar ` h ) ) ) <-> ( x ., y ) = if ( x = y , .1. , .0. ) ) )
24	23	2ralbidv 2989	. . . . . . . 8 |- ( h = W -> ( A. x e. b A. y e. b ( x ( .i ` h ) y ) = if ( x = y , ( 1r ` (Scalar ` h ) ) , ( 0g ` (Scalar ` h ) ) ) <-> A. x e. b A. y e. b ( x ., y ) = if ( x = y , .1. , .0. ) ) )
25		fveq2 6191	. . . . . . . . . . 11 |- ( h = W -> ( ocv ` h ) = ( ocv ` W ) )
26		isobs.o	. . . . . . . . . . 11 |- ._|_ = ( ocv ` W )
27	25, 26	syl6eqr 2674	. . . . . . . . . 10 |- ( h = W -> ( ocv ` h ) = ._|_ )
28	27	fveq1d 6193	. . . . . . . . 9 |- ( h = W -> ( ( ocv ` h ) ` b ) = ( ._|_ ` b ) )
29		fveq2 6191	. . . . . . . . . . 11 |- ( h = W -> ( 0g ` h ) = ( 0g ` W ) )
30		isobs.y	. . . . . . . . . . 11 |- Y = ( 0g ` W )
31	29, 30	syl6eqr 2674	. . . . . . . . . 10 |- ( h = W -> ( 0g ` h ) = Y )
32	31	sneqd 4189	. . . . . . . . 9 |- ( h = W -> { ( 0g ` h ) } = { Y } )
33	28, 32	eqeq12d 2637	. . . . . . . 8 |- ( h = W -> ( ( ( ocv ` h ) ` b ) = { ( 0g ` h ) } <-> ( ._|_ ` b ) = { Y } ) )
34	24, 33	anbi12d 747	. . . . . . 7 |- ( h = W -> ( ( A. x e. b A. y e. b ( x ( .i ` h ) y ) = if ( x = y , ( 1r ` (Scalar ` h ) ) , ( 0g ` (Scalar ` h ) ) ) /\ ( ( ocv ` h ) ` b ) = { ( 0g ` h ) } ) <-> ( A. x e. b A. y e. b ( x ., y ) = if ( x = y , .1. , .0. ) /\ ( ._|_ ` b ) = { Y } ) ) )
35	8, 34	rabeqbidv 3195	. . . . . 6 |- ( h = W -> { b e. ~P ( Base ` h ) | ( A. x e. b A. y e. b ( x ( .i ` h ) y ) = if ( x = y , ( 1r ` (Scalar ` h ) ) , ( 0g ` (Scalar ` h ) ) ) /\ ( ( ocv ` h ) ` b ) = { ( 0g ` h ) } ) } = { b e. ~P V | ( A. x e. b A. y e. b ( x ., y ) = if ( x = y , .1. , .0. ) /\ ( ._|_ ` b ) = { Y } ) } )
36		fvex 6201	. . . . . . . . 9 |- ( Base ` W ) e. _V
37	6, 36	eqeltri 2697	. . . . . . . 8 |- V e. _V
38	37	pwex 4848	. . . . . . 7 |- ~P V e. _V
39	38	rabex 4813	. . . . . 6 |- { b e. ~P V | ( A. x e. b A. y e. b ( x ., y ) = if ( x = y , .1. , .0. ) /\ ( ._|_ ` b ) = { Y } ) } e. _V
40	35, 1, 39	fvmpt 6282	. . . . 5 |- ( W e. PreHil -> (OBasis ` W ) = { b e. ~P V | ( A. x e. b A. y e. b ( x ., y ) = if ( x = y , .1. , .0. ) /\ ( ._|_ ` b ) = { Y } ) } )
41	40	eleq2d 2687	. . . 4 |- ( W e. PreHil -> ( B e. (OBasis ` W ) <-> B e. { b e. ~P V | ( A. x e. b A. y e. b ( x ., y ) = if ( x = y , .1. , .0. ) /\ ( ._|_ ` b ) = { Y } ) } ) )
42		raleq 3138	. . . . . . . 8 |- ( b = B -> ( A. y e. b ( x ., y ) = if ( x = y , .1. , .0. ) <-> A. y e. B ( x ., y ) = if ( x = y , .1. , .0. ) ) )
43	42	raleqbi1dv 3146	. . . . . . 7 |- ( b = B -> ( A. x e. b A. y e. b ( x ., y ) = if ( x = y , .1. , .0. ) <-> A. x e. B A. y e. B ( x ., y ) = if ( x = y , .1. , .0. ) ) )
44		fveq2 6191	. . . . . . . 8 |- ( b = B -> ( ._|_ ` b ) = ( ._|_ ` B ) )
45	44	eqeq1d 2624	. . . . . . 7 |- ( b = B -> ( ( ._|_ ` b ) = { Y } <-> ( ._|_ ` B ) = { Y } ) )
46	43, 45	anbi12d 747	. . . . . 6 |- ( b = B -> ( ( A. x e. b A. y e. b ( x ., y ) = if ( x = y , .1. , .0. ) /\ ( ._|_ ` b ) = { Y } ) <-> ( A. x e. B A. y e. B ( x ., y ) = if ( x = y , .1. , .0. ) /\ ( ._|_ ` B ) = { Y } ) ) )
47	46	elrab 3363	. . . . 5 |- ( B e. { b e. ~P V | ( A. x e. b A. y e. b ( x ., y ) = if ( x = y , .1. , .0. ) /\ ( ._|_ ` b ) = { Y } ) } <-> ( B e. ~P V /\ ( A. x e. B A. y e. B ( x ., y ) = if ( x = y , .1. , .0. ) /\ ( ._|_ ` B ) = { Y } ) ) )
48	37	elpw2 4828	. . . . . 6 |- ( B e. ~P V <-> B C_ V )
49	48	anbi1i 731	. . . . 5 |- ( ( B e. ~P V /\ ( A. x e. B A. y e. B ( x ., y ) = if ( x = y , .1. , .0. ) /\ ( ._|_ ` B ) = { Y } ) ) <-> ( B C_ V /\ ( A. x e. B A. y e. B ( x ., y ) = if ( x = y , .1. , .0. ) /\ ( ._|_ ` B ) = { Y } ) ) )
50	47, 49	bitri 264	. . . 4 |- ( B e. { b e. ~P V | ( A. x e. b A. y e. b ( x ., y ) = if ( x = y , .1. , .0. ) /\ ( ._|_ ` b ) = { Y } ) } <-> ( B C_ V /\ ( A. x e. B A. y e. B ( x ., y ) = if ( x = y , .1. , .0. ) /\ ( ._|_ ` B ) = { Y } ) ) )
51	41, 50	syl6bb 276	. . 3 |- ( W e. PreHil -> ( B e. (OBasis ` W ) <-> ( B C_ V /\ ( A. x e. B A. y e. B ( x ., y ) = if ( x = y , .1. , .0. ) /\ ( ._|_ ` B ) = { Y } ) ) ) )
52	4, 51	biadan2 674	. 2 |- ( B e. (OBasis ` W ) <-> ( W e. PreHil /\ ( B C_ V /\ ( A. x e. B A. y e. B ( x ., y ) = if ( x = y , .1. , .0. ) /\ ( ._|_ ` B ) = { Y } ) ) ) )
53		3anass 1042	. 2 |- ( ( W e. PreHil /\ B C_ V /\ ( A. x e. B A. y e. B ( x ., y ) = if ( x = y , .1. , .0. ) /\ ( ._|_ ` B ) = { Y } ) ) <-> ( W e. PreHil /\ ( B C_ V /\ ( A. x e. B A. y e. B ( x ., y ) = if ( x = y , .1. , .0. ) /\ ( ._|_ ` B ) = { Y } ) ) ) )
54	52, 53	bitr4i 267	1 |- ( B e. (OBasis ` W ) <-> ( W e. PreHil /\ B C_ V /\ ( A. x e. B A. y e. B ( x ., y ) = if ( x = y , .1. , .0. ) /\ ( ._|_ ` B ) = { Y } ) ) )

Syntax hints:   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   _Vcvv 3200   C_ wss 3574   ifcif 4086   ~Pcpw 4158   {csn 4177   dom cdm 5114   ` cfv 5888  (class class class)co 6650   Basecbs 15857  Scalarcsca 15944   .icip 15946   0gc0g 16100   1rcur 18501   PreHilcphl 19969   ocvcocv 20004  OBasiscobs 20046

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

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-fv 5896  df-ov 6653  df-obs 20049

This theorem is referenced by:  obsip  20065  obsrcl  20067  obsss  20068  obsocv  20070