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Theorem isobs 16910
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.
StepHypRef Expression
1 df-obs 16895 . . . . 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 ) } ) } )
21dmmptss 5333 . . . 4  |-  dom OBasis  C_  PreHil
3 elfvdm 5724 . . . 4  |-  ( B  e.  (OBasis `  W
)  ->  W  e.  dom OBasis )
42, 3sseldi 3314 . . 3  |-  ( B  e.  (OBasis `  W
)  ->  W  e.  PreHil )
5 fveq2 5695 . . . . . . . . 9  |-  ( h  =  W  ->  ( Base `  h )  =  ( Base `  W
) )
6 isobs.v . . . . . . . . 9  |-  V  =  ( Base `  W
)
75, 6syl6eqr 2462 . . . . . . . 8  |-  ( h  =  W  ->  ( Base `  h )  =  V )
87pweqd 3772 . . . . . . 7  |-  ( h  =  W  ->  ~P ( Base `  h )  =  ~P V )
9 fveq2 5695 . . . . . . . . . . . 12  |-  ( h  =  W  ->  ( .i `  h )  =  ( .i `  W
) )
10 isobs.h . . . . . . . . . . . 12  |-  .,  =  ( .i `  W )
119, 10syl6eqr 2462 . . . . . . . . . . 11  |-  ( h  =  W  ->  ( .i `  h )  = 
.,  )
1211oveqd 6065 . . . . . . . . . 10  |-  ( h  =  W  ->  (
x ( .i `  h ) y )  =  ( x  .,  y ) )
13 fveq2 5695 . . . . . . . . . . . . . 14  |-  ( h  =  W  ->  (Scalar `  h )  =  (Scalar `  W ) )
14 isobs.f . . . . . . . . . . . . . 14  |-  F  =  (Scalar `  W )
1513, 14syl6eqr 2462 . . . . . . . . . . . . 13  |-  ( h  =  W  ->  (Scalar `  h )  =  F )
1615fveq2d 5699 . . . . . . . . . . . 12  |-  ( h  =  W  ->  ( 1r `  (Scalar `  h
) )  =  ( 1r `  F ) )
17 isobs.u . . . . . . . . . . . 12  |-  .1.  =  ( 1r `  F )
1816, 17syl6eqr 2462 . . . . . . . . . . 11  |-  ( h  =  W  ->  ( 1r `  (Scalar `  h
) )  =  .1.  )
1915fveq2d 5699 . . . . . . . . . . . 12  |-  ( h  =  W  ->  ( 0g `  (Scalar `  h
) )  =  ( 0g `  F ) )
20 isobs.z . . . . . . . . . . . 12  |-  .0.  =  ( 0g `  F )
2119, 20syl6eqr 2462 . . . . . . . . . . 11  |-  ( h  =  W  ->  ( 0g `  (Scalar `  h
) )  =  .0.  )
2218, 21ifeq12d 3723 . . . . . . . . . 10  |-  ( h  =  W  ->  if ( x  =  y ,  ( 1r `  (Scalar `  h ) ) ,  ( 0g `  (Scalar `  h ) ) )  =  if ( x  =  y ,  .1.  ,  .0.  )
)
2312, 22eqeq12d 2426 . . . . . . . . 9  |-  ( h  =  W  ->  (
( x ( .i
`  h ) y )  =  if ( x  =  y ,  ( 1r `  (Scalar `  h ) ) ,  ( 0g `  (Scalar `  h ) ) )  <-> 
( x  .,  y
)  =  if ( x  =  y ,  .1.  ,  .0.  )
) )
24232ralbidv 2716 . . . . . . . 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 5695 . . . . . . . . . . 11  |-  ( h  =  W  ->  ( ocv `  h )  =  ( ocv `  W
) )
26 isobs.o . . . . . . . . . . 11  |-  ._|_  =  ( ocv `  W )
2725, 26syl6eqr 2462 . . . . . . . . . 10  |-  ( h  =  W  ->  ( ocv `  h )  = 
._|_  )
2827fveq1d 5697 . . . . . . . . 9  |-  ( h  =  W  ->  (
( ocv `  h
) `  b )  =  (  ._|_  `  b
) )
29 fveq2 5695 . . . . . . . . . . 11  |-  ( h  =  W  ->  ( 0g `  h )  =  ( 0g `  W
) )
30 isobs.y . . . . . . . . . . 11  |-  Y  =  ( 0g `  W
)
3129, 30syl6eqr 2462 . . . . . . . . . 10  |-  ( h  =  W  ->  ( 0g `  h )  =  Y )
3231sneqd 3795 . . . . . . . . 9  |-  ( h  =  W  ->  { ( 0g `  h ) }  =  { Y } )
3328, 32eqeq12d 2426 . . . . . . . 8  |-  ( h  =  W  ->  (
( ( ocv `  h
) `  b )  =  { ( 0g `  h ) }  <->  (  ._|_  `  b )  =  { Y } ) )
3424, 33anbi12d 692 . . . . . . 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 } ) ) )
358, 34rabeqbidv 2919 . . . . . 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 5709 . . . . . . . . 9  |-  ( Base `  W )  e.  _V
376, 36eqeltri 2482 . . . . . . . 8  |-  V  e. 
_V
3837pwex 4350 . . . . . . 7  |-  ~P V  e.  _V
3938rabex 4322 . . . . . 6  |-  { b  e.  ~P V  | 
( A. x  e.  b  A. y  e.  b  ( x  .,  y )  =  if ( x  =  y ,  .1.  ,  .0.  )  /\  (  ._|_  `  b
)  =  { Y } ) }  e.  _V
4035, 1, 39fvmpt 5773 . . . . 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 } ) } )
4140eleq2d 2479 . . . 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 2872 . . . . . . . 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.  ) ) )
4342raleqbi1dv 2880 . . . . . . 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 5695 . . . . . . . 8  |-  ( b  =  B  ->  (  ._|_  `  b )  =  (  ._|_  `  B ) )
4544eqeq1d 2420 . . . . . . 7  |-  ( b  =  B  ->  (
(  ._|_  `  b )  =  { Y }  <->  (  ._|_  `  B )  =  { Y } ) )
4643, 45anbi12d 692 . . . . . 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 } ) ) )
4746elrab 3060 . . . . 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 } ) ) )
4837elpw2 4332 . . . . . 6  |-  ( B  e.  ~P V  <->  B  C_  V
)
4948anbi1i 677 . . . . 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 } ) ) )
5047, 49bitri 241 . . . 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 } ) ) )
5141, 50syl6bb 253 . . 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 } ) ) ) )
524, 51biadan2 624 . 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 940 . 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 } ) ) ) )
5452, 53bitr4i 244 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 } ) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2674   {crab 2678   _Vcvv 2924    C_ wss 3288   ifcif 3707   ~Pcpw 3767   {csn 3782   dom cdm 4845   ` cfv 5421  (class class class)co 6048   Basecbs 13432  Scalarcsca 13495   .icip 13497   0gc0g 13686   1rcur 15625   PreHilcphl 16818   ocvcocv 16850  OBasiscobs 16892
This theorem is referenced by:  obsip  16911  obsrcl  16913  obsss  16914  obsocv  16916
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  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 5385  df-fun 5423  df-fv 5429  df-ov 6051  df-obs 16895
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