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Theorem isobs 16978
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 16963 . . . . 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 5395 . . . 4  |-  dom OBasis  C_  PreHil
3 elfvdm 5786 . . . 4  |-  ( B  e.  (OBasis `  W
)  ->  W  e.  dom OBasis )
42, 3sseldi 3332 . . 3  |-  ( B  e.  (OBasis `  W
)  ->  W  e.  PreHil )
5 fveq2 5757 . . . . . . . . 9  |-  ( h  =  W  ->  ( Base `  h )  =  ( Base `  W
) )
6 isobs.v . . . . . . . . 9  |-  V  =  ( Base `  W
)
75, 6syl6eqr 2492 . . . . . . . 8  |-  ( h  =  W  ->  ( Base `  h )  =  V )
87pweqd 3828 . . . . . . 7  |-  ( h  =  W  ->  ~P ( Base `  h )  =  ~P V )
9 fveq2 5757 . . . . . . . . . . . 12  |-  ( h  =  W  ->  ( .i `  h )  =  ( .i `  W
) )
10 isobs.h . . . . . . . . . . . 12  |-  .,  =  ( .i `  W )
119, 10syl6eqr 2492 . . . . . . . . . . 11  |-  ( h  =  W  ->  ( .i `  h )  = 
.,  )
1211oveqd 6127 . . . . . . . . . 10  |-  ( h  =  W  ->  (
x ( .i `  h ) y )  =  ( x  .,  y ) )
13 fveq2 5757 . . . . . . . . . . . . . 14  |-  ( h  =  W  ->  (Scalar `  h )  =  (Scalar `  W ) )
14 isobs.f . . . . . . . . . . . . . 14  |-  F  =  (Scalar `  W )
1513, 14syl6eqr 2492 . . . . . . . . . . . . 13  |-  ( h  =  W  ->  (Scalar `  h )  =  F )
1615fveq2d 5761 . . . . . . . . . . . 12  |-  ( h  =  W  ->  ( 1r `  (Scalar `  h
) )  =  ( 1r `  F ) )
17 isobs.u . . . . . . . . . . . 12  |-  .1.  =  ( 1r `  F )
1816, 17syl6eqr 2492 . . . . . . . . . . 11  |-  ( h  =  W  ->  ( 1r `  (Scalar `  h
) )  =  .1.  )
1915fveq2d 5761 . . . . . . . . . . . 12  |-  ( h  =  W  ->  ( 0g `  (Scalar `  h
) )  =  ( 0g `  F ) )
20 isobs.z . . . . . . . . . . . 12  |-  .0.  =  ( 0g `  F )
2119, 20syl6eqr 2492 . . . . . . . . . . 11  |-  ( h  =  W  ->  ( 0g `  (Scalar `  h
) )  =  .0.  )
2218, 21ifeq12d 3779 . . . . . . . . . 10  |-  ( h  =  W  ->  if ( x  =  y ,  ( 1r `  (Scalar `  h ) ) ,  ( 0g `  (Scalar `  h ) ) )  =  if ( x  =  y ,  .1.  ,  .0.  )
)
2312, 22eqeq12d 2456 . . . . . . . . 9  |-  ( h  =  W  ->  (
( x ( .i
`  h ) y )  =  if ( x  =  y ,  ( 1r `  (Scalar `  h ) ) ,  ( 0g `  (Scalar `  h ) ) )  <-> 
( x  .,  y
)  =  if ( x  =  y ,  .1.  ,  .0.  )
) )
24232ralbidv 2753 . . . . . . . 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 5757 . . . . . . . . . . 11  |-  ( h  =  W  ->  ( ocv `  h )  =  ( ocv `  W
) )
26 isobs.o . . . . . . . . . . 11  |-  ._|_  =  ( ocv `  W )
2725, 26syl6eqr 2492 . . . . . . . . . 10  |-  ( h  =  W  ->  ( ocv `  h )  = 
._|_  )
2827fveq1d 5759 . . . . . . . . 9  |-  ( h  =  W  ->  (
( ocv `  h
) `  b )  =  (  ._|_  `  b
) )
29 fveq2 5757 . . . . . . . . . . 11  |-  ( h  =  W  ->  ( 0g `  h )  =  ( 0g `  W
) )
30 isobs.y . . . . . . . . . . 11  |-  Y  =  ( 0g `  W
)
3129, 30syl6eqr 2492 . . . . . . . . . 10  |-  ( h  =  W  ->  ( 0g `  h )  =  Y )
3231sneqd 3851 . . . . . . . . 9  |-  ( h  =  W  ->  { ( 0g `  h ) }  =  { Y } )
3328, 32eqeq12d 2456 . . . . . . . 8  |-  ( h  =  W  ->  (
( ( ocv `  h
) `  b )  =  { ( 0g `  h ) }  <->  (  ._|_  `  b )  =  { Y } ) )
3424, 33anbi12d 693 . . . . . . 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 2957 . . . . . 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 5771 . . . . . . . . 9  |-  ( Base `  W )  e.  _V
376, 36eqeltri 2512 . . . . . . . 8  |-  V  e. 
_V
3837pwex 4411 . . . . . . 7  |-  ~P V  e.  _V
3938rabex 4383 . . . . . 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 5835 . . . . 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 2509 . . . 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 2910 . . . . . . . 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 2918 . . . . . . 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 5757 . . . . . . . 8  |-  ( b  =  B  ->  (  ._|_  `  b )  =  (  ._|_  `  B ) )
4544eqeq1d 2450 . . . . . . 7  |-  ( b  =  B  ->  (
(  ._|_  `  b )  =  { Y }  <->  (  ._|_  `  B )  =  { Y } ) )
4643, 45anbi12d 693 . . . . . 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 3098 . . . . 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 4393 . . . . . 6  |-  ( B  e.  ~P V  <->  B  C_  V
)
4948anbi1i 678 . . . . 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 242 . . . 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 254 . . 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 625 . 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 941 . 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 245 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 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1727   A.wral 2711   {crab 2715   _Vcvv 2962    C_ wss 3306   ifcif 3763   ~Pcpw 3823   {csn 3838   dom cdm 4907   ` cfv 5483  (class class class)co 6110   Basecbs 13500  Scalarcsca 13563   .icip 13565   0gc0g 13754   1rcur 15693   PreHilcphl 16886   ocvcocv 16918  OBasiscobs 16960
This theorem is referenced by:  obsip  16979  obsrcl  16981  obsss  16982  obsocv  16984
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-sbc 3168  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fv 5491  df-ov 6113  df-obs 16963
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