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Theorem obsip 16621
Description: The inner product of two elements of an orthonormal basis. (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 )
Assertion
Ref Expression
obsip  |-  ( ( B  e.  (OBasis `  W )  /\  P  e.  B  /\  Q  e.  B )  ->  ( P  .,  Q )  =  if ( P  =  Q ,  .1.  ,  .0.  ) )

Proof of Theorem obsip
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isobs.v . . . . . 6  |-  V  =  ( Base `  W
)
2 isobs.h . . . . . 6  |-  .,  =  ( .i `  W )
3 isobs.f . . . . . 6  |-  F  =  (Scalar `  W )
4 isobs.u . . . . . 6  |-  .1.  =  ( 1r `  F )
5 isobs.z . . . . . 6  |-  .0.  =  ( 0g `  F )
6 eqid 2283 . . . . . 6  |-  ( ocv `  W )  =  ( ocv `  W )
7 eqid 2283 . . . . . 6  |-  ( 0g
`  W )  =  ( 0g `  W
)
81, 2, 3, 4, 5, 6, 7isobs 16620 . . . . 5  |-  ( 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.  )  /\  ( ( ocv `  W ) `  B
)  =  { ( 0g `  W ) } ) ) )
98simp3bi 972 . . . 4  |-  ( B  e.  (OBasis `  W
)  ->  ( A. x  e.  B  A. y  e.  B  (
x  .,  y )  =  if ( x  =  y ,  .1.  ,  .0.  )  /\  (
( ocv `  W
) `  B )  =  { ( 0g `  W ) } ) )
109simpld 445 . . 3  |-  ( B  e.  (OBasis `  W
)  ->  A. x  e.  B  A. y  e.  B  ( x  .,  y )  =  if ( x  =  y ,  .1.  ,  .0.  ) )
11 oveq1 5865 . . . . 5  |-  ( x  =  P  ->  (
x  .,  y )  =  ( P  .,  y ) )
12 eqeq1 2289 . . . . . 6  |-  ( x  =  P  ->  (
x  =  y  <->  P  =  y ) )
1312ifbid 3583 . . . . 5  |-  ( x  =  P  ->  if ( x  =  y ,  .1.  ,  .0.  )  =  if ( P  =  y ,  .1.  ,  .0.  ) )
1411, 13eqeq12d 2297 . . . 4  |-  ( x  =  P  ->  (
( x  .,  y
)  =  if ( x  =  y ,  .1.  ,  .0.  )  <->  ( P  .,  y )  =  if ( P  =  y ,  .1.  ,  .0.  ) ) )
15 oveq2 5866 . . . . 5  |-  ( y  =  Q  ->  ( P  .,  y )  =  ( P  .,  Q
) )
16 eqeq2 2292 . . . . . 6  |-  ( y  =  Q  ->  ( P  =  y  <->  P  =  Q ) )
1716ifbid 3583 . . . . 5  |-  ( y  =  Q  ->  if ( P  =  y ,  .1.  ,  .0.  )  =  if ( P  =  Q ,  .1.  ,  .0.  ) )
1815, 17eqeq12d 2297 . . . 4  |-  ( y  =  Q  ->  (
( P  .,  y
)  =  if ( P  =  y ,  .1.  ,  .0.  )  <->  ( P  .,  Q )  =  if ( P  =  Q ,  .1.  ,  .0.  ) ) )
1914, 18rspc2v 2890 . . 3  |-  ( ( P  e.  B  /\  Q  e.  B )  ->  ( A. x  e.  B  A. y  e.  B  ( x  .,  y )  =  if ( x  =  y ,  .1.  ,  .0.  )  ->  ( P  .,  Q )  =  if ( P  =  Q ,  .1.  ,  .0.  ) ) )
2010, 19syl5com 26 . 2  |-  ( B  e.  (OBasis `  W
)  ->  ( ( P  e.  B  /\  Q  e.  B )  ->  ( P  .,  Q
)  =  if ( P  =  Q ,  .1.  ,  .0.  ) ) )
21203impib 1149 1  |-  ( ( B  e.  (OBasis `  W )  /\  P  e.  B  /\  Q  e.  B )  ->  ( P  .,  Q )  =  if ( P  =  Q ,  .1.  ,  .0.  ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543    C_ wss 3152   ifcif 3565   {csn 3640   ` cfv 5255  (class class class)co 5858   Basecbs 13148  Scalarcsca 13211   .icip 13213   0gc0g 13400   1rcur 15339   PreHilcphl 16528   ocvcocv 16560  OBasiscobs 16602
This theorem is referenced by:  obsipid  16622  obselocv  16628
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-obs 16605
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