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Theorem obsip 18270
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 2454 . . . . . 6  |-  ( ocv `  W )  =  ( ocv `  W )
7 eqid 2454 . . . . . 6  |-  ( 0g
`  W )  =  ( 0g `  W
)
81, 2, 3, 4, 5, 6, 7isobs 18269 . . . . 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 1005 . . . 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 459 . . 3  |-  ( B  e.  (OBasis `  W
)  ->  A. x  e.  B  A. y  e.  B  ( x  .,  y )  =  if ( x  =  y ,  .1.  ,  .0.  ) )
11 oveq1 6206 . . . . 5  |-  ( x  =  P  ->  (
x  .,  y )  =  ( P  .,  y ) )
12 eqeq1 2458 . . . . . 6  |-  ( x  =  P  ->  (
x  =  y  <->  P  =  y ) )
1312ifbid 3918 . . . . 5  |-  ( x  =  P  ->  if ( x  =  y ,  .1.  ,  .0.  )  =  if ( P  =  y ,  .1.  ,  .0.  ) )
1411, 13eqeq12d 2476 . . . 4  |-  ( x  =  P  ->  (
( x  .,  y
)  =  if ( x  =  y ,  .1.  ,  .0.  )  <->  ( P  .,  y )  =  if ( P  =  y ,  .1.  ,  .0.  ) ) )
15 oveq2 6207 . . . . 5  |-  ( y  =  Q  ->  ( P  .,  y )  =  ( P  .,  Q
) )
16 eqeq2 2469 . . . . . 6  |-  ( y  =  Q  ->  ( P  =  y  <->  P  =  Q ) )
1716ifbid 3918 . . . . 5  |-  ( y  =  Q  ->  if ( P  =  y ,  .1.  ,  .0.  )  =  if ( P  =  Q ,  .1.  ,  .0.  ) )
1815, 17eqeq12d 2476 . . . 4  |-  ( y  =  Q  ->  (
( P  .,  y
)  =  if ( P  =  y ,  .1.  ,  .0.  )  <->  ( P  .,  Q )  =  if ( P  =  Q ,  .1.  ,  .0.  ) ) )
1914, 18rspc2v 3184 . . 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 30 . 2  |-  ( B  e.  (OBasis `  W
)  ->  ( ( P  e.  B  /\  Q  e.  B )  ->  ( P  .,  Q
)  =  if ( P  =  Q ,  .1.  ,  .0.  ) ) )
21203impib 1186 1  |-  ( ( B  e.  (OBasis `  W )  /\  P  e.  B  /\  Q  e.  B )  ->  ( P  .,  Q )  =  if ( P  =  Q ,  .1.  ,  .0.  ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   A.wral 2798    C_ wss 3435   ifcif 3898   {csn 3984   ` cfv 5525  (class class class)co 6199   Basecbs 14291  Scalarcsca 14359   .icip 14361   0gc0g 14496   1rcur 16724   PreHilcphl 18177   ocvcocv 18209  OBasiscobs 18251
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-ral 2803  df-rex 2804  df-rab 2807  df-v 3078  df-sbc 3293  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-op 3991  df-uni 4199  df-br 4400  df-opab 4458  df-mpt 4459  df-id 4743  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fv 5533  df-ov 6202  df-obs 18254
This theorem is referenced by:  obsipid  18271  obselocv  18277
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