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Theorem obsip 18843
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 2382 . . . . . 6  |-  ( ocv `  W )  =  ( ocv `  W )
7 eqid 2382 . . . . . 6  |-  ( 0g
`  W )  =  ( 0g `  W
)
81, 2, 3, 4, 5, 6, 7isobs 18842 . . . . 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 1011 . . . 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 457 . . 3  |-  ( B  e.  (OBasis `  W
)  ->  A. x  e.  B  A. y  e.  B  ( x  .,  y )  =  if ( x  =  y ,  .1.  ,  .0.  ) )
11 oveq1 6203 . . . . 5  |-  ( x  =  P  ->  (
x  .,  y )  =  ( P  .,  y ) )
12 eqeq1 2386 . . . . . 6  |-  ( x  =  P  ->  (
x  =  y  <->  P  =  y ) )
1312ifbid 3879 . . . . 5  |-  ( x  =  P  ->  if ( x  =  y ,  .1.  ,  .0.  )  =  if ( P  =  y ,  .1.  ,  .0.  ) )
1411, 13eqeq12d 2404 . . . 4  |-  ( x  =  P  ->  (
( x  .,  y
)  =  if ( x  =  y ,  .1.  ,  .0.  )  <->  ( P  .,  y )  =  if ( P  =  y ,  .1.  ,  .0.  ) ) )
15 oveq2 6204 . . . . 5  |-  ( y  =  Q  ->  ( P  .,  y )  =  ( P  .,  Q
) )
16 eqeq2 2397 . . . . . 6  |-  ( y  =  Q  ->  ( P  =  y  <->  P  =  Q ) )
1716ifbid 3879 . . . . 5  |-  ( y  =  Q  ->  if ( P  =  y ,  .1.  ,  .0.  )  =  if ( P  =  Q ,  .1.  ,  .0.  ) )
1815, 17eqeq12d 2404 . . . 4  |-  ( y  =  Q  ->  (
( P  .,  y
)  =  if ( P  =  y ,  .1.  ,  .0.  )  <->  ( P  .,  Q )  =  if ( P  =  Q ,  .1.  ,  .0.  ) ) )
1914, 18rspc2v 3144 . . 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 1192 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 367    /\ w3a 971    = wceq 1399    e. wcel 1826   A.wral 2732    C_ wss 3389   ifcif 3857   {csn 3944   ` cfv 5496  (class class class)co 6196   Basecbs 14634  Scalarcsca 14705   .icip 14707   0gc0g 14847   1rcur 17266   PreHilcphl 18750   ocvcocv 18782  OBasiscobs 18824
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-sbc 3253  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fv 5504  df-ov 6199  df-obs 18827
This theorem is referenced by:  obsipid  18844  obselocv  18850
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