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Theorem obsip 16948
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 2436 . . . . . 6  |-  ( ocv `  W )  =  ( ocv `  W )
7 eqid 2436 . . . . . 6  |-  ( 0g
`  W )  =  ( 0g `  W
)
81, 2, 3, 4, 5, 6, 7isobs 16947 . . . . 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 974 . . . 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 446 . . 3  |-  ( B  e.  (OBasis `  W
)  ->  A. x  e.  B  A. y  e.  B  ( x  .,  y )  =  if ( x  =  y ,  .1.  ,  .0.  ) )
11 oveq1 6088 . . . . 5  |-  ( x  =  P  ->  (
x  .,  y )  =  ( P  .,  y ) )
12 eqeq1 2442 . . . . . 6  |-  ( x  =  P  ->  (
x  =  y  <->  P  =  y ) )
1312ifbid 3757 . . . . 5  |-  ( x  =  P  ->  if ( x  =  y ,  .1.  ,  .0.  )  =  if ( P  =  y ,  .1.  ,  .0.  ) )
1411, 13eqeq12d 2450 . . . 4  |-  ( x  =  P  ->  (
( x  .,  y
)  =  if ( x  =  y ,  .1.  ,  .0.  )  <->  ( P  .,  y )  =  if ( P  =  y ,  .1.  ,  .0.  ) ) )
15 oveq2 6089 . . . . 5  |-  ( y  =  Q  ->  ( P  .,  y )  =  ( P  .,  Q
) )
16 eqeq2 2445 . . . . . 6  |-  ( y  =  Q  ->  ( P  =  y  <->  P  =  Q ) )
1716ifbid 3757 . . . . 5  |-  ( y  =  Q  ->  if ( P  =  y ,  .1.  ,  .0.  )  =  if ( P  =  Q ,  .1.  ,  .0.  ) )
1815, 17eqeq12d 2450 . . . 4  |-  ( y  =  Q  ->  (
( P  .,  y
)  =  if ( P  =  y ,  .1.  ,  .0.  )  <->  ( P  .,  Q )  =  if ( P  =  Q ,  .1.  ,  .0.  ) ) )
1914, 18rspc2v 3058 . . 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 28 . 2  |-  ( B  e.  (OBasis `  W
)  ->  ( ( P  e.  B  /\  Q  e.  B )  ->  ( P  .,  Q
)  =  if ( P  =  Q ,  .1.  ,  .0.  ) ) )
21203impib 1151 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 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2705    C_ wss 3320   ifcif 3739   {csn 3814   ` cfv 5454  (class class class)co 6081   Basecbs 13469  Scalarcsca 13532   .icip 13534   0gc0g 13723   1rcur 15662   PreHilcphl 16855   ocvcocv 16887  OBasiscobs 16929
This theorem is referenced by:  obsipid  16949  obselocv  16955
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fv 5462  df-ov 6084  df-obs 16932
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