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Theorem obsip 16677
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 2316 . . . . . 6  |-  ( ocv `  W )  =  ( ocv `  W )
7 eqid 2316 . . . . . 6  |-  ( 0g
`  W )  =  ( 0g `  W
)
81, 2, 3, 4, 5, 6, 7isobs 16676 . . . . 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 5907 . . . . 5  |-  ( x  =  P  ->  (
x  .,  y )  =  ( P  .,  y ) )
12 eqeq1 2322 . . . . . 6  |-  ( x  =  P  ->  (
x  =  y  <->  P  =  y ) )
1312ifbid 3617 . . . . 5  |-  ( x  =  P  ->  if ( x  =  y ,  .1.  ,  .0.  )  =  if ( P  =  y ,  .1.  ,  .0.  ) )
1411, 13eqeq12d 2330 . . . 4  |-  ( x  =  P  ->  (
( x  .,  y
)  =  if ( x  =  y ,  .1.  ,  .0.  )  <->  ( P  .,  y )  =  if ( P  =  y ,  .1.  ,  .0.  ) ) )
15 oveq2 5908 . . . . 5  |-  ( y  =  Q  ->  ( P  .,  y )  =  ( P  .,  Q
) )
16 eqeq2 2325 . . . . . 6  |-  ( y  =  Q  ->  ( P  =  y  <->  P  =  Q ) )
1716ifbid 3617 . . . . 5  |-  ( y  =  Q  ->  if ( P  =  y ,  .1.  ,  .0.  )  =  if ( P  =  Q ,  .1.  ,  .0.  ) )
1815, 17eqeq12d 2330 . . . 4  |-  ( y  =  Q  ->  (
( P  .,  y
)  =  if ( P  =  y ,  .1.  ,  .0.  )  <->  ( P  .,  Q )  =  if ( P  =  Q ,  .1.  ,  .0.  ) ) )
1914, 18rspc2v 2924 . . 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 1633    e. wcel 1701   A.wral 2577    C_ wss 3186   ifcif 3599   {csn 3674   ` cfv 5292  (class class class)co 5900   Basecbs 13195  Scalarcsca 13258   .icip 13260   0gc0g 13449   1rcur 15388   PreHilcphl 16584   ocvcocv 16616  OBasiscobs 16658
This theorem is referenced by:  obsipid  16678  obselocv  16684
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fv 5300  df-ov 5903  df-obs 16661
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