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Theorem obsrcl 18514
Description: Reverse closure for an orthonormal basis. (Contributed by Mario Carneiro, 23-Oct-2015.)
Assertion
Ref Expression
obsrcl  |-  ( B  e.  (OBasis `  W
)  ->  W  e.  PreHil )

Proof of Theorem obsrcl
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2460 . . 3  |-  ( Base `  W )  =  (
Base `  W )
2 eqid 2460 . . 3  |-  ( .i
`  W )  =  ( .i `  W
)
3 eqid 2460 . . 3  |-  (Scalar `  W )  =  (Scalar `  W )
4 eqid 2460 . . 3  |-  ( 1r
`  (Scalar `  W )
)  =  ( 1r
`  (Scalar `  W )
)
5 eqid 2460 . . 3  |-  ( 0g
`  (Scalar `  W )
)  =  ( 0g
`  (Scalar `  W )
)
6 eqid 2460 . . 3  |-  ( ocv `  W )  =  ( ocv `  W )
7 eqid 2460 . . 3  |-  ( 0g
`  W )  =  ( 0g `  W
)
81, 2, 3, 4, 5, 6, 7isobs 18511 . 2  |-  ( B  e.  (OBasis `  W
)  <->  ( W  e. 
PreHil  /\  B  C_  ( Base `  W )  /\  ( A. x  e.  B  A. y  e.  B  ( x ( .i
`  W ) y )  =  if ( x  =  y ,  ( 1r `  (Scalar `  W ) ) ,  ( 0g `  (Scalar `  W ) ) )  /\  ( ( ocv `  W ) `  B
)  =  { ( 0g `  W ) } ) ) )
98simp1bi 1006 1  |-  ( B  e.  (OBasis `  W
)  ->  W  e.  PreHil )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762   A.wral 2807    C_ wss 3469   ifcif 3932   {csn 4020   ` cfv 5579  (class class class)co 6275   Basecbs 14479  Scalarcsca 14547   .icip 14549   0gc0g 14684   1rcur 16936   PreHilcphl 18419   ocvcocv 18451  OBasiscobs 18493
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fv 5587  df-ov 6278  df-obs 18496
This theorem is referenced by:  obsne0  18516  obs2ocv  18518  obselocv  18519  obslbs  18521
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