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Theorem obsrcl 16942
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 2435 . . 3  |-  ( Base `  W )  =  (
Base `  W )
2 eqid 2435 . . 3  |-  ( .i
`  W )  =  ( .i `  W
)
3 eqid 2435 . . 3  |-  (Scalar `  W )  =  (Scalar `  W )
4 eqid 2435 . . 3  |-  ( 1r
`  (Scalar `  W )
)  =  ( 1r
`  (Scalar `  W )
)
5 eqid 2435 . . 3  |-  ( 0g
`  (Scalar `  W )
)  =  ( 0g
`  (Scalar `  W )
)
6 eqid 2435 . . 3  |-  ( ocv `  W )  =  ( ocv `  W )
7 eqid 2435 . . 3  |-  ( 0g
`  W )  =  ( 0g `  W
)
81, 2, 3, 4, 5, 6, 7isobs 16939 . 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 972 1  |-  ( B  e.  (OBasis `  W
)  ->  W  e.  PreHil )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697    C_ wss 3312   ifcif 3731   {csn 3806   ` cfv 5446  (class class class)co 6073   Basecbs 13461  Scalarcsca 13524   .icip 13526   0gc0g 13715   1rcur 15654   PreHilcphl 16847   ocvcocv 16879  OBasiscobs 16921
This theorem is referenced by:  obsne0  16944  obs2ocv  16946  obselocv  16947  obslbs  16949
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 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fv 5454  df-ov 6076  df-obs 16924
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