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Theorem obsss 16640
Description: An orthonormal basis is a subset of the base set. (Contributed by Mario Carneiro, 23-Oct-2015.)
Hypothesis
Ref Expression
obsss.v  |-  V  =  ( Base `  W
)
Assertion
Ref Expression
obsss  |-  ( B  e.  (OBasis `  W
)  ->  B  C_  V
)

Proof of Theorem obsss
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 obsss.v . . 3  |-  V  =  ( Base `  W
)
2 eqid 2296 . . 3  |-  ( .i
`  W )  =  ( .i `  W
)
3 eqid 2296 . . 3  |-  (Scalar `  W )  =  (Scalar `  W )
4 eqid 2296 . . 3  |-  ( 1r
`  (Scalar `  W )
)  =  ( 1r
`  (Scalar `  W )
)
5 eqid 2296 . . 3  |-  ( 0g
`  (Scalar `  W )
)  =  ( 0g
`  (Scalar `  W )
)
6 eqid 2296 . . 3  |-  ( ocv `  W )  =  ( ocv `  W )
7 eqid 2296 . . 3  |-  ( 0g
`  W )  =  ( 0g `  W
)
81, 2, 3, 4, 5, 6, 7isobs 16636 . 2  |-  ( B  e.  (OBasis `  W
)  <->  ( W  e. 
PreHil  /\  B  C_  V  /\  ( 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 ) } ) ) )
98simp2bi 971 1  |-  ( B  e.  (OBasis `  W
)  ->  B  C_  V
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556    C_ wss 3165   ifcif 3578   {csn 3653   ` cfv 5271  (class class class)co 5874   Basecbs 13164  Scalarcsca 13227   .icip 13229   0gc0g 13416   1rcur 15355   PreHilcphl 16544   ocvcocv 16576  OBasiscobs 16618
This theorem is referenced by:  obsne0  16641  obselocv  16644  obslbs  16646
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-obs 16621
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