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Theorem bnsca 18761
Description: The scalar field of a complex Banach space is complete. (Contributed by NM, 8-Sep-2007.) (Revised by Mario Carneiro, 15-Oct-2015.)
Hypothesis
Ref Expression
isbn.1  |-  F  =  (Scalar `  W )
Assertion
Ref Expression
bnsca  |-  ( W  e. Ban  ->  F  e. CMetSp )

Proof of Theorem bnsca
StepHypRef Expression
1 isbn.1 . . 3  |-  F  =  (Scalar `  W )
21isbn 18760 . 2  |-  ( W  e. Ban 
<->  ( W  e. NrmVec  /\  W  e. CMetSp  /\  F  e. CMetSp )
)
32simp3bi 972 1  |-  ( W  e. Ban  ->  F  e. CMetSp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   ` cfv 5255  Scalarcsca 13211  NrmVeccnvc 18104  CMetSpccms 18754  Bancbn 18755
This theorem is referenced by:  lssbn  18773  hlprlem  18784
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-bn 18758
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