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Theorem bnsca 19284
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 19283 . 2  |-  ( W  e. Ban 
<->  ( W  e. NrmVec  /\  W  e. CMetSp  /\  F  e. CMetSp )
)
32simp3bi 974 1  |-  ( W  e. Ban  ->  F  e. CMetSp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   ` cfv 5446  Scalarcsca 13524  NrmVeccnvc 18621  CMetSpccms 19277  Bancbn 19278
This theorem is referenced by:  lssbn  19296  hlprlem  19313  sitgclbn  24649
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-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
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-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-iota 5410  df-fv 5454  df-bn 19281
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