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Theorem bnsca 23946
 Description: The scalar field of a 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 ∈ Ban → 𝐹 ∈ CMetSp)

Proof of Theorem bnsca
StepHypRef Expression
1 isbn.1 . . 3 𝐹 = (Scalar‘𝑊)
21isbn 23945 . 2 (𝑊 ∈ Ban ↔ (𝑊 ∈ NrmVec ∧ 𝑊 ∈ CMetSp ∧ 𝐹 ∈ CMetSp))
32simp3bi 1144 1 (𝑊 ∈ Ban → 𝐹 ∈ CMetSp)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2112  ‘cfv 6328  Scalarcsca 16563  NrmVeccnvc 23191  CMetSpccms 23939  Bancbn 23940 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-rab 3118  df-v 3446  df-un 3889  df-in 3891  df-ss 3901  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-iota 6287  df-fv 6336  df-bn 23943 This theorem is referenced by:  lssbn  23959  hlprlem  23974  bncssbn  23981  cmslsschl  23984
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