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Theorem bnsca 25211
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 = (Scalarβ€˜π‘Š)
Assertion
Ref Expression
bnsca (π‘Š ∈ Ban β†’ 𝐹 ∈ CMetSp)

Proof of Theorem bnsca
StepHypRef Expression
1 isbn.1 . . 3 𝐹 = (Scalarβ€˜π‘Š)
21isbn 25210 . 2 (π‘Š ∈ Ban ↔ (π‘Š ∈ NrmVec ∧ π‘Š ∈ CMetSp ∧ 𝐹 ∈ CMetSp))
32simp3bi 1144 1 (π‘Š ∈ Ban β†’ 𝐹 ∈ CMetSp)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1533   ∈ wcel 2098  β€˜cfv 6534  Scalarcsca 17205  NrmVeccnvc 24434  CMetSpccms 25204  Bancbn 25205
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-iota 6486  df-fv 6542  df-bn 25208
This theorem is referenced by:  lssbn  25224  hlprlem  25239  bncssbn  25246  cmslsschl  25249
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