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Theorem bnsca 24706
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 24705 . 2 (π‘Š ∈ Ban ↔ (π‘Š ∈ NrmVec ∧ π‘Š ∈ CMetSp ∧ 𝐹 ∈ CMetSp))
32simp3bi 1148 1 (π‘Š ∈ Ban β†’ 𝐹 ∈ CMetSp)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1542   ∈ wcel 2107  β€˜cfv 6497  Scalarcsca 17137  NrmVeccnvc 23940  CMetSpccms 24699  Bancbn 24700
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-iota 6449  df-fv 6505  df-bn 24703
This theorem is referenced by:  lssbn  24719  hlprlem  24734  bncssbn  24741  cmslsschl  24744
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