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Theorem bncms 25271
Description: A Banach space is a complete metric space. (Contributed by Mario Carneiro, 15-Oct-2015.)
Assertion
Ref Expression
bncms (π‘Š ∈ Ban β†’ π‘Š ∈ CMetSp)

Proof of Theorem bncms
StepHypRef Expression
1 eqid 2728 . . 3 (Scalarβ€˜π‘Š) = (Scalarβ€˜π‘Š)
21isbn 25265 . 2 (π‘Š ∈ Ban ↔ (π‘Š ∈ NrmVec ∧ π‘Š ∈ CMetSp ∧ (Scalarβ€˜π‘Š) ∈ CMetSp))
32simp2bi 1144 1 (π‘Š ∈ Ban β†’ π‘Š ∈ CMetSp)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∈ wcel 2099  β€˜cfv 6548  Scalarcsca 17235  NrmVeccnvc 24489  CMetSpccms 25259  Bancbn 25260
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-iota 6500  df-fv 6556  df-bn 25263
This theorem is referenced by:  bncmet  25274  lssbn  25279  hlcms  25293  bncssbn  25301  sitgclbn  33963
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