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Theorem bncms 25392
Description: A Banach space is a complete metric space. (Contributed by Mario Carneiro, 15-Oct-2015.)
Assertion
Ref Expression
bncms (𝑊 ∈ Ban → 𝑊 ∈ CMetSp)

Proof of Theorem bncms
StepHypRef Expression
1 eqid 2735 . . 3 (Scalar‘𝑊) = (Scalar‘𝑊)
21isbn 25386 . 2 (𝑊 ∈ Ban ↔ (𝑊 ∈ NrmVec ∧ 𝑊 ∈ CMetSp ∧ (Scalar‘𝑊) ∈ CMetSp))
32simp2bi 1145 1 (𝑊 ∈ Ban → 𝑊 ∈ CMetSp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  cfv 6563  Scalarcsca 17301  NrmVeccnvc 24610  CMetSpccms 25380  Bancbn 25381
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-bn 25384
This theorem is referenced by:  bncmet  25395  lssbn  25400  hlcms  25414  bncssbn  25422  sitgclbn  34325
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