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Theorem bnnvc 24656
Description: A Banach space is a normed vector space. (Contributed by Mario Carneiro, 15-Oct-2015.)
Assertion
Ref Expression
bnnvc (𝑊 ∈ Ban → 𝑊 ∈ NrmVec)

Proof of Theorem bnnvc
StepHypRef Expression
1 eqid 2738 . . 3 (Scalar‘𝑊) = (Scalar‘𝑊)
21isbn 24654 . 2 (𝑊 ∈ Ban ↔ (𝑊 ∈ NrmVec ∧ 𝑊 ∈ CMetSp ∧ (Scalar‘𝑊) ∈ CMetSp))
32simp1bi 1146 1 (𝑊 ∈ Ban → 𝑊 ∈ NrmVec)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  cfv 6494  Scalarcsca 17096  NrmVeccnvc 23889  CMetSpccms 24648  Bancbn 24649
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 2709
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 2716  df-cleq 2730  df-clel 2816  df-rab 3407  df-v 3446  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-br 5105  df-iota 6446  df-fv 6502  df-bn 24652
This theorem is referenced by:  bnnlm  24657  lssbn  24668  bncssbn  24690  cmslsschl  24693
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