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Theorem nvcnlm 24212
Description: A normed vector space is a normed module. (Contributed by Mario Carneiro, 4-Oct-2015.)
Assertion
Ref Expression
nvcnlm (𝑊 ∈ NrmVec → 𝑊 ∈ NrmMod)

Proof of Theorem nvcnlm
StepHypRef Expression
1 isnvc 24211 . 2 (𝑊 ∈ NrmVec ↔ (𝑊 ∈ NrmMod ∧ 𝑊 ∈ LVec))
21simplbi 498 1 (𝑊 ∈ NrmVec → 𝑊 ∈ NrmMod)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  LVecclvec 20712  NrmModcnlm 24088  NrmVeccnvc 24089
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-v 3476  df-in 3955  df-nvc 24095
This theorem is referenced by:  nvclmod  24214  nvctvc  24216  lssnvc  24218  ncvsprp  24668  ncvsm1  24670  ncvsdif  24671  ncvspi  24672  ncvs1  24673  ncvspds  24677  bnnlm  24857  cssbn  24891
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