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Theorem nvcnlm 24213
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 24212 . 2 (𝑊 ∈ NrmVec ↔ (𝑊 ∈ NrmMod ∧ 𝑊 ∈ LVec))
21simplbi 499 1 (𝑊 ∈ NrmVec → 𝑊 ∈ NrmMod)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  LVecclvec 20713  NrmModcnlm 24089  NrmVeccnvc 24090
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 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-in 3956  df-nvc 24096
This theorem is referenced by:  nvclmod  24215  nvctvc  24217  lssnvc  24219  ncvsprp  24669  ncvsm1  24671  ncvsdif  24672  ncvspi  24673  ncvs1  24674  ncvspds  24678  bnnlm  24858  cssbn  24892
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