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Theorem nvcnlm 23308
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 23307 . 2 (𝑊 ∈ NrmVec ↔ (𝑊 ∈ NrmMod ∧ 𝑊 ∈ LVec))
21simplbi 501 1 (𝑊 ∈ NrmVec → 𝑊 ∈ NrmMod)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2115  LVecclvec 19877  NrmModcnlm 23193  NrmVeccnvc 23194
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-v 3483  df-in 3927  df-nvc 23200
This theorem is referenced by:  nvclmod  23310  nvctvc  23312  lssnvc  23314  ncvsprp  23763  ncvsm1  23765  ncvsdif  23766  ncvspi  23767  ncvs1  23768  ncvspds  23772  bnnlm  23951  cssbn  23985
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