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Theorem nvcnlm 23232
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 23231 . 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 2105  LVecclvec 19803  NrmModcnlm 23117  NrmVeccnvc 23118
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-v 3494  df-in 3940  df-nvc 23124
This theorem is referenced by:  nvclmod  23234  nvctvc  23236  lssnvc  23238  ncvsprp  23683  ncvsm1  23685  ncvsdif  23686  ncvspi  23687  ncvs1  23688  ncvspds  23692  bnnlm  23871  cssbn  23905
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