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Theorem nvcnlm 22998
 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 22997 . 2 (𝑊 ∈ NrmVec ↔ (𝑊 ∈ NrmMod ∧ 𝑊 ∈ LVec))
21simplbi 490 1 (𝑊 ∈ NrmVec → 𝑊 ∈ NrmMod)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2048  LVecclvec 19586  NrmModcnlm 22883  NrmVeccnvc 22884 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-ext 2745 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-v 3411  df-in 3832  df-nvc 22890 This theorem is referenced by:  nvclmod  23000  nvctvc  23002  lssnvc  23004  ncvsprp  23449  ncvsm1  23451  ncvsdif  23452  ncvspi  23453  ncvs1  23454  ncvspds  23458  bnnlm  23637  cssbn  23671
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