Theorem nvcnlm 24834

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

Step	Hyp	Ref	Expression
1		isnvc 24833	. 2 ⊢ (𝑊 ∈ NrmVec ↔ (𝑊 ∈ NrmMod ∧ 𝑊 ∈ LVec))
2	1	simplbi 501	1 ⊢ (𝑊 ∈ NrmVec → 𝑊 ∈ NrmMod)

Syntax hints:   → wi 4   ∈ wcel 2143  LVecclvec 21204  NrmModcnlm 24718  NrmVeccnvc 24719

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-v 3457  df-in 3913  df-nvc 24725

This theorem is referenced by:  nvclmod  24836  nvctvc  24838  lssnvc  24840  ncvsprp  25292  ncvsm1  25294  ncvsdif  25295  ncvspi  25296  ncvs1  25297  ncvspds  25301  bnnlm  25481  cssbn  25515