Theorem nvcnlm 24661

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 24660	. 2 ⊢ (𝑊 ∈ NrmVec ↔ (𝑊 ∈ NrmMod ∧ 𝑊 ∈ LVec))
2	1	simplbi 496	1 ⊢ (𝑊 ∈ NrmVec → 𝑊 ∈ NrmMod)

Syntax hints:   → wi 4   ∈ wcel 2114  LVecclvec 21097  NrmModcnlm 24545  NrmVeccnvc 24546

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 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-v 3431  df-in 3896  df-nvc 24552

This theorem is referenced by:  nvclmod  24663  nvctvc  24665  lssnvc  24667  ncvsprp  25119  ncvsm1  25121  ncvsdif  25122  ncvspi  25123  ncvs1  25124  ncvspds  25128  bnnlm  25308  cssbn  25342