Theorem nvcnlm 25020

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 25019	. 2 ⊢ (𝑊 ∈ NrmVec ↔ (𝑊 ∈ NrmMod ∧ 𝑊 ∈ LVec))
2	1	simplbi 502	1 ⊢ (𝑊 ∈ NrmVec → 𝑊 ∈ NrmMod)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2144  LVecclvec 21382  NrmModcnlm 24904  NrmVeccnvc 24905

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-in 4091  df-nvc 24911

This theorem is used by:  nvclmod  25022  nvctvc  25024  lssnvc  25026  ncvsprp  25478  ncvsm1  25480  ncvsdif  25481  ncvspi  25482  ncvs1  25483  ncvspds  25487  bnnlm  25667  cssbn  25701