Theorem nvcnlm 24995

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 24994	. 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 2145  LVecclvec 21357  NrmModcnlm 24879  NrmVeccnvc 24880

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3453  df-in 3906  df-nvc 24886

This theorem is used by:  nvclmod  24997  nvctvc  24999  lssnvc  25001  ncvsprp  25453  ncvsm1  25455  ncvsdif  25456  ncvspi  25457  ncvs1  25458  ncvspds  25462  bnnlm  25642  cssbn  25676