Theorem nvcnlm 24591

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 24590	. 2 ⊢ (𝑊 ∈ NrmVec ↔ (𝑊 ∈ NrmMod ∧ 𝑊 ∈ LVec))
2	1	simplbi 497	1 ⊢ (𝑊 ∈ NrmVec → 𝑊 ∈ NrmMod)

Syntax hints:   → wi 4   ∈ wcel 2109  LVecclvec 21016  NrmModcnlm 24475  NrmVeccnvc 24476

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-v 3452  df-in 3924  df-nvc 24482

This theorem is referenced by:  nvclmod  24593  nvctvc  24595  lssnvc  24597  ncvsprp  25059  ncvsm1  25061  ncvsdif  25062  ncvspi  25063  ncvs1  25064  ncvspds  25068  bnnlm  25248  cssbn  25282