Theorem nvcnlm 24738

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 24737	. 2 ⊢ (𝑊 ∈ NrmVec ↔ (𝑊 ∈ NrmMod ∧ 𝑊 ∈ LVec))
2	1	simplbi 497	1 ⊢ (𝑊 ∈ NrmVec → 𝑊 ∈ NrmMod)

Syntax hints:   → wi 4   ∈ wcel 2108  LVecclvec 21124  NrmModcnlm 24614  NrmVeccnvc 24615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-v 3490  df-in 3983  df-nvc 24621

This theorem is referenced by:  nvclmod  24740  nvctvc  24742  lssnvc  24744  ncvsprp  25205  ncvsm1  25207  ncvsdif  25208  ncvspi  25209  ncvs1  25210  ncvspds  25214  bnnlm  25394  cssbn  25428