Theorem nvcnlm 24686

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 24685	. 2 ⊢ (𝑊 ∈ NrmVec ↔ (𝑊 ∈ NrmMod ∧ 𝑊 ∈ LVec))
2	1	simplbi 497	1 ⊢ (𝑊 ∈ NrmVec → 𝑊 ∈ NrmMod)

Syntax hints:   → wi 4   ∈ wcel 2119  LVecclvec 21099  NrmModcnlm 24570  NrmVeccnvc 24571

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-v 3434  df-in 3897  df-nvc 24577

This theorem is referenced by:  nvclmod  24688  nvctvc  24690  lssnvc  24692  ncvsprp  25144  ncvsm1  25146  ncvsdif  25147  ncvspi  25148  ncvs1  25149  ncvspds  25153  bnnlm  25333  cssbn  25367