Theorem nvcnlm 24890

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 24889	. 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 2146  LVecclvec 21260  NrmModcnlm 24774  NrmVeccnvc 24775

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 2148  ax-9 2156  ax-ext 2738

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-v 3460  df-in 3915  df-nvc 24781

This theorem is used by:  nvclmod  24892  nvctvc  24894  lssnvc  24896  ncvsprp  25348  ncvsm1  25350  ncvsdif  25351  ncvspi  25352  ncvs1  25353  ncvspds  25357  bnnlm  25537  cssbn  25571