Theorem isnvc 24889

Description: A normed vector space is just a normed module which is algebraically a vector space. (Contributed by Mario Carneiro, 4-Oct-2015.)

Assertion

Ref	Expression
isnvc	⊢ (𝑊 ∈ NrmVec ↔ (𝑊 ∈ NrmMod ∧ 𝑊 ∈ LVec))

Proof of Theorem isnvc

Step	Hyp	Ref	Expression
1		df-nvc 24781	. 2 ⊢ NrmVec = (NrmMod ∩ LVec)
2	1	elin2 4159	1 ⊢ (𝑊 ∈ NrmVec ↔ (𝑊 ∈ NrmMod ∧ 𝑊 ∈ LVec))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   ∈ 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:  nvcnlm  24890  nvclvec  24891  isnvc2  24893  rlmnvc  24897  isncvsngp  25345  cphnvc  25372