Theorem isnvc 24703

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 24587	. 2 ⊢ NrmVec = (NrmMod ∩ LVec)
2	1	elin2 4198	1 ⊢ (𝑊 ∈ NrmVec ↔ (𝑊 ∈ NrmMod ∧ 𝑊 ∈ LVec))

Syntax hints:   ↔ wb 205   ∧ wa 394   ∈ wcel 2099  LVecclvec 21080  NrmModcnlm 24580  NrmVeccnvc 24581

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-v 3464  df-in 3954  df-nvc 24587

This theorem is referenced by:  nvcnlm  24704  nvclvec  24705  isnvc2  24707  rlmnvc  24711  isncvsngp  25168  cphnvc  25195