Theorem isnvc 25047

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 24939	. 2 ⊢ NrmVec = (NrmMod ∩ LVec)
2	1	elin2 4103	1 ⊢ (𝑊 ∈ NrmVec ↔ (𝑊 ∈ NrmMod ∧ 𝑊 ∈ LVec))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   ∈ wcel 2147  LVecclvec 21410  NrmModcnlm 24932  NrmVeccnvc 24933

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456  df-in 4094  df-nvc 24939

This theorem is used by:  nvcnlm  25048  nvclvec  25049  isnvc2  25051  rlmnvc  25055  isncvsngp  25503  cphnvc  25530