Theorem isnvc 25020

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 24912	. 2 ⊢ NrmVec = (NrmMod ∩ LVec)
2	1	elin2 4100	1 ⊢ (𝑊 ∈ NrmVec ↔ (𝑊 ∈ NrmMod ∧ 𝑊 ∈ LVec))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   ∈ wcel 2144  LVecclvec 21383  NrmModcnlm 24905  NrmVeccnvc 24906

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-in 4091  df-nvc 24912

This theorem is used by:  nvcnlm  25021  nvclvec  25022  isnvc2  25024  rlmnvc  25028  isncvsngp  25476  cphnvc  25503