Theorem isnvc 24833

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 24725	. 2 ⊢ NrmVec = (NrmMod ∩ LVec)
2	1	elin2 4157	1 ⊢ (𝑊 ∈ NrmVec ↔ (𝑊 ∈ NrmMod ∧ 𝑊 ∈ LVec))

Syntax hints:   ↔ wb 209   ∧ wa 400   ∈ wcel 2143  LVecclvec 21204  NrmModcnlm 24718  NrmVeccnvc 24719

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-v 3457  df-in 3913  df-nvc 24725

This theorem is referenced by:  nvcnlm  24834  nvclvec  24835  isnvc2  24837  rlmnvc  24841  isncvsngp  25289  cphnvc  25316