Theorem isnvc 24731

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 24615	. 2 ⊢ NrmVec = (NrmMod ∩ LVec)
2	1	elin2 4212	1 ⊢ (𝑊 ∈ NrmVec ↔ (𝑊 ∈ NrmMod ∧ 𝑊 ∈ LVec))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∈ wcel 2105  LVecclvec 21118  NrmModcnlm 24608  NrmVeccnvc 24609

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1539  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-v 3479  df-in 3969  df-nvc 24615

This theorem is referenced by:  nvcnlm  24732  nvclvec  24733  isnvc2  24735  rlmnvc  24739  isncvsngp  25196  cphnvc  25223