Theorem isnvc 24670

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 24562	. 2 ⊢ NrmVec = (NrmMod ∩ LVec)
2	1	elin2 4144	1 ⊢ (𝑊 ∈ NrmVec ↔ (𝑊 ∈ NrmMod ∧ 𝑊 ∈ LVec))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∈ wcel 2114  LVecclvec 21089  NrmModcnlm 24555  NrmVeccnvc 24556

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3432  df-in 3897  df-nvc 24562

This theorem is referenced by:  nvcnlm  24671  nvclvec  24672  isnvc2  24674  rlmnvc  24678  isncvsngp  25126  cphnvc  25153