Theorem isnvc 24742

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 24634	. 2 ⊢ NrmVec = (NrmMod ∩ LVec)
2	1	elin2 4153	1 ⊢ (𝑊 ∈ NrmVec ↔ (𝑊 ∈ NrmMod ∧ 𝑊 ∈ LVec))

Syntax hints:   ↔ wb 208   ∧ wa 399   ∈ wcel 2141  LVecclvec 21156  NrmModcnlm 24627  NrmVeccnvc 24628

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1562  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3455  df-in 3909  df-nvc 24634

This theorem is referenced by:  nvcnlm  24743  nvclvec  24744  isnvc2  24746  rlmnvc  24750  isncvsngp  25198  cphnvc  25225