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Theorem isnvc 24082
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
StepHypRef Expression
1 df-nvc 23966 . 2 NrmVec = (NrmMod ∩ LVec)
21elin2 4161 1 (𝑊 ∈ NrmVec ↔ (𝑊 ∈ NrmMod ∧ 𝑊 ∈ LVec))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397  wcel 2107  LVecclvec 20607  NrmModcnlm 23959  NrmVeccnvc 23960
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3449  df-in 3921  df-nvc 23966
This theorem is referenced by:  nvcnlm  24083  nvclvec  24084  isnvc2  24086  rlmnvc  24090  isncvsngp  24536  cphnvc  24563
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