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Theorem isnvc 22422
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 22315 . 2 NrmVec = (NrmMod ∩ LVec)
21elin2 3784 1 (𝑊 ∈ NrmVec ↔ (𝑊 ∈ NrmMod ∧ 𝑊 ∈ LVec))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384  wcel 1987  LVecclvec 19034  NrmModcnlm 22308  NrmVeccnvc 22309
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3191  df-in 3566  df-nvc 22315
This theorem is referenced by:  nvcnlm  22423  nvclvec  22424  isnvc2  22426  rlmnvc  22430  isncvsngp  22872  cphnvc  22899
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