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Theorem nvcnlm 18733
Description: A normed vector space is a normed module. (Contributed by Mario Carneiro, 4-Oct-2015.)
Assertion
Ref Expression
nvcnlm  |-  ( W  e. NrmVec  ->  W  e. NrmMod )

Proof of Theorem nvcnlm
StepHypRef Expression
1 isnvc 18732 . 2  |-  ( W  e. NrmVec 
<->  ( W  e. NrmMod  /\  W  e.  LVec ) )
21simplbi 448 1  |-  ( W  e. NrmVec  ->  W  e. NrmMod )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1726   LVecclvec 16176  NrmModcnlm 18630  NrmVeccnvc 18631
This theorem is referenced by:  nvclmod  18735  nvctvc  18737  lssnvc  18739  bnnlm  19296
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-v 2960  df-in 3329  df-nvc 18637
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