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Theorem nvcnlm 18222
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 18221 . 2  |-  ( W  e. NrmVec 
<->  ( W  e. NrmMod  /\  W  e.  LVec ) )
21simplbi 446 1  |-  ( W  e. NrmVec  ->  W  e. NrmMod )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1696   LVecclvec 15871  NrmModcnlm 18119  NrmVeccnvc 18120
This theorem is referenced by:  nvclmod  18224  nvctvc  18226  lssnvc  18228  bnnlm  18779
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-in 3172  df-nvc 18126
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