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Theorem isnvc2 18209
Description: A normed vector space is just a normed module whose scalar ring is a division ring. (Contributed by Mario Carneiro, 4-Oct-2015.)
Hypothesis
Ref Expression
isnvc2.1  |-  F  =  (Scalar `  W )
Assertion
Ref Expression
isnvc2  |-  ( W  e. NrmVec 
<->  ( W  e. NrmMod  /\  F  e.  DivRing ) )

Proof of Theorem isnvc2
StepHypRef Expression
1 isnvc 18205 . 2  |-  ( W  e. NrmVec 
<->  ( W  e. NrmMod  /\  W  e.  LVec ) )
2 nlmlmod 18189 . . . 4  |-  ( W  e. NrmMod  ->  W  e.  LMod )
3 isnvc2.1 . . . . . 6  |-  F  =  (Scalar `  W )
43islvec 15857 . . . . 5  |-  ( W  e.  LVec  <->  ( W  e. 
LMod  /\  F  e.  DivRing ) )
54baib 871 . . . 4  |-  ( W  e.  LMod  ->  ( W  e.  LVec  <->  F  e.  DivRing ) )
62, 5syl 15 . . 3  |-  ( W  e. NrmMod  ->  ( W  e. 
LVec 
<->  F  e.  DivRing ) )
76pm5.32i 618 . 2  |-  ( ( W  e. NrmMod  /\  W  e. 
LVec )  <->  ( W  e. NrmMod  /\  F  e.  DivRing ) )
81, 7bitri 240 1  |-  ( W  e. NrmVec 
<->  ( W  e. NrmMod  /\  F  e.  DivRing ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   ` cfv 5255  Scalarcsca 13211   DivRingcdr 15512   LModclmod 15627   LVecclvec 15855  NrmModcnlm 18103  NrmVeccnvc 18104
This theorem is referenced by:  lssnvc  18212  srabn  18777
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-nul 4149
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ov 5861  df-lvec 15856  df-nlm 18109  df-nvc 18110
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