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Theorem isnvc2 18736
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 18732 . 2  |-  ( W  e. NrmVec 
<->  ( W  e. NrmMod  /\  W  e.  LVec ) )
2 nlmlmod 18716 . . . 4  |-  ( W  e. NrmMod  ->  W  e.  LMod )
3 isnvc2.1 . . . . . 6  |-  F  =  (Scalar `  W )
43islvec 16178 . . . . 5  |-  ( W  e.  LVec  <->  ( W  e. 
LMod  /\  F  e.  DivRing ) )
54baib 873 . . . 4  |-  ( W  e.  LMod  ->  ( W  e.  LVec  <->  F  e.  DivRing ) )
62, 5syl 16 . . 3  |-  ( W  e. NrmMod  ->  ( W  e. 
LVec 
<->  F  e.  DivRing ) )
76pm5.32i 620 . 2  |-  ( ( W  e. NrmMod  /\  W  e. 
LVec )  <->  ( W  e. NrmMod  /\  F  e.  DivRing ) )
81, 7bitri 242 1  |-  ( W  e. NrmVec 
<->  ( W  e. NrmMod  /\  F  e.  DivRing ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   ` cfv 5456  Scalarcsca 13534   DivRingcdr 15837   LModclmod 15952   LVecclvec 16176  NrmModcnlm 18630  NrmVeccnvc 18631
This theorem is referenced by:  lssnvc  18739  srabn  19316
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  ax-nul 4340
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-iota 5420  df-fv 5464  df-ov 6086  df-lvec 16177  df-nlm 18636  df-nvc 18637
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