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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 Scalar
Assertion
Ref Expression
isnvc2 NrmVec NrmMod

Proof of Theorem isnvc2
StepHypRef Expression
1 isnvc 18732 . 2 NrmVec NrmMod
2 nlmlmod 18716 . . . 4 NrmMod
3 isnvc2.1 . . . . . 6 Scalar
43islvec 16178 . . . . 5
54baib 873 . . . 4
62, 5syl 16 . . 3 NrmMod
76pm5.32i 620 . 2 NrmMod NrmMod
81, 7bitri 242 1 NrmVec NrmMod
 Colors of variables: wff set class Syntax hints:   wb 178   wa 360   wceq 1653   wcel 1726  cfv 5456  Scalarcsca 13534  cdr 15837  clmod 15952  clvec 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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