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Theorem nlmngp2 18207
Description: The scalar component of a left module is a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.)
Hypothesis
Ref Expression
nlmnrg.1  |-  F  =  (Scalar `  W )
Assertion
Ref Expression
nlmngp2  |-  ( W  e. NrmMod  ->  F  e. NrmGrp )

Proof of Theorem nlmngp2
StepHypRef Expression
1 nlmnrg.1 . . 3  |-  F  =  (Scalar `  W )
21nlmnrg 18206 . 2  |-  ( W  e. NrmMod  ->  F  e. NrmRing )
3 nrgngp 18189 . 2  |-  ( F  e. NrmRing  ->  F  e. NrmGrp )
42, 3syl 15 1  |-  ( W  e. NrmMod  ->  F  e. NrmGrp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   ` cfv 5271  Scalarcsca 13227  NrmGrpcngp 18116  NrmRingcnrg 18118  NrmModcnlm 18119
This theorem is referenced by:  nlmdsdir  18209  nlmmul0or  18210  nlmvscnlem2  18212  nlmvscnlem1  18213  nlmvscn  18214
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  ax-nul 4165
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-ov 5877  df-nrg 18124  df-nlm 18125
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