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Theorem nlmngp2 18191
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 18190 . 2  |-  ( W  e. NrmMod  ->  F  e. NrmRing )
3 nrgngp 18173 . 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 1623    e. wcel 1684   ` cfv 5255  Scalarcsca 13211  NrmGrpcngp 18100  NrmRingcnrg 18102  NrmModcnlm 18103
This theorem is referenced by:  nlmdsdir  18193  nlmmul0or  18194  nlmvscnlem2  18196  nlmvscnlem1  18197  nlmvscn  18198
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-nrg 18108  df-nlm 18109
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