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Theorem nlmngp2 24641
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 ∈ NrmMod → 𝐹 ∈ NrmGrp)

Proof of Theorem nlmngp2
StepHypRef Expression
1 nlmnrg.1 . . 3 𝐹 = (Scalar‘𝑊)
21nlmnrg 24640 . 2 (𝑊 ∈ NrmMod → 𝐹 ∈ NrmRing)
3 nrgngp 24623 . 2 (𝐹 ∈ NrmRing → 𝐹 ∈ NrmGrp)
42, 3syl 17 1 (𝑊 ∈ NrmMod → 𝐹 ∈ NrmGrp)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  cfv 6549  Scalarcsca 17239  NrmGrpcngp 24530  NrmRingcnrg 24532  NrmModcnlm 24533
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-nul 5307
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2930  df-ral 3051  df-rab 3419  df-v 3463  df-sbc 3774  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-iota 6501  df-fv 6557  df-ov 7422  df-nrg 24538  df-nlm 24539
This theorem is referenced by:  nlmdsdir  24643  nlmmul0or  24644  nlmvscnlem2  24646  nlmvscnlem1  24647  nlmvscn  24648
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