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Theorem nlmngp2 22531
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 22530 . 2 (𝑊 ∈ NrmMod → 𝐹 ∈ NrmRing)
3 nrgngp 22513 . 2 (𝐹 ∈ NrmRing → 𝐹 ∈ NrmGrp)
42, 3syl 17 1 (𝑊 ∈ NrmMod → 𝐹 ∈ NrmGrp)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  wcel 2030  cfv 5926  Scalarcsca 15991  NrmGrpcngp 22429  NrmRingcnrg 22431  NrmModcnlm 22432
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-nul 4822
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-iota 5889  df-fv 5934  df-ov 6693  df-nrg 22437  df-nlm 22438
This theorem is referenced by:  nlmdsdir  22533  nlmmul0or  22534  nlmvscnlem2  22536  nlmvscnlem1  22537  nlmvscn  22538
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