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Theorem nlmngp2 24716
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 24715 . 2 (𝑊 ∈ NrmMod → 𝐹 ∈ NrmRing)
3 nrgngp 24698 . 2 (𝐹 ∈ NrmRing → 𝐹 ∈ NrmGrp)
42, 3syl 17 1 (𝑊 ∈ NrmMod → 𝐹 ∈ NrmGrp)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1536  wcel 2105  cfv 6562  Scalarcsca 17300  NrmGrpcngp 24605  NrmRingcnrg 24607  NrmModcnlm 24608
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-nul 5311
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2938  df-ral 3059  df-rab 3433  df-v 3479  df-sbc 3791  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-iota 6515  df-fv 6570  df-ov 7433  df-nrg 24613  df-nlm 24614
This theorem is referenced by:  nlmdsdir  24718  nlmmul0or  24719  nlmvscnlem2  24721  nlmvscnlem1  24722  nlmvscn  24723
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