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Theorem nlmngp2 24067
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 = (Scalarβ€˜π‘Š)
Assertion
Ref Expression
nlmngp2 (π‘Š ∈ NrmMod β†’ 𝐹 ∈ NrmGrp)

Proof of Theorem nlmngp2
StepHypRef Expression
1 nlmnrg.1 . . 3 𝐹 = (Scalarβ€˜π‘Š)
21nlmnrg 24066 . 2 (π‘Š ∈ NrmMod β†’ 𝐹 ∈ NrmRing)
3 nrgngp 24049 . 2 (𝐹 ∈ NrmRing β†’ 𝐹 ∈ NrmGrp)
42, 3syl 17 1 (π‘Š ∈ NrmMod β†’ 𝐹 ∈ NrmGrp)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1542   ∈ wcel 2107  β€˜cfv 6500  Scalarcsca 17144  NrmGrpcngp 23956  NrmRingcnrg 23958  NrmModcnlm 23959
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-nul 5267
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-ral 3062  df-rab 3407  df-v 3449  df-sbc 3744  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-iota 6452  df-fv 6508  df-ov 7364  df-nrg 23964  df-nlm 23965
This theorem is referenced by:  nlmdsdir  24069  nlmmul0or  24070  nlmvscnlem2  24072  nlmvscnlem1  24073  nlmvscn  24074
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