Theorem nlmngp2 24196

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

Step	Hyp	Ref	Expression
1		nlmnrg.1	. . 3 ⊢ 𝐹 = (Scalar‘𝑊)
2	1	nlmnrg 24195	. 2 ⊢ (𝑊 ∈ NrmMod → 𝐹 ∈ NrmRing)
3		nrgngp 24178	. 2 ⊢ (𝐹 ∈ NrmRing → 𝐹 ∈ NrmGrp)
4	2, 3	syl 17	1 ⊢ (𝑊 ∈ NrmMod → 𝐹 ∈ NrmGrp)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2106  ‘cfv 6543  Scalarcsca 17199  NrmGrpcngp 24085  NrmRingcnrg 24087  NrmModcnlm 24088

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-nul 5306

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rab 3433  df-v 3476  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-iota 6495  df-fv 6551  df-ov 7411  df-nrg 24093  df-nlm 24094

This theorem is referenced by:  nlmdsdir  24198  nlmmul0or  24199  nlmvscnlem2  24201  nlmvscnlem1  24202  nlmvscn  24203