Theorem nlmngp2 25026

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 25025	. 2 ⊢ (𝑊 ∈ NrmMod → 𝐹 ∈ NrmRing)
3		nrgngp 25008	. 2 ⊢ (𝐹 ∈ NrmRing → 𝐹 ∈ NrmGrp)
4	2, 3	syl 18	1 ⊢ (𝑊 ∈ NrmMod → 𝐹 ∈ NrmGrp)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  ‘cfv 6555  Scalarcsca 17457  NrmGrpcngp 24923  NrmRingcnrg 24925  NrmModcnlm 24926

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-nul 5263

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-iota 6509  df-fv 6563  df-ov 7443  df-nrg 24931  df-nlm 24932

This theorem is used by:  nlmdsdir  25028  nlmmul0or  25029  nlmvscnlem2  25031  nlmvscnlem1  25032  nlmvscn  25033