Theorem nlmngp2 24664

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 24663	. 2 ⊢ (𝑊 ∈ NrmMod → 𝐹 ∈ NrmRing)
3		nrgngp 24646	. 2 ⊢ (𝐹 ∈ NrmRing → 𝐹 ∈ NrmGrp)
4	2, 3	syl 17	1 ⊢ (𝑊 ∈ NrmMod → 𝐹 ∈ NrmGrp)

Syntax hints:   → wi 4   = wceq 1547   ∈ wcel 2119  ‘cfv 6486  Scalarcsca 17215  NrmGrpcngp 24561  NrmRingcnrg 24563  NrmModcnlm 24564

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-nul 5229

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ne 2935  df-ral 3054  df-rab 3392  df-v 3433  df-sbc 3724  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4263  df-if 4456  df-sn 4557  df-pr 4559  df-op 4563  df-uni 4840  df-br 5074  df-iota 6442  df-fv 6494  df-ov 7360  df-nrg 24569  df-nlm 24570

This theorem is referenced by:  nlmdsdir  24666  nlmmul0or  24667  nlmvscnlem2  24669  nlmvscnlem1  24670  nlmvscn  24671