Theorem nlmngp2 24701

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 24700	. 2 ⊢ (𝑊 ∈ NrmMod → 𝐹 ∈ NrmRing)
3		nrgngp 24683	. 2 ⊢ (𝐹 ∈ NrmRing → 𝐹 ∈ NrmGrp)
4	2, 3	syl 17	1 ⊢ (𝑊 ∈ NrmMod → 𝐹 ∈ NrmGrp)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2108  ‘cfv 6561  Scalarcsca 17300  NrmGrpcngp 24590  NrmRingcnrg 24592  NrmModcnlm 24593

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-nul 5306

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-ov 7434  df-nrg 24598  df-nlm 24599

This theorem is referenced by:  nlmdsdir  24703  nlmmul0or  24704  nlmvscnlem2  24706  nlmvscnlem1  24707  nlmvscn  24708