Theorem nlmngp2 24626

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 24625	. 2 ⊢ (𝑊 ∈ NrmMod → 𝐹 ∈ NrmRing)
3		nrgngp 24608	. 2 ⊢ (𝐹 ∈ NrmRing → 𝐹 ∈ NrmGrp)
4	2, 3	syl 17	1 ⊢ (𝑊 ∈ NrmMod → 𝐹 ∈ NrmGrp)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  ‘cfv 6491  Scalarcsca 17182  NrmGrpcngp 24523  NrmRingcnrg 24525  NrmModcnlm 24526

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2707  ax-nul 5250

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2714  df-cleq 2727  df-clel 2810  df-ne 2932  df-ral 3051  df-rab 3399  df-v 3441  df-sbc 3740  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4285  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-iota 6447  df-fv 6499  df-ov 7361  df-nrg 24531  df-nlm 24532

This theorem is referenced by:  nlmdsdir  24628  nlmmul0or  24629  nlmvscnlem2  24631  nlmvscnlem1  24632  nlmvscn  24633