Theorem clmfgrp 24972

Description: The scalar ring of a subcomplex module is a group. (Contributed by Mario Carneiro, 16-Oct-2015.)

Hypothesis

Ref	Expression
clm0.f	⊢ 𝐹 = (Scalar‘𝑊)

Assertion

Ref	Expression
clmfgrp	⊢ (𝑊 ∈ ℂMod → 𝐹 ∈ Grp)

Proof of Theorem clmfgrp

Step	Hyp	Ref	Expression
1		clmlmod 24968	. 2 ⊢ (𝑊 ∈ ℂMod → 𝑊 ∈ LMod)
2		clm0.f	. . 3 ⊢ 𝐹 = (Scalar‘𝑊)
3	2	lmodfgrp 20734	. 2 ⊢ (𝑊 ∈ LMod → 𝐹 ∈ Grp)
4	1, 3	syl 17	1 ⊢ (𝑊 ∈ ℂMod → 𝐹 ∈ Grp)

Syntax hints:   → wi 4   = wceq 1534   ∈ wcel 2099  ‘cfv 6542  Scalarcsca 17221  Grpcgrp 18875  LModclmod 20725  ℂModcclm 24963

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2698  ax-nul 5300

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-ne 2936  df-ral 3057  df-rab 3428  df-v 3471  df-sbc 3775  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-iota 6494  df-fv 6550  df-ov 7417  df-ring 20159  df-lmod 20727  df-clm 24964

This theorem is referenced by:  ncvspi  25058