Theorem clmfgrp 25191

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 25187	. 2 ⊢ (𝑊 ∈ ℂMod → 𝑊 ∈ LMod)
2		clm0.f	. . 3 ⊢ 𝐹 = (Scalar‘𝑊)
3	2	lmodfgrp 20959	. 2 ⊢ (𝑊 ∈ LMod → 𝐹 ∈ Grp)
4	1, 3	syl 18	1 ⊢ (𝑊 ∈ ℂMod → 𝐹 ∈ Grp)

Syntax hints:   → wi 4   = wceq 1563   ∈ wcel 2145  ‘cfv 6525  Scalarcsca 17303  Grpcgrp 18990  LModclmod 20950  ℂModcclm 25182

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-nul 5261

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3080  df-rab 3418  df-v 3459  df-sbc 3748  df-dif 3910  df-un 3912  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-iota 6481  df-fv 6533  df-ov 7403  df-ring 20308  df-lmod 20952  df-clm 25183

This theorem is referenced by:  ncvspi  25276