Theorem clmfgrp 25118

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 25114	. 2 ⊢ (𝑊 ∈ ℂMod → 𝑊 ∈ LMod)
2		clm0.f	. . 3 ⊢ 𝐹 = (Scalar‘𝑊)
3	2	lmodfgrp 20884	. 2 ⊢ (𝑊 ∈ LMod → 𝐹 ∈ Grp)
4	1, 3	syl 17	1 ⊢ (𝑊 ∈ ℂMod → 𝐹 ∈ Grp)

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2106  ‘cfv 6563  Scalarcsca 17301  Grpcgrp 18964  LModclmod 20875  ℂModcclm 25109

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-nul 5312

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-ov 7434  df-ring 20253  df-lmod 20877  df-clm 25110

This theorem is referenced by:  ncvspi  25204