Theorem clmfgrp 25111

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 25107	. 2 ⊢ (𝑊 ∈ ℂMod → 𝑊 ∈ LMod)
2		clm0.f	. . 3 ⊢ 𝐹 = (Scalar‘𝑊)
3	2	lmodfgrp 20914	. 2 ⊢ (𝑊 ∈ LMod → 𝐹 ∈ Grp)
4	1, 3	syl 17	1 ⊢ (𝑊 ∈ ℂMod → 𝐹 ∈ Grp)

Syntax hints:   → wi 4   = wceq 1559   ∈ wcel 2141  ‘cfv 6515  Scalarcsca 17270  Grpcgrp 18956  LModclmod 20905  ℂModcclm 25102

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-nul 5255

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3076  df-rab 3414  df-v 3455  df-sbc 3745  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-iota 6471  df-fv 6523  df-ov 7393  df-ring 20262  df-lmod 20907  df-clm 25103

This theorem is referenced by:  ncvspi  25196