Theorem clmfgrp 25372

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 25368	. 2 ⊢ (𝑊 ∈ ℂMod → 𝑊 ∈ LMod)
2		clm0.f	. . 3 ⊢ 𝐹 = (Scalar‘𝑊)
3	2	lmodfgrp 21124	. 2 ⊢ (𝑊 ∈ LMod → 𝐹 ∈ Grp)
4	1, 3	syl 18	1 ⊢ (𝑊 ∈ ℂMod → 𝐹 ∈ Grp)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  ‘cfv 6531  Scalarcsca 17411  Grpcgrp 19124  LModclmod 21115  ℂModcclm 25363

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-nul 5260

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-rab 3414  df-v 3453  df-sbc 3740  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-iota 6487  df-fv 6539  df-ov 7415  df-ring 20441  df-lmod 21117  df-clm 25364

This theorem is used by:  ncvspi  25457