Theorem lmodfgrp 21124

Description: The scalar component of a left module is an additive group. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)

Hypothesis

Ref	Expression
lmodring.1	⊢ 𝐹 = (Scalar‘𝑊)

Assertion

Ref	Expression
lmodfgrp	⊢ (𝑊 ∈ LMod → 𝐹 ∈ Grp)

Proof of Theorem lmodfgrp

Step	Hyp	Ref	Expression
1		lmodring.1	. . 3 ⊢ 𝐹 = (Scalar‘𝑊)
2	1	lmodring 21123	. 2 ⊢ (𝑊 ∈ LMod → 𝐹 ∈ Ring)
3		ringgrp 20444	. 2 ⊢ (𝐹 ∈ Ring → 𝐹 ∈ Grp)
4	2, 3	syl 18	1 ⊢ (𝑊 ∈ LMod → 𝐹 ∈ Grp)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  ‘cfv 6531  Scalarcsca 17411  Grpcgrp 19124  Ringcrg 20439  LModclmod 21115

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

This theorem is used by:  lmodacl  21127  lmodsn0  21129  lmodvneg1  21160  lssvsubcl  21199  lspsnneg  21261  lvecvscan2  21370  lspexch  21387  lspsolvlem  21400  ipsubdir  21928  ipsubdi  21929  ip2eq  21939  ocvlss  21958  lsmcss  21978  islindf4  22124  ascl0  22172  clmfgrp  25372  lmodvslmhm  33593  lflmul  40093  lkrlss  40120  eqlkr  40124  lkrlsp  40127  lshpkrlem1  40135  ldualvsubval  40182  lcfrlem1  42567  lcdvsubval  42643  lmodvsmdi  49435  lincsum  49485  lincsumcl  49487  lincext1  49510  lindslinindsimp1  49513  lindslinindimp2lem1  49514  lindslinindsimp2lem5  49518  ldepsprlem  49528  ldepspr  49529  lincresunit3lem3  49530  lincresunit3lem1  49535  lincresunit3lem2  49536  lincresunit3  49537