Theorem lmodfgrp 20959

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 20958	. 2 ⊢ (𝑊 ∈ LMod → 𝐹 ∈ Ring)
3		ringgrp 20311	. 2 ⊢ (𝐹 ∈ Ring → 𝐹 ∈ Grp)
4	2, 3	syl 18	1 ⊢ (𝑊 ∈ LMod → 𝐹 ∈ Grp)

Syntax hints:   → wi 4   = wceq 1563   ∈ wcel 2145  ‘cfv 6525  Scalarcsca 17303  Grpcgrp 18990  Ringcrg 20306  LModclmod 20950

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

This theorem is referenced by:  lmodacl  20962  lmodsn0  20964  lmodvneg1  20995  lssvsubcl  21034  lspsnneg  21096  lvecvscan2  21205  lspexch  21222  lspsolvlem  21235  ipsubdir  21752  ipsubdi  21753  ip2eq  21763  ocvlss  21782  lsmcss  21802  islindf4  21948  ascl0  21994  clmfgrp  25191  lmodvslmhm  33283  lflmul  39704  lkrlss  39731  eqlkr  39735  lkrlsp  39738  lshpkrlem1  39746  ldualvsubval  39793  lcfrlem1  42178  lcdvsubval  42254  lmodvsmdi  49010  lincsum  49060  lincsumcl  49062  lincext1  49085  lindslinindsimp1  49088  lindslinindimp2lem1  49089  lindslinindsimp2lem5  49093  ldepsprlem  49103  ldepspr  49104  lincresunit3lem3  49105  lincresunit3lem1  49110  lincresunit3lem2  49111  lincresunit3  49112