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Theorem lmodfgrp 19908
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
StepHypRef Expression
1 lmodring.1 . . 3 𝐹 = (Scalar‘𝑊)
21lmodring 19907 . 2 (𝑊 ∈ LMod → 𝐹 ∈ Ring)
3 ringgrp 19567 . 2 (𝐹 ∈ Ring → 𝐹 ∈ Grp)
42, 3syl 17 1 (𝑊 ∈ LMod → 𝐹 ∈ Grp)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1543  wcel 2110  cfv 6380  Scalarcsca 16805  Grpcgrp 18365  Ringcrg 19562  LModclmod 19899
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-nul 5199
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-sbc 3695  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-iota 6338  df-fv 6388  df-ov 7216  df-ring 19564  df-lmod 19901
This theorem is referenced by:  lmodacl  19910  lmodsn0  19912  lmodvneg1  19942  lssvsubcl  19980  lspsnneg  20043  lvecvscan2  20149  lspexch  20166  lspsolvlem  20179  ipsubdir  20604  ipsubdi  20605  ip2eq  20615  ocvlss  20634  lsmcss  20654  islindf4  20800  ascl0  20843  clmfgrp  23968  lmodvslmhm  31029  lflmul  36819  lkrlss  36846  eqlkr  36850  lkrlsp  36853  lshpkrlem1  36861  ldualvsubval  36908  lcfrlem1  39293  lcdvsubval  39369  lmodvsmdi  45391  lincsum  45443  lincsumcl  45445  lincext1  45468  lindslinindsimp1  45471  lindslinindimp2lem1  45472  lindslinindsimp2lem5  45476  ldepsprlem  45486  ldepspr  45487  lincresunit3lem3  45488  lincresunit3lem1  45493  lincresunit3lem2  45494  lincresunit3  45495
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