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Theorem lmodfgrp 18644
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 18643 . 2 (𝑊 ∈ LMod → 𝐹 ∈ Ring)
3 ringgrp 18324 . 2 (𝐹 ∈ Ring → 𝐹 ∈ Grp)
42, 3syl 17 1 (𝑊 ∈ LMod → 𝐹 ∈ Grp)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  wcel 1976  cfv 5790  Scalarcsca 15720  Grpcgrp 17194  Ringcrg 18319  LModclmod 18635
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-nul 4712
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-iota 5754  df-fv 5798  df-ov 6530  df-ring 18321  df-lmod 18637
This theorem is referenced by:  lmodacl  18646  lmodsn0  18648  lmodvneg1  18678  lssvsubcl  18714  lspsnneg  18776  lvecvscan2  18882  lspexch  18899  lspsolvlem  18912  ipsubdir  19754  ipsubdi  19755  ip2eq  19765  ocvlss  19783  lsmcss  19803  islindf4  19944  clmfgrp  22627  lflmul  33197  lkrlss  33224  eqlkr  33228  lkrlsp  33231  lshpkrlem1  33239  ldualvsubval  33286  lcfrlem1  35673  lcdvsubval  35749  lmodvsmdi  41979  ascl0  41981  lincsum  42034  lincsumcl  42036  lincext1  42059  lindslinindsimp1  42062  lindslinindimp2lem1  42063  lindslinindsimp2lem5  42067  ldepsprlem  42077  ldepspr  42078  lincresunit3lem3  42079  lincresunit3lem1  42084  lincresunit3lem2  42085  lincresunit3  42086
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