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Theorem lmodfgrp 18920
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 18919 . 2 (𝑊 ∈ LMod → 𝐹 ∈ Ring)
3 ringgrp 18598 . 2 (𝐹 ∈ Ring → 𝐹 ∈ Grp)
42, 3syl 17 1 (𝑊 ∈ LMod → 𝐹 ∈ Grp)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  wcel 2030  cfv 5926  Scalarcsca 15991  Grpcgrp 17469  Ringcrg 18593  LModclmod 18911
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-nul 4822
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-iota 5889  df-fv 5934  df-ov 6693  df-ring 18595  df-lmod 18913
This theorem is referenced by:  lmodacl  18922  lmodsn0  18924  lmodvneg1  18954  lssvsubcl  18992  lspsnneg  19054  lvecvscan2  19160  lspexch  19177  lspsolvlem  19190  ipsubdir  20035  ipsubdi  20036  ip2eq  20046  ocvlss  20064  lsmcss  20084  islindf4  20225  clmfgrp  22917  lflmul  34673  lkrlss  34700  eqlkr  34704  lkrlsp  34707  lshpkrlem1  34715  ldualvsubval  34762  lcfrlem1  37148  lcdvsubval  37224  lmodvsmdi  42488  ascl0  42490  lincsum  42543  lincsumcl  42545  lincext1  42568  lindslinindsimp1  42571  lindslinindimp2lem1  42572  lindslinindsimp2lem5  42576  ldepsprlem  42586  ldepspr  42587  lincresunit3lem3  42588  lincresunit3lem1  42593  lincresunit3lem2  42594  lincresunit3  42595
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