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Theorem lmghm 21047
Description: A homomorphism of left modules is a homomorphism of groups. (Contributed by Stefan O'Rear, 1-Jan-2015.)
Assertion
Ref Expression
lmghm (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇))

Proof of Theorem lmghm
StepHypRef Expression
1 eqid 2734 . . 3 (Scalar‘𝑆) = (Scalar‘𝑆)
2 eqid 2734 . . 3 (Scalar‘𝑇) = (Scalar‘𝑇)
31, 2lmhmlem 21045 . 2 (𝐹 ∈ (𝑆 LMHom 𝑇) → ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ (Scalar‘𝑇) = (Scalar‘𝑆))))
43simprld 772 1 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1536  wcel 2105  cfv 6562  (class class class)co 7430  Scalarcsca 17300   GrpHom cghm 19242  LModclmod 20874   LMHom clmhm 21035
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-iota 6515  df-fun 6564  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-lmhm 21038
This theorem is referenced by:  lmhmf  21050  islmhm2  21054  lmhmco  21059  lmhmplusg  21060  lmhmvsca  21061  lmhmf1o  21062  lmhmima  21063  lmhmpreima  21064  reslmhm  21068  reslmhm2  21069  reslmhm2b  21070  lmhmeql  21071  lmimgim  21081  ip0l  21671  ipdir  21674  islindf5  21876  isnmhm2  24788  nmoleub2lem  25160  nmoleub2lem2  25162  nmhmcn  25166  lmhmghmd  33024  lmhmqusker  33424  dimkerim  33654  kercvrlsm  43071  pwssplit4  43077  mendring  43176
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