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Theorem lmghm 20915
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) β†’ 𝐹 ∈ (𝑆 GrpHom 𝑇))

Proof of Theorem lmghm
StepHypRef Expression
1 eqid 2728 . . 3 (Scalarβ€˜π‘†) = (Scalarβ€˜π‘†)
2 eqid 2728 . . 3 (Scalarβ€˜π‘‡) = (Scalarβ€˜π‘‡)
31, 2lmhmlem 20913 . 2 (𝐹 ∈ (𝑆 LMHom 𝑇) β†’ ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ (Scalarβ€˜π‘‡) = (Scalarβ€˜π‘†))))
43simprld 771 1 (𝐹 ∈ (𝑆 LMHom 𝑇) β†’ 𝐹 ∈ (𝑆 GrpHom 𝑇))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1534   ∈ wcel 2099  β€˜cfv 6548  (class class class)co 7420  Scalarcsca 17235   GrpHom cghm 19166  LModclmod 20742   LMHom clmhm 20903
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-iota 6500  df-fun 6550  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-lmhm 20906
This theorem is referenced by:  lmhmf  20918  islmhm2  20922  lmhmco  20927  lmhmplusg  20928  lmhmvsca  20929  lmhmf1o  20930  lmhmima  20931  lmhmpreima  20932  reslmhm  20936  reslmhm2  20937  reslmhm2b  20938  lmhmeql  20939  lmimgim  20949  ip0l  21567  ipdir  21570  islindf5  21772  isnmhm2  24668  nmoleub2lem  25040  nmoleub2lem2  25042  nmhmcn  25046  lmhmghmd  32757  lmhmqusker  33127  dimkerim  33321  kercvrlsm  42507  pwssplit4  42513  mendring  42616
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