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Theorem lmghm 20875
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 2724 . . 3 (Scalarβ€˜π‘†) = (Scalarβ€˜π‘†)
2 eqid 2724 . . 3 (Scalarβ€˜π‘‡) = (Scalarβ€˜π‘‡)
31, 2lmhmlem 20873 . 2 (𝐹 ∈ (𝑆 LMHom 𝑇) β†’ ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ (Scalarβ€˜π‘‡) = (Scalarβ€˜π‘†))))
43simprld 769 1 (𝐹 ∈ (𝑆 LMHom 𝑇) β†’ 𝐹 ∈ (𝑆 GrpHom 𝑇))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  β€˜cfv 6534  (class class class)co 7402  Scalarcsca 17205   GrpHom cghm 19134  LModclmod 20702   LMHom clmhm 20863
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3771  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-iota 6486  df-fun 6536  df-fv 6542  df-ov 7405  df-oprab 7406  df-mpo 7407  df-lmhm 20866
This theorem is referenced by:  lmhmf  20878  islmhm2  20882  lmhmco  20887  lmhmplusg  20888  lmhmvsca  20889  lmhmf1o  20890  lmhmima  20891  lmhmpreima  20892  reslmhm  20896  reslmhm2  20897  reslmhm2b  20898  lmhmeql  20899  lmimgim  20909  ip0l  21518  ipdir  21521  islindf5  21723  isnmhm2  24613  nmoleub2lem  24985  nmoleub2lem2  24987  nmhmcn  24991  lmhmghmd  32688  lmhmqusker  33029  dimkerim  33219  kercvrlsm  42375  pwssplit4  42381  mendring  42484
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