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Theorem lmghm 19795
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 2819 . . 3 (Scalar‘𝑆) = (Scalar‘𝑆)
2 eqid 2819 . . 3 (Scalar‘𝑇) = (Scalar‘𝑇)
31, 2lmhmlem 19793 . 2 (𝐹 ∈ (𝑆 LMHom 𝑇) → ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ (Scalar‘𝑇) = (Scalar‘𝑆))))
43simprld 770 1 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1531  wcel 2108  cfv 6348  (class class class)co 7148  Scalarcsca 16560   GrpHom cghm 18347  LModclmod 19626   LMHom clmhm 19783
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 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7151  df-oprab 7152  df-mpo 7153  df-lmhm 19786
This theorem is referenced by:  lmhmf  19798  islmhm2  19802  lmhmco  19807  lmhmplusg  19808  lmhmvsca  19809  lmhmf1o  19810  lmhmima  19811  lmhmpreima  19812  reslmhm  19816  reslmhm2  19817  reslmhm2b  19818  lmhmeql  19819  lmimgim  19829  ip0l  20772  ipdir  20775  islindf5  20975  isnmhm2  23353  nmoleub2lem  23710  nmoleub2lem2  23712  nmhmcn  23716  dimkerim  31016  kercvrlsm  39674  pwssplit4  39680  mendring  39783
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