MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  lmghm Structured version   Visualization version   GIF version

Theorem lmghm 21030
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 2737 . . 3 (Scalar‘𝑆) = (Scalar‘𝑆)
2 eqid 2737 . . 3 (Scalar‘𝑇) = (Scalar‘𝑇)
31, 2lmhmlem 21028 . 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 1540  wcel 2108  cfv 6561  (class class class)co 7431  Scalarcsca 17300   GrpHom cghm 19230  LModclmod 20858   LMHom clmhm 21018
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-lmhm 21021
This theorem is referenced by:  lmhmf  21033  islmhm2  21037  lmhmco  21042  lmhmplusg  21043  lmhmvsca  21044  lmhmf1o  21045  lmhmima  21046  lmhmpreima  21047  reslmhm  21051  reslmhm2  21052  reslmhm2b  21053  lmhmeql  21054  lmimgim  21064  ip0l  21654  ipdir  21657  islindf5  21859  isnmhm2  24773  nmoleub2lem  25147  nmoleub2lem2  25149  nmhmcn  25153  lmhmghmd  33042  lmhmqusker  33445  dimkerim  33678  kercvrlsm  43095  pwssplit4  43101  mendring  43200
  Copyright terms: Public domain W3C validator