MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  lmhmlmod2 Structured version   Visualization version   GIF version

Theorem lmhmlmod2 21031
Description: A homomorphism of left modules has a left module as codomain. (Contributed by Stefan O'Rear, 1-Jan-2015.)
Assertion
Ref Expression
lmhmlmod2 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑇 ∈ LMod)

Proof of Theorem lmhmlmod2
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))))
43simplrd 770 1 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑇 ∈ LMod)
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:  lmhmco  21042  lmhmplusg  21043  lmhmvsca  21044  lmhmf1o  21045  lmhmima  21046  lmhmpreima  21047  lmhmlsp  21048  lmhmkerlss  21050  reslmhm  21051  islmim  21061  lmicrcl  21070  lmhmlvec  21109  lindfmm  21847  lindsmm  21848  lmhmclm  25120  lmhmqusker  33445  lmhmlvec2  33670  dimkerim  33678  lmhmfgima  43096  lnmepi  43097  lmhmfgsplit  43098  lmhmlnmsplit  43099
  Copyright terms: Public domain W3C validator