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Theorem lmhmsca 18851
Description: A homomorphism of left modules constrains both modules to the same ring of scalars. (Contributed by Stefan O'Rear, 1-Jan-2015.)
Hypotheses
Ref Expression
lmhmlem.k 𝐾 = (Scalar‘𝑆)
lmhmlem.l 𝐿 = (Scalar‘𝑇)
Assertion
Ref Expression
lmhmsca (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐿 = 𝐾)

Proof of Theorem lmhmsca
StepHypRef Expression
1 lmhmlem.k . . 3 𝐾 = (Scalar‘𝑆)
2 lmhmlem.l . . 3 𝐿 = (Scalar‘𝑇)
31, 2lmhmlem 18850 . 2 (𝐹 ∈ (𝑆 LMHom 𝑇) → ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐿 = 𝐾)))
43simprrd 793 1 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐿 = 𝐾)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  wcel 1977  cfv 5804  (class class class)co 6549  Scalarcsca 15771   GrpHom cghm 17480  LModclmod 18686   LMHom clmhm 18840
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-lmhm 18843
This theorem is referenced by:  islmhm2  18859  lmhmco  18864  lmhmplusg  18865  lmhmvsca  18866  lmhmf1o  18867  lmhmima  18868  lmhmpreima  18869  reslmhm  18873  reslmhm2  18874  reslmhm2b  18875  lindfmm  19985  lmhmclm  22695  nmoleub2lem3  22723  nmoleub3  22727
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