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

Theorem lmhmsca 20922
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 = (Scalarβ€˜π‘†)
lmhmlem.l 𝐿 = (Scalarβ€˜π‘‡)
Assertion
Ref Expression
lmhmsca (𝐹 ∈ (𝑆 LMHom 𝑇) β†’ 𝐿 = 𝐾)

Proof of Theorem lmhmsca
StepHypRef Expression
1 lmhmlem.k . . 3 𝐾 = (Scalarβ€˜π‘†)
2 lmhmlem.l . . 3 𝐿 = (Scalarβ€˜π‘‡)
31, 2lmhmlem 20921 . 2 (𝐹 ∈ (𝑆 LMHom 𝑇) β†’ ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐿 = 𝐾)))
43simprrd 772 1 (𝐹 ∈ (𝑆 LMHom 𝑇) β†’ 𝐿 = 𝐾)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  β€˜cfv 6553  (class class class)co 7426  Scalarcsca 17243   GrpHom cghm 19174  LModclmod 20750   LMHom clmhm 20911
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 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-iota 6505  df-fun 6555  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-lmhm 20914
This theorem is referenced by:  islmhm2  20930  lmhmco  20935  lmhmplusg  20936  lmhmvsca  20937  lmhmf1o  20938  lmhmima  20939  lmhmpreima  20940  reslmhm  20944  reslmhm2  20945  reslmhm2b  20946  lmhmlvec  21002  lindfmm  21768  lmhmclm  25034  nmoleub2lem3  25062  nmoleub3  25066  lmhmqusker  33152  lmhmlvec2  33350
  Copyright terms: Public domain W3C validator