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Theorem lmhmsca 19794
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 19793 . 2 (𝐹 ∈ (𝑆 LMHom 𝑇) → ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐿 = 𝐾)))
43simprrd 772 1 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐿 = 𝐾)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1530  wcel 2107  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 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  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 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  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:  islmhm2  19802  lmhmco  19807  lmhmplusg  19808  lmhmvsca  19809  lmhmf1o  19810  lmhmima  19811  lmhmpreima  19812  reslmhm  19816  reslmhm2  19817  reslmhm2b  19818  lindfmm  20963  lmhmclm  23683  nmoleub2lem3  23711  nmoleub3  23715  lmhmlvec2  31005  lmhmlvec  39133
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