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Theorem lmhmsca 19355
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 19354 . 2 (𝐹 ∈ (𝑆 LMHom 𝑇) → ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐿 = 𝐾)))
43simprrd 791 1 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐿 = 𝐾)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385   = wceq 1653  wcel 2157  cfv 6105  (class class class)co 6882  Scalarcsca 16274   GrpHom cghm 17974  LModclmod 19185   LMHom clmhm 19344
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2379  ax-ext 2781  ax-sep 4979  ax-nul 4987  ax-pow 5039  ax-pr 5101
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2593  df-eu 2611  df-clab 2790  df-cleq 2796  df-clel 2799  df-nfc 2934  df-ral 3098  df-rex 3099  df-rab 3102  df-v 3391  df-sbc 3638  df-dif 3776  df-un 3778  df-in 3780  df-ss 3787  df-nul 4120  df-if 4282  df-sn 4373  df-pr 4375  df-op 4379  df-uni 4633  df-br 4848  df-opab 4910  df-id 5224  df-xp 5322  df-rel 5323  df-cnv 5324  df-co 5325  df-dm 5326  df-iota 6068  df-fun 6107  df-fv 6113  df-ov 6885  df-oprab 6886  df-mpt2 6887  df-lmhm 19347
This theorem is referenced by:  islmhm2  19363  lmhmco  19368  lmhmplusg  19369  lmhmvsca  19370  lmhmf1o  19371  lmhmima  19372  lmhmpreima  19373  reslmhm  19377  reslmhm2  19378  reslmhm2b  19379  lindfmm  20495  lmhmclm  23218  nmoleub2lem3  23246  nmoleub3  23250
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