Theorem lmhmsca 19785

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

Step	Hyp	Ref	Expression
1		lmhmlem.k	. . 3 ⊢ 𝐾 = (Scalar‘𝑆)
2		lmhmlem.l	. . 3 ⊢ 𝐿 = (Scalar‘𝑇)
3	1, 2	lmhmlem 19784	. 2 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐿 = 𝐾)))
4	3	simprrd 772	1 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐿 = 𝐾)

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ‘cfv 6341  (class class class)co 7142  Scalarcsca 16551   GrpHom cghm 18338  LModclmod 19617   LMHom clmhm 19774

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5189  ax-nul 5196  ax-pow 5252  ax-pr 5316

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3488  df-sbc 3764  df-dif 3927  df-un 3929  df-in 3931  df-ss 3940  df-nul 4280  df-if 4454  df-sn 4554  df-pr 4556  df-op 4560  df-uni 4825  df-br 5053  df-opab 5115  df-id 5446  df-xp 5547  df-rel 5548  df-cnv 5549  df-co 5550  df-dm 5551  df-iota 6300  df-fun 6343  df-fv 6349  df-ov 7145  df-oprab 7146  df-mpo 7147  df-lmhm 19777

This theorem is referenced by:  islmhm2  19793  lmhmco  19798  lmhmplusg  19799  lmhmvsca  19800  lmhmf1o  19801  lmhmima  19802  lmhmpreima  19803  reslmhm  19807  reslmhm2  19808  reslmhm2b  19809  lindfmm  20954  lmhmclm  23674  nmoleub2lem3  23702  nmoleub3  23706  lmhmlvec2  31027  lmhmlvec  39223