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Theorem reldmlmhm 19797
Description: Lemma for module homomorphisms. (Contributed by Stefan O'Rear, 31-Dec-2014.)
Assertion
Ref Expression
reldmlmhm Rel dom LMHom

Proof of Theorem reldmlmhm
Dummy variables 𝑓 𝑠 𝑡 𝑤 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-lmhm 19794 . 2 LMHom = (𝑠 ∈ LMod, 𝑡 ∈ LMod ↦ {𝑓 ∈ (𝑠 GrpHom 𝑡) ∣ [(Scalar‘𝑠) / 𝑤]((Scalar‘𝑡) = 𝑤 ∧ ∀𝑥 ∈ (Base‘𝑤)∀𝑦 ∈ (Base‘𝑠)(𝑓‘(𝑥( ·𝑠𝑠)𝑦)) = (𝑥( ·𝑠𝑡)(𝑓𝑦)))})
21reldmmpo 7278 1 Rel dom LMHom
Colors of variables: wff setvar class
Syntax hints:  wa 399   = wceq 1538  wral 3133  {crab 3137  [wsbc 3758  dom cdm 5542  Rel wrel 5547  cfv 6343  (class class class)co 7149  Basecbs 16483  Scalarcsca 16568   ·𝑠 cvsca 16569   GrpHom cghm 18355  LModclmod 19634   LMHom clmhm 19791
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-br 5053  df-opab 5115  df-xp 5548  df-rel 5549  df-dm 5552  df-oprab 7153  df-mpo 7154  df-lmhm 19794
This theorem is referenced by:  mendbas  40044
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