Theorem reldmlmhm 19790

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.

Step	Hyp	Ref	Expression
1		df-lmhm 19787	. 2 ⊢ LMHom = (𝑠 ∈ LMod, 𝑡 ∈ LMod ↦ {𝑓 ∈ (𝑠 GrpHom 𝑡) ∣ [(Scalar‘𝑠) / 𝑤]((Scalar‘𝑡) = 𝑤 ∧ ∀𝑥 ∈ (Base‘𝑤)∀𝑦 ∈ (Base‘𝑠)(𝑓‘(𝑥( ·𝑠 ‘𝑠)𝑦)) = (𝑥( ·𝑠 ‘𝑡)(𝑓‘𝑦)))})
2	1	reldmmpo 7264	1 ⊢ Rel dom LMHom

Syntax hints:   ∧ wa 399   = wceq 1538  ∀wral 3106  {crab 3110  [wsbc 3720  dom cdm 5519  Rel wrel 5524  ‘cfv 6324  (class class class)co 7135  Basecbs 16475  Scalarcsca 16560   ·_𝑠 cvsca 16561   GrpHom cghm 18347  LModclmod 19627   LMHom clmhm 19784

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

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 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-opab 5093  df-xp 5525  df-rel 5526  df-dm 5529  df-oprab 7139  df-mpo 7140  df-lmhm 19787

This theorem is referenced by:  mendbas  40128