Theorem reldmlmhm 21015

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 21012	. 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 7495	1 ⊢ Rel dom LMHom

Syntax hints:   ∧ wa 395   = wceq 1542  ∀wral 3052  {crab 3390  [wsbc 3729  dom cdm 5625  Rel wrel 5630  ‘cfv 6493  (class class class)co 7361  Basecbs 17173  Scalarcsca 17217   ·_𝑠 cvsca 17218   GrpHom cghm 19181  LModclmod 20849   LMHom clmhm 21009

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 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-xp 5631  df-rel 5632  df-dm 5635  df-oprab 7365  df-mpo 7366  df-lmhm 21012

This theorem is referenced by:  mendbas  43629