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Theorem reldmlmhm 16106
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 16103 . 2  |- LMHom  =  ( s  e.  LMod ,  t  e.  LMod  |->  { f  e.  ( s  GrpHom  t )  |  [. (Scalar `  s )  /  w ]. ( (Scalar `  t
)  =  w  /\  A. x  e.  ( Base `  w ) A. y  e.  ( Base `  s
) ( f `  ( x ( .s
`  s ) y ) )  =  ( x ( .s `  t ) ( f `
 y ) ) ) } )
21reldmmpt2 6184 1  |-  Rel  dom LMHom
Colors of variables: wff set class
Syntax hints:    /\ wa 360    = wceq 1653   A.wral 2707   {crab 2711   [.wsbc 3163   dom cdm 4881   Rel wrel 4886   ` cfv 5457  (class class class)co 6084   Basecbs 13474  Scalarcsca 13537   .scvsca 13538    GrpHom cghm 15008   LModclmod 15955   LMHom clmhm 16100
This theorem is referenced by:  mendbas  27482
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4216  df-opab 4270  df-xp 4887  df-rel 4888  df-dm 4891  df-oprab 6088  df-mpt2 6089  df-lmhm 16103
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