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Theorem reldmlmhm 15782
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 15779 . 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 5955 1  |-  Rel  dom LMHom
Colors of variables: wff set class
Syntax hints:    /\ wa 358    = wceq 1623   A.wral 2543   {crab 2547   [.wsbc 2991   dom cdm 4689   Rel wrel 4694   ` cfv 5255  (class class class)co 5858   Basecbs 13148  Scalarcsca 13211   .scvsca 13212    GrpHom cghm 14680   LModclmod 15627   LMHom clmhm 15776
This theorem is referenced by:  mendbas  27492
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-dm 4699  df-oprab 5862  df-mpt2 5863  df-lmhm 15779
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