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Theorem reldmlmhm 16093
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 16090 . 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 6173 1  |-  Rel  dom LMHom
Colors of variables: wff set class
Syntax hints:    /\ wa 359    = wceq 1652   A.wral 2697   {crab 2701   [.wsbc 3153   dom cdm 4870   Rel wrel 4875   ` cfv 5446  (class class class)co 6073   Basecbs 13461  Scalarcsca 13524   .scvsca 13525    GrpHom cghm 14995   LModclmod 15942   LMHom clmhm 16087
This theorem is referenced by:  mendbas  27460
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-xp 4876  df-rel 4877  df-dm 4880  df-oprab 6077  df-mpt2 6078  df-lmhm 16090
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