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Theorem reldmlmhm 16029
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 16026 . 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 6121 1  |-  Rel  dom LMHom
Colors of variables: wff set class
Syntax hints:    /\ wa 359    = wceq 1649   A.wral 2650   {crab 2654   [.wsbc 3105   dom cdm 4819   Rel wrel 4824   ` cfv 5395  (class class class)co 6021   Basecbs 13397  Scalarcsca 13460   .scvsca 13461    GrpHom cghm 14931   LModclmod 15878   LMHom clmhm 16023
This theorem is referenced by:  mendbas  27162
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pr 4345
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-br 4155  df-opab 4209  df-xp 4825  df-rel 4826  df-dm 4829  df-oprab 6025  df-mpt2 6026  df-lmhm 16026
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