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Theorem reldmdsmm 27199
Description: The direct sum is a well-behaved binary operator. (Contributed by Stefan O'Rear, 7-Jan-2015.)
Assertion
Ref Expression
reldmdsmm  |-  Rel  dom  (+)m

Proof of Theorem reldmdsmm
Dummy variables  s 
r  f  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-dsmm 27198 . 2  |-  (+)m  =  ( s  e.  _V , 
r  e.  _V  |->  ( ( s X_s r )s  { f  e.  X_ x  e.  dom  r (
Base `  ( r `  x ) )  |  { x  e.  dom  r  |  ( f `  x )  =/=  ( 0g `  ( r `  x ) ) }  e.  Fin } ) )
21reldmmpt2 5955 1  |-  Rel  dom  (+)m
Colors of variables: wff set class
Syntax hints:    e. wcel 1684    =/= wne 2446   {crab 2547   _Vcvv 2788   dom cdm 4689   Rel wrel 4694   ` cfv 5255  (class class class)co 5858   X_cixp 6817   Fincfn 6863   Basecbs 13148   ↾s cress 13149   X_scprds 13346   0gc0g 13400    (+)m cdsmm 27197
This theorem is referenced by:  dsmmval  27200  dsmmval2  27202
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-dsmm 27198
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