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Theorem reldmdsmm 18276
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 18275 . 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 6304 1  |-  Rel  dom  (+)m
Colors of variables: wff setvar class
Syntax hints:    e. wcel 1758    =/= wne 2644   {crab 2799   _Vcvv 3071   dom cdm 4941   Rel wrel 4946   ` cfv 5519  (class class class)co 6193   X_cixp 7366   Fincfn 7413   Basecbs 14285   ↾s cress 14286   0gc0g 14489   X_scprds 14495    (+)m cdsmm 18274
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pr 4632
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-op 3985  df-br 4394  df-opab 4452  df-xp 4947  df-rel 4948  df-dm 4951  df-oprab 6197  df-mpt2 6198  df-dsmm 18275
This theorem is referenced by:  dsmmval  18277  dsmmval2  18279
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