Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  reldmdsmm Unicode version

Theorem reldmdsmm 26347
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 26346 . 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 5997 1  |-  Rel  dom  (+)m
Colors of variables: wff set class
Syntax hints:    e. wcel 1701    =/= wne 2479   {crab 2581   _Vcvv 2822   dom cdm 4726   Rel wrel 4731   ` cfv 5292  (class class class)co 5900   X_cixp 6860   Fincfn 6906   Basecbs 13195   ↾s cress 13196   X_scprds 13395   0gc0g 13449    (+)m cdsmm 26345
This theorem is referenced by:  dsmmval  26348  dsmmval2  26350
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pr 4251
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-br 4061  df-opab 4115  df-xp 4732  df-rel 4733  df-dm 4736  df-oprab 5904  df-mpt2 5905  df-dsmm 26346
  Copyright terms: Public domain W3C validator