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.

Step	Hyp	Ref	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 } ) )
2	1	reldmmpt2 6304	1 |- Rel dom (+)m

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