Theorem reldmdsmm 20077

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 20076	. 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 6771	1 |- Rel dom (+)m

Syntax hints:   e. wcel 1990   =/= wne 2794   {crab 2916   _Vcvv 3200   dom cdm 5114   Rel wrel 5119   ` cfv 5888  (class class class)co 6650   X_cixp 7908   Fincfn 7955   Basecbs 15857   ↾_s cress 15858   0gc0g 16100   X__scprds 16106   (+)_m cdsmm 20075

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-dm 5124  df-oprab 6654  df-mpt2 6655  df-dsmm 20076

This theorem is referenced by:  dsmmval  20078  dsmmval2  20080