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Definition df-dsmm 27301
Description: The direct sum of a family of Abelian groups or left modules is the induced group structure on finite linear combinations of elements, here represented as functions with finite support. (Contributed by Stefan O'Rear, 7-Jan-2015.)
Assertion
Ref Expression
df-dsmm  |-  (+)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 } ) )
Distinct variable group:    s, r, f, x

Detailed syntax breakdown of Definition df-dsmm
StepHypRef Expression
1 cdsmm 27300 . 2  class  (+)m
2 vs . . 3  set  s
3 vr . . 3  set  r
4 cvv 2801 . . 3  class  _V
52cv 1631 . . . . 5  class  s
63cv 1631 . . . . 5  class  r
7 cprds 13362 . . . . 5  class  X_s
85, 6, 7co 5874 . . . 4  class  ( s
X_s r )
9 vx . . . . . . . . . 10  set  x
109cv 1631 . . . . . . . . 9  class  x
11 vf . . . . . . . . . 10  set  f
1211cv 1631 . . . . . . . . 9  class  f
1310, 12cfv 5271 . . . . . . . 8  class  ( f `
 x )
1410, 6cfv 5271 . . . . . . . . 9  class  ( r `
 x )
15 c0g 13416 . . . . . . . . 9  class  0g
1614, 15cfv 5271 . . . . . . . 8  class  ( 0g
`  ( r `  x ) )
1713, 16wne 2459 . . . . . . 7  wff  ( f `
 x )  =/=  ( 0g `  (
r `  x )
)
186cdm 4705 . . . . . . 7  class  dom  r
1917, 9, 18crab 2560 . . . . . 6  class  { x  e.  dom  r  |  ( f `  x )  =/=  ( 0g `  ( r `  x
) ) }
20 cfn 6879 . . . . . 6  class  Fin
2119, 20wcel 1696 . . . . 5  wff  { x  e.  dom  r  |  ( f `  x )  =/=  ( 0g `  ( r `  x
) ) }  e.  Fin
22 cbs 13164 . . . . . . 7  class  Base
2314, 22cfv 5271 . . . . . 6  class  ( Base `  ( r `  x
) )
249, 18, 23cixp 6833 . . . . 5  class  X_ x  e.  dom  r ( Base `  ( r `  x
) )
2521, 11, 24crab 2560 . . . 4  class  { f  e.  X_ x  e.  dom  r ( Base `  (
r `  x )
)  |  { x  e.  dom  r  |  ( f `  x )  =/=  ( 0g `  ( r `  x
) ) }  e.  Fin }
26 cress 13165 . . . 4  classs
278, 25, 26co 5874 . . 3  class  ( ( 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 } )
282, 3, 4, 4, 27cmpt2 5876 . 2  class  ( 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 } ) )
291, 28wceq 1632 1  wff  (+)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 } ) )
Colors of variables: wff set class
This definition is referenced by:  reldmdsmm  27302  dsmmval  27303
  Copyright terms: Public domain W3C validator