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Theorem gsumzres 15194
Description: Extend a finite group sum by padding outside with zeroes. (Contributed by Mario Carneiro, 24-Apr-2016.)
Hypotheses
Ref Expression
gsumzcl.b  |-  B  =  ( Base `  G
)
gsumzcl.0  |-  .0.  =  ( 0g `  G )
gsumzcl.z  |-  Z  =  (Cntz `  G )
gsumzcl.g  |-  ( ph  ->  G  e.  Mnd )
gsumzcl.a  |-  ( ph  ->  A  e.  V )
gsumzcl.f  |-  ( ph  ->  F : A --> B )
gsumzcl.c  |-  ( ph  ->  ran  F  C_  ( Z `  ran  F ) )
gsumzres.s  |-  ( ph  ->  ( `' F "
( _V  \  {  .0.  } ) )  C_  W )
gsumzres.w  |-  ( ph  ->  ( `' F "
( _V  \  {  .0.  } ) )  e. 
Fin )
Assertion
Ref Expression
gsumzres  |-  ( ph  ->  ( G  gsumg  ( F  |`  W ) )  =  ( G 
gsumg  F ) )

Proof of Theorem gsumzres
Dummy variables  f 
k are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 gsumzcl.g . . . . . . 7  |-  ( ph  ->  G  e.  Mnd )
2 gsumzcl.a . . . . . . . 8  |-  ( ph  ->  A  e.  V )
3 inex1g 4157 . . . . . . . 8  |-  ( A  e.  V  ->  ( A  i^i  W )  e. 
_V )
42, 3syl 15 . . . . . . 7  |-  ( ph  ->  ( A  i^i  W
)  e.  _V )
5 gsumzcl.0 . . . . . . . 8  |-  .0.  =  ( 0g `  G )
65gsumz 14458 . . . . . . 7  |-  ( ( G  e.  Mnd  /\  ( A  i^i  W )  e.  _V )  -> 
( G  gsumg  ( k  e.  ( A  i^i  W ) 
|->  .0.  ) )  =  .0.  )
71, 4, 6syl2anc 642 . . . . . 6  |-  ( ph  ->  ( G  gsumg  ( k  e.  ( A  i^i  W ) 
|->  .0.  ) )  =  .0.  )
85gsumz 14458 . . . . . . 7  |-  ( ( G  e.  Mnd  /\  A  e.  V )  ->  ( G  gsumg  ( k  e.  A  |->  .0.  ) )  =  .0.  )
91, 2, 8syl2anc 642 . . . . . 6  |-  ( ph  ->  ( G  gsumg  ( k  e.  A  |->  .0.  ) )  =  .0.  )
107, 9eqtr4d 2318 . . . . 5  |-  ( ph  ->  ( G  gsumg  ( k  e.  ( A  i^i  W ) 
|->  .0.  ) )  =  ( G  gsumg  ( k  e.  A  |->  .0.  ) ) )
1110adantr 451 . . . 4  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  ( G  gsumg  ( k  e.  ( A  i^i  W ) 
|->  .0.  ) )  =  ( G  gsumg  ( k  e.  A  |->  .0.  ) ) )
12 resres 4968 . . . . . . . 8  |-  ( ( F  |`  A )  |`  W )  =  ( F  |`  ( A  i^i  W ) )
13 gsumzcl.f . . . . . . . . . 10  |-  ( ph  ->  F : A --> B )
14 ffn 5389 . . . . . . . . . 10  |-  ( F : A --> B  ->  F  Fn  A )
15 fnresdm 5353 . . . . . . . . . 10  |-  ( F  Fn  A  ->  ( F  |`  A )  =  F )
1613, 14, 153syl 18 . . . . . . . . 9  |-  ( ph  ->  ( F  |`  A )  =  F )
1716reseq1d 4954 . . . . . . . 8  |-  ( ph  ->  ( ( F  |`  A )  |`  W )  =  ( F  |`  W ) )
1812, 17syl5eqr 2329 . . . . . . 7  |-  ( ph  ->  ( F  |`  ( A  i^i  W ) )  =  ( F  |`  W ) )
1918adantr 451 . . . . . 6  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  ( F  |`  ( A  i^i  W ) )  =  ( F  |`  W )
)
20 ssid 3197 . . . . . . . . . 10  |-  ( `' F " ( _V 
\  {  .0.  }
) )  C_  ( `' F " ( _V 
\  {  .0.  }
) )
2120a1i 10 . . . . . . . . 9  |-  ( ph  ->  ( `' F "
( _V  \  {  .0.  } ) )  C_  ( `' F " ( _V 
\  {  .0.  }
) ) )
2213, 21gsumcllem 15193 . . . . . . . 8  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  F  =  ( k  e.  A  |->  .0.  ) )
2322reseq1d 4954 . . . . . . 7  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  ( F  |`  ( A  i^i  W ) )  =  ( ( k  e.  A  |->  .0.  )  |`  ( A  i^i  W ) ) )
24 inss1 3389 . . . . . . . 8  |-  ( A  i^i  W )  C_  A
25 resmpt 5000 . . . . . . . 8  |-  ( ( A  i^i  W ) 
C_  A  ->  (
( k  e.  A  |->  .0.  )  |`  ( A  i^i  W ) )  =  ( k  e.  ( A  i^i  W
)  |->  .0.  ) )
2624, 25ax-mp 8 . . . . . . 7  |-  ( ( k  e.  A  |->  .0.  )  |`  ( A  i^i  W ) )  =  ( k  e.  ( A  i^i  W ) 
|->  .0.  )
2723, 26syl6eq 2331 . . . . . 6  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  ( F  |`  ( A  i^i  W ) )  =  ( k  e.  ( A  i^i  W )  |->  .0.  ) )
2819, 27eqtr3d 2317 . . . . 5  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  ( F  |`  W )  =  ( k  e.  ( A  i^i  W ) 
|->  .0.  ) )
2928oveq2d 5874 . . . 4  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  ( G  gsumg  ( F  |`  W ) )  =  ( G 
gsumg  ( k  e.  ( A  i^i  W ) 
|->  .0.  ) ) )
3022oveq2d 5874 . . . 4  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  ( G  gsumg  F )  =  ( G  gsumg  ( k  e.  A  |->  .0.  ) ) )
3111, 29, 303eqtr4d 2325 . . 3  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  ( G  gsumg  ( F  |`  W ) )  =  ( G 
gsumg  F ) )
3231ex 423 . 2  |-  ( ph  ->  ( ( `' F " ( _V  \  {  .0.  } ) )  =  (/)  ->  ( G  gsumg  ( F  |`  W ) )  =  ( G  gsumg  F ) ) )
33 f1ofo 5479 . . . . . . . . . . . 12  |-  ( f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) )  ->  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) )
-onto-> ( `' F "
( _V  \  {  .0.  } ) ) )
34 forn 5454 . . . . . . . . . . . 12  |-  ( f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) )
-onto-> ( `' F "
( _V  \  {  .0.  } ) )  ->  ran  f  =  ( `' F " ( _V 
\  {  .0.  }
) ) )
3533, 34syl 15 . . . . . . . . . . 11  |-  ( f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) )  ->  ran  f  =  ( `' F " ( _V  \  {  .0.  } ) ) )
3635ad2antll 709 . . . . . . . . . 10  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ran  f  =  ( `' F " ( _V 
\  {  .0.  }
) ) )
37 gsumzres.s . . . . . . . . . . 11  |-  ( ph  ->  ( `' F "
( _V  \  {  .0.  } ) )  C_  W )
3837adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( `' F " ( _V  \  {  .0.  } ) )  C_  W )
3936, 38eqsstrd 3212 . . . . . . . . 9  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ran  f  C_  W )
40 cores 5176 . . . . . . . . 9  |-  ( ran  f  C_  W  ->  ( ( F  |`  W )  o.  f )  =  ( F  o.  f
) )
4139, 40syl 15 . . . . . . . 8  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( ( F  |`  W )  o.  f
)  =  ( F  o.  f ) )
4241seqeq3d 11054 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  seq  1 ( ( +g  `  G
) ,  ( ( F  |`  W )  o.  f ) )  =  seq  1 ( ( +g  `  G ) ,  ( F  o.  f ) ) )
4342fveq1d 5527 . . . . . 6  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  (  seq  1
( ( +g  `  G
) ,  ( ( F  |`  W )  o.  f ) ) `  ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) )  =  (  seq  1
( ( +g  `  G
) ,  ( F  o.  f ) ) `
 ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) )
44 gsumzcl.b . . . . . . 7  |-  B  =  ( Base `  G
)
45 eqid 2283 . . . . . . 7  |-  ( +g  `  G )  =  ( +g  `  G )
46 gsumzcl.z . . . . . . 7  |-  Z  =  (Cntz `  G )
471adantr 451 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  G  e.  Mnd )
484adantr 451 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( A  i^i  W )  e.  _V )
4913adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  F : A --> B )
50 fssres 5408 . . . . . . . . 9  |-  ( ( F : A --> B  /\  ( A  i^i  W ) 
C_  A )  -> 
( F  |`  ( A  i^i  W ) ) : ( A  i^i  W ) --> B )
5149, 24, 50sylancl 643 . . . . . . . 8  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( F  |`  ( A  i^i  W ) ) : ( A  i^i  W ) --> B )
5218feq1d 5379 . . . . . . . . 9  |-  ( ph  ->  ( ( F  |`  ( A  i^i  W ) ) : ( A  i^i  W ) --> B  <-> 
( F  |`  W ) : ( A  i^i  W ) --> B ) )
5352biimpa 470 . . . . . . . 8  |-  ( (
ph  /\  ( F  |`  ( A  i^i  W
) ) : ( A  i^i  W ) --> B )  ->  ( F  |`  W ) : ( A  i^i  W
) --> B )
5451, 53syldan 456 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( F  |`  W ) : ( A  i^i  W ) --> B )
55 gsumzcl.c . . . . . . . . 9  |-  ( ph  ->  ran  F  C_  ( Z `  ran  F ) )
56 resss 4979 . . . . . . . . . 10  |-  ( F  |`  W )  C_  F
57 rnss 4907 . . . . . . . . . 10  |-  ( ( F  |`  W )  C_  F  ->  ran  ( F  |`  W )  C_  ran  F )
5856, 57ax-mp 8 . . . . . . . . 9  |-  ran  ( F  |`  W )  C_  ran  F
5946cntzidss 14813 . . . . . . . . 9  |-  ( ( ran  F  C_  ( Z `  ran  F )  /\  ran  ( F  |`  W )  C_  ran  F )  ->  ran  ( F  |`  W )  C_  ( Z `  ran  ( F  |`  W ) ) )
6055, 58, 59sylancl 643 . . . . . . . 8  |-  ( ph  ->  ran  ( F  |`  W )  C_  ( Z `  ran  ( F  |`  W ) ) )
6160adantr 451 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ran  ( F  |`  W )  C_  ( Z `  ran  ( F  |`  W ) ) )
62 simprl 732 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) )  e.  NN )
63 f1of1 5471 . . . . . . . . 9  |-  ( f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) )  ->  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) )
-1-1-> ( `' F "
( _V  \  {  .0.  } ) ) )
6463ad2antll 709 . . . . . . . 8  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  f : ( 1 ... ( # `  ( `' F "
( _V  \  {  .0.  } ) ) ) ) -1-1-> ( `' F " ( _V  \  {  .0.  } ) ) )
65 cnvimass 5033 . . . . . . . . . . 11  |-  ( `' F " ( _V 
\  {  .0.  }
) )  C_  dom  F
66 fdm 5393 . . . . . . . . . . . 12  |-  ( F : A --> B  ->  dom  F  =  A )
6713, 66syl 15 . . . . . . . . . . 11  |-  ( ph  ->  dom  F  =  A )
6865, 67syl5sseq 3226 . . . . . . . . . 10  |-  ( ph  ->  ( `' F "
( _V  \  {  .0.  } ) )  C_  A )
6968, 37ssind 3393 . . . . . . . . 9  |-  ( ph  ->  ( `' F "
( _V  \  {  .0.  } ) )  C_  ( A  i^i  W ) )
7069adantr 451 . . . . . . . 8  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( `' F " ( _V  \  {  .0.  } ) )  C_  ( A  i^i  W ) )
71 f1ss 5442 . . . . . . . 8  |-  ( ( f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) )
-1-1-> ( `' F "
( _V  \  {  .0.  } ) )  /\  ( `' F " ( _V 
\  {  .0.  }
) )  C_  ( A  i^i  W ) )  ->  f : ( 1 ... ( # `  ( `' F "
( _V  \  {  .0.  } ) ) ) ) -1-1-> ( A  i^i  W ) )
7264, 70, 71syl2anc 642 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  f : ( 1 ... ( # `  ( `' F "
( _V  \  {  .0.  } ) ) ) ) -1-1-> ( A  i^i  W ) )
73 cnvss 4854 . . . . . . . . 9  |-  ( ( F  |`  W )  C_  F  ->  `' ( F  |`  W )  C_  `' F )
74 imass1 5048 . . . . . . . . 9  |-  ( `' ( F  |`  W ) 
C_  `' F  -> 
( `' ( F  |`  W ) " ( _V  \  {  .0.  }
) )  C_  ( `' F " ( _V 
\  {  .0.  }
) ) )
7556, 73, 74mp2b 9 . . . . . . . 8  |-  ( `' ( F  |`  W )
" ( _V  \  {  .0.  } ) ) 
C_  ( `' F " ( _V  \  {  .0.  } ) )
7675, 36syl5sseqr 3227 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( `' ( F  |`  W ) " ( _V  \  {  .0.  } ) ) 
C_  ran  f )
77 eqid 2283 . . . . . . 7  |-  ( `' ( ( F  |`  W )  o.  f
) " ( _V 
\  {  .0.  }
) )  =  ( `' ( ( F  |`  W )  o.  f
) " ( _V 
\  {  .0.  }
) )
7844, 5, 45, 46, 47, 48, 54, 61, 62, 72, 76, 77gsumval3 15191 . . . . . 6  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( G  gsumg  ( F  |`  W ) )  =  (  seq  1 ( ( +g  `  G
) ,  ( ( F  |`  W )  o.  f ) ) `  ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) )
792adantr 451 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  A  e.  V
)
8055adantr 451 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ran  F  C_  ( Z `  ran  F ) )
8168adantr 451 . . . . . . . 8  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( `' F " ( _V  \  {  .0.  } ) )  C_  A )
82 f1ss 5442 . . . . . . . 8  |-  ( ( f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) )
-1-1-> ( `' F "
( _V  \  {  .0.  } ) )  /\  ( `' F " ( _V 
\  {  .0.  }
) )  C_  A
)  ->  f :
( 1 ... ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) ) ) -1-1-> A )
8364, 81, 82syl2anc 642 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  f : ( 1 ... ( # `  ( `' F "
( _V  \  {  .0.  } ) ) ) ) -1-1-> A )
8420, 36syl5sseqr 3227 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( `' F " ( _V  \  {  .0.  } ) )  C_  ran  f )
85 eqid 2283 . . . . . . 7  |-  ( `' ( F  o.  f
) " ( _V 
\  {  .0.  }
) )  =  ( `' ( F  o.  f ) " ( _V  \  {  .0.  }
) )
8644, 5, 45, 46, 47, 79, 49, 80, 62, 83, 84, 85gsumval3 15191 . . . . . 6  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( G  gsumg  F )  =  (  seq  1
( ( +g  `  G
) ,  ( F  o.  f ) ) `
 ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) )
8743, 78, 863eqtr4d 2325 . . . . 5  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( G  gsumg  ( F  |`  W ) )  =  ( G  gsumg  F ) )
8887expr 598 . . . 4  |-  ( (
ph  /\  ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) )  e.  NN )  ->  (
f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) )  ->  ( G  gsumg  ( F  |`  W ) )  =  ( G 
gsumg  F ) ) )
8988exlimdv 1664 . . 3  |-  ( (
ph  /\  ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) )  e.  NN )  ->  ( E. f  f :
( 1 ... ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) ) ) -1-1-onto-> ( `' F "
( _V  \  {  .0.  } ) )  -> 
( G  gsumg  ( F  |`  W ) )  =  ( G 
gsumg  F ) ) )
9089expimpd 586 . 2  |-  ( ph  ->  ( ( ( # `  ( `' F "
( _V  \  {  .0.  } ) ) )  e.  NN  /\  E. f  f : ( 1 ... ( # `  ( `' F "
( _V  \  {  .0.  } ) ) ) ) -1-1-onto-> ( `' F "
( _V  \  {  .0.  } ) ) )  ->  ( G  gsumg  ( F  |`  W ) )  =  ( G  gsumg  F ) ) )
91 gsumzres.w . . 3  |-  ( ph  ->  ( `' F "
( _V  \  {  .0.  } ) )  e. 
Fin )
92 fz1f1o 12183 . . 3  |-  ( ( `' F " ( _V 
\  {  .0.  }
) )  e.  Fin  ->  ( ( `' F " ( _V  \  {  .0.  } ) )  =  (/)  \/  ( ( # `  ( `' F "
( _V  \  {  .0.  } ) ) )  e.  NN  /\  E. f  f : ( 1 ... ( # `  ( `' F "
( _V  \  {  .0.  } ) ) ) ) -1-1-onto-> ( `' F "
( _V  \  {  .0.  } ) ) ) ) )
9391, 92syl 15 . 2  |-  ( ph  ->  ( ( `' F " ( _V  \  {  .0.  } ) )  =  (/)  \/  ( ( # `  ( `' F "
( _V  \  {  .0.  } ) ) )  e.  NN  /\  E. f  f : ( 1 ... ( # `  ( `' F "
( _V  \  {  .0.  } ) ) ) ) -1-1-onto-> ( `' F "
( _V  \  {  .0.  } ) ) ) ) )
9432, 90, 93mpjaod 370 1  |-  ( ph  ->  ( G  gsumg  ( F  |`  W ) )  =  ( G 
gsumg  F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   _Vcvv 2788    \ cdif 3149    i^i cin 3151    C_ wss 3152   (/)c0 3455   {csn 3640    e. cmpt 4077   `'ccnv 4688   dom cdm 4689   ran crn 4690    |` cres 4691   "cima 4692    o. ccom 4693    Fn wfn 5250   -->wf 5251   -1-1->wf1 5252   -onto->wfo 5253   -1-1-onto->wf1o 5254   ` cfv 5255  (class class class)co 5858   Fincfn 6863   1c1 8738   NNcn 9746   ...cfz 10782    seq cseq 11046   #chash 11337   Basecbs 13148   +g cplusg 13208   0gc0g 13400    gsumg cgsu 13401   Mndcmnd 14361  Cntzccntz 14791
This theorem is referenced by:  gsumres  15197  gsumzsplit  15206  gsumpt  15222  dmdprdsplitlem  15272  dpjidcl  15293
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-fzo 10871  df-seq 11047  df-hash 11338  df-0g 13404  df-gsum 13405  df-mnd 14367  df-cntz 14793
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