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Theorem gsumzresOLD 16897
Description: Extend a finite group sum by padding outside with zeroes. (Contributed by Mario Carneiro, 24-Apr-2016.) Obsolete version of gsumzres 16893 as of 31-May-2019. (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
gsumzclOLD.b  |-  B  =  ( Base `  G
)
gsumzclOLD.0  |-  .0.  =  ( 0g `  G )
gsumzclOLD.z  |-  Z  =  (Cntz `  G )
gsumzclOLD.g  |-  ( ph  ->  G  e.  Mnd )
gsumzclOLD.a  |-  ( ph  ->  A  e.  V )
gsumzclOLD.f  |-  ( ph  ->  F : A --> B )
gsumzclOLD.c  |-  ( ph  ->  ran  F  C_  ( Z `  ran  F ) )
gsumzresOLD.s  |-  ( ph  ->  ( `' F "
( _V  \  {  .0.  } ) )  C_  W )
gsumzresOLD.w  |-  ( ph  ->  ( `' F "
( _V  \  {  .0.  } ) )  e. 
Fin )
Assertion
Ref Expression
gsumzresOLD  |-  ( ph  ->  ( G  gsumg  ( F  |`  W ) )  =  ( G 
gsumg  F ) )

Proof of Theorem gsumzresOLD
Dummy variables  f 
k are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 gsumzclOLD.g . . . . . . 7  |-  ( ph  ->  G  e.  Mnd )
2 gsumzclOLD.a . . . . . . . 8  |-  ( ph  ->  A  e.  V )
3 inex1g 4580 . . . . . . . 8  |-  ( A  e.  V  ->  ( A  i^i  W )  e. 
_V )
42, 3syl 16 . . . . . . 7  |-  ( ph  ->  ( A  i^i  W
)  e.  _V )
5 gsumzclOLD.0 . . . . . . . 8  |-  .0.  =  ( 0g `  G )
65gsumz 15984 . . . . . . 7  |-  ( ( G  e.  Mnd  /\  ( A  i^i  W )  e.  _V )  -> 
( G  gsumg  ( k  e.  ( A  i^i  W ) 
|->  .0.  ) )  =  .0.  )
71, 4, 6syl2anc 661 . . . . . 6  |-  ( ph  ->  ( G  gsumg  ( k  e.  ( A  i^i  W ) 
|->  .0.  ) )  =  .0.  )
85gsumz 15984 . . . . . . 7  |-  ( ( G  e.  Mnd  /\  A  e.  V )  ->  ( G  gsumg  ( k  e.  A  |->  .0.  ) )  =  .0.  )
91, 2, 8syl2anc 661 . . . . . 6  |-  ( ph  ->  ( G  gsumg  ( k  e.  A  |->  .0.  ) )  =  .0.  )
107, 9eqtr4d 2487 . . . . 5  |-  ( ph  ->  ( G  gsumg  ( k  e.  ( A  i^i  W ) 
|->  .0.  ) )  =  ( G  gsumg  ( k  e.  A  |->  .0.  ) ) )
1110adantr 465 . . . 4  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  ( G  gsumg  ( k  e.  ( A  i^i  W ) 
|->  .0.  ) )  =  ( G  gsumg  ( k  e.  A  |->  .0.  ) ) )
12 resres 5276 . . . . . . . 8  |-  ( ( F  |`  A )  |`  W )  =  ( F  |`  ( A  i^i  W ) )
13 gsumzclOLD.f . . . . . . . . . 10  |-  ( ph  ->  F : A --> B )
14 ffn 5721 . . . . . . . . . 10  |-  ( F : A --> B  ->  F  Fn  A )
15 fnresdm 5680 . . . . . . . . . 10  |-  ( F  Fn  A  ->  ( F  |`  A )  =  F )
1613, 14, 153syl 20 . . . . . . . . 9  |-  ( ph  ->  ( F  |`  A )  =  F )
1716reseq1d 5262 . . . . . . . 8  |-  ( ph  ->  ( ( F  |`  A )  |`  W )  =  ( F  |`  W ) )
1812, 17syl5eqr 2498 . . . . . . 7  |-  ( ph  ->  ( F  |`  ( A  i^i  W ) )  =  ( F  |`  W ) )
1918adantr 465 . . . . . 6  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  ( F  |`  ( A  i^i  W ) )  =  ( F  |`  W )
)
20 ssid 3508 . . . . . . . . . 10  |-  ( `' F " ( _V 
\  {  .0.  }
) )  C_  ( `' F " ( _V 
\  {  .0.  }
) )
2120a1i 11 . . . . . . . . 9  |-  ( ph  ->  ( `' F "
( _V  \  {  .0.  } ) )  C_  ( `' F " ( _V 
\  {  .0.  }
) ) )
2213, 21gsumcllemOLD 16892 . . . . . . . 8  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  F  =  ( k  e.  A  |->  .0.  ) )
2322reseq1d 5262 . . . . . . 7  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  ( F  |`  ( A  i^i  W ) )  =  ( ( k  e.  A  |->  .0.  )  |`  ( A  i^i  W ) ) )
24 inss1 3703 . . . . . . . 8  |-  ( A  i^i  W )  C_  A
25 resmpt 5313 . . . . . . . 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 5 . . . . . . 7  |-  ( ( k  e.  A  |->  .0.  )  |`  ( A  i^i  W ) )  =  ( k  e.  ( A  i^i  W ) 
|->  .0.  )
2723, 26syl6eq 2500 . . . . . 6  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  ( F  |`  ( A  i^i  W ) )  =  ( k  e.  ( A  i^i  W )  |->  .0.  ) )
2819, 27eqtr3d 2486 . . . . 5  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  ( F  |`  W )  =  ( k  e.  ( A  i^i  W ) 
|->  .0.  ) )
2928oveq2d 6297 . . . 4  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  ( G  gsumg  ( F  |`  W ) )  =  ( G 
gsumg  ( k  e.  ( A  i^i  W ) 
|->  .0.  ) ) )
3022oveq2d 6297 . . . 4  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  ( G  gsumg  F )  =  ( G  gsumg  ( k  e.  A  |->  .0.  ) ) )
3111, 29, 303eqtr4d 2494 . . 3  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  ( G  gsumg  ( F  |`  W ) )  =  ( G 
gsumg  F ) )
3231ex 434 . 2  |-  ( ph  ->  ( ( `' F " ( _V  \  {  .0.  } ) )  =  (/)  ->  ( G  gsumg  ( F  |`  W ) )  =  ( G  gsumg  F ) ) )
33 f1ofo 5813 . . . . . . . . . . . 12  |-  ( f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) )  ->  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) )
-onto-> ( `' F "
( _V  \  {  .0.  } ) ) )
34 forn 5788 . . . . . . . . . . . 12  |-  ( f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) )
-onto-> ( `' F "
( _V  \  {  .0.  } ) )  ->  ran  f  =  ( `' F " ( _V 
\  {  .0.  }
) ) )
3533, 34syl 16 . . . . . . . . . . 11  |-  ( f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) )  ->  ran  f  =  ( `' F " ( _V  \  {  .0.  } ) ) )
3635ad2antll 728 . . . . . . . . . 10  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ran  f  =  ( `' F " ( _V 
\  {  .0.  }
) ) )
37 gsumzresOLD.s . . . . . . . . . . 11  |-  ( ph  ->  ( `' F "
( _V  \  {  .0.  } ) )  C_  W )
3837adantr 465 . . . . . . . . . 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 3523 . . . . . . . . 9  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ran  f  C_  W )
40 cores 5500 . . . . . . . . 9  |-  ( ran  f  C_  W  ->  ( ( F  |`  W )  o.  f )  =  ( F  o.  f
) )
4139, 40syl 16 . . . . . . . 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 12097 . . . . . . 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 5858 . . . . . 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 gsumzclOLD.b . . . . . . 7  |-  B  =  ( Base `  G
)
45 eqid 2443 . . . . . . 7  |-  ( +g  `  G )  =  ( +g  `  G )
46 gsumzclOLD.z . . . . . . 7  |-  Z  =  (Cntz `  G )
471adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  G  e.  Mnd )
484adantr 465 . . . . . . 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 465 . . . . . . . . 9  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  F : A --> B )
50 fssres 5741 . . . . . . . . 9  |-  ( ( F : A --> B  /\  ( A  i^i  W ) 
C_  A )  -> 
( F  |`  ( A  i^i  W ) ) : ( A  i^i  W ) --> B )
5149, 24, 50sylancl 662 . . . . . . . 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 5707 . . . . . . . . 9  |-  ( ph  ->  ( ( F  |`  ( A  i^i  W ) ) : ( A  i^i  W ) --> B  <-> 
( F  |`  W ) : ( A  i^i  W ) --> B ) )
5352biimpa 484 . . . . . . . 8  |-  ( (
ph  /\  ( F  |`  ( A  i^i  W
) ) : ( A  i^i  W ) --> B )  ->  ( F  |`  W ) : ( A  i^i  W
) --> B )
5451, 53syldan 470 . . . . . . 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 gsumzclOLD.c . . . . . . . . 9  |-  ( ph  ->  ran  F  C_  ( Z `  ran  F ) )
56 resss 5287 . . . . . . . . . 10  |-  ( F  |`  W )  C_  F
57 rnss 5221 . . . . . . . . . 10  |-  ( ( F  |`  W )  C_  F  ->  ran  ( F  |`  W )  C_  ran  F )
5856, 57ax-mp 5 . . . . . . . . 9  |-  ran  ( F  |`  W )  C_  ran  F
5946cntzidss 16354 . . . . . . . . 9  |-  ( ( ran  F  C_  ( Z `  ran  F )  /\  ran  ( F  |`  W )  C_  ran  F )  ->  ran  ( F  |`  W )  C_  ( Z `  ran  ( F  |`  W ) ) )
6055, 58, 59sylancl 662 . . . . . . . 8  |-  ( ph  ->  ran  ( F  |`  W )  C_  ( Z `  ran  ( F  |`  W ) ) )
6160adantr 465 . . . . . . 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 756 . . . . . . 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 5805 . . . . . . . . 9  |-  ( f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) )  ->  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) )
-1-1-> ( `' F "
( _V  \  {  .0.  } ) ) )
6463ad2antll 728 . . . . . . . 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 5347 . . . . . . . . . . 11  |-  ( `' F " ( _V 
\  {  .0.  }
) )  C_  dom  F
66 fdm 5725 . . . . . . . . . . . 12  |-  ( F : A --> B  ->  dom  F  =  A )
6713, 66syl 16 . . . . . . . . . . 11  |-  ( ph  ->  dom  F  =  A )
6865, 67syl5sseq 3537 . . . . . . . . . 10  |-  ( ph  ->  ( `' F "
( _V  \  {  .0.  } ) )  C_  A )
6968, 37ssind 3707 . . . . . . . . 9  |-  ( ph  ->  ( `' F "
( _V  \  {  .0.  } ) )  C_  ( A  i^i  W ) )
7069adantr 465 . . . . . . . 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 5776 . . . . . . . 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 661 . . . . . . 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 5165 . . . . . . . . 9  |-  ( ( F  |`  W )  C_  F  ->  `' ( F  |`  W )  C_  `' F )
74 imass1 5361 . . . . . . . . 9  |-  ( `' ( F  |`  W ) 
C_  `' F  -> 
( `' ( F  |`  W ) " ( _V  \  {  .0.  }
) )  C_  ( `' F " ( _V 
\  {  .0.  }
) ) )
7556, 73, 74mp2b 10 . . . . . . . 8  |-  ( `' ( F  |`  W )
" ( _V  \  {  .0.  } ) ) 
C_  ( `' F " ( _V  \  {  .0.  } ) )
7675, 36syl5sseqr 3538 . . . . . . 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 2443 . . . . . . 7  |-  ( `' ( ( F  |`  W )  o.  f
) " ( _V 
\  {  .0.  }
) )  =  ( `' ( ( F  |`  W )  o.  f
) " ( _V 
\  {  .0.  }
) )
7844, 5, 45, 46, 47, 48, 54, 61, 62, 72, 76, 77gsumval3OLD 16887 . . . . . 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 465 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  A  e.  V
)
8055adantr 465 . . . . . . 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 465 . . . . . . . 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 5776 . . . . . . . 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 661 . . . . . . 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 3538 . . . . . . 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 2443 . . . . . . 7  |-  ( `' ( F  o.  f
) " ( _V 
\  {  .0.  }
) )  =  ( `' ( F  o.  f ) " ( _V  \  {  .0.  }
) )
8644, 5, 45, 46, 47, 79, 49, 80, 62, 83, 84, 85gsumval3OLD 16887 . . . . . 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 2494 . . . . 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 615 . . . 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 1711 . . 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 603 . 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 gsumzresOLD.w . . 3  |-  ( ph  ->  ( `' F "
( _V  \  {  .0.  } ) )  e. 
Fin )
92 fz1f1o 13514 . . 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 16 . 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 381 1  |-  ( ph  ->  ( G  gsumg  ( F  |`  W ) )  =  ( G 
gsumg  F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1383   E.wex 1599    e. wcel 1804   _Vcvv 3095    \ cdif 3458    i^i cin 3460    C_ wss 3461   (/)c0 3770   {csn 4014    |-> cmpt 4495   `'ccnv 4988   dom cdm 4989   ran crn 4990    |` cres 4991   "cima 4992    o. ccom 4993    Fn wfn 5573   -->wf 5574   -1-1->wf1 5575   -onto->wfo 5576   -1-1-onto->wf1o 5577   ` cfv 5578  (class class class)co 6281   Fincfn 7518   1c1 9496   NNcn 10543   ...cfz 11683    seqcseq 12089   #chash 12387   Basecbs 14614   +g cplusg 14679   0gc0g 14819    gsumg cgsu 14820   Mndcmnd 15898  Cntzccntz 16332
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-oadd 7136  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-oi 7938  df-card 8323  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10544  df-n0 10803  df-z 10872  df-uz 11093  df-fz 11684  df-fzo 11807  df-seq 12090  df-hash 12388  df-0g 14821  df-gsum 14822  df-mgm 15851  df-sgrp 15890  df-mnd 15900  df-cntz 16334
This theorem is referenced by:  gsumresOLD  16904  gsumzsplitOLD  16924  gsumptOLD  16968  dmdprdsplitlemOLD  17064  dpjidclOLD  17093
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