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Theorem gsumzf1o 15474
Description: Re-index a finite group sum using a bijection. (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 ) )
gsumzcl.w  |-  ( ph  ->  ( `' F "
( _V  \  {  .0.  } ) )  e. 
Fin )
gsumzf1o.h  |-  ( ph  ->  H : C -1-1-onto-> A )
Assertion
Ref Expression
gsumzf1o  |-  ( ph  ->  ( G  gsumg  F )  =  ( G  gsumg  ( F  o.  H
) ) )

Proof of Theorem gsumzf1o
Dummy variables  f 
k  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 gsumzcl.g . . . . . . 7  |-  ( ph  ->  G  e.  Mnd )
2 gsumzcl.a . . . . . . 7  |-  ( ph  ->  A  e.  V )
3 gsumzcl.0 . . . . . . . 8  |-  .0.  =  ( 0g `  G )
43gsumz 14736 . . . . . . 7  |-  ( ( G  e.  Mnd  /\  A  e.  V )  ->  ( G  gsumg  ( k  e.  A  |->  .0.  ) )  =  .0.  )
51, 2, 4syl2anc 643 . . . . . 6  |-  ( ph  ->  ( G  gsumg  ( k  e.  A  |->  .0.  ) )  =  .0.  )
6 gsumzf1o.h . . . . . . . . 9  |-  ( ph  ->  H : C -1-1-onto-> A )
7 f1of1 5632 . . . . . . . . 9  |-  ( H : C -1-1-onto-> A  ->  H : C -1-1-> A )
86, 7syl 16 . . . . . . . 8  |-  ( ph  ->  H : C -1-1-> A
)
9 f1dmex 5930 . . . . . . . 8  |-  ( ( H : C -1-1-> A  /\  A  e.  V
)  ->  C  e.  _V )
108, 2, 9syl2anc 643 . . . . . . 7  |-  ( ph  ->  C  e.  _V )
113gsumz 14736 . . . . . . 7  |-  ( ( G  e.  Mnd  /\  C  e.  _V )  ->  ( G  gsumg  ( x  e.  C  |->  .0.  ) )  =  .0.  )
121, 10, 11syl2anc 643 . . . . . 6  |-  ( ph  ->  ( G  gsumg  ( x  e.  C  |->  .0.  ) )  =  .0.  )
135, 12eqtr4d 2439 . . . . 5  |-  ( ph  ->  ( G  gsumg  ( k  e.  A  |->  .0.  ) )  =  ( G  gsumg  ( x  e.  C  |->  .0.  ) ) )
1413adantr 452 . . . 4  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  ( G  gsumg  ( k  e.  A  |->  .0.  ) )  =  ( G  gsumg  ( x  e.  C  |->  .0.  ) ) )
15 gsumzcl.f . . . . . 6  |-  ( ph  ->  F : A --> B )
16 ssid 3327 . . . . . . 7  |-  ( `' F " ( _V 
\  {  .0.  }
) )  C_  ( `' F " ( _V 
\  {  .0.  }
) )
1716a1i 11 . . . . . 6  |-  ( ph  ->  ( `' F "
( _V  \  {  .0.  } ) )  C_  ( `' F " ( _V 
\  {  .0.  }
) ) )
1815, 17gsumcllem 15471 . . . . 5  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  F  =  ( k  e.  A  |->  .0.  ) )
1918oveq2d 6056 . . . 4  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  ( G  gsumg  F )  =  ( G  gsumg  ( k  e.  A  |->  .0.  ) ) )
20 f1of 5633 . . . . . . . . 9  |-  ( H : C -1-1-onto-> A  ->  H : C
--> A )
216, 20syl 16 . . . . . . . 8  |-  ( ph  ->  H : C --> A )
2221adantr 452 . . . . . . 7  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  H : C --> A )
2322ffvelrnda 5829 . . . . . 6  |-  ( ( ( ph  /\  ( `' F " ( _V 
\  {  .0.  }
) )  =  (/) )  /\  x  e.  C
)  ->  ( H `  x )  e.  A
)
2422feqmptd 5738 . . . . . 6  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  H  =  ( x  e.  C  |->  ( H `  x ) ) )
25 eqidd 2405 . . . . . 6  |-  ( k  =  ( H `  x )  ->  .0.  =  .0.  )
2623, 24, 18, 25fmptco 5860 . . . . 5  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  ( F  o.  H )  =  ( x  e.  C  |->  .0.  ) )
2726oveq2d 6056 . . . 4  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  ( G  gsumg  ( F  o.  H
) )  =  ( G  gsumg  ( x  e.  C  |->  .0.  ) ) )
2814, 19, 273eqtr4d 2446 . . 3  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  ( G  gsumg  F )  =  ( G  gsumg  ( F  o.  H
) ) )
2928ex 424 . 2  |-  ( ph  ->  ( ( `' F " ( _V  \  {  .0.  } ) )  =  (/)  ->  ( G  gsumg  F )  =  ( G  gsumg  ( F  o.  H ) ) ) )
30 coass 5347 . . . . . . . . . . 11  |-  ( ( H  o.  `' H
)  o.  f )  =  ( H  o.  ( `' H  o.  f
) )
316adantr 452 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  H : C -1-1-onto-> A
)
32 f1ococnv2 5661 . . . . . . . . . . . . . 14  |-  ( H : C -1-1-onto-> A  ->  ( H  o.  `' H )  =  (  _I  |`  A )
)
3331, 32syl 16 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( H  o.  `' H )  =  (  _I  |`  A )
)
3433coeq1d 4993 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( ( H  o.  `' H )  o.  f )  =  ( (  _I  |`  A )  o.  f ) )
35 f1of1 5632 . . . . . . . . . . . . . . 15  |-  ( f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) )  ->  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) )
-1-1-> ( `' F "
( _V  \  {  .0.  } ) ) )
3635ad2antll 710 . . . . . . . . . . . . . 14  |-  ( (
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.  } ) ) )
37 cnvimass 5183 . . . . . . . . . . . . . . . 16  |-  ( `' F " ( _V 
\  {  .0.  }
) )  C_  dom  F
38 fdm 5554 . . . . . . . . . . . . . . . . 17  |-  ( F : A --> B  ->  dom  F  =  A )
3915, 38syl 16 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  dom  F  =  A )
4037, 39syl5sseq 3356 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( `' F "
( _V  \  {  .0.  } ) )  C_  A )
4140adantr 452 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( `' F " ( _V  \  {  .0.  } ) )  C_  A )
42 f1ss 5603 . . . . . . . . . . . . . 14  |-  ( ( f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) )
-1-1-> ( `' F "
( _V  \  {  .0.  } ) )  /\  ( `' F " ( _V 
\  {  .0.  }
) )  C_  A
)  ->  f :
( 1 ... ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) ) ) -1-1-> A )
4336, 41, 42syl2anc 643 . . . . . . . . . . . . 13  |-  ( (
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 )
44 f1f 5598 . . . . . . . . . . . . 13  |-  ( f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) )
-1-1-> A  ->  f :
( 1 ... ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) ) ) --> A )
45 fcoi2 5577 . . . . . . . . . . . . 13  |-  ( f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) --> A  ->  ( (  _I  |`  A )  o.  f )  =  f )
4643, 44, 453syl 19 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( (  _I  |`  A )  o.  f
)  =  f )
4734, 46eqtrd 2436 . . . . . . . . . . 11  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( ( H  o.  `' H )  o.  f )  =  f )
4830, 47syl5reqr 2451 . . . . . . . . . 10  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  f  =  ( H  o.  ( `' H  o.  f ) ) )
4948coeq2d 4994 . . . . . . . . 9  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( F  o.  f )  =  ( F  o.  ( H  o.  ( `' H  o.  f ) ) ) )
50 coass 5347 . . . . . . . . 9  |-  ( ( F  o.  H )  o.  ( `' H  o.  f ) )  =  ( F  o.  ( H  o.  ( `' H  o.  f )
) )
5149, 50syl6eqr 2454 . . . . . . . 8  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( F  o.  f )  =  ( ( F  o.  H
)  o.  ( `' H  o.  f ) ) )
5251seqeq3d 11286 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  seq  1 ( ( +g  `  G
) ,  ( F  o.  f ) )  =  seq  1 ( ( +g  `  G
) ,  ( ( F  o.  H )  o.  ( `' H  o.  f ) ) ) )
5352fveq1d 5689 . . . . . 6  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  (  seq  1
( ( +g  `  G
) ,  ( F  o.  f ) ) `
 ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) )  =  (  seq  1
( ( +g  `  G
) ,  ( ( F  o.  H )  o.  ( `' H  o.  f ) ) ) `
 ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) )
54 gsumzcl.b . . . . . . 7  |-  B  =  ( Base `  G
)
55 eqid 2404 . . . . . . 7  |-  ( +g  `  G )  =  ( +g  `  G )
56 gsumzcl.z . . . . . . 7  |-  Z  =  (Cntz `  G )
571adantr 452 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  G  e.  Mnd )
582adantr 452 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  A  e.  V
)
5915adantr 452 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  F : A --> B )
60 gsumzcl.c . . . . . . . 8  |-  ( ph  ->  ran  F  C_  ( Z `  ran  F ) )
6160adantr 452 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ran  F  C_  ( Z `  ran  F ) )
62 simprl 733 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) )  e.  NN )
63 f1ofo 5640 . . . . . . . . . 10  |-  ( f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) )  ->  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) )
-onto-> ( `' F "
( _V  \  {  .0.  } ) ) )
64 forn 5615 . . . . . . . . . 10  |-  ( f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) )
-onto-> ( `' F "
( _V  \  {  .0.  } ) )  ->  ran  f  =  ( `' F " ( _V 
\  {  .0.  }
) ) )
6563, 64syl 16 . . . . . . . . 9  |-  ( f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) )  ->  ran  f  =  ( `' F " ( _V  \  {  .0.  } ) ) )
6665ad2antll 710 . . . . . . . 8  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ran  f  =  ( `' F " ( _V 
\  {  .0.  }
) ) )
6716, 66syl5sseqr 3357 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( `' F " ( _V  \  {  .0.  } ) )  C_  ran  f )
68 eqid 2404 . . . . . . 7  |-  ( `' ( F  o.  f
) " ( _V 
\  {  .0.  }
) )  =  ( `' ( F  o.  f ) " ( _V  \  {  .0.  }
) )
6954, 3, 55, 56, 57, 58, 59, 61, 62, 43, 67, 68gsumval3 15469 . . . . . 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.  }
) ) ) ) )
7010adantr 452 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  C  e.  _V )
71 fco 5559 . . . . . . . . 9  |-  ( ( F : A --> B  /\  H : C --> A )  ->  ( F  o.  H ) : C --> B )
7215, 21, 71syl2anc 643 . . . . . . . 8  |-  ( ph  ->  ( F  o.  H
) : C --> B )
7372adantr 452 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( F  o.  H ) : C --> B )
74 rncoss 5095 . . . . . . . . 9  |-  ran  ( F  o.  H )  C_ 
ran  F
7556cntzidss 15091 . . . . . . . . 9  |-  ( ( ran  F  C_  ( Z `  ran  F )  /\  ran  ( F  o.  H )  C_  ran  F )  ->  ran  ( F  o.  H
)  C_  ( Z `  ran  ( F  o.  H ) ) )
7660, 74, 75sylancl 644 . . . . . . . 8  |-  ( ph  ->  ran  ( F  o.  H )  C_  ( Z `  ran  ( F  o.  H ) ) )
7776adantr 452 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ran  ( F  o.  H )  C_  ( Z `  ran  ( F  o.  H ) ) )
78 f1ocnv 5646 . . . . . . . . . 10  |-  ( H : C -1-1-onto-> A  ->  `' H : A -1-1-onto-> C )
79 f1of1 5632 . . . . . . . . . 10  |-  ( `' H : A -1-1-onto-> C  ->  `' H : A -1-1-> C
)
806, 78, 793syl 19 . . . . . . . . 9  |-  ( ph  ->  `' H : A -1-1-> C
)
8180adantr 452 . . . . . . . 8  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  `' H : A -1-1-> C )
82 f1co 5607 . . . . . . . 8  |-  ( ( `' H : A -1-1-> C  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) )
-1-1-> A )  ->  ( `' H  o.  f
) : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) )
-1-1-> C )
8381, 43, 82syl2anc 643 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( `' H  o.  f ) : ( 1 ... ( # `  ( `' F "
( _V  \  {  .0.  } ) ) ) ) -1-1-> C )
84 imass2 5199 . . . . . . . . 9  |-  ( ( `' F " ( _V 
\  {  .0.  }
) )  C_  ran  f  ->  ( `' H " ( `' F "
( _V  \  {  .0.  } ) ) ) 
C_  ( `' H " ran  f ) )
8567, 84syl 16 . . . . . . . 8  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( `' H " ( `' F "
( _V  \  {  .0.  } ) ) ) 
C_  ( `' H " ran  f ) )
86 cnvco 5015 . . . . . . . . . 10  |-  `' ( F  o.  H )  =  ( `' H  o.  `' F )
8786imaeq1i 5159 . . . . . . . . 9  |-  ( `' ( F  o.  H
) " ( _V 
\  {  .0.  }
) )  =  ( ( `' H  o.  `' F ) " ( _V  \  {  .0.  }
) )
88 imaco 5334 . . . . . . . . 9  |-  ( ( `' H  o.  `' F ) " ( _V  \  {  .0.  }
) )  =  ( `' H " ( `' F " ( _V 
\  {  .0.  }
) ) )
8987, 88eqtri 2424 . . . . . . . 8  |-  ( `' ( F  o.  H
) " ( _V 
\  {  .0.  }
) )  =  ( `' H " ( `' F " ( _V 
\  {  .0.  }
) ) )
90 rnco2 5336 . . . . . . . 8  |-  ran  ( `' H  o.  f
)  =  ( `' H " ran  f
)
9185, 89, 903sstr4g 3349 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( `' ( F  o.  H )
" ( _V  \  {  .0.  } ) ) 
C_  ran  ( `' H  o.  f )
)
92 eqid 2404 . . . . . . 7  |-  ( `' ( ( F  o.  H )  o.  ( `' H  o.  f
) ) " ( _V  \  {  .0.  }
) )  =  ( `' ( ( F  o.  H )  o.  ( `' H  o.  f ) ) "
( _V  \  {  .0.  } ) )
9354, 3, 55, 56, 57, 70, 73, 77, 62, 83, 91, 92gsumval3 15469 . . . . . 6  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( G  gsumg  ( F  o.  H ) )  =  (  seq  1
( ( +g  `  G
) ,  ( ( F  o.  H )  o.  ( `' H  o.  f ) ) ) `
 ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) )
9453, 69, 933eqtr4d 2446 . . . . 5  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( G  gsumg  F )  =  ( G  gsumg  ( F  o.  H ) ) )
9594expr 599 . . . 4  |-  ( (
ph  /\  ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) )  e.  NN )  ->  (
f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) )  ->  ( G  gsumg  F )  =  ( G  gsumg  ( F  o.  H
) ) ) )
9695exlimdv 1643 . . 3  |-  ( (
ph  /\  ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) )  e.  NN )  ->  ( E. f  f :
( 1 ... ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) ) ) -1-1-onto-> ( `' F "
( _V  \  {  .0.  } ) )  -> 
( G  gsumg  F )  =  ( G  gsumg  ( F  o.  H
) ) ) )
9796expimpd 587 . 2  |-  ( ph  ->  ( ( ( # `  ( `' F "
( _V  \  {  .0.  } ) ) )  e.  NN  /\  E. f  f : ( 1 ... ( # `  ( `' F "
( _V  \  {  .0.  } ) ) ) ) -1-1-onto-> ( `' F "
( _V  \  {  .0.  } ) ) )  ->  ( G  gsumg  F )  =  ( G  gsumg  ( F  o.  H ) ) ) )
98 gsumzcl.w . . 3  |-  ( ph  ->  ( `' F "
( _V  \  {  .0.  } ) )  e. 
Fin )
99 fz1f1o 12459 . . 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.  } ) ) ) ) )
10098, 99syl 16 . 2  |-  ( ph  ->  ( ( `' F " ( _V  \  {  .0.  } ) )  =  (/)  \/  ( ( # `  ( `' F "
( _V  \  {  .0.  } ) ) )  e.  NN  /\  E. f  f : ( 1 ... ( # `  ( `' F "
( _V  \  {  .0.  } ) ) ) ) -1-1-onto-> ( `' F "
( _V  \  {  .0.  } ) ) ) ) )
10129, 97, 100mpjaod 371 1  |-  ( ph  ->  ( G  gsumg  F )  =  ( G  gsumg  ( F  o.  H
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 358    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1721   _Vcvv 2916    \ cdif 3277    C_ wss 3280   (/)c0 3588   {csn 3774    e. cmpt 4226    _I cid 4453   `'ccnv 4836   dom cdm 4837   ran crn 4838    |` cres 4839   "cima 4840    o. ccom 4841   -->wf 5409   -1-1->wf1 5410   -onto->wfo 5411   -1-1-onto->wf1o 5412   ` cfv 5413  (class class class)co 6040   Fincfn 7068   1c1 8947   NNcn 9956   ...cfz 10999    seq cseq 11278   #chash 11573   Basecbs 13424   +g cplusg 13484   0gc0g 13678    gsumg cgsu 13679   Mndcmnd 14639  Cntzccntz 15069
This theorem is referenced by:  gsumf1o  15477
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-oi 7435  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-n0 10178  df-z 10239  df-uz 10445  df-fz 11000  df-fzo 11091  df-seq 11279  df-hash 11574  df-0g 13682  df-gsum 13683  df-mnd 14645  df-cntz 15071
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