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Theorem gsumzf1o 17545
Description: Re-index a finite group sum using a bijection. (Contributed by Mario Carneiro, 24-Apr-2016.) (Revised by AV, 2-Jun-2019.)
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 finSupp  .0.  )
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 16620 . . . . . . 7  |-  ( ( G  e.  Mnd  /\  A  e.  V )  ->  ( G  gsumg  ( k  e.  A  |->  .0.  ) )  =  .0.  )
51, 2, 4syl2anc 665 . . . . . 6  |-  ( ph  ->  ( G  gsumg  ( k  e.  A  |->  .0.  ) )  =  .0.  )
6 gsumzf1o.h . . . . . . . . 9  |-  ( ph  ->  H : C -1-1-onto-> A )
7 f1of1 5830 . . . . . . . . 9  |-  ( H : C -1-1-onto-> A  ->  H : C -1-1-> A )
86, 7syl 17 . . . . . . . 8  |-  ( ph  ->  H : C -1-1-> A
)
9 f1dmex 6777 . . . . . . . 8  |-  ( ( H : C -1-1-> A  /\  A  e.  V
)  ->  C  e.  _V )
108, 2, 9syl2anc 665 . . . . . . 7  |-  ( ph  ->  C  e.  _V )
113gsumz 16620 . . . . . . 7  |-  ( ( G  e.  Mnd  /\  C  e.  _V )  ->  ( G  gsumg  ( x  e.  C  |->  .0.  ) )  =  .0.  )
121, 10, 11syl2anc 665 . . . . . 6  |-  ( ph  ->  ( G  gsumg  ( x  e.  C  |->  .0.  ) )  =  .0.  )
135, 12eqtr4d 2466 . . . . 5  |-  ( ph  ->  ( G  gsumg  ( k  e.  A  |->  .0.  ) )  =  ( G  gsumg  ( x  e.  C  |->  .0.  ) ) )
1413adantr 466 . . . 4  |-  ( (
ph  /\  ( F supp  .0.  )  =  (/) )  -> 
( G  gsumg  ( k  e.  A  |->  .0.  ) )  =  ( G  gsumg  ( x  e.  C  |->  .0.  ) ) )
15 gsumzcl.f . . . . . 6  |-  ( ph  ->  F : A --> B )
16 fvex 5891 . . . . . . . 8  |-  ( 0g
`  G )  e. 
_V
173, 16eqeltri 2503 . . . . . . 7  |-  .0.  e.  _V
1817a1i 11 . . . . . 6  |-  ( ph  ->  .0.  e.  _V )
19 ssid 3483 . . . . . . 7  |-  ( F supp 
.0.  )  C_  ( F supp  .0.  )
2019a1i 11 . . . . . 6  |-  ( ph  ->  ( F supp  .0.  )  C_  ( F supp  .0.  )
)
2115, 2, 18, 20gsumcllem 17541 . . . . 5  |-  ( (
ph  /\  ( F supp  .0.  )  =  (/) )  ->  F  =  ( k  e.  A  |->  .0.  )
)
2221oveq2d 6321 . . . 4  |-  ( (
ph  /\  ( F supp  .0.  )  =  (/) )  -> 
( G  gsumg  F )  =  ( G  gsumg  ( k  e.  A  |->  .0.  ) ) )
23 f1of 5831 . . . . . . . . 9  |-  ( H : C -1-1-onto-> A  ->  H : C
--> A )
246, 23syl 17 . . . . . . . 8  |-  ( ph  ->  H : C --> A )
2524adantr 466 . . . . . . 7  |-  ( (
ph  /\  ( F supp  .0.  )  =  (/) )  ->  H : C --> A )
2625ffvelrnda 6037 . . . . . 6  |-  ( ( ( ph  /\  ( F supp  .0.  )  =  (/) )  /\  x  e.  C
)  ->  ( H `  x )  e.  A
)
2725feqmptd 5934 . . . . . 6  |-  ( (
ph  /\  ( F supp  .0.  )  =  (/) )  ->  H  =  ( x  e.  C  |->  ( H `
 x ) ) )
28 eqidd 2423 . . . . . 6  |-  ( k  =  ( H `  x )  ->  .0.  =  .0.  )
2926, 27, 21, 28fmptco 6071 . . . . 5  |-  ( (
ph  /\  ( F supp  .0.  )  =  (/) )  -> 
( F  o.  H
)  =  ( x  e.  C  |->  .0.  )
)
3029oveq2d 6321 . . . 4  |-  ( (
ph  /\  ( F supp  .0.  )  =  (/) )  -> 
( G  gsumg  ( F  o.  H
) )  =  ( G  gsumg  ( x  e.  C  |->  .0.  ) ) )
3114, 22, 303eqtr4d 2473 . . 3  |-  ( (
ph  /\  ( F supp  .0.  )  =  (/) )  -> 
( G  gsumg  F )  =  ( G  gsumg  ( F  o.  H
) ) )
3231ex 435 . 2  |-  ( ph  ->  ( ( F supp  .0.  )  =  (/)  ->  ( G  gsumg  F )  =  ( G  gsumg  ( F  o.  H
) ) ) )
33 coass 5373 . . . . . . . . . . 11  |-  ( ( H  o.  `' H
)  o.  f )  =  ( H  o.  ( `' H  o.  f
) )
346adantr 466 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( ( # `
 ( F supp  .0.  ) )  e.  NN  /\  f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
)  ->  H : C
-1-1-onto-> A )
35 f1ococnv2 5857 . . . . . . . . . . . . . 14  |-  ( H : C -1-1-onto-> A  ->  ( H  o.  `' H )  =  (  _I  |`  A )
)
3634, 35syl 17 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( ( # `
 ( F supp  .0.  ) )  e.  NN  /\  f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
)  ->  ( H  o.  `' H )  =  (  _I  |`  A )
)
3736coeq1d 5015 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( ( # `
 ( F supp  .0.  ) )  e.  NN  /\  f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
)  ->  ( ( H  o.  `' H
)  o.  f )  =  ( (  _I  |`  A )  o.  f
) )
38 f1of1 5830 . . . . . . . . . . . . . . 15  |-  ( f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  )  ->  f : ( 1 ... ( # `  ( F supp  .0.  ) ) )
-1-1-> ( F supp  .0.  )
)
3938ad2antll 733 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( ( # `
 ( F supp  .0.  ) )  e.  NN  /\  f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
)  ->  f :
( 1 ... ( # `
 ( F supp  .0.  ) ) ) -1-1-> ( F supp  .0.  ) )
40 suppssdm 6938 . . . . . . . . . . . . . . . 16  |-  ( F supp 
.0.  )  C_  dom  F
41 fdm 5750 . . . . . . . . . . . . . . . . 17  |-  ( F : A --> B  ->  dom  F  =  A )
4215, 41syl 17 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  dom  F  =  A )
4340, 42syl5sseq 3512 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( F supp  .0.  )  C_  A )
4443adantr 466 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( ( # `
 ( F supp  .0.  ) )  e.  NN  /\  f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
)  ->  ( F supp  .0.  )  C_  A )
45 f1ss 5801 . . . . . . . . . . . . . 14  |-  ( ( f : ( 1 ... ( # `  ( F supp  .0.  ) ) )
-1-1-> ( F supp  .0.  )  /\  ( F supp  .0.  )  C_  A )  ->  f : ( 1 ... ( # `  ( F supp  .0.  ) ) )
-1-1-> A )
4639, 44, 45syl2anc 665 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( ( # `
 ( F supp  .0.  ) )  e.  NN  /\  f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
)  ->  f :
( 1 ... ( # `
 ( F supp  .0.  ) ) ) -1-1-> A
)
47 f1f 5796 . . . . . . . . . . . . 13  |-  ( f : ( 1 ... ( # `  ( F supp  .0.  ) ) )
-1-1-> A  ->  f :
( 1 ... ( # `
 ( F supp  .0.  ) ) ) --> A )
48 fcoi2 5775 . . . . . . . . . . . . 13  |-  ( f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) --> A  ->  ( (  _I  |`  A )  o.  f )  =  f )
4946, 47, 483syl 18 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( ( # `
 ( F supp  .0.  ) )  e.  NN  /\  f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
)  ->  ( (  _I  |`  A )  o.  f )  =  f )
5037, 49eqtrd 2463 . . . . . . . . . . 11  |-  ( (
ph  /\  ( ( # `
 ( F supp  .0.  ) )  e.  NN  /\  f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
)  ->  ( ( H  o.  `' H
)  o.  f )  =  f )
5133, 50syl5reqr 2478 . . . . . . . . . 10  |-  ( (
ph  /\  ( ( # `
 ( F supp  .0.  ) )  e.  NN  /\  f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
)  ->  f  =  ( H  o.  ( `' H  o.  f
) ) )
5251coeq2d 5016 . . . . . . . . 9  |-  ( (
ph  /\  ( ( # `
 ( F supp  .0.  ) )  e.  NN  /\  f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
)  ->  ( F  o.  f )  =  ( F  o.  ( H  o.  ( `' H  o.  f ) ) ) )
53 coass 5373 . . . . . . . . 9  |-  ( ( F  o.  H )  o.  ( `' H  o.  f ) )  =  ( F  o.  ( H  o.  ( `' H  o.  f )
) )
5452, 53syl6eqr 2481 . . . . . . . 8  |-  ( (
ph  /\  ( ( # `
 ( F supp  .0.  ) )  e.  NN  /\  f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
)  ->  ( F  o.  f )  =  ( ( F  o.  H
)  o.  ( `' H  o.  f ) ) )
5554seqeq3d 12227 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 ( F supp  .0.  ) )  e.  NN  /\  f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
)  ->  seq 1
( ( +g  `  G
) ,  ( F  o.  f ) )  =  seq 1 ( ( +g  `  G
) ,  ( ( F  o.  H )  o.  ( `' H  o.  f ) ) ) )
5655fveq1d 5883 . . . . . 6  |-  ( (
ph  /\  ( ( # `
 ( F supp  .0.  ) )  e.  NN  /\  f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
)  ->  (  seq 1 ( ( +g  `  G ) ,  ( F  o.  f ) ) `  ( # `  ( F supp  .0.  )
) )  =  (  seq 1 ( ( +g  `  G ) ,  ( ( F  o.  H )  o.  ( `' H  o.  f ) ) ) `
 ( # `  ( F supp  .0.  ) ) ) )
57 gsumzcl.b . . . . . . 7  |-  B  =  ( Base `  G
)
58 eqid 2422 . . . . . . 7  |-  ( +g  `  G )  =  ( +g  `  G )
59 gsumzcl.z . . . . . . 7  |-  Z  =  (Cntz `  G )
601adantr 466 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 ( F supp  .0.  ) )  e.  NN  /\  f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
)  ->  G  e.  Mnd )
612adantr 466 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 ( F supp  .0.  ) )  e.  NN  /\  f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
)  ->  A  e.  V )
6215adantr 466 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 ( F supp  .0.  ) )  e.  NN  /\  f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
)  ->  F : A
--> B )
63 gsumzcl.c . . . . . . . 8  |-  ( ph  ->  ran  F  C_  ( Z `  ran  F ) )
6463adantr 466 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 ( F supp  .0.  ) )  e.  NN  /\  f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
)  ->  ran  F  C_  ( Z `  ran  F
) )
65 simprl 762 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 ( F supp  .0.  ) )  e.  NN  /\  f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
)  ->  ( # `  ( F supp  .0.  ) )  e.  NN )
66 f1ofo 5838 . . . . . . . . . 10  |-  ( f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  )  ->  f : ( 1 ... ( # `  ( F supp  .0.  ) ) )
-onto-> ( F supp  .0.  )
)
67 forn 5813 . . . . . . . . . 10  |-  ( f : ( 1 ... ( # `  ( F supp  .0.  ) ) )
-onto-> ( F supp  .0.  )  ->  ran  f  =  ( F supp  .0.  ) )
6866, 67syl 17 . . . . . . . . 9  |-  ( f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  )  ->  ran  f  =  ( F supp 
.0.  ) )
6968ad2antll 733 . . . . . . . 8  |-  ( (
ph  /\  ( ( # `
 ( F supp  .0.  ) )  e.  NN  /\  f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
)  ->  ran  f  =  ( F supp  .0.  )
)
7019, 69syl5sseqr 3513 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 ( F supp  .0.  ) )  e.  NN  /\  f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
)  ->  ( F supp  .0.  )  C_  ran  f )
71 eqid 2422 . . . . . . 7  |-  ( ( F  o.  f ) supp 
.0.  )  =  ( ( F  o.  f
) supp  .0.  )
7257, 3, 58, 59, 60, 61, 62, 64, 65, 46, 70, 71gsumval3 17540 . . . . . 6  |-  ( (
ph  /\  ( ( # `
 ( F supp  .0.  ) )  e.  NN  /\  f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
)  ->  ( G  gsumg  F )  =  (  seq 1 ( ( +g  `  G ) ,  ( F  o.  f ) ) `  ( # `  ( F supp  .0.  )
) ) )
7310adantr 466 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 ( F supp  .0.  ) )  e.  NN  /\  f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
)  ->  C  e.  _V )
74 fco 5756 . . . . . . . . 9  |-  ( ( F : A --> B  /\  H : C --> A )  ->  ( F  o.  H ) : C --> B )
7515, 24, 74syl2anc 665 . . . . . . . 8  |-  ( ph  ->  ( F  o.  H
) : C --> B )
7675adantr 466 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 ( F supp  .0.  ) )  e.  NN  /\  f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
)  ->  ( F  o.  H ) : C --> B )
77 rncoss 5114 . . . . . . . . 9  |-  ran  ( F  o.  H )  C_ 
ran  F
7859cntzidss 16990 . . . . . . . . 9  |-  ( ( ran  F  C_  ( Z `  ran  F )  /\  ran  ( F  o.  H )  C_  ran  F )  ->  ran  ( F  o.  H
)  C_  ( Z `  ran  ( F  o.  H ) ) )
7963, 77, 78sylancl 666 . . . . . . . 8  |-  ( ph  ->  ran  ( F  o.  H )  C_  ( Z `  ran  ( F  o.  H ) ) )
8079adantr 466 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 ( F supp  .0.  ) )  e.  NN  /\  f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
)  ->  ran  ( F  o.  H )  C_  ( Z `  ran  ( F  o.  H )
) )
81 f1ocnv 5843 . . . . . . . . . 10  |-  ( H : C -1-1-onto-> A  ->  `' H : A -1-1-onto-> C )
82 f1of1 5830 . . . . . . . . . 10  |-  ( `' H : A -1-1-onto-> C  ->  `' H : A -1-1-> C
)
836, 81, 823syl 18 . . . . . . . . 9  |-  ( ph  ->  `' H : A -1-1-> C
)
8483adantr 466 . . . . . . . 8  |-  ( (
ph  /\  ( ( # `
 ( F supp  .0.  ) )  e.  NN  /\  f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
)  ->  `' H : A -1-1-> C )
85 f1co 5805 . . . . . . . 8  |-  ( ( `' H : A -1-1-> C  /\  f : ( 1 ... ( # `  ( F supp  .0.  ) ) )
-1-1-> A )  ->  ( `' H  o.  f
) : ( 1 ... ( # `  ( F supp  .0.  ) ) )
-1-1-> C )
8684, 46, 85syl2anc 665 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 ( F supp  .0.  ) )  e.  NN  /\  f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
)  ->  ( `' H  o.  f ) : ( 1 ... ( # `  ( F supp  .0.  ) ) )
-1-1-> C )
8719a1i 11 . . . . . . . . . . 11  |-  ( (
ph  /\  ( ( # `
 ( F supp  .0.  ) )  e.  NN  /\  f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
)  ->  ( F supp  .0.  )  C_  ( F supp  .0.  ) )
88 fex 6153 . . . . . . . . . . . . . . 15  |-  ( ( F : A --> B  /\  A  e.  V )  ->  F  e.  _V )
8915, 2, 88syl2anc 665 . . . . . . . . . . . . . 14  |-  ( ph  ->  F  e.  _V )
90 suppimacnv 6936 . . . . . . . . . . . . . 14  |-  ( ( F  e.  _V  /\  .0.  e.  _V )  -> 
( F supp  .0.  )  =  ( `' F " ( _V  \  {  .0.  } ) ) )
9189, 17, 90sylancl 666 . . . . . . . . . . . . 13  |-  ( ph  ->  ( F supp  .0.  )  =  ( `' F " ( _V  \  {  .0.  } ) ) )
9291eqcomd 2430 . . . . . . . . . . . 12  |-  ( ph  ->  ( `' F "
( _V  \  {  .0.  } ) )  =  ( F supp  .0.  )
)
9392adantr 466 . . . . . . . . . . 11  |-  ( (
ph  /\  ( ( # `
 ( F supp  .0.  ) )  e.  NN  /\  f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
)  ->  ( `' F " ( _V  \  {  .0.  } ) )  =  ( F supp  .0.  ) )
9487, 93, 693sstr4d 3507 . . . . . . . . . 10  |-  ( (
ph  /\  ( ( # `
 ( F supp  .0.  ) )  e.  NN  /\  f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
)  ->  ( `' F " ( _V  \  {  .0.  } ) ) 
C_  ran  f )
95 imass2 5223 . . . . . . . . . 10  |-  ( ( `' F " ( _V 
\  {  .0.  }
) )  C_  ran  f  ->  ( `' H " ( `' F "
( _V  \  {  .0.  } ) ) ) 
C_  ( `' H " ran  f ) )
9694, 95syl 17 . . . . . . . . 9  |-  ( (
ph  /\  ( ( # `
 ( F supp  .0.  ) )  e.  NN  /\  f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
)  ->  ( `' H " ( `' F " ( _V  \  {  .0.  } ) ) ) 
C_  ( `' H " ran  f ) )
97 cnvco 5039 . . . . . . . . . . 11  |-  `' ( F  o.  H )  =  ( `' H  o.  `' F )
9897imaeq1i 5184 . . . . . . . . . 10  |-  ( `' ( F  o.  H
) " ( _V 
\  {  .0.  }
) )  =  ( ( `' H  o.  `' F ) " ( _V  \  {  .0.  }
) )
99 imaco 5359 . . . . . . . . . 10  |-  ( ( `' H  o.  `' F ) " ( _V  \  {  .0.  }
) )  =  ( `' H " ( `' F " ( _V 
\  {  .0.  }
) ) )
10098, 99eqtri 2451 . . . . . . . . 9  |-  ( `' ( F  o.  H
) " ( _V 
\  {  .0.  }
) )  =  ( `' H " ( `' F " ( _V 
\  {  .0.  }
) ) )
101 rnco2 5361 . . . . . . . . 9  |-  ran  ( `' H  o.  f
)  =  ( `' H " ran  f
)
10296, 100, 1013sstr4g 3505 . . . . . . . 8  |-  ( (
ph  /\  ( ( # `
 ( F supp  .0.  ) )  e.  NN  /\  f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
)  ->  ( `' ( F  o.  H
) " ( _V 
\  {  .0.  }
) )  C_  ran  ( `' H  o.  f
) )
103 f1oexrnex 6756 . . . . . . . . . . . . 13  |-  ( ( H : C -1-1-onto-> A  /\  A  e.  V )  ->  H  e.  _V )
1046, 2, 103syl2anc 665 . . . . . . . . . . . 12  |-  ( ph  ->  H  e.  _V )
105 coexg 6758 . . . . . . . . . . . 12  |-  ( ( F  e.  _V  /\  H  e.  _V )  ->  ( F  o.  H
)  e.  _V )
10689, 104, 105syl2anc 665 . . . . . . . . . . 11  |-  ( ph  ->  ( F  o.  H
)  e.  _V )
107 suppimacnv 6936 . . . . . . . . . . 11  |-  ( ( ( F  o.  H
)  e.  _V  /\  .0.  e.  _V )  -> 
( ( F  o.  H ) supp  .0.  )  =  ( `' ( F  o.  H )
" ( _V  \  {  .0.  } ) ) )
108106, 17, 107sylancl 666 . . . . . . . . . 10  |-  ( ph  ->  ( ( F  o.  H ) supp  .0.  )  =  ( `' ( F  o.  H )
" ( _V  \  {  .0.  } ) ) )
109108sseq1d 3491 . . . . . . . . 9  |-  ( ph  ->  ( ( ( F  o.  H ) supp  .0.  )  C_  ran  ( `' H  o.  f )  <-> 
( `' ( F  o.  H ) "
( _V  \  {  .0.  } ) )  C_  ran  ( `' H  o.  f ) ) )
110109adantr 466 . . . . . . . 8  |-  ( (
ph  /\  ( ( # `
 ( F supp  .0.  ) )  e.  NN  /\  f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
)  ->  ( (
( F  o.  H
) supp  .0.  )  C_  ran  ( `' H  o.  f )  <->  ( `' ( F  o.  H
) " ( _V 
\  {  .0.  }
) )  C_  ran  ( `' H  o.  f
) ) )
111102, 110mpbird 235 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 ( F supp  .0.  ) )  e.  NN  /\  f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
)  ->  ( ( F  o.  H ) supp  .0.  )  C_  ran  ( `' H  o.  f
) )
112 eqid 2422 . . . . . . 7  |-  ( ( ( F  o.  H
)  o.  ( `' H  o.  f ) ) supp  .0.  )  =  ( ( ( F  o.  H )  o.  ( `' H  o.  f ) ) supp  .0.  )
11357, 3, 58, 59, 60, 73, 76, 80, 65, 86, 111, 112gsumval3 17540 . . . . . 6  |-  ( (
ph  /\  ( ( # `
 ( F supp  .0.  ) )  e.  NN  /\  f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
)  ->  ( G  gsumg  ( F  o.  H ) )  =  (  seq 1 ( ( +g  `  G ) ,  ( ( F  o.  H
)  o.  ( `' H  o.  f ) ) ) `  ( # `
 ( F supp  .0.  ) ) ) )
11456, 72, 1133eqtr4d 2473 . . . . 5  |-  ( (
ph  /\  ( ( # `
 ( F supp  .0.  ) )  e.  NN  /\  f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
)  ->  ( G  gsumg  F )  =  ( G 
gsumg  ( F  o.  H
) ) )
115114expr 618 . . . 4  |-  ( (
ph  /\  ( # `  ( F supp  .0.  ) )  e.  NN )  ->  (
f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  )  ->  ( G  gsumg  F )  =  ( G  gsumg  ( F  o.  H
) ) ) )
116115exlimdv 1772 . . 3  |-  ( (
ph  /\  ( # `  ( F supp  .0.  ) )  e.  NN )  ->  ( E. f  f :
( 1 ... ( # `
 ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp 
.0.  )  ->  ( G  gsumg  F )  =  ( G  gsumg  ( F  o.  H
) ) ) )
117116expimpd 606 . 2  |-  ( ph  ->  ( ( ( # `  ( F supp  .0.  )
)  e.  NN  /\  E. f  f : ( 1 ... ( # `  ( F supp  .0.  )
) ) -1-1-onto-> ( F supp  .0.  )
)  ->  ( G  gsumg  F )  =  ( G 
gsumg  ( F  o.  H
) ) ) )
118 gsumzcl.w . . 3  |-  ( ph  ->  F finSupp  .0.  )
119 fsuppimp 7898 . . . 4  |-  ( F finSupp  .0.  ->  ( Fun  F  /\  ( F supp  .0.  )  e.  Fin ) )
120119simprd 464 . . 3  |-  ( F finSupp  .0.  ->  ( F supp  .0.  )  e.  Fin )
121 fz1f1o 13775 . . 3  |-  ( ( F supp  .0.  )  e.  Fin  ->  ( ( F supp 
.0.  )  =  (/)  \/  ( ( # `  ( F supp  .0.  ) )  e.  NN  /\  E. f 
f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
) )
122118, 120, 1213syl 18 . 2  |-  ( ph  ->  ( ( F supp  .0.  )  =  (/)  \/  (
( # `  ( F supp 
.0.  ) )  e.  NN  /\  E. f 
f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
) )
12332, 117, 122mpjaod 382 1  |-  ( ph  ->  ( G  gsumg  F )  =  ( G  gsumg  ( F  o.  H
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    \/ wo 369    /\ wa 370    = wceq 1437   E.wex 1657    e. wcel 1872   _Vcvv 3080    \ cdif 3433    C_ wss 3436   (/)c0 3761   {csn 3998   class class class wbr 4423    |-> cmpt 4482    _I cid 4763   `'ccnv 4852   dom cdm 4853   ran crn 4854    |` cres 4855   "cima 4856    o. ccom 4857   Fun wfun 5595   -->wf 5597   -1-1->wf1 5598   -onto->wfo 5599   -1-1-onto->wf1o 5600   ` cfv 5601  (class class class)co 6305   supp csupp 6925   Fincfn 7580   finSupp cfsupp 7892   1c1 9547   NNcn 10616   ...cfz 11791    seqcseq 12219   #chash 12521   Basecbs 15120   +g cplusg 15189   0gc0g 15337    gsumg cgsu 15338   Mndcmnd 16534  Cntzccntz 16968
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597  ax-cnex 9602  ax-resscn 9603  ax-1cn 9604  ax-icn 9605  ax-addcl 9606  ax-addrcl 9607  ax-mulcl 9608  ax-mulrcl 9609  ax-mulcom 9610  ax-addass 9611  ax-mulass 9612  ax-distr 9613  ax-i2m1 9614  ax-1ne0 9615  ax-1rid 9616  ax-rnegex 9617  ax-rrecex 9618  ax-cnre 9619  ax-pre-lttri 9620  ax-pre-lttrn 9621  ax-pre-ltadd 9622  ax-pre-mulgt0 9623
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rmo 2779  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-int 4256  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-se 4813  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-supp 6926  df-wrecs 7039  df-recs 7101  df-rdg 7139  df-1o 7193  df-oadd 7197  df-er 7374  df-en 7581  df-dom 7582  df-sdom 7583  df-fin 7584  df-fsupp 7893  df-oi 8034  df-card 8381  df-pnf 9684  df-mnf 9685  df-xr 9686  df-ltxr 9687  df-le 9688  df-sub 9869  df-neg 9870  df-nn 10617  df-n0 10877  df-z 10945  df-uz 11167  df-fz 11792  df-fzo 11923  df-seq 12220  df-hash 12522  df-0g 15339  df-gsum 15340  df-mgm 16487  df-sgrp 16526  df-mnd 16536  df-cntz 16970
This theorem is referenced by:  gsumf1o  17549  smadiadetlem3  19691
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