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Theorem gsumzf1o 16786
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 15874 . . . . . . 7  |-  ( ( G  e.  Mnd  /\  A  e.  V )  ->  ( G  gsumg  ( k  e.  A  |->  .0.  ) )  =  .0.  )
51, 2, 4syl2anc 661 . . . . . 6  |-  ( ph  ->  ( G  gsumg  ( k  e.  A  |->  .0.  ) )  =  .0.  )
6 gsumzf1o.h . . . . . . . . 9  |-  ( ph  ->  H : C -1-1-onto-> A )
7 f1of1 5801 . . . . . . . . 9  |-  ( H : C -1-1-onto-> A  ->  H : C -1-1-> A )
86, 7syl 16 . . . . . . . 8  |-  ( ph  ->  H : C -1-1-> A
)
9 f1dmex 6751 . . . . . . . 8  |-  ( ( H : C -1-1-> A  /\  A  e.  V
)  ->  C  e.  _V )
108, 2, 9syl2anc 661 . . . . . . 7  |-  ( ph  ->  C  e.  _V )
113gsumz 15874 . . . . . . 7  |-  ( ( G  e.  Mnd  /\  C  e.  _V )  ->  ( G  gsumg  ( x  e.  C  |->  .0.  ) )  =  .0.  )
121, 10, 11syl2anc 661 . . . . . 6  |-  ( ph  ->  ( G  gsumg  ( x  e.  C  |->  .0.  ) )  =  .0.  )
135, 12eqtr4d 2485 . . . . 5  |-  ( ph  ->  ( G  gsumg  ( k  e.  A  |->  .0.  ) )  =  ( G  gsumg  ( x  e.  C  |->  .0.  ) ) )
1413adantr 465 . . . 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 5862 . . . . . . . 8  |-  ( 0g
`  G )  e. 
_V
173, 16eqeltri 2525 . . . . . . 7  |-  .0.  e.  _V
1817a1i 11 . . . . . 6  |-  ( ph  ->  .0.  e.  _V )
19 ssid 3505 . . . . . . 7  |-  ( F supp 
.0.  )  C_  ( F supp  .0.  )
2019a1i 11 . . . . . 6  |-  ( ph  ->  ( F supp  .0.  )  C_  ( F supp  .0.  )
)
2115, 2, 18, 20gsumcllem 16781 . . . . 5  |-  ( (
ph  /\  ( F supp  .0.  )  =  (/) )  ->  F  =  ( k  e.  A  |->  .0.  )
)
2221oveq2d 6293 . . . 4  |-  ( (
ph  /\  ( F supp  .0.  )  =  (/) )  -> 
( G  gsumg  F )  =  ( G  gsumg  ( k  e.  A  |->  .0.  ) ) )
23 f1of 5802 . . . . . . . . 9  |-  ( H : C -1-1-onto-> A  ->  H : C
--> A )
246, 23syl 16 . . . . . . . 8  |-  ( ph  ->  H : C --> A )
2524adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( F supp  .0.  )  =  (/) )  ->  H : C --> A )
2625ffvelrnda 6012 . . . . . 6  |-  ( ( ( ph  /\  ( F supp  .0.  )  =  (/) )  /\  x  e.  C
)  ->  ( H `  x )  e.  A
)
2725feqmptd 5907 . . . . . 6  |-  ( (
ph  /\  ( F supp  .0.  )  =  (/) )  ->  H  =  ( x  e.  C  |->  ( H `
 x ) ) )
28 eqidd 2442 . . . . . 6  |-  ( k  =  ( H `  x )  ->  .0.  =  .0.  )
2926, 27, 21, 28fmptco 6045 . . . . 5  |-  ( (
ph  /\  ( F supp  .0.  )  =  (/) )  -> 
( F  o.  H
)  =  ( x  e.  C  |->  .0.  )
)
3029oveq2d 6293 . . . 4  |-  ( (
ph  /\  ( F supp  .0.  )  =  (/) )  -> 
( G  gsumg  ( F  o.  H
) )  =  ( G  gsumg  ( x  e.  C  |->  .0.  ) ) )
3114, 22, 303eqtr4d 2492 . . 3  |-  ( (
ph  /\  ( F supp  .0.  )  =  (/) )  -> 
( G  gsumg  F )  =  ( G  gsumg  ( F  o.  H
) ) )
3231ex 434 . 2  |-  ( ph  ->  ( ( F supp  .0.  )  =  (/)  ->  ( G  gsumg  F )  =  ( G  gsumg  ( F  o.  H
) ) ) )
33 coass 5512 . . . . . . . . . . 11  |-  ( ( H  o.  `' H
)  o.  f )  =  ( H  o.  ( `' H  o.  f
) )
346adantr 465 . . . . . . . . . . . . . 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 5828 . . . . . . . . . . . . . 14  |-  ( H : C -1-1-onto-> A  ->  ( H  o.  `' H )  =  (  _I  |`  A )
)
3634, 35syl 16 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( ( # `
 ( F supp  .0.  ) )  e.  NN  /\  f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
)  ->  ( H  o.  `' H )  =  (  _I  |`  A )
)
3736coeq1d 5150 . . . . . . . . . . . 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 5801 . . . . . . . . . . . . . . 15  |-  ( f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  )  ->  f : ( 1 ... ( # `  ( F supp  .0.  ) ) )
-1-1-> ( F supp  .0.  )
)
3938ad2antll 728 . . . . . . . . . . . . . 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 6912 . . . . . . . . . . . . . . . 16  |-  ( F supp 
.0.  )  C_  dom  F
41 fdm 5721 . . . . . . . . . . . . . . . . 17  |-  ( F : A --> B  ->  dom  F  =  A )
4215, 41syl 16 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  dom  F  =  A )
4340, 42syl5sseq 3534 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( F supp  .0.  )  C_  A )
4443adantr 465 . . . . . . . . . . . . . 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 5772 . . . . . . . . . . . . . 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 661 . . . . . . . . . . . . 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 5767 . . . . . . . . . . . . 13  |-  ( f : ( 1 ... ( # `  ( F supp  .0.  ) ) )
-1-1-> A  ->  f :
( 1 ... ( # `
 ( F supp  .0.  ) ) ) --> A )
48 fcoi2 5746 . . . . . . . . . . . . 13  |-  ( f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) --> A  ->  ( (  _I  |`  A )  o.  f )  =  f )
4946, 47, 483syl 20 . . . . . . . . . . . 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 2482 . . . . . . . . . . 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 2497 . . . . . . . . . 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 5151 . . . . . . . . 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 5512 . . . . . . . . 9  |-  ( ( F  o.  H )  o.  ( `' H  o.  f ) )  =  ( F  o.  ( H  o.  ( `' H  o.  f )
) )
5452, 53syl6eqr 2500 . . . . . . . 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 12089 . . . . . . 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 5854 . . . . . 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 2441 . . . . . . 7  |-  ( +g  `  G )  =  ( +g  `  G )
59 gsumzcl.z . . . . . . 7  |-  Z  =  (Cntz `  G )
601adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 ( F supp  .0.  ) )  e.  NN  /\  f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
)  ->  G  e.  Mnd )
612adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 ( F supp  .0.  ) )  e.  NN  /\  f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
)  ->  A  e.  V )
6215adantr 465 . . . . . . 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 465 . . . . . . 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 755 . . . . . . 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 5809 . . . . . . . . . 10  |-  ( f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  )  ->  f : ( 1 ... ( # `  ( F supp  .0.  ) ) )
-onto-> ( F supp  .0.  )
)
67 forn 5784 . . . . . . . . . 10  |-  ( f : ( 1 ... ( # `  ( F supp  .0.  ) ) )
-onto-> ( F supp  .0.  )  ->  ran  f  =  ( F supp  .0.  ) )
6866, 67syl 16 . . . . . . . . 9  |-  ( f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  )  ->  ran  f  =  ( F supp 
.0.  ) )
6968ad2antll 728 . . . . . . . 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 3535 . . . . . . 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 2441 . . . . . . 7  |-  ( ( F  o.  f ) supp 
.0.  )  =  ( ( F  o.  f
) supp  .0.  )
7257, 3, 58, 59, 60, 61, 62, 64, 65, 46, 70, 71gsumval3 16780 . . . . . 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 465 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 ( F supp  .0.  ) )  e.  NN  /\  f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
)  ->  C  e.  _V )
74 fco 5727 . . . . . . . . 9  |-  ( ( F : A --> B  /\  H : C --> A )  ->  ( F  o.  H ) : C --> B )
7515, 24, 74syl2anc 661 . . . . . . . 8  |-  ( ph  ->  ( F  o.  H
) : C --> B )
7675adantr 465 . . . . . . 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 5249 . . . . . . . . 9  |-  ran  ( F  o.  H )  C_ 
ran  F
7859cntzidss 16244 . . . . . . . . 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 662 . . . . . . . 8  |-  ( ph  ->  ran  ( F  o.  H )  C_  ( Z `  ran  ( F  o.  H ) ) )
8079adantr 465 . . . . . . 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 5814 . . . . . . . . . 10  |-  ( H : C -1-1-onto-> A  ->  `' H : A -1-1-onto-> C )
82 f1of1 5801 . . . . . . . . . 10  |-  ( `' H : A -1-1-onto-> C  ->  `' H : A -1-1-> C
)
836, 81, 823syl 20 . . . . . . . . 9  |-  ( ph  ->  `' H : A -1-1-> C
)
8483adantr 465 . . . . . . . 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 5776 . . . . . . . 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 661 . . . . . . 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 6126 . . . . . . . . . . . . . . 15  |-  ( ( F : A --> B  /\  A  e.  V )  ->  F  e.  _V )
8915, 2, 88syl2anc 661 . . . . . . . . . . . . . 14  |-  ( ph  ->  F  e.  _V )
90 suppimacnv 6910 . . . . . . . . . . . . . 14  |-  ( ( F  e.  _V  /\  .0.  e.  _V )  -> 
( F supp  .0.  )  =  ( `' F " ( _V  \  {  .0.  } ) ) )
9189, 17, 90sylancl 662 . . . . . . . . . . . . 13  |-  ( ph  ->  ( F supp  .0.  )  =  ( `' F " ( _V  \  {  .0.  } ) ) )
9291eqcomd 2449 . . . . . . . . . . . 12  |-  ( ph  ->  ( `' F "
( _V  \  {  .0.  } ) )  =  ( F supp  .0.  )
)
9392adantr 465 . . . . . . . . . . 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 3529 . . . . . . . . . 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 5358 . . . . . . . . . 10  |-  ( ( `' F " ( _V 
\  {  .0.  }
) )  C_  ran  f  ->  ( `' H " ( `' F "
( _V  \  {  .0.  } ) ) ) 
C_  ( `' H " ran  f ) )
9694, 95syl 16 . . . . . . . . 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 5174 . . . . . . . . . . 11  |-  `' ( F  o.  H )  =  ( `' H  o.  `' F )
9897imaeq1i 5320 . . . . . . . . . 10  |-  ( `' ( F  o.  H
) " ( _V 
\  {  .0.  }
) )  =  ( ( `' H  o.  `' F ) " ( _V  \  {  .0.  }
) )
99 imaco 5498 . . . . . . . . . 10  |-  ( ( `' H  o.  `' F ) " ( _V  \  {  .0.  }
) )  =  ( `' H " ( `' F " ( _V 
\  {  .0.  }
) ) )
10098, 99eqtri 2470 . . . . . . . . 9  |-  ( `' ( F  o.  H
) " ( _V 
\  {  .0.  }
) )  =  ( `' H " ( `' F " ( _V 
\  {  .0.  }
) ) )
101 rnco2 5500 . . . . . . . . 9  |-  ran  ( `' H  o.  f
)  =  ( `' H " ran  f
)
10296, 100, 1013sstr4g 3527 . . . . . . . 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 6730 . . . . . . . . . . . . 13  |-  ( ( H : C -1-1-onto-> A  /\  A  e.  V )  ->  H  e.  _V )
1046, 2, 103syl2anc 661 . . . . . . . . . . . 12  |-  ( ph  ->  H  e.  _V )
105 coexg 6732 . . . . . . . . . . . 12  |-  ( ( F  e.  _V  /\  H  e.  _V )  ->  ( F  o.  H
)  e.  _V )
10689, 104, 105syl2anc 661 . . . . . . . . . . 11  |-  ( ph  ->  ( F  o.  H
)  e.  _V )
107 suppimacnv 6910 . . . . . . . . . . 11  |-  ( ( ( F  o.  H
)  e.  _V  /\  .0.  e.  _V )  -> 
( ( F  o.  H ) supp  .0.  )  =  ( `' ( F  o.  H )
" ( _V  \  {  .0.  } ) ) )
108106, 17, 107sylancl 662 . . . . . . . . . 10  |-  ( ph  ->  ( ( F  o.  H ) supp  .0.  )  =  ( `' ( F  o.  H )
" ( _V  \  {  .0.  } ) ) )
109108sseq1d 3513 . . . . . . . . 9  |-  ( ph  ->  ( ( ( F  o.  H ) supp  .0.  )  C_  ran  ( `' H  o.  f )  <-> 
( `' ( F  o.  H ) "
( _V  \  {  .0.  } ) )  C_  ran  ( `' H  o.  f ) ) )
110109adantr 465 . . . . . . . 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 232 . . . . . . 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 2441 . . . . . . 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 16780 . . . . . 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 2492 . . . . 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 615 . . . 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 1709 . . 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 603 . 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 7833 . . . 4  |-  ( F finSupp  .0.  ->  ( Fun  F  /\  ( F supp  .0.  )  e.  Fin ) )
120119simprd 463 . . 3  |-  ( F finSupp  .0.  ->  ( F supp  .0.  )  e.  Fin )
121 fz1f1o 13506 . . 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 20 . 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 381 1  |-  ( ph  ->  ( G  gsumg  F )  =  ( G  gsumg  ( F  o.  H
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1381   E.wex 1597    e. wcel 1802   _Vcvv 3093    \ cdif 3455    C_ wss 3458   (/)c0 3767   {csn 4010   class class class wbr 4433    |-> cmpt 4491    _I cid 4776   `'ccnv 4984   dom cdm 4985   ran crn 4986    |` cres 4987   "cima 4988    o. ccom 4989   Fun wfun 5568   -->wf 5570   -1-1->wf1 5571   -onto->wfo 5572   -1-1-onto->wf1o 5573   ` cfv 5574  (class class class)co 6277   supp csupp 6899   Fincfn 7514   finSupp cfsupp 7827   1c1 9491   NNcn 10537   ...cfz 11676    seqcseq 12081   #chash 12379   Basecbs 14504   +g cplusg 14569   0gc0g 14709    gsumg cgsu 14710   Mndcmnd 15788  Cntzccntz 16222
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-int 4268  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-se 4825  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-isom 5583  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-1st 6781  df-2nd 6782  df-supp 6900  df-recs 7040  df-rdg 7074  df-1o 7128  df-oadd 7132  df-er 7309  df-en 7515  df-dom 7516  df-sdom 7517  df-fin 7518  df-fsupp 7828  df-oi 7933  df-card 8318  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-nn 10538  df-n0 10797  df-z 10866  df-uz 11086  df-fz 11677  df-fzo 11799  df-seq 12082  df-hash 12380  df-0g 14711  df-gsum 14712  df-mgm 15741  df-sgrp 15780  df-mnd 15790  df-cntz 16224
This theorem is referenced by:  gsumf1o  16793  smadiadetlem3  19037
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