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Theorem gsumzf1o 16412
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 15532 . . . . . . 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 5661 . . . . . . . . 9  |-  ( H : C -1-1-onto-> A  ->  H : C -1-1-> A )
86, 7syl 16 . . . . . . . 8  |-  ( ph  ->  H : C -1-1-> A
)
9 f1dmex 6568 . . . . . . . 8  |-  ( ( H : C -1-1-> A  /\  A  e.  V
)  ->  C  e.  _V )
108, 2, 9syl2anc 661 . . . . . . 7  |-  ( ph  ->  C  e.  _V )
113gsumz 15532 . . . . . . 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 2478 . . . . 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 5722 . . . . . . . 8  |-  ( 0g
`  G )  e. 
_V
173, 16eqeltri 2513 . . . . . . 7  |-  .0.  e.  _V
1817a1i 11 . . . . . 6  |-  ( ph  ->  .0.  e.  _V )
19 ssid 3396 . . . . . . 7  |-  ( F supp 
.0.  )  C_  ( F supp  .0.  )
2019a1i 11 . . . . . 6  |-  ( ph  ->  ( F supp  .0.  )  C_  ( F supp  .0.  )
)
2115, 2, 18, 20gsumcllem 16407 . . . . 5  |-  ( (
ph  /\  ( F supp  .0.  )  =  (/) )  ->  F  =  ( k  e.  A  |->  .0.  )
)
2221oveq2d 6128 . . . 4  |-  ( (
ph  /\  ( F supp  .0.  )  =  (/) )  -> 
( G  gsumg  F )  =  ( G  gsumg  ( k  e.  A  |->  .0.  ) ) )
23 f1of 5662 . . . . . . . . 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 5864 . . . . . 6  |-  ( ( ( ph  /\  ( F supp  .0.  )  =  (/) )  /\  x  e.  C
)  ->  ( H `  x )  e.  A
)
2725feqmptd 5765 . . . . . 6  |-  ( (
ph  /\  ( F supp  .0.  )  =  (/) )  ->  H  =  ( x  e.  C  |->  ( H `
 x ) ) )
28 eqidd 2444 . . . . . 6  |-  ( k  =  ( H `  x )  ->  .0.  =  .0.  )
2926, 27, 21, 28fmptco 5897 . . . . 5  |-  ( (
ph  /\  ( F supp  .0.  )  =  (/) )  -> 
( F  o.  H
)  =  ( x  e.  C  |->  .0.  )
)
3029oveq2d 6128 . . . 4  |-  ( (
ph  /\  ( F supp  .0.  )  =  (/) )  -> 
( G  gsumg  ( F  o.  H
) )  =  ( G  gsumg  ( x  e.  C  |->  .0.  ) ) )
3114, 22, 303eqtr4d 2485 . . 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 5377 . . . . . . . . . . 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 5688 . . . . . . . . . . . . . 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 5022 . . . . . . . . . . . 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 5661 . . . . . . . . . . . . . . 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 6724 . . . . . . . . . . . . . . . 16  |-  ( F supp 
.0.  )  C_  dom  F
41 fdm 5584 . . . . . . . . . . . . . . . . 17  |-  ( F : A --> B  ->  dom  F  =  A )
4215, 41syl 16 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  dom  F  =  A )
4340, 42syl5sseq 3425 . . . . . . . . . . . . . . 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 5632 . . . . . . . . . . . . . 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 5627 . . . . . . . . . . . . 13  |-  ( f : ( 1 ... ( # `  ( F supp  .0.  ) ) )
-1-1-> A  ->  f :
( 1 ... ( # `
 ( F supp  .0.  ) ) ) --> A )
48 fcoi2 5607 . . . . . . . . . . . . 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 2475 . . . . . . . . . . 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 2490 . . . . . . . . . 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 5023 . . . . . . . . 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 5377 . . . . . . . . 9  |-  ( ( F  o.  H )  o.  ( `' H  o.  f ) )  =  ( F  o.  ( H  o.  ( `' H  o.  f )
) )
5452, 53syl6eqr 2493 . . . . . . . 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 11835 . . . . . . 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 5714 . . . . . 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 2443 . . . . . . 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 5669 . . . . . . . . . 10  |-  ( f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  )  ->  f : ( 1 ... ( # `  ( F supp  .0.  ) ) )
-onto-> ( F supp  .0.  )
)
67 forn 5644 . . . . . . . . . 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 3426 . . . . . . 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 2443 . . . . . . 7  |-  ( ( F  o.  f ) supp 
.0.  )  =  ( ( F  o.  f
) supp  .0.  )
7257, 3, 58, 59, 60, 61, 62, 64, 65, 46, 70, 71gsumval3 16406 . . . . . 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 5589 . . . . . . . . 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 5121 . . . . . . . . 9  |-  ran  ( F  o.  H )  C_ 
ran  F
7859cntzidss 15876 . . . . . . . . 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 5674 . . . . . . . . . 10  |-  ( H : C -1-1-onto-> A  ->  `' H : A -1-1-onto-> C )
82 f1of1 5661 . . . . . . . . . 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 5636 . . . . . . . 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 5971 . . . . . . . . . . . . . . 15  |-  ( ( F : A --> B  /\  A  e.  V )  ->  F  e.  _V )
8915, 2, 88syl2anc 661 . . . . . . . . . . . . . 14  |-  ( ph  ->  F  e.  _V )
90 suppimacnv 6722 . . . . . . . . . . . . . 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 2448 . . . . . . . . . . . 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 3420 . . . . . . . . . 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 5225 . . . . . . . . . 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 5046 . . . . . . . . . . 11  |-  `' ( F  o.  H )  =  ( `' H  o.  `' F )
9897imaeq1i 5187 . . . . . . . . . 10  |-  ( `' ( F  o.  H
) " ( _V 
\  {  .0.  }
) )  =  ( ( `' H  o.  `' F ) " ( _V  \  {  .0.  }
) )
99 imaco 5364 . . . . . . . . . 10  |-  ( ( `' H  o.  `' F ) " ( _V  \  {  .0.  }
) )  =  ( `' H " ( `' F " ( _V 
\  {  .0.  }
) ) )
10098, 99eqtri 2463 . . . . . . . . 9  |-  ( `' ( F  o.  H
) " ( _V 
\  {  .0.  }
) )  =  ( `' H " ( `' F " ( _V 
\  {  .0.  }
) ) )
101 rnco2 5366 . . . . . . . . 9  |-  ran  ( `' H  o.  f
)  =  ( `' H " ran  f
)
10296, 100, 1013sstr4g 3418 . . . . . . . 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 6548 . . . . . . . . . . . . 13  |-  ( ( H : C -1-1-onto-> A  /\  A  e.  V )  ->  H  e.  _V )
1046, 2, 103syl2anc 661 . . . . . . . . . . . 12  |-  ( ph  ->  H  e.  _V )
105 coexg 6549 . . . . . . . . . . . 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 6722 . . . . . . . . . . 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 3404 . . . . . . . . 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 2443 . . . . . . 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 16406 . . . . . 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 2485 . . . . 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 1690 . . 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 7647 . . . 4  |-  ( F finSupp  .0.  ->  ( Fun  F  /\  ( F supp  .0.  )  e.  Fin ) )
120119simprd 463 . . 3  |-  ( F finSupp  .0.  ->  ( F supp  .0.  )  e.  Fin )
121 fz1f1o 13208 . . 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 1369   E.wex 1586    e. wcel 1756   _Vcvv 2993    \ cdif 3346    C_ wss 3349   (/)c0 3658   {csn 3898   class class class wbr 4313    e. cmpt 4371    _I cid 4652   `'ccnv 4860   dom cdm 4861   ran crn 4862    |` cres 4863   "cima 4864    o. ccom 4865   Fun wfun 5433   -->wf 5435   -1-1->wf1 5436   -onto->wfo 5437   -1-1-onto->wf1o 5438   ` cfv 5439  (class class class)co 6112   supp csupp 6711   Fincfn 7331   finSupp cfsupp 7641   1c1 9304   NNcn 10343   ...cfz 11458    seqcseq 11827   #chash 12124   Basecbs 14195   +g cplusg 14259   0gc0g 14399    gsumg cgsu 14400   Mndcmnd 15430  Cntzccntz 15854
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-int 4150  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-se 4701  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-isom 5448  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-om 6498  df-1st 6598  df-2nd 6599  df-supp 6712  df-recs 6853  df-rdg 6887  df-1o 6941  df-oadd 6945  df-er 7122  df-en 7332  df-dom 7333  df-sdom 7334  df-fin 7335  df-fsupp 7642  df-oi 7745  df-card 8130  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-nn 10344  df-n0 10601  df-z 10668  df-uz 10883  df-fz 11459  df-fzo 11570  df-seq 11828  df-hash 12125  df-0g 14401  df-gsum 14402  df-mnd 15436  df-cntz 15856
This theorem is referenced by:  gsumf1o  16419  smadiadetlem3  18496
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