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Theorem gsumzf1o 17624
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 16699 . . . . . . 7  |-  ( ( G  e.  Mnd  /\  A  e.  V )  ->  ( G  gsumg  ( k  e.  A  |->  .0.  ) )  =  .0.  )
51, 2, 4syl2anc 673 . . . . . 6  |-  ( ph  ->  ( G  gsumg  ( k  e.  A  |->  .0.  ) )  =  .0.  )
6 gsumzf1o.h . . . . . . . . 9  |-  ( ph  ->  H : C -1-1-onto-> A )
7 f1of1 5827 . . . . . . . . 9  |-  ( H : C -1-1-onto-> A  ->  H : C -1-1-> A )
86, 7syl 17 . . . . . . . 8  |-  ( ph  ->  H : C -1-1-> A
)
9 f1dmex 6782 . . . . . . . 8  |-  ( ( H : C -1-1-> A  /\  A  e.  V
)  ->  C  e.  _V )
108, 2, 9syl2anc 673 . . . . . . 7  |-  ( ph  ->  C  e.  _V )
113gsumz 16699 . . . . . . 7  |-  ( ( G  e.  Mnd  /\  C  e.  _V )  ->  ( G  gsumg  ( x  e.  C  |->  .0.  ) )  =  .0.  )
121, 10, 11syl2anc 673 . . . . . 6  |-  ( ph  ->  ( G  gsumg  ( x  e.  C  |->  .0.  ) )  =  .0.  )
135, 12eqtr4d 2508 . . . . 5  |-  ( ph  ->  ( G  gsumg  ( k  e.  A  |->  .0.  ) )  =  ( G  gsumg  ( x  e.  C  |->  .0.  ) ) )
1413adantr 472 . . . 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 5889 . . . . . . . 8  |-  ( 0g
`  G )  e. 
_V
173, 16eqeltri 2545 . . . . . . 7  |-  .0.  e.  _V
1817a1i 11 . . . . . 6  |-  ( ph  ->  .0.  e.  _V )
19 ssid 3437 . . . . . . 7  |-  ( F supp 
.0.  )  C_  ( F supp  .0.  )
2019a1i 11 . . . . . 6  |-  ( ph  ->  ( F supp  .0.  )  C_  ( F supp  .0.  )
)
2115, 2, 18, 20gsumcllem 17620 . . . . 5  |-  ( (
ph  /\  ( F supp  .0.  )  =  (/) )  ->  F  =  ( k  e.  A  |->  .0.  )
)
2221oveq2d 6324 . . . 4  |-  ( (
ph  /\  ( F supp  .0.  )  =  (/) )  -> 
( G  gsumg  F )  =  ( G  gsumg  ( k  e.  A  |->  .0.  ) ) )
23 f1of 5828 . . . . . . . . 9  |-  ( H : C -1-1-onto-> A  ->  H : C
--> A )
246, 23syl 17 . . . . . . . 8  |-  ( ph  ->  H : C --> A )
2524adantr 472 . . . . . . 7  |-  ( (
ph  /\  ( F supp  .0.  )  =  (/) )  ->  H : C --> A )
2625ffvelrnda 6037 . . . . . 6  |-  ( ( ( ph  /\  ( F supp  .0.  )  =  (/) )  /\  x  e.  C
)  ->  ( H `  x )  e.  A
)
2725feqmptd 5932 . . . . . 6  |-  ( (
ph  /\  ( F supp  .0.  )  =  (/) )  ->  H  =  ( x  e.  C  |->  ( H `
 x ) ) )
28 eqidd 2472 . . . . . 6  |-  ( k  =  ( H `  x )  ->  .0.  =  .0.  )
2926, 27, 21, 28fmptco 6072 . . . . 5  |-  ( (
ph  /\  ( F supp  .0.  )  =  (/) )  -> 
( F  o.  H
)  =  ( x  e.  C  |->  .0.  )
)
3029oveq2d 6324 . . . 4  |-  ( (
ph  /\  ( F supp  .0.  )  =  (/) )  -> 
( G  gsumg  ( F  o.  H
) )  =  ( G  gsumg  ( x  e.  C  |->  .0.  ) ) )
3114, 22, 303eqtr4d 2515 . . 3  |-  ( (
ph  /\  ( F supp  .0.  )  =  (/) )  -> 
( G  gsumg  F )  =  ( G  gsumg  ( F  o.  H
) ) )
3231ex 441 . 2  |-  ( ph  ->  ( ( F supp  .0.  )  =  (/)  ->  ( G  gsumg  F )  =  ( G  gsumg  ( F  o.  H
) ) ) )
33 coass 5361 . . . . . . . . . . 11  |-  ( ( H  o.  `' H
)  o.  f )  =  ( H  o.  ( `' H  o.  f
) )
346adantr 472 . . . . . . . . . . . . . 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 5854 . . . . . . . . . . . . . 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 5001 . . . . . . . . . . . 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 5827 . . . . . . . . . . . . . . 15  |-  ( f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  )  ->  f : ( 1 ... ( # `  ( F supp  .0.  ) ) )
-1-1-> ( F supp  .0.  )
)
3938ad2antll 743 . . . . . . . . . . . . . 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 6946 . . . . . . . . . . . . . . . 16  |-  ( F supp 
.0.  )  C_  dom  F
41 fdm 5745 . . . . . . . . . . . . . . . . 17  |-  ( F : A --> B  ->  dom  F  =  A )
4215, 41syl 17 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  dom  F  =  A )
4340, 42syl5sseq 3466 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( F supp  .0.  )  C_  A )
4443adantr 472 . . . . . . . . . . . . . 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 5797 . . . . . . . . . . . . . 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 673 . . . . . . . . . . . . 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 5792 . . . . . . . . . . . . 13  |-  ( f : ( 1 ... ( # `  ( F supp  .0.  ) ) )
-1-1-> A  ->  f :
( 1 ... ( # `
 ( F supp  .0.  ) ) ) --> A )
48 fcoi2 5770 . . . . . . . . . . . . 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 2505 . . . . . . . . . . 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 2520 . . . . . . . . . 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 5002 . . . . . . . . 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 5361 . . . . . . . . 9  |-  ( ( F  o.  H )  o.  ( `' H  o.  f ) )  =  ( F  o.  ( H  o.  ( `' H  o.  f )
) )
5452, 53syl6eqr 2523 . . . . . . . 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 12259 . . . . . . 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 5881 . . . . . 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 2471 . . . . . . 7  |-  ( +g  `  G )  =  ( +g  `  G )
59 gsumzcl.z . . . . . . 7  |-  Z  =  (Cntz `  G )
601adantr 472 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 ( F supp  .0.  ) )  e.  NN  /\  f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
)  ->  G  e.  Mnd )
612adantr 472 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 ( F supp  .0.  ) )  e.  NN  /\  f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
)  ->  A  e.  V )
6215adantr 472 . . . . . . 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 472 . . . . . . 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 772 . . . . . . 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 5835 . . . . . . . . . 10  |-  ( f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  )  ->  f : ( 1 ... ( # `  ( F supp  .0.  ) ) )
-onto-> ( F supp  .0.  )
)
67 forn 5809 . . . . . . . . . 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 743 . . . . . . . 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 3467 . . . . . . 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 2471 . . . . . . 7  |-  ( ( F  o.  f ) supp 
.0.  )  =  ( ( F  o.  f
) supp  .0.  )
7257, 3, 58, 59, 60, 61, 62, 64, 65, 46, 70, 71gsumval3 17619 . . . . . 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 472 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 ( F supp  .0.  ) )  e.  NN  /\  f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
)  ->  C  e.  _V )
74 fco 5751 . . . . . . . . 9  |-  ( ( F : A --> B  /\  H : C --> A )  ->  ( F  o.  H ) : C --> B )
7515, 24, 74syl2anc 673 . . . . . . . 8  |-  ( ph  ->  ( F  o.  H
) : C --> B )
7675adantr 472 . . . . . . 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 5101 . . . . . . . . 9  |-  ran  ( F  o.  H )  C_ 
ran  F
7859cntzidss 17069 . . . . . . . . 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 675 . . . . . . . 8  |-  ( ph  ->  ran  ( F  o.  H )  C_  ( Z `  ran  ( F  o.  H ) ) )
8079adantr 472 . . . . . . 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 5840 . . . . . . . . . 10  |-  ( H : C -1-1-onto-> A  ->  `' H : A -1-1-onto-> C )
82 f1of1 5827 . . . . . . . . . 10  |-  ( `' H : A -1-1-onto-> C  ->  `' H : A -1-1-> C
)
836, 81, 823syl 18 . . . . . . . . 9  |-  ( ph  ->  `' H : A -1-1-> C
)
8483adantr 472 . . . . . . . 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 5801 . . . . . . . 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 673 . . . . . . 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 6155 . . . . . . . . . . . . . . 15  |-  ( ( F : A --> B  /\  A  e.  V )  ->  F  e.  _V )
8915, 2, 88syl2anc 673 . . . . . . . . . . . . . 14  |-  ( ph  ->  F  e.  _V )
90 suppimacnv 6944 . . . . . . . . . . . . . 14  |-  ( ( F  e.  _V  /\  .0.  e.  _V )  -> 
( F supp  .0.  )  =  ( `' F " ( _V  \  {  .0.  } ) ) )
9189, 17, 90sylancl 675 . . . . . . . . . . . . 13  |-  ( ph  ->  ( F supp  .0.  )  =  ( `' F " ( _V  \  {  .0.  } ) ) )
9291eqcomd 2477 . . . . . . . . . . . 12  |-  ( ph  ->  ( `' F "
( _V  \  {  .0.  } ) )  =  ( F supp  .0.  )
)
9392adantr 472 . . . . . . . . . . 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 3461 . . . . . . . . . 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 5210 . . . . . . . . . 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 5025 . . . . . . . . . . 11  |-  `' ( F  o.  H )  =  ( `' H  o.  `' F )
9897imaeq1i 5171 . . . . . . . . . 10  |-  ( `' ( F  o.  H
) " ( _V 
\  {  .0.  }
) )  =  ( ( `' H  o.  `' F ) " ( _V  \  {  .0.  }
) )
99 imaco 5347 . . . . . . . . . 10  |-  ( ( `' H  o.  `' F ) " ( _V  \  {  .0.  }
) )  =  ( `' H " ( `' F " ( _V 
\  {  .0.  }
) ) )
10098, 99eqtri 2493 . . . . . . . . 9  |-  ( `' ( F  o.  H
) " ( _V 
\  {  .0.  }
) )  =  ( `' H " ( `' F " ( _V 
\  {  .0.  }
) ) )
101 rnco2 5349 . . . . . . . . 9  |-  ran  ( `' H  o.  f
)  =  ( `' H " ran  f
)
10296, 100, 1013sstr4g 3459 . . . . . . . 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 6761 . . . . . . . . . . . . 13  |-  ( ( H : C -1-1-onto-> A  /\  A  e.  V )  ->  H  e.  _V )
1046, 2, 103syl2anc 673 . . . . . . . . . . . 12  |-  ( ph  ->  H  e.  _V )
105 coexg 6763 . . . . . . . . . . . 12  |-  ( ( F  e.  _V  /\  H  e.  _V )  ->  ( F  o.  H
)  e.  _V )
10689, 104, 105syl2anc 673 . . . . . . . . . . 11  |-  ( ph  ->  ( F  o.  H
)  e.  _V )
107 suppimacnv 6944 . . . . . . . . . . 11  |-  ( ( ( F  o.  H
)  e.  _V  /\  .0.  e.  _V )  -> 
( ( F  o.  H ) supp  .0.  )  =  ( `' ( F  o.  H )
" ( _V  \  {  .0.  } ) ) )
108106, 17, 107sylancl 675 . . . . . . . . . 10  |-  ( ph  ->  ( ( F  o.  H ) supp  .0.  )  =  ( `' ( F  o.  H )
" ( _V  \  {  .0.  } ) ) )
109108sseq1d 3445 . . . . . . . . 9  |-  ( ph  ->  ( ( ( F  o.  H ) supp  .0.  )  C_  ran  ( `' H  o.  f )  <-> 
( `' ( F  o.  H ) "
( _V  \  {  .0.  } ) )  C_  ran  ( `' H  o.  f ) ) )
110109adantr 472 . . . . . . . 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 240 . . . . . . 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 2471 . . . . . . 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 17619 . . . . . 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 2515 . . . . 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 626 . . . 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 1787 . . 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 614 . 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 7907 . . . 4  |-  ( F finSupp  .0.  ->  ( Fun  F  /\  ( F supp  .0.  )  e.  Fin ) )
120119simprd 470 . . 3  |-  ( F finSupp  .0.  ->  ( F supp  .0.  )  e.  Fin )
121 fz1f1o 13853 . . 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 388 1  |-  ( ph  ->  ( G  gsumg  F )  =  ( G  gsumg  ( F  o.  H
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    \/ wo 375    /\ wa 376    = wceq 1452   E.wex 1671    e. wcel 1904   _Vcvv 3031    \ cdif 3387    C_ wss 3390   (/)c0 3722   {csn 3959   class class class wbr 4395    |-> cmpt 4454    _I cid 4749   `'ccnv 4838   dom cdm 4839   ran crn 4840    |` cres 4841   "cima 4842    o. ccom 4843   Fun wfun 5583   -->wf 5585   -1-1->wf1 5586   -onto->wfo 5587   -1-1-onto->wf1o 5588   ` cfv 5589  (class class class)co 6308   supp csupp 6933   Fincfn 7587   finSupp cfsupp 7901   1c1 9558   NNcn 10631   ...cfz 11810    seqcseq 12251   #chash 12553   Basecbs 15199   +g cplusg 15268   0gc0g 15416    gsumg cgsu 15417   Mndcmnd 16613  Cntzccntz 17047
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-oi 8043  df-card 8391  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-fzo 11943  df-seq 12252  df-hash 12554  df-0g 15418  df-gsum 15419  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-cntz 17049
This theorem is referenced by:  gsumf1o  17628  smadiadetlem3  19770
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