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Theorem gsumzres 16381
Description: Extend a finite group sum by padding outside with zeroes. (Contributed by Mario Carneiro, 24-Apr-2016.) (Revised by AV, 31-May-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 ) )
gsumzres.s  |-  ( ph  ->  ( F supp  .0.  )  C_  W )
gsumzres.w  |-  ( ph  ->  F finSupp  .0.  )
Assertion
Ref Expression
gsumzres  |-  ( ph  ->  ( G  gsumg  ( F  |`  W ) )  =  ( G 
gsumg  F ) )

Proof of Theorem gsumzres
Dummy variables  f 
k are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 gsumzcl.g . . . . . . 7  |-  ( ph  ->  G  e.  Mnd )
2 gsumzcl.a . . . . . . . 8  |-  ( ph  ->  A  e.  V )
3 inex1g 4432 . . . . . . . 8  |-  ( A  e.  V  ->  ( A  i^i  W )  e. 
_V )
42, 3syl 16 . . . . . . 7  |-  ( ph  ->  ( A  i^i  W
)  e.  _V )
5 gsumzcl.0 . . . . . . . 8  |-  .0.  =  ( 0g `  G )
65gsumz 15504 . . . . . . 7  |-  ( ( G  e.  Mnd  /\  ( A  i^i  W )  e.  _V )  -> 
( G  gsumg  ( k  e.  ( A  i^i  W ) 
|->  .0.  ) )  =  .0.  )
71, 4, 6syl2anc 656 . . . . . 6  |-  ( ph  ->  ( G  gsumg  ( k  e.  ( A  i^i  W ) 
|->  .0.  ) )  =  .0.  )
85gsumz 15504 . . . . . . 7  |-  ( ( G  e.  Mnd  /\  A  e.  V )  ->  ( G  gsumg  ( k  e.  A  |->  .0.  ) )  =  .0.  )
91, 2, 8syl2anc 656 . . . . . 6  |-  ( ph  ->  ( G  gsumg  ( k  e.  A  |->  .0.  ) )  =  .0.  )
107, 9eqtr4d 2476 . . . . 5  |-  ( ph  ->  ( G  gsumg  ( k  e.  ( A  i^i  W ) 
|->  .0.  ) )  =  ( G  gsumg  ( k  e.  A  |->  .0.  ) ) )
1110adantr 462 . . . 4  |-  ( (
ph  /\  ( F supp  .0.  )  =  (/) )  -> 
( G  gsumg  ( k  e.  ( A  i^i  W ) 
|->  .0.  ) )  =  ( G  gsumg  ( k  e.  A  |->  .0.  ) ) )
12 resres 5120 . . . . . . . 8  |-  ( ( F  |`  A )  |`  W )  =  ( F  |`  ( A  i^i  W ) )
13 gsumzcl.f . . . . . . . . . 10  |-  ( ph  ->  F : A --> B )
14 ffn 5556 . . . . . . . . . 10  |-  ( F : A --> B  ->  F  Fn  A )
15 fnresdm 5517 . . . . . . . . . 10  |-  ( F  Fn  A  ->  ( F  |`  A )  =  F )
1613, 14, 153syl 20 . . . . . . . . 9  |-  ( ph  ->  ( F  |`  A )  =  F )
1716reseq1d 5105 . . . . . . . 8  |-  ( ph  ->  ( ( F  |`  A )  |`  W )  =  ( F  |`  W ) )
1812, 17syl5eqr 2487 . . . . . . 7  |-  ( ph  ->  ( F  |`  ( A  i^i  W ) )  =  ( F  |`  W ) )
1918adantr 462 . . . . . 6  |-  ( (
ph  /\  ( F supp  .0.  )  =  (/) )  -> 
( F  |`  ( A  i^i  W ) )  =  ( F  |`  W ) )
20 fvex 5698 . . . . . . . . . . 11  |-  ( 0g
`  G )  e. 
_V
215, 20eqeltri 2511 . . . . . . . . . 10  |-  .0.  e.  _V
2221a1i 11 . . . . . . . . 9  |-  ( ph  ->  .0.  e.  _V )
23 ssid 3372 . . . . . . . . . 10  |-  ( F supp 
.0.  )  C_  ( F supp  .0.  )
2423a1i 11 . . . . . . . . 9  |-  ( ph  ->  ( F supp  .0.  )  C_  ( F supp  .0.  )
)
2513, 2, 22, 24gsumcllem 16379 . . . . . . . 8  |-  ( (
ph  /\  ( F supp  .0.  )  =  (/) )  ->  F  =  ( k  e.  A  |->  .0.  )
)
2625reseq1d 5105 . . . . . . 7  |-  ( (
ph  /\  ( F supp  .0.  )  =  (/) )  -> 
( F  |`  ( A  i^i  W ) )  =  ( ( k  e.  A  |->  .0.  )  |`  ( A  i^i  W
) ) )
27 inss1 3567 . . . . . . . 8  |-  ( A  i^i  W )  C_  A
28 resmpt 5153 . . . . . . . 8  |-  ( ( A  i^i  W ) 
C_  A  ->  (
( k  e.  A  |->  .0.  )  |`  ( A  i^i  W ) )  =  ( k  e.  ( A  i^i  W
)  |->  .0.  ) )
2927, 28ax-mp 5 . . . . . . 7  |-  ( ( k  e.  A  |->  .0.  )  |`  ( A  i^i  W ) )  =  ( k  e.  ( A  i^i  W ) 
|->  .0.  )
3026, 29syl6eq 2489 . . . . . 6  |-  ( (
ph  /\  ( F supp  .0.  )  =  (/) )  -> 
( F  |`  ( A  i^i  W ) )  =  ( k  e.  ( A  i^i  W
)  |->  .0.  ) )
3119, 30eqtr3d 2475 . . . . 5  |-  ( (
ph  /\  ( F supp  .0.  )  =  (/) )  -> 
( F  |`  W )  =  ( k  e.  ( A  i^i  W
)  |->  .0.  ) )
3231oveq2d 6106 . . . 4  |-  ( (
ph  /\  ( F supp  .0.  )  =  (/) )  -> 
( G  gsumg  ( F  |`  W ) )  =  ( G 
gsumg  ( k  e.  ( A  i^i  W ) 
|->  .0.  ) ) )
3325oveq2d 6106 . . . 4  |-  ( (
ph  /\  ( F supp  .0.  )  =  (/) )  -> 
( G  gsumg  F )  =  ( G  gsumg  ( k  e.  A  |->  .0.  ) ) )
3411, 32, 333eqtr4d 2483 . . 3  |-  ( (
ph  /\  ( F supp  .0.  )  =  (/) )  -> 
( G  gsumg  ( F  |`  W ) )  =  ( G 
gsumg  F ) )
3534ex 434 . 2  |-  ( ph  ->  ( ( F supp  .0.  )  =  (/)  ->  ( G  gsumg  ( F  |`  W ) )  =  ( G 
gsumg  F ) ) )
36 f1ofo 5645 . . . . . . . . . . . 12  |-  ( f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  )  ->  f : ( 1 ... ( # `  ( F supp  .0.  ) ) )
-onto-> ( F supp  .0.  )
)
37 forn 5620 . . . . . . . . . . . 12  |-  ( f : ( 1 ... ( # `  ( F supp  .0.  ) ) )
-onto-> ( F supp  .0.  )  ->  ran  f  =  ( F supp  .0.  ) )
3836, 37syl 16 . . . . . . . . . . 11  |-  ( f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  )  ->  ran  f  =  ( F supp 
.0.  ) )
3938ad2antll 723 . . . . . . . . . 10  |-  ( (
ph  /\  ( ( # `
 ( F supp  .0.  ) )  e.  NN  /\  f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
)  ->  ran  f  =  ( F supp  .0.  )
)
40 gsumzres.s . . . . . . . . . . 11  |-  ( ph  ->  ( F supp  .0.  )  C_  W )
4140adantr 462 . . . . . . . . . 10  |-  ( (
ph  /\  ( ( # `
 ( F supp  .0.  ) )  e.  NN  /\  f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
)  ->  ( F supp  .0.  )  C_  W )
4239, 41eqsstrd 3387 . . . . . . . . 9  |-  ( (
ph  /\  ( ( # `
 ( F supp  .0.  ) )  e.  NN  /\  f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
)  ->  ran  f  C_  W )
43 cores 5338 . . . . . . . . 9  |-  ( ran  f  C_  W  ->  ( ( F  |`  W )  o.  f )  =  ( F  o.  f
) )
4442, 43syl 16 . . . . . . . 8  |-  ( (
ph  /\  ( ( # `
 ( F supp  .0.  ) )  e.  NN  /\  f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
)  ->  ( ( F  |`  W )  o.  f )  =  ( F  o.  f ) )
4544seqeq3d 11810 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 ( F supp  .0.  ) )  e.  NN  /\  f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
)  ->  seq 1
( ( +g  `  G
) ,  ( ( F  |`  W )  o.  f ) )  =  seq 1 ( ( +g  `  G ) ,  ( F  o.  f ) ) )
4645fveq1d 5690 . . . . . 6  |-  ( (
ph  /\  ( ( # `
 ( F supp  .0.  ) )  e.  NN  /\  f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
)  ->  (  seq 1 ( ( +g  `  G ) ,  ( ( F  |`  W )  o.  f ) ) `
 ( # `  ( F supp  .0.  ) ) )  =  (  seq 1
( ( +g  `  G
) ,  ( F  o.  f ) ) `
 ( # `  ( F supp  .0.  ) ) ) )
47 gsumzcl.b . . . . . . 7  |-  B  =  ( Base `  G
)
48 eqid 2441 . . . . . . 7  |-  ( +g  `  G )  =  ( +g  `  G )
49 gsumzcl.z . . . . . . 7  |-  Z  =  (Cntz `  G )
501adantr 462 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 ( F supp  .0.  ) )  e.  NN  /\  f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
)  ->  G  e.  Mnd )
514adantr 462 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 ( F supp  .0.  ) )  e.  NN  /\  f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
)  ->  ( A  i^i  W )  e.  _V )
5213adantr 462 . . . . . . . . 9  |-  ( (
ph  /\  ( ( # `
 ( F supp  .0.  ) )  e.  NN  /\  f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
)  ->  F : A
--> B )
53 fssres 5575 . . . . . . . . 9  |-  ( ( F : A --> B  /\  ( A  i^i  W ) 
C_  A )  -> 
( F  |`  ( A  i^i  W ) ) : ( A  i^i  W ) --> B )
5452, 27, 53sylancl 657 . . . . . . . 8  |-  ( (
ph  /\  ( ( # `
 ( F supp  .0.  ) )  e.  NN  /\  f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
)  ->  ( F  |`  ( A  i^i  W
) ) : ( A  i^i  W ) --> B )
5518feq1d 5543 . . . . . . . . 9  |-  ( ph  ->  ( ( F  |`  ( A  i^i  W ) ) : ( A  i^i  W ) --> B  <-> 
( F  |`  W ) : ( A  i^i  W ) --> B ) )
5655biimpa 481 . . . . . . . 8  |-  ( (
ph  /\  ( F  |`  ( A  i^i  W
) ) : ( A  i^i  W ) --> B )  ->  ( F  |`  W ) : ( A  i^i  W
) --> B )
5754, 56syldan 467 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 ( F supp  .0.  ) )  e.  NN  /\  f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
)  ->  ( F  |`  W ) : ( A  i^i  W ) --> B )
58 gsumzcl.c . . . . . . . . 9  |-  ( ph  ->  ran  F  C_  ( Z `  ran  F ) )
59 resss 5131 . . . . . . . . . 10  |-  ( F  |`  W )  C_  F
60 rnss 5064 . . . . . . . . . 10  |-  ( ( F  |`  W )  C_  F  ->  ran  ( F  |`  W )  C_  ran  F )
6159, 60ax-mp 5 . . . . . . . . 9  |-  ran  ( F  |`  W )  C_  ran  F
6249cntzidss 15848 . . . . . . . . 9  |-  ( ( ran  F  C_  ( Z `  ran  F )  /\  ran  ( F  |`  W )  C_  ran  F )  ->  ran  ( F  |`  W )  C_  ( Z `  ran  ( F  |`  W ) ) )
6358, 61, 62sylancl 657 . . . . . . . 8  |-  ( ph  ->  ran  ( F  |`  W )  C_  ( Z `  ran  ( F  |`  W ) ) )
6463adantr 462 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 ( F supp  .0.  ) )  e.  NN  /\  f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
)  ->  ran  ( F  |`  W )  C_  ( Z `  ran  ( F  |`  W ) ) )
65 simprl 750 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 ( F supp  .0.  ) )  e.  NN  /\  f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
)  ->  ( # `  ( F supp  .0.  ) )  e.  NN )
66 f1of1 5637 . . . . . . . . 9  |-  ( f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  )  ->  f : ( 1 ... ( # `  ( F supp  .0.  ) ) )
-1-1-> ( F supp  .0.  )
)
6766ad2antll 723 . . . . . . . 8  |-  ( (
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.  ) )
68 suppssdm 6702 . . . . . . . . . . 11  |-  ( F supp 
.0.  )  C_  dom  F
69 fdm 5560 . . . . . . . . . . . 12  |-  ( F : A --> B  ->  dom  F  =  A )
7013, 69syl 16 . . . . . . . . . . 11  |-  ( ph  ->  dom  F  =  A )
7168, 70syl5sseq 3401 . . . . . . . . . 10  |-  ( ph  ->  ( F supp  .0.  )  C_  A )
7271, 40ssind 3571 . . . . . . . . 9  |-  ( ph  ->  ( F supp  .0.  )  C_  ( A  i^i  W
) )
7372adantr 462 . . . . . . . 8  |-  ( (
ph  /\  ( ( # `
 ( F supp  .0.  ) )  e.  NN  /\  f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
)  ->  ( F supp  .0.  )  C_  ( A  i^i  W ) )
74 f1ss 5608 . . . . . . . 8  |-  ( ( f : ( 1 ... ( # `  ( F supp  .0.  ) ) )
-1-1-> ( F supp  .0.  )  /\  ( F supp  .0.  )  C_  ( A  i^i  W
) )  ->  f : ( 1 ... ( # `  ( F supp  .0.  ) ) )
-1-1-> ( A  i^i  W
) )
7567, 73, 74syl2anc 656 . . . . . . 7  |-  ( (
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  i^i  W ) )
76 fex 5947 . . . . . . . . . . . . 13  |-  ( ( F : A --> B  /\  A  e.  V )  ->  F  e.  _V )
7713, 2, 76syl2anc 656 . . . . . . . . . . . 12  |-  ( ph  ->  F  e.  _V )
78 ressuppss 6707 . . . . . . . . . . . 12  |-  ( ( F  e.  _V  /\  .0.  e.  _V )  -> 
( ( F  |`  W ) supp  .0.  )  C_  ( F supp  .0.  )
)
7977, 21, 78sylancl 657 . . . . . . . . . . 11  |-  ( ph  ->  ( ( F  |`  W ) supp  .0.  )  C_  ( F supp  .0.  )
)
80 sseq2 3375 . . . . . . . . . . 11  |-  ( ran  f  =  ( F supp 
.0.  )  ->  (
( ( F  |`  W ) supp  .0.  )  C_ 
ran  f  <->  ( ( F  |`  W ) supp  .0.  )  C_  ( F supp  .0.  ) ) )
8179, 80syl5ibr 221 . . . . . . . . . 10  |-  ( ran  f  =  ( F supp 
.0.  )  ->  ( ph  ->  ( ( F  |`  W ) supp  .0.  )  C_ 
ran  f ) )
8236, 37, 813syl 20 . . . . . . . . 9  |-  ( f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  )  ->  (
ph  ->  ( ( F  |`  W ) supp  .0.  )  C_ 
ran  f ) )
8382adantl 463 . . . . . . . 8  |-  ( ( ( # `  ( F supp  .0.  ) )  e.  NN  /\  f : ( 1 ... ( # `
 ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp 
.0.  ) )  -> 
( ph  ->  ( ( F  |`  W ) supp  .0.  )  C_  ran  f
) )
8483impcom 430 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 ( F supp  .0.  ) )  e.  NN  /\  f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
)  ->  ( ( F  |`  W ) supp  .0.  )  C_  ran  f )
85 eqid 2441 . . . . . . 7  |-  ( ( ( F  |`  W )  o.  f ) supp  .0.  )  =  ( (
( F  |`  W )  o.  f ) supp  .0.  )
8647, 5, 48, 49, 50, 51, 57, 64, 65, 75, 84, 85gsumval3 16378 . . . . . 6  |-  ( (
ph  /\  ( ( # `
 ( F supp  .0.  ) )  e.  NN  /\  f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
)  ->  ( G  gsumg  ( F  |`  W )
)  =  (  seq 1 ( ( +g  `  G ) ,  ( ( F  |`  W )  o.  f ) ) `
 ( # `  ( F supp  .0.  ) ) ) )
872adantr 462 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 ( F supp  .0.  ) )  e.  NN  /\  f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
)  ->  A  e.  V )
8858adantr 462 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 ( F supp  .0.  ) )  e.  NN  /\  f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
)  ->  ran  F  C_  ( Z `  ran  F
) )
8971adantr 462 . . . . . . . 8  |-  ( (
ph  /\  ( ( # `
 ( F supp  .0.  ) )  e.  NN  /\  f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
)  ->  ( F supp  .0.  )  C_  A )
90 f1ss 5608 . . . . . . . 8  |-  ( ( f : ( 1 ... ( # `  ( F supp  .0.  ) ) )
-1-1-> ( F supp  .0.  )  /\  ( F supp  .0.  )  C_  A )  ->  f : ( 1 ... ( # `  ( F supp  .0.  ) ) )
-1-1-> A )
9167, 89, 90syl2anc 656 . . . . . . 7  |-  ( (
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
)
9223, 39syl5sseqr 3402 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 ( F supp  .0.  ) )  e.  NN  /\  f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
)  ->  ( F supp  .0.  )  C_  ran  f )
93 eqid 2441 . . . . . . 7  |-  ( ( F  o.  f ) supp 
.0.  )  =  ( ( F  o.  f
) supp  .0.  )
9447, 5, 48, 49, 50, 87, 52, 88, 65, 91, 92, 93gsumval3 16378 . . . . . 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.  )
) ) )
9546, 86, 943eqtr4d 2483 . . . . 5  |-  ( (
ph  /\  ( ( # `
 ( F supp  .0.  ) )  e.  NN  /\  f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
)  ->  ( G  gsumg  ( F  |`  W )
)  =  ( G 
gsumg  F ) )
9695expr 612 . . . 4  |-  ( (
ph  /\  ( # `  ( F supp  .0.  ) )  e.  NN )  ->  (
f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  )  ->  ( G  gsumg  ( F  |`  W ) )  =  ( G 
gsumg  F ) ) )
9796exlimdv 1695 . . 3  |-  ( (
ph  /\  ( # `  ( F supp  .0.  ) )  e.  NN )  ->  ( E. f  f :
( 1 ... ( # `
 ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp 
.0.  )  ->  ( G  gsumg  ( F  |`  W ) )  =  ( G 
gsumg  F ) ) )
9897expimpd 600 . 2  |-  ( ph  ->  ( ( ( # `  ( F supp  .0.  )
)  e.  NN  /\  E. f  f : ( 1 ... ( # `  ( F supp  .0.  )
) ) -1-1-onto-> ( F supp  .0.  )
)  ->  ( G  gsumg  ( F  |`  W )
)  =  ( G 
gsumg  F ) ) )
99 gsumzres.w . . 3  |-  ( ph  ->  F finSupp  .0.  )
100 fsuppimp 7622 . . . 4  |-  ( F finSupp  .0.  ->  ( Fun  F  /\  ( F supp  .0.  )  e.  Fin ) )
101100simprd 460 . . 3  |-  ( F finSupp  .0.  ->  ( F supp  .0.  )  e.  Fin )
102 fz1f1o 13183 . . 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.  ) )
) )
10399, 101, 1023syl 20 . 2  |-  ( ph  ->  ( ( F supp  .0.  )  =  (/)  \/  (
( # `  ( F supp 
.0.  ) )  e.  NN  /\  E. f 
f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
) )
10435, 98, 103mpjaod 381 1  |-  ( ph  ->  ( G  gsumg  ( F  |`  W ) )  =  ( G 
gsumg  F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1364   E.wex 1591    e. wcel 1761   _Vcvv 2970    i^i cin 3324    C_ wss 3325   (/)c0 3634   class class class wbr 4289    e. cmpt 4347   dom cdm 4836   ran crn 4837    |` cres 4838    o. ccom 4840   Fun wfun 5409    Fn wfn 5410   -->wf 5411   -1-1->wf1 5412   -onto->wfo 5413   -1-1-onto->wf1o 5414   ` cfv 5415  (class class class)co 6090   supp csupp 6689   Fincfn 7306   finSupp cfsupp 7616   1c1 9279   NNcn 10318   ...cfz 11433    seqcseq 11802   #chash 12099   Basecbs 14170   +g cplusg 14234   0gc0g 14374    gsumg cgsu 14375   Mndcmnd 15405  Cntzccntz 15826
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-supp 6690  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-fsupp 7617  df-oi 7720  df-card 8105  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-n0 10576  df-z 10643  df-uz 10858  df-fz 11434  df-fzo 11545  df-seq 11803  df-hash 12100  df-0g 14376  df-gsum 14377  df-mnd 15411  df-cntz 15828
This theorem is referenced by:  gsumres  16388  gsumzsplit  16411  gsumpt  16445  dmdprdsplitlem  16524  dpjidcl  16547
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