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Theorem gsumzres 16729
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 4590 . . . . . . . 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 15836 . . . . . . 7  |-  ( ( G  e.  Mnd  /\  ( A  i^i  W )  e.  _V )  -> 
( G  gsumg  ( k  e.  ( A  i^i  W ) 
|->  .0.  ) )  =  .0.  )
71, 4, 6syl2anc 661 . . . . . 6  |-  ( ph  ->  ( G  gsumg  ( k  e.  ( A  i^i  W ) 
|->  .0.  ) )  =  .0.  )
85gsumz 15836 . . . . . . 7  |-  ( ( G  e.  Mnd  /\  A  e.  V )  ->  ( G  gsumg  ( k  e.  A  |->  .0.  ) )  =  .0.  )
91, 2, 8syl2anc 661 . . . . . 6  |-  ( ph  ->  ( G  gsumg  ( k  e.  A  |->  .0.  ) )  =  .0.  )
107, 9eqtr4d 2511 . . . . 5  |-  ( ph  ->  ( G  gsumg  ( k  e.  ( A  i^i  W ) 
|->  .0.  ) )  =  ( G  gsumg  ( k  e.  A  |->  .0.  ) ) )
1110adantr 465 . . . 4  |-  ( (
ph  /\  ( F supp  .0.  )  =  (/) )  -> 
( G  gsumg  ( k  e.  ( A  i^i  W ) 
|->  .0.  ) )  =  ( G  gsumg  ( k  e.  A  |->  .0.  ) ) )
12 resres 5286 . . . . . . . 8  |-  ( ( F  |`  A )  |`  W )  =  ( F  |`  ( A  i^i  W ) )
13 gsumzcl.f . . . . . . . . . 10  |-  ( ph  ->  F : A --> B )
14 ffn 5731 . . . . . . . . . 10  |-  ( F : A --> B  ->  F  Fn  A )
15 fnresdm 5690 . . . . . . . . . 10  |-  ( F  Fn  A  ->  ( F  |`  A )  =  F )
1613, 14, 153syl 20 . . . . . . . . 9  |-  ( ph  ->  ( F  |`  A )  =  F )
1716reseq1d 5272 . . . . . . . 8  |-  ( ph  ->  ( ( F  |`  A )  |`  W )  =  ( F  |`  W ) )
1812, 17syl5eqr 2522 . . . . . . 7  |-  ( ph  ->  ( F  |`  ( A  i^i  W ) )  =  ( F  |`  W ) )
1918adantr 465 . . . . . 6  |-  ( (
ph  /\  ( F supp  .0.  )  =  (/) )  -> 
( F  |`  ( A  i^i  W ) )  =  ( F  |`  W ) )
20 fvex 5876 . . . . . . . . . . 11  |-  ( 0g
`  G )  e. 
_V
215, 20eqeltri 2551 . . . . . . . . . 10  |-  .0.  e.  _V
2221a1i 11 . . . . . . . . 9  |-  ( ph  ->  .0.  e.  _V )
23 ssid 3523 . . . . . . . . . 10  |-  ( F supp 
.0.  )  C_  ( F supp  .0.  )
2423a1i 11 . . . . . . . . 9  |-  ( ph  ->  ( F supp  .0.  )  C_  ( F supp  .0.  )
)
2513, 2, 22, 24gsumcllem 16727 . . . . . . . 8  |-  ( (
ph  /\  ( F supp  .0.  )  =  (/) )  ->  F  =  ( k  e.  A  |->  .0.  )
)
2625reseq1d 5272 . . . . . . 7  |-  ( (
ph  /\  ( F supp  .0.  )  =  (/) )  -> 
( F  |`  ( A  i^i  W ) )  =  ( ( k  e.  A  |->  .0.  )  |`  ( A  i^i  W
) ) )
27 inss1 3718 . . . . . . . 8  |-  ( A  i^i  W )  C_  A
28 resmpt 5323 . . . . . . . 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 2524 . . . . . 6  |-  ( (
ph  /\  ( F supp  .0.  )  =  (/) )  -> 
( F  |`  ( A  i^i  W ) )  =  ( k  e.  ( A  i^i  W
)  |->  .0.  ) )
3119, 30eqtr3d 2510 . . . . 5  |-  ( (
ph  /\  ( F supp  .0.  )  =  (/) )  -> 
( F  |`  W )  =  ( k  e.  ( A  i^i  W
)  |->  .0.  ) )
3231oveq2d 6301 . . . 4  |-  ( (
ph  /\  ( F supp  .0.  )  =  (/) )  -> 
( G  gsumg  ( F  |`  W ) )  =  ( G 
gsumg  ( k  e.  ( A  i^i  W ) 
|->  .0.  ) ) )
3325oveq2d 6301 . . . 4  |-  ( (
ph  /\  ( F supp  .0.  )  =  (/) )  -> 
( G  gsumg  F )  =  ( G  gsumg  ( k  e.  A  |->  .0.  ) ) )
3411, 32, 333eqtr4d 2518 . . 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 5823 . . . . . . . . . . . 12  |-  ( f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  )  ->  f : ( 1 ... ( # `  ( F supp  .0.  ) ) )
-onto-> ( F supp  .0.  )
)
37 forn 5798 . . . . . . . . . . . 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 728 . . . . . . . . . 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 465 . . . . . . . . . 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 3538 . . . . . . . . 9  |-  ( (
ph  /\  ( ( # `
 ( F supp  .0.  ) )  e.  NN  /\  f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
)  ->  ran  f  C_  W )
43 cores 5510 . . . . . . . . 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 12084 . . . . . . 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 5868 . . . . . 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 2467 . . . . . . 7  |-  ( +g  `  G )  =  ( +g  `  G )
49 gsumzcl.z . . . . . . 7  |-  Z  =  (Cntz `  G )
501adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 ( F supp  .0.  ) )  e.  NN  /\  f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
)  ->  G  e.  Mnd )
514adantr 465 . . . . . . 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 465 . . . . . . . . 9  |-  ( (
ph  /\  ( ( # `
 ( F supp  .0.  ) )  e.  NN  /\  f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
)  ->  F : A
--> B )
53 fssres 5751 . . . . . . . . 9  |-  ( ( F : A --> B  /\  ( A  i^i  W ) 
C_  A )  -> 
( F  |`  ( A  i^i  W ) ) : ( A  i^i  W ) --> B )
5452, 27, 53sylancl 662 . . . . . . . 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 5717 . . . . . . . . 9  |-  ( ph  ->  ( ( F  |`  ( A  i^i  W ) ) : ( A  i^i  W ) --> B  <-> 
( F  |`  W ) : ( A  i^i  W ) --> B ) )
5655biimpa 484 . . . . . . . 8  |-  ( (
ph  /\  ( F  |`  ( A  i^i  W
) ) : ( A  i^i  W ) --> B )  ->  ( F  |`  W ) : ( A  i^i  W
) --> B )
5754, 56syldan 470 . . . . . . 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 5297 . . . . . . . . . 10  |-  ( F  |`  W )  C_  F
60 rnss 5231 . . . . . . . . . 10  |-  ( ( F  |`  W )  C_  F  ->  ran  ( F  |`  W )  C_  ran  F )
6159, 60ax-mp 5 . . . . . . . . 9  |-  ran  ( F  |`  W )  C_  ran  F
6249cntzidss 16189 . . . . . . . . 9  |-  ( ( ran  F  C_  ( Z `  ran  F )  /\  ran  ( F  |`  W )  C_  ran  F )  ->  ran  ( F  |`  W )  C_  ( Z `  ran  ( F  |`  W ) ) )
6358, 61, 62sylancl 662 . . . . . . . 8  |-  ( ph  ->  ran  ( F  |`  W )  C_  ( Z `  ran  ( F  |`  W ) ) )
6463adantr 465 . . . . . . 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 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 f1of1 5815 . . . . . . . . 9  |-  ( f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  )  ->  f : ( 1 ... ( # `  ( F supp  .0.  ) ) )
-1-1-> ( F supp  .0.  )
)
6766ad2antll 728 . . . . . . . 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 6915 . . . . . . . . . . 11  |-  ( F supp 
.0.  )  C_  dom  F
69 fdm 5735 . . . . . . . . . . . 12  |-  ( F : A --> B  ->  dom  F  =  A )
7013, 69syl 16 . . . . . . . . . . 11  |-  ( ph  ->  dom  F  =  A )
7168, 70syl5sseq 3552 . . . . . . . . . 10  |-  ( ph  ->  ( F supp  .0.  )  C_  A )
7271, 40ssind 3722 . . . . . . . . 9  |-  ( ph  ->  ( F supp  .0.  )  C_  ( A  i^i  W
) )
7372adantr 465 . . . . . . . 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 5786 . . . . . . . 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 661 . . . . . . 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 6134 . . . . . . . . . . . . 13  |-  ( ( F : A --> B  /\  A  e.  V )  ->  F  e.  _V )
7713, 2, 76syl2anc 661 . . . . . . . . . . . 12  |-  ( ph  ->  F  e.  _V )
78 ressuppss 6920 . . . . . . . . . . . 12  |-  ( ( F  e.  _V  /\  .0.  e.  _V )  -> 
( ( F  |`  W ) supp  .0.  )  C_  ( F supp  .0.  )
)
7977, 21, 78sylancl 662 . . . . . . . . . . 11  |-  ( ph  ->  ( ( F  |`  W ) supp  .0.  )  C_  ( F supp  .0.  )
)
80 sseq2 3526 . . . . . . . . . . 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 466 . . . . . . . 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 2467 . . . . . . 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 16726 . . . . . 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 465 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 ( F supp  .0.  ) )  e.  NN  /\  f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
)  ->  A  e.  V )
8858adantr 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
) )
8971adantr 465 . . . . . . . 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 5786 . . . . . . . 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 661 . . . . . . 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 3553 . . . . . . 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 2467 . . . . . . 7  |-  ( ( F  o.  f ) supp 
.0.  )  =  ( ( F  o.  f
) supp  .0.  )
9447, 5, 48, 49, 50, 87, 52, 88, 65, 91, 92, 93gsumval3 16726 . . . . . 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 2518 . . . . 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 615 . . . 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 1700 . . 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 603 . 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 7836 . . . 4  |-  ( F finSupp  .0.  ->  ( Fun  F  /\  ( F supp  .0.  )  e.  Fin ) )
101100simprd 463 . . 3  |-  ( F finSupp  .0.  ->  ( F supp  .0.  )  e.  Fin )
102 fz1f1o 13498 . . 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 1379   E.wex 1596    e. wcel 1767   _Vcvv 3113    i^i cin 3475    C_ wss 3476   (/)c0 3785   class class class wbr 4447    |-> cmpt 4505   dom cdm 4999   ran crn 5000    |` cres 5001    o. ccom 5003   Fun wfun 5582    Fn wfn 5583   -->wf 5584   -1-1->wf1 5585   -onto->wfo 5586   -1-1-onto->wf1o 5587   ` cfv 5588  (class class class)co 6285   supp csupp 6902   Fincfn 7517   finSupp cfsupp 7830   1c1 9494   NNcn 10537   ...cfz 11673    seqcseq 12076   #chash 12374   Basecbs 14493   +g cplusg 14558   0gc0g 14698    gsumg cgsu 14699   Mndcmnd 15729  Cntzccntz 16167
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-om 6686  df-1st 6785  df-2nd 6786  df-supp 6903  df-recs 7043  df-rdg 7077  df-1o 7131  df-oadd 7135  df-er 7312  df-en 7518  df-dom 7519  df-sdom 7520  df-fin 7521  df-fsupp 7831  df-oi 7936  df-card 8321  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-nn 10538  df-n0 10797  df-z 10866  df-uz 11084  df-fz 11674  df-fzo 11794  df-seq 12077  df-hash 12375  df-0g 14700  df-gsum 14701  df-mnd 15735  df-cntz 16169
This theorem is referenced by:  gsumres  16736  gsumzsplit  16759  gsumpt  16803  dmdprdsplitlem  16898  dpjidcl  16921  mplcoe5  17942
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