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Theorem gsumzres 17543
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 4546 . . . . . . . 8  |-  ( A  e.  V  ->  ( A  i^i  W )  e. 
_V )
42, 3syl 17 . . . . . . 7  |-  ( ph  ->  ( A  i^i  W
)  e.  _V )
5 gsumzcl.0 . . . . . . . 8  |-  .0.  =  ( 0g `  G )
65gsumz 16621 . . . . . . 7  |-  ( ( G  e.  Mnd  /\  ( A  i^i  W )  e.  _V )  -> 
( G  gsumg  ( k  e.  ( A  i^i  W ) 
|->  .0.  ) )  =  .0.  )
71, 4, 6syl2anc 667 . . . . . 6  |-  ( ph  ->  ( G  gsumg  ( k  e.  ( A  i^i  W ) 
|->  .0.  ) )  =  .0.  )
85gsumz 16621 . . . . . . 7  |-  ( ( G  e.  Mnd  /\  A  e.  V )  ->  ( G  gsumg  ( k  e.  A  |->  .0.  ) )  =  .0.  )
91, 2, 8syl2anc 667 . . . . . 6  |-  ( ph  ->  ( G  gsumg  ( k  e.  A  |->  .0.  ) )  =  .0.  )
107, 9eqtr4d 2488 . . . . 5  |-  ( ph  ->  ( G  gsumg  ( k  e.  ( A  i^i  W ) 
|->  .0.  ) )  =  ( G  gsumg  ( k  e.  A  |->  .0.  ) ) )
1110adantr 467 . . . 4  |-  ( (
ph  /\  ( F supp  .0.  )  =  (/) )  -> 
( G  gsumg  ( k  e.  ( A  i^i  W ) 
|->  .0.  ) )  =  ( G  gsumg  ( k  e.  A  |->  .0.  ) ) )
12 resres 5117 . . . . . . . 8  |-  ( ( F  |`  A )  |`  W )  =  ( F  |`  ( A  i^i  W ) )
13 gsumzcl.f . . . . . . . . . 10  |-  ( ph  ->  F : A --> B )
14 ffn 5728 . . . . . . . . . 10  |-  ( F : A --> B  ->  F  Fn  A )
15 fnresdm 5685 . . . . . . . . . 10  |-  ( F  Fn  A  ->  ( F  |`  A )  =  F )
1613, 14, 153syl 18 . . . . . . . . 9  |-  ( ph  ->  ( F  |`  A )  =  F )
1716reseq1d 5104 . . . . . . . 8  |-  ( ph  ->  ( ( F  |`  A )  |`  W )  =  ( F  |`  W ) )
1812, 17syl5eqr 2499 . . . . . . 7  |-  ( ph  ->  ( F  |`  ( A  i^i  W ) )  =  ( F  |`  W ) )
1918adantr 467 . . . . . 6  |-  ( (
ph  /\  ( F supp  .0.  )  =  (/) )  -> 
( F  |`  ( A  i^i  W ) )  =  ( F  |`  W ) )
20 fvex 5875 . . . . . . . . . . 11  |-  ( 0g
`  G )  e. 
_V
215, 20eqeltri 2525 . . . . . . . . . 10  |-  .0.  e.  _V
2221a1i 11 . . . . . . . . 9  |-  ( ph  ->  .0.  e.  _V )
23 ssid 3451 . . . . . . . . . 10  |-  ( F supp 
.0.  )  C_  ( F supp  .0.  )
2423a1i 11 . . . . . . . . 9  |-  ( ph  ->  ( F supp  .0.  )  C_  ( F supp  .0.  )
)
2513, 2, 22, 24gsumcllem 17542 . . . . . . . 8  |-  ( (
ph  /\  ( F supp  .0.  )  =  (/) )  ->  F  =  ( k  e.  A  |->  .0.  )
)
2625reseq1d 5104 . . . . . . 7  |-  ( (
ph  /\  ( F supp  .0.  )  =  (/) )  -> 
( F  |`  ( A  i^i  W ) )  =  ( ( k  e.  A  |->  .0.  )  |`  ( A  i^i  W
) ) )
27 inss1 3652 . . . . . . . 8  |-  ( A  i^i  W )  C_  A
28 resmpt 5154 . . . . . . . 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 2501 . . . . . 6  |-  ( (
ph  /\  ( F supp  .0.  )  =  (/) )  -> 
( F  |`  ( A  i^i  W ) )  =  ( k  e.  ( A  i^i  W
)  |->  .0.  ) )
3119, 30eqtr3d 2487 . . . . 5  |-  ( (
ph  /\  ( F supp  .0.  )  =  (/) )  -> 
( F  |`  W )  =  ( k  e.  ( A  i^i  W
)  |->  .0.  ) )
3231oveq2d 6306 . . . 4  |-  ( (
ph  /\  ( F supp  .0.  )  =  (/) )  -> 
( G  gsumg  ( F  |`  W ) )  =  ( G 
gsumg  ( k  e.  ( A  i^i  W ) 
|->  .0.  ) ) )
3325oveq2d 6306 . . . 4  |-  ( (
ph  /\  ( F supp  .0.  )  =  (/) )  -> 
( G  gsumg  F )  =  ( G  gsumg  ( k  e.  A  |->  .0.  ) ) )
3411, 32, 333eqtr4d 2495 . . 3  |-  ( (
ph  /\  ( F supp  .0.  )  =  (/) )  -> 
( G  gsumg  ( F  |`  W ) )  =  ( G 
gsumg  F ) )
3534ex 436 . 2  |-  ( ph  ->  ( ( F supp  .0.  )  =  (/)  ->  ( G  gsumg  ( F  |`  W ) )  =  ( G 
gsumg  F ) ) )
36 f1ofo 5821 . . . . . . . . . . . 12  |-  ( f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  )  ->  f : ( 1 ... ( # `  ( F supp  .0.  ) ) )
-onto-> ( F supp  .0.  )
)
37 forn 5796 . . . . . . . . . . . 12  |-  ( f : ( 1 ... ( # `  ( F supp  .0.  ) ) )
-onto-> ( F supp  .0.  )  ->  ran  f  =  ( F supp  .0.  ) )
3836, 37syl 17 . . . . . . . . . . 11  |-  ( f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  )  ->  ran  f  =  ( F supp 
.0.  ) )
3938ad2antll 735 . . . . . . . . . 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 467 . . . . . . . . . 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 3466 . . . . . . . . 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 17 . . . . . . . 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 12221 . . . . . . 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 5867 . . . . . 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 2451 . . . . . . 7  |-  ( +g  `  G )  =  ( +g  `  G )
49 gsumzcl.z . . . . . . 7  |-  Z  =  (Cntz `  G )
501adantr 467 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 ( F supp  .0.  ) )  e.  NN  /\  f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
)  ->  G  e.  Mnd )
514adantr 467 . . . . . . 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 467 . . . . . . . . 9  |-  ( (
ph  /\  ( ( # `
 ( F supp  .0.  ) )  e.  NN  /\  f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
)  ->  F : A
--> B )
53 fssres 5749 . . . . . . . . 9  |-  ( ( F : A --> B  /\  ( A  i^i  W ) 
C_  A )  -> 
( F  |`  ( A  i^i  W ) ) : ( A  i^i  W ) --> B )
5452, 27, 53sylancl 668 . . . . . . . 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 5714 . . . . . . . . 9  |-  ( ph  ->  ( ( F  |`  ( A  i^i  W ) ) : ( A  i^i  W ) --> B  <-> 
( F  |`  W ) : ( A  i^i  W ) --> B ) )
5655biimpa 487 . . . . . . . 8  |-  ( (
ph  /\  ( F  |`  ( A  i^i  W
) ) : ( A  i^i  W ) --> B )  ->  ( F  |`  W ) : ( A  i^i  W
) --> B )
5754, 56syldan 473 . . . . . . 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 5128 . . . . . . . . . 10  |-  ( F  |`  W )  C_  F
60 rnss 5063 . . . . . . . . . 10  |-  ( ( F  |`  W )  C_  F  ->  ran  ( F  |`  W )  C_  ran  F )
6159, 60ax-mp 5 . . . . . . . . 9  |-  ran  ( F  |`  W )  C_  ran  F
6249cntzidss 16991 . . . . . . . . 9  |-  ( ( ran  F  C_  ( Z `  ran  F )  /\  ran  ( F  |`  W )  C_  ran  F )  ->  ran  ( F  |`  W )  C_  ( Z `  ran  ( F  |`  W ) ) )
6358, 61, 62sylancl 668 . . . . . . . 8  |-  ( ph  ->  ran  ( F  |`  W )  C_  ( Z `  ran  ( F  |`  W ) ) )
6463adantr 467 . . . . . . 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 764 . . . . . . 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 5813 . . . . . . . . 9  |-  ( f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  )  ->  f : ( 1 ... ( # `  ( F supp  .0.  ) ) )
-1-1-> ( F supp  .0.  )
)
6766ad2antll 735 . . . . . . . 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 6927 . . . . . . . . . . 11  |-  ( F supp 
.0.  )  C_  dom  F
69 fdm 5733 . . . . . . . . . . . 12  |-  ( F : A --> B  ->  dom  F  =  A )
7013, 69syl 17 . . . . . . . . . . 11  |-  ( ph  ->  dom  F  =  A )
7168, 70syl5sseq 3480 . . . . . . . . . 10  |-  ( ph  ->  ( F supp  .0.  )  C_  A )
7271, 40ssind 3656 . . . . . . . . 9  |-  ( ph  ->  ( F supp  .0.  )  C_  ( A  i^i  W
) )
7372adantr 467 . . . . . . . 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 5784 . . . . . . . 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 667 . . . . . . 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 6138 . . . . . . . . . . . . 13  |-  ( ( F : A --> B  /\  A  e.  V )  ->  F  e.  _V )
7713, 2, 76syl2anc 667 . . . . . . . . . . . 12  |-  ( ph  ->  F  e.  _V )
78 ressuppss 6934 . . . . . . . . . . . 12  |-  ( ( F  e.  _V  /\  .0.  e.  _V )  -> 
( ( F  |`  W ) supp  .0.  )  C_  ( F supp  .0.  )
)
7977, 21, 78sylancl 668 . . . . . . . . . . 11  |-  ( ph  ->  ( ( F  |`  W ) supp  .0.  )  C_  ( F supp  .0.  )
)
80 sseq2 3454 . . . . . . . . . . 11  |-  ( ran  f  =  ( F supp 
.0.  )  ->  (
( ( F  |`  W ) supp  .0.  )  C_ 
ran  f  <->  ( ( F  |`  W ) supp  .0.  )  C_  ( F supp  .0.  ) ) )
8179, 80syl5ibr 225 . . . . . . . . . 10  |-  ( ran  f  =  ( F supp 
.0.  )  ->  ( ph  ->  ( ( F  |`  W ) supp  .0.  )  C_ 
ran  f ) )
8236, 37, 813syl 18 . . . . . . . . 9  |-  ( f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  )  ->  (
ph  ->  ( ( F  |`  W ) supp  .0.  )  C_ 
ran  f ) )
8382adantl 468 . . . . . . . 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 432 . . . . . . 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 2451 . . . . . . 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 17541 . . . . . 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 467 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 ( F supp  .0.  ) )  e.  NN  /\  f : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
)  ->  A  e.  V )
8858adantr 467 . . . . . . 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 467 . . . . . . . 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 5784 . . . . . . . 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 667 . . . . . . 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 3481 . . . . . . 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 2451 . . . . . . 7  |-  ( ( F  o.  f ) supp 
.0.  )  =  ( ( F  o.  f
) supp  .0.  )
9447, 5, 48, 49, 50, 87, 52, 88, 65, 91, 92, 93gsumval3 17541 . . . . . 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 2495 . . . . 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 620 . . . 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 1779 . . 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 608 . 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 7889 . . . 4  |-  ( F finSupp  .0.  ->  ( Fun  F  /\  ( F supp  .0.  )  e.  Fin ) )
101100simprd 465 . . 3  |-  ( F finSupp  .0.  ->  ( F supp  .0.  )  e.  Fin )
102 fz1f1o 13776 . . 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 18 . 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 383 1  |-  ( ph  ->  ( G  gsumg  ( F  |`  W ) )  =  ( G 
gsumg  F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 370    /\ wa 371    = wceq 1444   E.wex 1663    e. wcel 1887   _Vcvv 3045    i^i cin 3403    C_ wss 3404   (/)c0 3731   class class class wbr 4402    |-> cmpt 4461   dom cdm 4834   ran crn 4835    |` cres 4836    o. ccom 4838   Fun wfun 5576    Fn wfn 5577   -->wf 5578   -1-1->wf1 5579   -onto->wfo 5580   -1-1-onto->wf1o 5581   ` cfv 5582  (class class class)co 6290   supp csupp 6914   Fincfn 7569   finSupp cfsupp 7883   1c1 9540   NNcn 10609   ...cfz 11784    seqcseq 12213   #chash 12515   Basecbs 15121   +g cplusg 15190   0gc0g 15338    gsumg cgsu 15339   Mndcmnd 16535  Cntzccntz 16969
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-supp 6915  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-fsupp 7884  df-oi 8025  df-card 8373  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11785  df-fzo 11916  df-seq 12214  df-hash 12516  df-0g 15340  df-gsum 15341  df-mgm 16488  df-sgrp 16527  df-mnd 16537  df-cntz 16971
This theorem is referenced by:  gsumres  17547  gsumzsplit  17560  gsumpt  17594  dmdprdsplitlem  17670  dpjidcl  17691  mplcoe5  18692
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