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Theorem gsumval3 16695
Description: Value of the group sum operation over an arbitrary finite set. (Contributed by Mario Carneiro, 15-Dec-2014.) (Revised by AV, 31-May-2019.)
Hypotheses
Ref Expression
gsumval3.b  |-  B  =  ( Base `  G
)
gsumval3.0  |-  .0.  =  ( 0g `  G )
gsumval3.p  |-  .+  =  ( +g  `  G )
gsumval3.z  |-  Z  =  (Cntz `  G )
gsumval3.g  |-  ( ph  ->  G  e.  Mnd )
gsumval3.a  |-  ( ph  ->  A  e.  V )
gsumval3.f  |-  ( ph  ->  F : A --> B )
gsumval3.c  |-  ( ph  ->  ran  F  C_  ( Z `  ran  F ) )
gsumval3.m  |-  ( ph  ->  M  e.  NN )
gsumval3.h  |-  ( ph  ->  H : ( 1 ... M ) -1-1-> A
)
gsumval3.n  |-  ( ph  ->  ( F supp  .0.  )  C_ 
ran  H )
gsumval3.w  |-  W  =  ( ( F  o.  H ) supp  .0.  )
Assertion
Ref Expression
gsumval3  |-  ( ph  ->  ( G  gsumg  F )  =  (  seq 1 (  .+  ,  ( F  o.  H ) ) `  M ) )

Proof of Theorem gsumval3
Dummy variables  f 
k  m  n  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 gsumval3.g . . . . 5  |-  ( ph  ->  G  e.  Mnd )
2 gsumval3.a . . . . 5  |-  ( ph  ->  A  e.  V )
3 gsumval3.0 . . . . . 6  |-  .0.  =  ( 0g `  G )
43gsumz 15817 . . . . 5  |-  ( ( G  e.  Mnd  /\  A  e.  V )  ->  ( G  gsumg  ( x  e.  A  |->  .0.  ) )  =  .0.  )
51, 2, 4syl2anc 661 . . . 4  |-  ( ph  ->  ( G  gsumg  ( x  e.  A  |->  .0.  ) )  =  .0.  )
65adantr 465 . . 3  |-  ( (
ph  /\  W  =  (/) )  ->  ( G  gsumg  ( x  e.  A  |->  .0.  ) )  =  .0.  )
7 gsumval3.f . . . . . . 7  |-  ( ph  ->  F : A --> B )
87feqmptd 5911 . . . . . 6  |-  ( ph  ->  F  =  ( x  e.  A  |->  ( F `
 x ) ) )
98adantr 465 . . . . 5  |-  ( (
ph  /\  W  =  (/) )  ->  F  =  ( x  e.  A  |->  ( F `  x
) ) )
10 gsumval3.h . . . . . . . . . . . . . 14  |-  ( ph  ->  H : ( 1 ... M ) -1-1-> A
)
11 f1f 5772 . . . . . . . . . . . . . 14  |-  ( H : ( 1 ... M ) -1-1-> A  ->  H : ( 1 ... M ) --> A )
1210, 11syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  H : ( 1 ... M ) --> A )
1312ad2antrr 725 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  W  =  (/) )  /\  x  e.  ran  H )  ->  H : ( 1 ... M ) --> A )
14 f1f1orn 5818 . . . . . . . . . . . . . . . 16  |-  ( H : ( 1 ... M ) -1-1-> A  ->  H : ( 1 ... M ) -1-1-onto-> ran  H )
1510, 14syl 16 . . . . . . . . . . . . . . 15  |-  ( ph  ->  H : ( 1 ... M ) -1-1-onto-> ran  H
)
1615adantr 465 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  W  =  (/) )  ->  H :
( 1 ... M
)
-1-1-onto-> ran  H )
17 f1ocnv 5819 . . . . . . . . . . . . . 14  |-  ( H : ( 1 ... M ) -1-1-onto-> ran  H  ->  `' H : ran  H -1-1-onto-> ( 1 ... M ) )
18 f1of 5807 . . . . . . . . . . . . . 14  |-  ( `' H : ran  H -1-1-onto-> (
1 ... M )  ->  `' H : ran  H --> ( 1 ... M
) )
1916, 17, 183syl 20 . . . . . . . . . . . . 13  |-  ( (
ph  /\  W  =  (/) )  ->  `' H : ran  H --> ( 1 ... M ) )
2019ffvelrnda 6012 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  W  =  (/) )  /\  x  e.  ran  H )  -> 
( `' H `  x )  e.  ( 1 ... M ) )
21 fvco3 5935 . . . . . . . . . . . 12  |-  ( ( H : ( 1 ... M ) --> A  /\  ( `' H `  x )  e.  ( 1 ... M ) )  ->  ( ( F  o.  H ) `  ( `' H `  x ) )  =  ( F `  ( H `  ( `' H `  x )
) ) )
2213, 20, 21syl2anc 661 . . . . . . . . . . 11  |-  ( ( ( ph  /\  W  =  (/) )  /\  x  e.  ran  H )  -> 
( ( F  o.  H ) `  ( `' H `  x ) )  =  ( F `
 ( H `  ( `' H `  x ) ) ) )
23 simpr 461 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  W  =  (/) )  ->  W  =  (/) )
2423difeq2d 3615 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  W  =  (/) )  ->  ( (
1 ... M )  \  W )  =  ( ( 1 ... M
)  \  (/) ) )
25 dif0 3890 . . . . . . . . . . . . . . 15  |-  ( ( 1 ... M ) 
\  (/) )  =  ( 1 ... M )
2624, 25syl6eq 2517 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  W  =  (/) )  ->  ( (
1 ... M )  \  W )  =  ( 1 ... M ) )
2726adantr 465 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  W  =  (/) )  /\  x  e.  ran  H )  -> 
( ( 1 ... M )  \  W
)  =  ( 1 ... M ) )
2820, 27eleqtrrd 2551 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  W  =  (/) )  /\  x  e.  ran  H )  -> 
( `' H `  x )  e.  ( ( 1 ... M
)  \  W )
)
29 fco 5732 . . . . . . . . . . . . . . 15  |-  ( ( F : A --> B  /\  H : ( 1 ... M ) --> A )  ->  ( F  o.  H ) : ( 1 ... M ) --> B )
307, 12, 29syl2anc 661 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( F  o.  H
) : ( 1 ... M ) --> B )
3130adantr 465 . . . . . . . . . . . . 13  |-  ( (
ph  /\  W  =  (/) )  ->  ( F  o.  H ) : ( 1 ... M ) --> B )
32 gsumval3.w . . . . . . . . . . . . . . 15  |-  W  =  ( ( F  o.  H ) supp  .0.  )
3332eqimss2i 3552 . . . . . . . . . . . . . 14  |-  ( ( F  o.  H ) supp 
.0.  )  C_  W
3433a1i 11 . . . . . . . . . . . . 13  |-  ( (
ph  /\  W  =  (/) )  ->  ( ( F  o.  H ) supp  .0.  )  C_  W )
35 ovex 6300 . . . . . . . . . . . . . 14  |-  ( 1 ... M )  e. 
_V
3635a1i 11 . . . . . . . . . . . . 13  |-  ( (
ph  /\  W  =  (/) )  ->  ( 1 ... M )  e. 
_V )
37 fvex 5867 . . . . . . . . . . . . . . 15  |-  ( 0g
`  G )  e. 
_V
383, 37eqeltri 2544 . . . . . . . . . . . . . 14  |-  .0.  e.  _V
3938a1i 11 . . . . . . . . . . . . 13  |-  ( (
ph  /\  W  =  (/) )  ->  .0.  e.  _V )
4031, 34, 36, 39suppssr 6921 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  W  =  (/) )  /\  ( `' H `  x )  e.  ( ( 1 ... M )  \  W ) )  -> 
( ( F  o.  H ) `  ( `' H `  x ) )  =  .0.  )
4128, 40syldan 470 . . . . . . . . . . 11  |-  ( ( ( ph  /\  W  =  (/) )  /\  x  e.  ran  H )  -> 
( ( F  o.  H ) `  ( `' H `  x ) )  =  .0.  )
42 f1ocnvfv2 6162 . . . . . . . . . . . . 13  |-  ( ( H : ( 1 ... M ) -1-1-onto-> ran  H  /\  x  e.  ran  H )  ->  ( H `  ( `' H `  x ) )  =  x )
4316, 42sylan 471 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  W  =  (/) )  /\  x  e.  ran  H )  -> 
( H `  ( `' H `  x ) )  =  x )
4443fveq2d 5861 . . . . . . . . . . 11  |-  ( ( ( ph  /\  W  =  (/) )  /\  x  e.  ran  H )  -> 
( F `  ( H `  ( `' H `  x )
) )  =  ( F `  x ) )
4522, 41, 443eqtr3rd 2510 . . . . . . . . . 10  |-  ( ( ( ph  /\  W  =  (/) )  /\  x  e.  ran  H )  -> 
( F `  x
)  =  .0.  )
46 fvex 5867 . . . . . . . . . . 11  |-  ( F `
 x )  e. 
_V
4746elsnc 4044 . . . . . . . . . 10  |-  ( ( F `  x )  e.  {  .0.  }  <->  ( F `  x )  =  .0.  )
4845, 47sylibr 212 . . . . . . . . 9  |-  ( ( ( ph  /\  W  =  (/) )  /\  x  e.  ran  H )  -> 
( F `  x
)  e.  {  .0.  } )
4948adantlr 714 . . . . . . . 8  |-  ( ( ( ( ph  /\  W  =  (/) )  /\  x  e.  A )  /\  x  e.  ran  H )  ->  ( F `  x )  e.  {  .0.  } )
50 eldif 3479 . . . . . . . . . . 11  |-  ( x  e.  ( A  \  ran  H )  <->  ( x  e.  A  /\  -.  x  e.  ran  H ) )
51 gsumval3.n . . . . . . . . . . . . 13  |-  ( ph  ->  ( F supp  .0.  )  C_ 
ran  H )
5238a1i 11 . . . . . . . . . . . . 13  |-  ( ph  ->  .0.  e.  _V )
537, 51, 2, 52suppssr 6921 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( A  \  ran  H
) )  ->  ( F `  x )  =  .0.  )
5453, 47sylibr 212 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( A  \  ran  H
) )  ->  ( F `  x )  e.  {  .0.  } )
5550, 54sylan2br 476 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  A  /\  -.  x  e.  ran  H ) )  ->  ( F `  x )  e.  {  .0.  } )
5655adantlr 714 . . . . . . . . 9  |-  ( ( ( ph  /\  W  =  (/) )  /\  (
x  e.  A  /\  -.  x  e.  ran  H ) )  ->  ( F `  x )  e.  {  .0.  } )
5756anassrs 648 . . . . . . . 8  |-  ( ( ( ( ph  /\  W  =  (/) )  /\  x  e.  A )  /\  -.  x  e.  ran  H )  ->  ( F `  x )  e.  {  .0.  } )
5849, 57pm2.61dan 789 . . . . . . 7  |-  ( ( ( ph  /\  W  =  (/) )  /\  x  e.  A )  ->  ( F `  x )  e.  {  .0.  } )
5958, 47sylib 196 . . . . . 6  |-  ( ( ( ph  /\  W  =  (/) )  /\  x  e.  A )  ->  ( F `  x )  =  .0.  )
6059mpteq2dva 4526 . . . . 5  |-  ( (
ph  /\  W  =  (/) )  ->  ( x  e.  A  |->  ( F `
 x ) )  =  ( x  e.  A  |->  .0.  ) )
619, 60eqtrd 2501 . . . 4  |-  ( (
ph  /\  W  =  (/) )  ->  F  =  ( x  e.  A  |->  .0.  ) )
6261oveq2d 6291 . . 3  |-  ( (
ph  /\  W  =  (/) )  ->  ( G  gsumg  F )  =  ( G 
gsumg  ( x  e.  A  |->  .0.  ) ) )
63 gsumval3.b . . . . . . . 8  |-  B  =  ( Base `  G
)
6463, 3mndidcl 15745 . . . . . . 7  |-  ( G  e.  Mnd  ->  .0.  e.  B )
651, 64syl 16 . . . . . 6  |-  ( ph  ->  .0.  e.  B )
66 gsumval3.p . . . . . . 7  |-  .+  =  ( +g  `  G )
6763, 66, 3mndlid 15747 . . . . . 6  |-  ( ( G  e.  Mnd  /\  .0.  e.  B )  -> 
(  .0.  .+  .0.  )  =  .0.  )
681, 65, 67syl2anc 661 . . . . 5  |-  ( ph  ->  (  .0.  .+  .0.  )  =  .0.  )
6968adantr 465 . . . 4  |-  ( (
ph  /\  W  =  (/) )  ->  (  .0.  .+  .0.  )  =  .0.  )
70 gsumval3.m . . . . . 6  |-  ( ph  ->  M  e.  NN )
71 nnuz 11106 . . . . . 6  |-  NN  =  ( ZZ>= `  1 )
7270, 71syl6eleq 2558 . . . . 5  |-  ( ph  ->  M  e.  ( ZZ>= ` 
1 ) )
7372adantr 465 . . . 4  |-  ( (
ph  /\  W  =  (/) )  ->  M  e.  ( ZZ>= `  1 )
)
7426eleq2d 2530 . . . . . 6  |-  ( (
ph  /\  W  =  (/) )  ->  ( x  e.  ( ( 1 ... M )  \  W
)  <->  x  e.  (
1 ... M ) ) )
7574biimpar 485 . . . . 5  |-  ( ( ( ph  /\  W  =  (/) )  /\  x  e.  ( 1 ... M
) )  ->  x  e.  ( ( 1 ... M )  \  W
) )
7631, 34, 36, 39suppssr 6921 . . . . 5  |-  ( ( ( ph  /\  W  =  (/) )  /\  x  e.  ( ( 1 ... M )  \  W
) )  ->  (
( F  o.  H
) `  x )  =  .0.  )
7775, 76syldan 470 . . . 4  |-  ( ( ( ph  /\  W  =  (/) )  /\  x  e.  ( 1 ... M
) )  ->  (
( F  o.  H
) `  x )  =  .0.  )
7869, 73, 77seqid3 12107 . . 3  |-  ( (
ph  /\  W  =  (/) )  ->  (  seq 1 (  .+  , 
( F  o.  H
) ) `  M
)  =  .0.  )
796, 62, 783eqtr4d 2511 . 2  |-  ( (
ph  /\  W  =  (/) )  ->  ( G  gsumg  F )  =  (  seq 1 (  .+  , 
( F  o.  H
) ) `  M
) )
80 fzf 11665 . . . . 5  |-  ... :
( ZZ  X.  ZZ )
--> ~P ZZ
81 ffn 5722 . . . . 5  |-  ( ...
: ( ZZ  X.  ZZ ) --> ~P ZZ  ->  ... 
Fn  ( ZZ  X.  ZZ ) )
82 ovelrn 6426 . . . . 5  |-  ( ... 
Fn  ( ZZ  X.  ZZ )  ->  ( A  e.  ran  ...  <->  E. m  e.  ZZ  E. n  e.  ZZ  A  =  ( m ... n ) ) )
8380, 81, 82mp2b 10 . . . 4  |-  ( A  e.  ran  ...  <->  E. m  e.  ZZ  E. n  e.  ZZ  A  =  ( m ... n ) )
841ad2antrr 725 . . . . . . . . 9  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  A  =  ( m ... n ) )  ->  G  e.  Mnd )
85 simpr 461 . . . . . . . . . . 11  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  A  =  ( m ... n ) )  ->  A  =  ( m ... n ) )
86 frel 5725 . . . . . . . . . . . . . . . . 17  |-  ( F : A --> B  ->  Rel  F )
87 reldm0 5211 . . . . . . . . . . . . . . . . 17  |-  ( Rel 
F  ->  ( F  =  (/)  <->  dom  F  =  (/) ) )
887, 86, 873syl 20 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( F  =  (/)  <->  dom  F  =  (/) ) )
89 fdm 5726 . . . . . . . . . . . . . . . . . 18  |-  ( F : A --> B  ->  dom  F  =  A )
907, 89syl 16 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  dom  F  =  A )
9190eqeq1d 2462 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( dom  F  =  (/) 
<->  A  =  (/) ) )
9288, 91bitrd 253 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( F  =  (/)  <->  A  =  (/) ) )
93 coeq1 5151 . . . . . . . . . . . . . . . . . . 19  |-  ( F  =  (/)  ->  ( F  o.  H )  =  ( (/)  o.  H
) )
94 co01 5513 . . . . . . . . . . . . . . . . . . 19  |-  ( (/)  o.  H )  =  (/)
9593, 94syl6eq 2517 . . . . . . . . . . . . . . . . . 18  |-  ( F  =  (/)  ->  ( F  o.  H )  =  (/) )
9695oveq1d 6290 . . . . . . . . . . . . . . . . 17  |-  ( F  =  (/)  ->  ( ( F  o.  H ) supp 
.0.  )  =  (
(/) supp  .0.  ) )
97 supp0 6896 . . . . . . . . . . . . . . . . . 18  |-  (  .0. 
e.  _V  ->  ( (/) supp  .0.  )  =  (/) )
9838, 97ax-mp 5 . . . . . . . . . . . . . . . . 17  |-  ( (/) supp  .0.  )  =  (/)
9996, 98syl6eq 2517 . . . . . . . . . . . . . . . 16  |-  ( F  =  (/)  ->  ( ( F  o.  H ) supp 
.0.  )  =  (/) )
10032, 99syl5eq 2513 . . . . . . . . . . . . . . 15  |-  ( F  =  (/)  ->  W  =  (/) )
10192, 100syl6bir 229 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( A  =  (/)  ->  W  =  (/) ) )
102101necon3d 2684 . . . . . . . . . . . . 13  |-  ( ph  ->  ( W  =/=  (/)  ->  A  =/=  (/) ) )
103102imp 429 . . . . . . . . . . . 12  |-  ( (
ph  /\  W  =/=  (/) )  ->  A  =/=  (/) )
104103adantr 465 . . . . . . . . . . 11  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  A  =  ( m ... n ) )  ->  A  =/=  (/) )
10585, 104eqnetrrd 2754 . . . . . . . . . 10  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  A  =  ( m ... n ) )  -> 
( m ... n
)  =/=  (/) )
106 fzn0 11689 . . . . . . . . . 10  |-  ( ( m ... n )  =/=  (/)  <->  n  e.  ( ZZ>=
`  m ) )
107105, 106sylib 196 . . . . . . . . 9  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  A  =  ( m ... n ) )  ->  n  e.  ( ZZ>= `  m ) )
1087ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  A  =  ( m ... n ) )  ->  F : A --> B )
10985feq2d 5709 . . . . . . . . . 10  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  A  =  ( m ... n ) )  -> 
( F : A --> B 
<->  F : ( m ... n ) --> B ) )
110108, 109mpbid 210 . . . . . . . . 9  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  A  =  ( m ... n ) )  ->  F : ( m ... n ) --> B )
11163, 66, 84, 107, 110gsumval2 15819 . . . . . . . 8  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  A  =  ( m ... n ) )  -> 
( G  gsumg  F )  =  (  seq m (  .+  ,  F ) `  n
) )
112 frn 5728 . . . . . . . . . . . . . . 15  |-  ( H : ( 1 ... M ) --> A  ->  ran  H  C_  A )
11310, 11, 1123syl 20 . . . . . . . . . . . . . 14  |-  ( ph  ->  ran  H  C_  A
)
114113ad2antrr 725 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  A  =  ( m ... n ) )  ->  ran  H  C_  A )
115114, 85sseqtrd 3533 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  A  =  ( m ... n ) )  ->  ran  H  C_  ( m ... n ) )
116 fzssuz 11713 . . . . . . . . . . . . 13  |-  ( m ... n )  C_  ( ZZ>= `  m )
117 uzssz 11090 . . . . . . . . . . . . . 14  |-  ( ZZ>= `  m )  C_  ZZ
118 zssre 10860 . . . . . . . . . . . . . 14  |-  ZZ  C_  RR
119117, 118sstri 3506 . . . . . . . . . . . . 13  |-  ( ZZ>= `  m )  C_  RR
120116, 119sstri 3506 . . . . . . . . . . . 12  |-  ( m ... n )  C_  RR
121115, 120syl6ss 3509 . . . . . . . . . . 11  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  A  =  ( m ... n ) )  ->  ran  H  C_  RR )
122 ltso 9654 . . . . . . . . . . 11  |-  <  Or  RR
123 soss 4811 . . . . . . . . . . 11  |-  ( ran 
H  C_  RR  ->  (  <  Or  RR  ->  < 
Or  ran  H )
)
124121, 122, 123mpisyl 18 . . . . . . . . . 10  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  A  =  ( m ... n ) )  ->  <  Or  ran  H )
125 fzfi 12038 . . . . . . . . . . . 12  |-  ( 1 ... M )  e. 
Fin
126125a1i 11 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( 1 ... M
)  e.  Fin )
127 fex2 6729 . . . . . . . . . . . . . . 15  |-  ( ( H : ( 1 ... M ) --> A  /\  ( 1 ... M )  e.  Fin  /\  A  e.  V )  ->  H  e.  _V )
12812, 126, 2, 127syl3anc 1223 . . . . . . . . . . . . . 14  |-  ( ph  ->  H  e.  _V )
129 f1oen3g 7521 . . . . . . . . . . . . . 14  |-  ( ( H  e.  _V  /\  H : ( 1 ... M ) -1-1-onto-> ran  H )  -> 
( 1 ... M
)  ~~  ran  H )
130128, 15, 129syl2anc 661 . . . . . . . . . . . . 13  |-  ( ph  ->  ( 1 ... M
)  ~~  ran  H )
131 enfi 7726 . . . . . . . . . . . . 13  |-  ( ( 1 ... M ) 
~~  ran  H  ->  ( ( 1 ... M
)  e.  Fin  <->  ran  H  e. 
Fin ) )
132130, 131syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( 1 ... M )  e.  Fin  <->  ran  H  e.  Fin ) )
133125, 132mpbii 211 . . . . . . . . . . 11  |-  ( ph  ->  ran  H  e.  Fin )
134133ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  A  =  ( m ... n ) )  ->  ran  H  e.  Fin )
135 fz1iso 12464 . . . . . . . . . 10  |-  ( (  <  Or  ran  H  /\  ran  H  e.  Fin )  ->  E. f  f  Isom  <  ,  <  ( ( 1 ... ( # `  ran  H ) ) ,  ran  H ) )
136124, 134, 135syl2anc 661 . . . . . . . . 9  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  A  =  ( m ... n ) )  ->  E. f  f  Isom  <  ,  <  ( ( 1 ... ( # `  ran  H ) ) ,  ran  H ) )
13770nnnn0d 10841 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  M  e.  NN0 )
138 hashfz1 12374 . . . . . . . . . . . . . . . 16  |-  ( M  e.  NN0  ->  ( # `  ( 1 ... M
) )  =  M )
139137, 138syl 16 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( # `  (
1 ... M ) )  =  M )
140 hashen 12375 . . . . . . . . . . . . . . . . 17  |-  ( ( ( 1 ... M
)  e.  Fin  /\  ran  H  e.  Fin )  ->  ( ( # `  (
1 ... M ) )  =  ( # `  ran  H )  <->  ( 1 ... M )  ~~  ran  H ) )
141125, 133, 140sylancr 663 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( ( # `  (
1 ... M ) )  =  ( # `  ran  H )  <->  ( 1 ... M )  ~~  ran  H ) )
142130, 141mpbird 232 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( # `  (
1 ... M ) )  =  ( # `  ran  H ) )
143139, 142eqtr3d 2503 . . . . . . . . . . . . . 14  |-  ( ph  ->  M  =  ( # `  ran  H ) )
144143ad2antrr 725 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  ->  M  =  ( # `  ran  H ) )
145144fveq2d 5861 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  -> 
(  seq 1 (  .+  ,  ( F  o.  f ) ) `  M )  =  (  seq 1 (  .+  ,  ( F  o.  f ) ) `  ( # `  ran  H
) ) )
1461ad2antrr 725 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  ->  G  e.  Mnd )
14763, 66mndcl 15726 . . . . . . . . . . . . . . 15  |-  ( ( G  e.  Mnd  /\  x  e.  B  /\  y  e.  B )  ->  ( x  .+  y
)  e.  B )
1481473expb 1192 . . . . . . . . . . . . . 14  |-  ( ( G  e.  Mnd  /\  ( x  e.  B  /\  y  e.  B
) )  ->  (
x  .+  y )  e.  B )
149146, 148sylan 471 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  /\  ( x  e.  B  /\  y  e.  B
) )  ->  (
x  .+  y )  e.  B )
150 gsumval3.c . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ran  F  C_  ( Z `  ran  F ) )
151150ad2antrr 725 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  ->  ran  F  C_  ( Z `  ran  F ) )
152151sselda 3497 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  /\  x  e.  ran  F )  ->  x  e.  ( Z `  ran  F
) )
153 gsumval3.z . . . . . . . . . . . . . . . 16  |-  Z  =  (Cntz `  G )
15466, 153cntzi 16155 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  ( Z `
 ran  F )  /\  y  e.  ran  F )  ->  ( x  .+  y )  =  ( y  .+  x ) )
155152, 154sylan 471 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  (
m ... n )  /\  f  Isom  <  ,  <  ( ( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  /\  x  e.  ran  F )  /\  y  e.  ran  F )  ->  ( x  .+  y )  =  ( y  .+  x ) )
156155anasss 647 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  /\  ( x  e.  ran  F  /\  y  e.  ran  F ) )  ->  (
x  .+  y )  =  ( y  .+  x ) )
15763, 66mndass 15727 . . . . . . . . . . . . . 14  |-  ( ( G  e.  Mnd  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B
) )  ->  (
( x  .+  y
)  .+  z )  =  ( x  .+  ( y  .+  z
) ) )
158146, 157sylan 471 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B
) )  ->  (
( x  .+  y
)  .+  z )  =  ( x  .+  ( y  .+  z
) ) )
15972ad2antrr 725 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  ->  M  e.  ( ZZ>= ` 
1 ) )
1607ad2antrr 725 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  ->  F : A --> B )
161 frn 5728 . . . . . . . . . . . . . 14  |-  ( F : A --> B  ->  ran  F  C_  B )
162160, 161syl 16 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  ->  ran  F  C_  B )
163 simprr 756 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  -> 
f  Isom  <  ,  <  ( ( 1 ... ( # `
 ran  H )
) ,  ran  H
) )
164 isof1o 6200 . . . . . . . . . . . . . . . . 17  |-  ( f 
Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
)  ->  f :
( 1 ... ( # `
 ran  H )
)
-1-1-onto-> ran  H )
165163, 164syl 16 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  -> 
f : ( 1 ... ( # `  ran  H ) ) -1-1-onto-> ran  H )
166144oveq2d 6291 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  -> 
( 1 ... M
)  =  ( 1 ... ( # `  ran  H ) ) )
167 f1oeq2 5799 . . . . . . . . . . . . . . . . 17  |-  ( ( 1 ... M )  =  ( 1 ... ( # `  ran  H ) )  ->  (
f : ( 1 ... M ) -1-1-onto-> ran  H  <->  f : ( 1 ... ( # `  ran  H ) ) -1-1-onto-> ran  H ) )
168166, 167syl 16 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  -> 
( f : ( 1 ... M ) -1-1-onto-> ran 
H  <->  f : ( 1 ... ( # `  ran  H ) ) -1-1-onto-> ran 
H ) )
169165, 168mpbird 232 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  -> 
f : ( 1 ... M ) -1-1-onto-> ran  H
)
170 f1ocnv 5819 . . . . . . . . . . . . . . 15  |-  ( f : ( 1 ... M ) -1-1-onto-> ran  H  ->  `' f : ran  H -1-1-onto-> ( 1 ... M ) )
171169, 170syl 16 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  ->  `' f : ran  H -1-1-onto-> ( 1 ... M ) )
17215ad2antrr 725 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  ->  H : ( 1 ... M ) -1-1-onto-> ran  H )
173 f1oco 5829 . . . . . . . . . . . . . 14  |-  ( ( `' f : ran  H -1-1-onto-> ( 1 ... M )  /\  H : ( 1 ... M ) -1-1-onto-> ran 
H )  ->  ( `' f  o.  H
) : ( 1 ... M ) -1-1-onto-> ( 1 ... M ) )
174171, 172, 173syl2anc 661 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  -> 
( `' f  o.  H ) : ( 1 ... M ) -1-1-onto-> ( 1 ... M ) )
175 ffn 5722 . . . . . . . . . . . . . . . . 17  |-  ( F : A --> B  ->  F  Fn  A )
176 dffn4 5792 . . . . . . . . . . . . . . . . 17  |-  ( F  Fn  A  <->  F : A -onto-> ran  F )
177175, 176sylib 196 . . . . . . . . . . . . . . . 16  |-  ( F : A --> B  ->  F : A -onto-> ran  F
)
178 fof 5786 . . . . . . . . . . . . . . . 16  |-  ( F : A -onto-> ran  F  ->  F : A --> ran  F
)
179160, 177, 1783syl 20 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  ->  F : A --> ran  F
)
180 f1of 5807 . . . . . . . . . . . . . . . . 17  |-  ( f : ( 1 ... M ) -1-1-onto-> ran  H  ->  f : ( 1 ... M ) --> ran  H
)
181169, 180syl 16 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  -> 
f : ( 1 ... M ) --> ran 
H )
182113ad2antrr 725 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  ->  ran  H  C_  A )
183 fss 5730 . . . . . . . . . . . . . . . 16  |-  ( ( f : ( 1 ... M ) --> ran 
H  /\  ran  H  C_  A )  ->  f : ( 1 ... M ) --> A )
184181, 182, 183syl2anc 661 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  -> 
f : ( 1 ... M ) --> A )
185 fco 5732 . . . . . . . . . . . . . . 15  |-  ( ( F : A --> ran  F  /\  f : ( 1 ... M ) --> A )  ->  ( F  o.  f ) : ( 1 ... M ) --> ran  F )
186179, 184, 185syl2anc 661 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  -> 
( F  o.  f
) : ( 1 ... M ) --> ran 
F )
187186ffvelrnda 6012 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  /\  x  e.  ( 1 ... M ) )  ->  ( ( F  o.  f ) `  x )  e.  ran  F )
188 f1ococnv2 5833 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( f : ( 1 ... M ) -1-1-onto-> ran  H  ->  (
f  o.  `' f )  =  (  _I  |`  ran  H ) )
189169, 188syl 16 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  -> 
( f  o.  `' f )  =  (  _I  |`  ran  H ) )
190189coeq1d 5155 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  -> 
( ( f  o.  `' f )  o.  H )  =  ( (  _I  |`  ran  H
)  o.  H ) )
191 f1of 5807 . . . . . . . . . . . . . . . . . . . . 21  |-  ( H : ( 1 ... M ) -1-1-onto-> ran  H  ->  H : ( 1 ... M ) --> ran  H
)
192 fcoi2 5751 . . . . . . . . . . . . . . . . . . . . 21  |-  ( H : ( 1 ... M ) --> ran  H  ->  ( (  _I  |`  ran  H
)  o.  H )  =  H )
193172, 191, 1923syl 20 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  -> 
( (  _I  |`  ran  H
)  o.  H )  =  H )
194190, 193eqtr2d 2502 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  ->  H  =  ( (
f  o.  `' f )  o.  H ) )
195 coass 5517 . . . . . . . . . . . . . . . . . . 19  |-  ( ( f  o.  `' f )  o.  H )  =  ( f  o.  ( `' f  o.  H ) )
196194, 195syl6eq 2517 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  ->  H  =  ( f  o.  ( `' f  o.  H ) ) )
197196coeq2d 5156 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  -> 
( F  o.  H
)  =  ( F  o.  ( f  o.  ( `' f  o.  H ) ) ) )
198 coass 5517 . . . . . . . . . . . . . . . . 17  |-  ( ( F  o.  f )  o.  ( `' f  o.  H ) )  =  ( F  o.  ( f  o.  ( `' f  o.  H
) ) )
199197, 198syl6eqr 2519 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  -> 
( F  o.  H
)  =  ( ( F  o.  f )  o.  ( `' f  o.  H ) ) )
200199fveq1d 5859 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  -> 
( ( F  o.  H ) `  k
)  =  ( ( ( F  o.  f
)  o.  ( `' f  o.  H ) ) `  k ) )
201200adantr 465 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  /\  k  e.  ( 1 ... M ) )  ->  ( ( F  o.  H ) `  k )  =  ( ( ( F  o.  f )  o.  ( `' f  o.  H
) ) `  k
) )
202 f1of 5807 . . . . . . . . . . . . . . . . 17  |-  ( `' f : ran  H -1-1-onto-> (
1 ... M )  ->  `' f : ran  H --> ( 1 ... M
) )
203169, 170, 2023syl 20 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  ->  `' f : ran  H --> ( 1 ... M
) )
204172, 191syl 16 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  ->  H : ( 1 ... M ) --> ran  H
)
205 fco 5732 . . . . . . . . . . . . . . . 16  |-  ( ( `' f : ran  H --> ( 1 ... M
)  /\  H :
( 1 ... M
) --> ran  H )  ->  ( `' f  o.  H ) : ( 1 ... M ) --> ( 1 ... M
) )
206203, 204, 205syl2anc 661 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  -> 
( `' f  o.  H ) : ( 1 ... M ) --> ( 1 ... M
) )
207 fvco3 5935 . . . . . . . . . . . . . . 15  |-  ( ( ( `' f  o.  H ) : ( 1 ... M ) --> ( 1 ... M
)  /\  k  e.  ( 1 ... M
) )  ->  (
( ( F  o.  f )  o.  ( `' f  o.  H
) ) `  k
)  =  ( ( F  o.  f ) `
 ( ( `' f  o.  H ) `
 k ) ) )
208206, 207sylan 471 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  /\  k  e.  ( 1 ... M ) )  ->  ( ( ( F  o.  f )  o.  ( `' f  o.  H ) ) `
 k )  =  ( ( F  o.  f ) `  (
( `' f  o.  H ) `  k
) ) )
209201, 208eqtrd 2501 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  /\  k  e.  ( 1 ... M ) )  ->  ( ( F  o.  H ) `  k )  =  ( ( F  o.  f
) `  ( ( `' f  o.  H
) `  k )
) )
210149, 156, 158, 159, 162, 174, 187, 209seqf1o 12104 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  -> 
(  seq 1 (  .+  ,  ( F  o.  H ) ) `  M )  =  (  seq 1 (  .+  ,  ( F  o.  f ) ) `  M ) )
21163, 66, 3mndlid 15747 . . . . . . . . . . . . . 14  |-  ( ( G  e.  Mnd  /\  x  e.  B )  ->  (  .0.  .+  x
)  =  x )
212146, 211sylan 471 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  /\  x  e.  B )  ->  (  .0.  .+  x
)  =  x )
21363, 66, 3mndrid 15748 . . . . . . . . . . . . . 14  |-  ( ( G  e.  Mnd  /\  x  e.  B )  ->  ( x  .+  .0.  )  =  x )
214146, 213sylan 471 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  /\  x  e.  B )  ->  ( x  .+  .0.  )  =  x )
215146, 64syl 16 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  ->  .0.  e.  B )
216 fdm 5726 . . . . . . . . . . . . . . . . 17  |-  ( H : ( 1 ... M ) --> A  ->  dom  H  =  ( 1 ... M ) )
21710, 11, 2163syl 20 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  dom  H  =  ( 1 ... M ) )
218 eluzfz1 11682 . . . . . . . . . . . . . . . . 17  |-  ( M  e.  ( ZZ>= `  1
)  ->  1  e.  ( 1 ... M
) )
219 ne0i 3784 . . . . . . . . . . . . . . . . 17  |-  ( 1  e.  ( 1 ... M )  ->  (
1 ... M )  =/=  (/) )
22072, 218, 2193syl 20 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( 1 ... M
)  =/=  (/) )
221217, 220eqnetrd 2753 . . . . . . . . . . . . . . 15  |-  ( ph  ->  dom  H  =/=  (/) )
222 dm0rn0 5210 . . . . . . . . . . . . . . . 16  |-  ( dom 
H  =  (/)  <->  ran  H  =  (/) )
223222necon3bii 2728 . . . . . . . . . . . . . . 15  |-  ( dom 
H  =/=  (/)  <->  ran  H  =/=  (/) )
224221, 223sylib 196 . . . . . . . . . . . . . 14  |-  ( ph  ->  ran  H  =/=  (/) )
225224ad2antrr 725 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  ->  ran  H  =/=  (/) )
226115adantrr 716 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  ->  ran  H  C_  ( m ... n ) )
227 simprl 755 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  ->  A  =  ( m ... n ) )
228227eleq2d 2530 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  -> 
( x  e.  A  <->  x  e.  ( m ... n ) ) )
229228biimpar 485 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  /\  x  e.  ( m ... n ) )  ->  x  e.  A )
230160ffvelrnda 6012 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  /\  x  e.  A )  ->  ( F `  x
)  e.  B )
231229, 230syldan 470 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  /\  x  e.  ( m ... n ) )  -> 
( F `  x
)  e.  B )
232227difeq1d 3614 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  -> 
( A  \  ran  H )  =  ( ( m ... n ) 
\  ran  H )
)
233232eleq2d 2530 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  -> 
( x  e.  ( A  \  ran  H
)  <->  x  e.  (
( m ... n
)  \  ran  H ) ) )
234233biimpar 485 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  /\  x  e.  ( (
m ... n )  \  ran  H ) )  ->  x  e.  ( A  \  ran  H ) )
235 simpll 753 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  ->  ph )
236235, 53sylan 471 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  /\  x  e.  ( A  \  ran  H ) )  ->  ( F `  x )  =  .0.  )
237234, 236syldan 470 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  /\  x  e.  ( (
m ... n )  \  ran  H ) )  -> 
( F `  x
)  =  .0.  )
238 f1of 5807 . . . . . . . . . . . . . . 15  |-  ( f : ( 1 ... ( # `  ran  H ) ) -1-1-onto-> ran  H  ->  f : ( 1 ... ( # `  ran  H ) ) --> ran  H
)
239163, 164, 2383syl 20 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  -> 
f : ( 1 ... ( # `  ran  H ) ) --> ran  H
)
240 fvco3 5935 . . . . . . . . . . . . . 14  |-  ( ( f : ( 1 ... ( # `  ran  H ) ) --> ran  H  /\  y  e.  (
1 ... ( # `  ran  H ) ) )  -> 
( ( F  o.  f ) `  y
)  =  ( F `
 ( f `  y ) ) )
241239, 240sylan 471 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  /\  y  e.  ( 1 ... ( # `  ran  H ) ) )  -> 
( ( F  o.  f ) `  y
)  =  ( F `
 ( f `  y ) ) )
242212, 214, 149, 215, 163, 225, 226, 231, 237, 241seqcoll2 12466 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  -> 
(  seq m (  .+  ,  F ) `  n
)  =  (  seq 1 (  .+  , 
( F  o.  f
) ) `  ( # `
 ran  H )
) )
243145, 210, 2423eqtr4d 2511 . . . . . . . . . . 11  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  -> 
(  seq 1 (  .+  ,  ( F  o.  H ) ) `  M )  =  (  seq m (  .+  ,  F ) `  n
) )
244243expr 615 . . . . . . . . . 10  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  A  =  ( m ... n ) )  -> 
( f  Isom  <  ,  <  ( ( 1 ... ( # `  ran  H ) ) ,  ran  H )  ->  (  seq 1 (  .+  , 
( F  o.  H
) ) `  M
)  =  (  seq m (  .+  ,  F ) `  n
) ) )
245244exlimdv 1695 . . . . . . . . 9  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  A  =  ( m ... n ) )  -> 
( E. f  f 
Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
)  ->  (  seq 1 (  .+  , 
( F  o.  H
) ) `  M
)  =  (  seq m (  .+  ,  F ) `  n
) ) )
246136, 245mpd 15 . . . . . . . 8  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  A  =  ( m ... n ) )  -> 
(  seq 1 (  .+  ,  ( F  o.  H ) ) `  M )  =  (  seq m (  .+  ,  F ) `  n
) )
247111, 246eqtr4d 2504 . . . . . . 7  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  A  =  ( m ... n ) )  -> 
( G  gsumg  F )  =  (  seq 1 (  .+  ,  ( F  o.  H ) ) `  M ) )
248247ex 434 . . . . . 6  |-  ( (
ph  /\  W  =/=  (/) )  ->  ( A  =  ( m ... n )  ->  ( G  gsumg  F )  =  (  seq 1 (  .+  ,  ( F  o.  H ) ) `  M ) ) )
249248rexlimdvw 2951 . . . . 5  |-  ( (
ph  /\  W  =/=  (/) )  ->  ( E. n  e.  ZZ  A  =  ( m ... n )  ->  ( G  gsumg  F )  =  (  seq 1 (  .+  ,  ( F  o.  H ) ) `  M ) ) )
250249rexlimdvw 2951 . . . 4  |-  ( (
ph  /\  W  =/=  (/) )  ->  ( E. m  e.  ZZ  E. n  e.  ZZ  A  =  ( m ... n )  ->  ( G  gsumg  F )  =  (  seq 1
(  .+  ,  ( F  o.  H )
) `  M )
) )
25183, 250syl5bi 217 . . 3  |-  ( (
ph  /\  W  =/=  (/) )  ->  ( A  e.  ran  ...  ->  ( G 
gsumg  F )  =  (  seq 1 (  .+  ,  ( F  o.  H ) ) `  M ) ) )
252 suppssdm 6904 . . . . . . . . . . 11  |-  ( ( F  o.  H ) supp 
.0.  )  C_  dom  ( F  o.  H
)
25332, 252eqsstri 3527 . . . . . . . . . 10  |-  W  C_  dom  ( F  o.  H
)
254 fdm 5726 . . . . . . . . . . 11  |-  ( ( F  o.  H ) : ( 1 ... M ) --> B  ->  dom  ( F  o.  H
)  =  ( 1 ... M ) )
25530, 254syl 16 . . . . . . . . . 10  |-  ( ph  ->  dom  ( F  o.  H )  =  ( 1 ... M ) )
256253, 255syl5sseq 3545 . . . . . . . . 9  |-  ( ph  ->  W  C_  ( 1 ... M ) )
257 fzssuz 11713 . . . . . . . . . . 11  |-  ( 1 ... M )  C_  ( ZZ>= `  1 )
258257, 71sseqtr4i 3530 . . . . . . . . . 10  |-  ( 1 ... M )  C_  NN
259 nnssre 10529 . . . . . . . . . 10  |-  NN  C_  RR
260258, 259sstri 3506 . . . . . . . . 9  |-  ( 1 ... M )  C_  RR
261256, 260syl6ss 3509 . . . . . . . 8  |-  ( ph  ->  W  C_  RR )
262 soss 4811 . . . . . . . 8  |-  ( W 
C_  RR  ->  (  < 
Or  RR  ->  <  Or  W ) )
263261, 122, 262mpisyl 18 . . . . . . 7  |-  ( ph  ->  <  Or  W )
264 ssfi 7730 . . . . . . . 8  |-  ( ( ( 1 ... M
)  e.  Fin  /\  W  C_  ( 1 ... M ) )  ->  W  e.  Fin )
265125, 256, 264sylancr 663 . . . . . . 7  |-  ( ph  ->  W  e.  Fin )
266 fz1iso 12464 . . . . . . 7  |-  ( (  <  Or  W  /\  W  e.  Fin )  ->  E. f  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) )
267263, 265, 266syl2anc 661 . . . . . 6  |-  ( ph  ->  E. f  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) )
268267ad2antrr 725 . . . . 5  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  -.  A  e.  ran  ... )  ->  E. f  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) )
26963, 3, 66, 153, 1, 2, 7, 150, 70, 10, 51, 32gsumval3lem2 16694 . . . . . . . 8  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  -> 
( G  gsumg  F )  =  (  seq 1 (  .+  ,  ( F  o.  ( H  o.  f
) ) ) `  ( # `  W ) ) )
2701ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  ->  G  e.  Mnd )
271270, 211sylan 471 . . . . . . . . 9  |-  ( ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  /\  x  e.  B )  ->  (  .0.  .+  x
)  =  x )
272270, 213sylan 471 . . . . . . . . 9  |-  ( ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  /\  x  e.  B )  ->  ( x  .+  .0.  )  =  x )
273270, 148sylan 471 . . . . . . . . 9  |-  ( ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  /\  ( x  e.  B  /\  y  e.  B
) )  ->  (
x  .+  y )  e.  B )
274270, 64syl 16 . . . . . . . . 9  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  ->  .0.  e.  B )
275 simprr 756 . . . . . . . . 9  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  -> 
f  Isom  <  ,  <  ( ( 1 ... ( # `
 W ) ) ,  W ) )
276 simplr 754 . . . . . . . . 9  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  ->  W  =/=  (/) )
277256ad2antrr 725 . . . . . . . . 9  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  ->  W  C_  ( 1 ... M ) )
27830ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  -> 
( F  o.  H
) : ( 1 ... M ) --> B )
279278ffvelrnda 6012 . . . . . . . . 9  |-  ( ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  /\  x  e.  ( 1 ... M ) )  ->  ( ( F  o.  H ) `  x )  e.  B
)
28033a1i 11 . . . . . . . . . 10  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  -> 
( ( F  o.  H ) supp  .0.  )  C_  W )
28135a1i 11 . . . . . . . . . 10  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  -> 
( 1 ... M
)  e.  _V )
28238a1i 11 . . . . . . . . . 10  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  ->  .0.  e.  _V )
283278, 280, 281, 282suppssr 6921 . . . . . . . . 9  |-  ( ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  /\  x  e.  ( (
1 ... M )  \  W ) )  -> 
( ( F  o.  H ) `  x
)  =  .0.  )
284 coass 5517 . . . . . . . . . . 11  |-  ( ( F  o.  H )  o.  f )  =  ( F  o.  ( H  o.  f )
)
285284fveq1i 5858 . . . . . . . . . 10  |-  ( ( ( F  o.  H
)  o.  f ) `
 y )  =  ( ( F  o.  ( H  o.  f
) ) `  y
)
286 isof1o 6200 . . . . . . . . . . . 12  |-  ( f 
Isom  <  ,  <  (
( 1 ... ( # `
 W ) ) ,  W )  -> 
f : ( 1 ... ( # `  W
) ) -1-1-onto-> W )
287 f1of 5807 . . . . . . . . . . . 12  |-  ( f : ( 1 ... ( # `  W
) ) -1-1-onto-> W  ->  f :
( 1 ... ( # `
 W ) ) --> W )
288275, 286, 2873syl 20 . . . . . . . . . . 11  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  -> 
f : ( 1 ... ( # `  W
) ) --> W )
289 fvco3 5935 . . . . . . . . . . 11  |-  ( ( f : ( 1 ... ( # `  W
) ) --> W  /\  y  e.  ( 1 ... ( # `  W
) ) )  -> 
( ( ( F  o.  H )  o.  f ) `  y
)  =  ( ( F  o.  H ) `
 ( f `  y ) ) )
290288, 289sylan 471 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  /\  y  e.  ( 1 ... ( # `  W
) ) )  -> 
( ( ( F  o.  H )  o.  f ) `  y
)  =  ( ( F  o.  H ) `
 ( f `  y ) ) )
291285, 290syl5eqr 2515 . . . . . . . . 9  |-  ( ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  /\  y  e.  ( 1 ... ( # `  W
) ) )  -> 
( ( F  o.  ( H  o.  f
) ) `  y
)  =  ( ( F  o.  H ) `
 ( f `  y ) ) )
292271, 272, 273, 274, 275, 276, 277, 279, 283, 291seqcoll2 12466 . . . . . . . 8  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  -> 
(  seq 1 (  .+  ,  ( F  o.  H ) ) `  M )  =  (  seq 1 (  .+  ,  ( F  o.  ( H  o.  f
) ) ) `  ( # `  W ) ) )
293269, 292eqtr4d 2504 . . . . . . 7  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  -> 
( G  gsumg  F )  =  (  seq 1 (  .+  ,  ( F  o.  H ) ) `  M ) )
294293expr 615 . . . . . 6  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  -.  A  e.  ran  ... )  ->  ( f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
)  ->  ( G  gsumg  F )  =  (  seq 1 (  .+  , 
( F  o.  H
) ) `  M
) ) )
295294exlimdv 1695 . . . . 5  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  -.  A  e.  ran  ... )  ->  ( E. f  f 
Isom  <  ,  <  (
( 1 ... ( # `
 W ) ) ,  W )  -> 
( G  gsumg  F )  =  (  seq 1 (  .+  ,  ( F  o.  H ) ) `  M ) ) )
296268, 295mpd 15 . . . 4  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  -.  A  e.  ran  ... )  ->  ( G  gsumg  F )  =  (  seq 1 (  .+  ,  ( F  o.  H ) ) `  M ) )
297296ex 434 . . 3  |-  ( (
ph  /\  W  =/=  (/) )  ->  ( -.  A  e.  ran  ...  ->  ( G  gsumg  F )  =  (  seq 1 (  .+  ,  ( F  o.  H ) ) `  M ) ) )
298251, 297pm2.61d 158 . 2  |-  ( (
ph  /\  W  =/=  (/) )  ->  ( G  gsumg  F )  =  (  seq 1 (  .+  , 
( F  o.  H
) ) `  M
) )
29979, 298pm2.61dane 2778 1  |-  ( ph  ->  ( G  gsumg  F )  =  (  seq 1 (  .+  ,  ( F  o.  H ) ) `  M ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 968    = wceq 1374   E.wex 1591    e. wcel 1762    =/= wne 2655   E.wrex 2808   _Vcvv 3106    \ cdif 3466    C_ wss 3469   (/)c0 3778   ~Pcpw 4003   {csn 4020   class class class wbr 4440    |-> cmpt 4498    _I cid 4783    Or wor 4792    X. cxp 4990   `'ccnv 4991   dom cdm 4992   ran crn 4993    |` cres 4994    o. ccom 4996   Rel wrel 4997    Fn wfn 5574   -->wf 5575   -1-1->wf1 5576   -onto->wfo 5577   -1-1-onto->wf1o 5578   ` cfv 5579    Isom wiso 5580  (class class class)co 6275   supp csupp 6891    ~~ cen 7503   Fincfn 7506   RRcr 9480   1c1 9482    < clt 9617   NNcn 10525   NN0cn0 10784   ZZcz 10853   ZZ>=cuz 11071   ...cfz 11661    seqcseq 12063   #chash 12360   Basecbs 14479   +g cplusg 14544   0gc0g 14684    gsumg cgsu 14685   Mndcmnd 15715  Cntzccntz 16141
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 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-se 4832  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-isom 5588  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-supp 6892  df-recs 7032  df-rdg 7066  df-1o 7120  df-oadd 7124  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-oi 7924  df-card 8309  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-n0 10785  df-z 10854  df-uz 11072  df-fz 11662  df-fzo 11782  df-seq 12064  df-hash 12361  df-0g 14686  df-gsum 14687  df-mnd 15721  df-cntz 16143
This theorem is referenced by:  gsumzres  16698  gsumzcl2  16699  gsumzf1o  16701  gsumzaddlem  16718  gsumconst  16738  gsumzmhm  16741  gsumzoppg  16751  gsumfsum  18245  wilthlem3  23065
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