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Theorem gsumval3 17476
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 16572 . . . . 5  |-  ( ( G  e.  Mnd  /\  A  e.  V )  ->  ( G  gsumg  ( x  e.  A  |->  .0.  ) )  =  .0.  )
51, 2, 4syl2anc 665 . . . 4  |-  ( ph  ->  ( G  gsumg  ( x  e.  A  |->  .0.  ) )  =  .0.  )
65adantr 466 . . 3  |-  ( (
ph  /\  W  =  (/) )  ->  ( G  gsumg  ( x  e.  A  |->  .0.  ) )  =  .0.  )
7 gsumval3.f . . . . . . 7  |-  ( ph  ->  F : A --> B )
87feqmptd 5934 . . . . . 6  |-  ( ph  ->  F  =  ( x  e.  A  |->  ( F `
 x ) ) )
98adantr 466 . . . . 5  |-  ( (
ph  /\  W  =  (/) )  ->  F  =  ( x  e.  A  |->  ( F `  x
) ) )
10 gsumval3.h . . . . . . . . . . . . . 14  |-  ( ph  ->  H : ( 1 ... M ) -1-1-> A
)
11 f1f 5796 . . . . . . . . . . . . . 14  |-  ( H : ( 1 ... M ) -1-1-> A  ->  H : ( 1 ... M ) --> A )
1210, 11syl 17 . . . . . . . . . . . . 13  |-  ( ph  ->  H : ( 1 ... M ) --> A )
1312ad2antrr 730 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  W  =  (/) )  /\  x  e.  ran  H )  ->  H : ( 1 ... M ) --> A )
14 f1f1orn 5842 . . . . . . . . . . . . . . . 16  |-  ( H : ( 1 ... M ) -1-1-> A  ->  H : ( 1 ... M ) -1-1-onto-> ran  H )
1510, 14syl 17 . . . . . . . . . . . . . . 15  |-  ( ph  ->  H : ( 1 ... M ) -1-1-onto-> ran  H
)
1615adantr 466 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  W  =  (/) )  ->  H :
( 1 ... M
)
-1-1-onto-> ran  H )
17 f1ocnv 5843 . . . . . . . . . . . . . 14  |-  ( H : ( 1 ... M ) -1-1-onto-> ran  H  ->  `' H : ran  H -1-1-onto-> ( 1 ... M ) )
18 f1of 5831 . . . . . . . . . . . . . 14  |-  ( `' H : ran  H -1-1-onto-> (
1 ... M )  ->  `' H : ran  H --> ( 1 ... M
) )
1916, 17, 183syl 18 . . . . . . . . . . . . 13  |-  ( (
ph  /\  W  =  (/) )  ->  `' H : ran  H --> ( 1 ... M ) )
2019ffvelrnda 6037 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  W  =  (/) )  /\  x  e.  ran  H )  -> 
( `' H `  x )  e.  ( 1 ... M ) )
21 fvco3 5958 . . . . . . . . . . . 12  |-  ( ( H : ( 1 ... M ) --> A  /\  ( `' H `  x )  e.  ( 1 ... M ) )  ->  ( ( F  o.  H ) `  ( `' H `  x ) )  =  ( F `  ( H `  ( `' H `  x )
) ) )
2213, 20, 21syl2anc 665 . . . . . . . . . . 11  |-  ( ( ( ph  /\  W  =  (/) )  /\  x  e.  ran  H )  -> 
( ( F  o.  H ) `  ( `' H `  x ) )  =  ( F `
 ( H `  ( `' H `  x ) ) ) )
23 simpr 462 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  W  =  (/) )  ->  W  =  (/) )
2423difeq2d 3589 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  W  =  (/) )  ->  ( (
1 ... M )  \  W )  =  ( ( 1 ... M
)  \  (/) ) )
25 dif0 3871 . . . . . . . . . . . . . . 15  |-  ( ( 1 ... M ) 
\  (/) )  =  ( 1 ... M )
2624, 25syl6eq 2486 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  W  =  (/) )  ->  ( (
1 ... M )  \  W )  =  ( 1 ... M ) )
2726adantr 466 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  W  =  (/) )  /\  x  e.  ran  H )  -> 
( ( 1 ... M )  \  W
)  =  ( 1 ... M ) )
2820, 27eleqtrrd 2520 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  W  =  (/) )  /\  x  e.  ran  H )  -> 
( `' H `  x )  e.  ( ( 1 ... M
)  \  W )
)
29 fco 5756 . . . . . . . . . . . . . . 15  |-  ( ( F : A --> B  /\  H : ( 1 ... M ) --> A )  ->  ( F  o.  H ) : ( 1 ... M ) --> B )
307, 12, 29syl2anc 665 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( F  o.  H
) : ( 1 ... M ) --> B )
3130adantr 466 . . . . . . . . . . . . 13  |-  ( (
ph  /\  W  =  (/) )  ->  ( F  o.  H ) : ( 1 ... M ) --> B )
32 gsumval3.w . . . . . . . . . . . . . . 15  |-  W  =  ( ( F  o.  H ) supp  .0.  )
3332eqimss2i 3525 . . . . . . . . . . . . . 14  |-  ( ( F  o.  H ) supp 
.0.  )  C_  W
3433a1i 11 . . . . . . . . . . . . 13  |-  ( (
ph  /\  W  =  (/) )  ->  ( ( F  o.  H ) supp  .0.  )  C_  W )
35 ovex 6333 . . . . . . . . . . . . . 14  |-  ( 1 ... M )  e. 
_V
3635a1i 11 . . . . . . . . . . . . 13  |-  ( (
ph  /\  W  =  (/) )  ->  ( 1 ... M )  e. 
_V )
37 fvex 5891 . . . . . . . . . . . . . . 15  |-  ( 0g
`  G )  e. 
_V
383, 37eqeltri 2513 . . . . . . . . . . . . . 14  |-  .0.  e.  _V
3938a1i 11 . . . . . . . . . . . . 13  |-  ( (
ph  /\  W  =  (/) )  ->  .0.  e.  _V )
4031, 34, 36, 39suppssr 6957 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  W  =  (/) )  /\  ( `' H `  x )  e.  ( ( 1 ... M )  \  W ) )  -> 
( ( F  o.  H ) `  ( `' H `  x ) )  =  .0.  )
4128, 40syldan 472 . . . . . . . . . . 11  |-  ( ( ( ph  /\  W  =  (/) )  /\  x  e.  ran  H )  -> 
( ( F  o.  H ) `  ( `' H `  x ) )  =  .0.  )
42 f1ocnvfv2 6191 . . . . . . . . . . . . 13  |-  ( ( H : ( 1 ... M ) -1-1-onto-> ran  H  /\  x  e.  ran  H )  ->  ( H `  ( `' H `  x ) )  =  x )
4316, 42sylan 473 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  W  =  (/) )  /\  x  e.  ran  H )  -> 
( H `  ( `' H `  x ) )  =  x )
4443fveq2d 5885 . . . . . . . . . . 11  |-  ( ( ( ph  /\  W  =  (/) )  /\  x  e.  ran  H )  -> 
( F `  ( H `  ( `' H `  x )
) )  =  ( F `  x ) )
4522, 41, 443eqtr3rd 2479 . . . . . . . . . 10  |-  ( ( ( ph  /\  W  =  (/) )  /\  x  e.  ran  H )  -> 
( F `  x
)  =  .0.  )
46 fvex 5891 . . . . . . . . . . 11  |-  ( F `
 x )  e. 
_V
4746elsnc 4026 . . . . . . . . . 10  |-  ( ( F `  x )  e.  {  .0.  }  <->  ( F `  x )  =  .0.  )
4845, 47sylibr 215 . . . . . . . . 9  |-  ( ( ( ph  /\  W  =  (/) )  /\  x  e.  ran  H )  -> 
( F `  x
)  e.  {  .0.  } )
4948adantlr 719 . . . . . . . 8  |-  ( ( ( ( ph  /\  W  =  (/) )  /\  x  e.  A )  /\  x  e.  ran  H )  ->  ( F `  x )  e.  {  .0.  } )
50 eldif 3452 . . . . . . . . . . 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 6957 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( A  \  ran  H
) )  ->  ( F `  x )  =  .0.  )
5453, 47sylibr 215 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( A  \  ran  H
) )  ->  ( F `  x )  e.  {  .0.  } )
5550, 54sylan2br 478 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  A  /\  -.  x  e.  ran  H ) )  ->  ( F `  x )  e.  {  .0.  } )
5655adantlr 719 . . . . . . . . 9  |-  ( ( ( ph  /\  W  =  (/) )  /\  (
x  e.  A  /\  -.  x  e.  ran  H ) )  ->  ( F `  x )  e.  {  .0.  } )
5756anassrs 652 . . . . . . . 8  |-  ( ( ( ( ph  /\  W  =  (/) )  /\  x  e.  A )  /\  -.  x  e.  ran  H )  ->  ( F `  x )  e.  {  .0.  } )
5849, 57pm2.61dan 798 . . . . . . 7  |-  ( ( ( ph  /\  W  =  (/) )  /\  x  e.  A )  ->  ( F `  x )  e.  {  .0.  } )
5958, 47sylib 199 . . . . . 6  |-  ( ( ( ph  /\  W  =  (/) )  /\  x  e.  A )  ->  ( F `  x )  =  .0.  )
6059mpteq2dva 4512 . . . . 5  |-  ( (
ph  /\  W  =  (/) )  ->  ( x  e.  A  |->  ( F `
 x ) )  =  ( x  e.  A  |->  .0.  ) )
619, 60eqtrd 2470 . . . 4  |-  ( (
ph  /\  W  =  (/) )  ->  F  =  ( x  e.  A  |->  .0.  ) )
6261oveq2d 6321 . . 3  |-  ( (
ph  /\  W  =  (/) )  ->  ( G  gsumg  F )  =  ( G 
gsumg  ( x  e.  A  |->  .0.  ) ) )
63 gsumval3.b . . . . . . . 8  |-  B  =  ( Base `  G
)
6463, 3mndidcl 16505 . . . . . . 7  |-  ( G  e.  Mnd  ->  .0.  e.  B )
651, 64syl 17 . . . . . 6  |-  ( ph  ->  .0.  e.  B )
66 gsumval3.p . . . . . . 7  |-  .+  =  ( +g  `  G )
6763, 66, 3mndlid 16508 . . . . . 6  |-  ( ( G  e.  Mnd  /\  .0.  e.  B )  -> 
(  .0.  .+  .0.  )  =  .0.  )
681, 65, 67syl2anc 665 . . . . 5  |-  ( ph  ->  (  .0.  .+  .0.  )  =  .0.  )
6968adantr 466 . . . 4  |-  ( (
ph  /\  W  =  (/) )  ->  (  .0.  .+  .0.  )  =  .0.  )
70 gsumval3.m . . . . . 6  |-  ( ph  ->  M  e.  NN )
71 nnuz 11194 . . . . . 6  |-  NN  =  ( ZZ>= `  1 )
7270, 71syl6eleq 2527 . . . . 5  |-  ( ph  ->  M  e.  ( ZZ>= ` 
1 ) )
7372adantr 466 . . . 4  |-  ( (
ph  /\  W  =  (/) )  ->  M  e.  ( ZZ>= `  1 )
)
7426eleq2d 2499 . . . . . 6  |-  ( (
ph  /\  W  =  (/) )  ->  ( x  e.  ( ( 1 ... M )  \  W
)  <->  x  e.  (
1 ... M ) ) )
7574biimpar 487 . . . . 5  |-  ( ( ( ph  /\  W  =  (/) )  /\  x  e.  ( 1 ... M
) )  ->  x  e.  ( ( 1 ... M )  \  W
) )
7631, 34, 36, 39suppssr 6957 . . . . 5  |-  ( ( ( ph  /\  W  =  (/) )  /\  x  e.  ( ( 1 ... M )  \  W
) )  ->  (
( F  o.  H
) `  x )  =  .0.  )
7775, 76syldan 472 . . . 4  |-  ( ( ( ph  /\  W  =  (/) )  /\  x  e.  ( 1 ... M
) )  ->  (
( F  o.  H
) `  x )  =  .0.  )
7869, 73, 77seqid3 12254 . . 3  |-  ( (
ph  /\  W  =  (/) )  ->  (  seq 1 (  .+  , 
( F  o.  H
) ) `  M
)  =  .0.  )
796, 62, 783eqtr4d 2480 . 2  |-  ( (
ph  /\  W  =  (/) )  ->  ( G  gsumg  F )  =  (  seq 1 (  .+  , 
( F  o.  H
) ) `  M
) )
80 fzf 11786 . . . . 5  |-  ... :
( ZZ  X.  ZZ )
--> ~P ZZ
81 ffn 5746 . . . . 5  |-  ( ...
: ( ZZ  X.  ZZ ) --> ~P ZZ  ->  ... 
Fn  ( ZZ  X.  ZZ ) )
82 ovelrn 6459 . . . . 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 730 . . . . . . . . 9  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  A  =  ( m ... n ) )  ->  G  e.  Mnd )
85 simpr 462 . . . . . . . . . . 11  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  A  =  ( m ... n ) )  ->  A  =  ( m ... n ) )
86 frel 5749 . . . . . . . . . . . . . . . . 17  |-  ( F : A --> B  ->  Rel  F )
87 reldm0 5072 . . . . . . . . . . . . . . . . 17  |-  ( Rel 
F  ->  ( F  =  (/)  <->  dom  F  =  (/) ) )
887, 86, 873syl 18 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( F  =  (/)  <->  dom  F  =  (/) ) )
89 fdm 5750 . . . . . . . . . . . . . . . . . 18  |-  ( F : A --> B  ->  dom  F  =  A )
907, 89syl 17 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  dom  F  =  A )
9190eqeq1d 2431 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( dom  F  =  (/) 
<->  A  =  (/) ) )
9288, 91bitrd 256 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( F  =  (/)  <->  A  =  (/) ) )
93 coeq1 5012 . . . . . . . . . . . . . . . . . . 19  |-  ( F  =  (/)  ->  ( F  o.  H )  =  ( (/)  o.  H
) )
94 co01 5370 . . . . . . . . . . . . . . . . . . 19  |-  ( (/)  o.  H )  =  (/)
9593, 94syl6eq 2486 . . . . . . . . . . . . . . . . . 18  |-  ( F  =  (/)  ->  ( F  o.  H )  =  (/) )
9695oveq1d 6320 . . . . . . . . . . . . . . . . 17  |-  ( F  =  (/)  ->  ( ( F  o.  H ) supp 
.0.  )  =  (
(/) supp  .0.  ) )
97 supp0 6930 . . . . . . . . . . . . . . . . . 18  |-  (  .0. 
e.  _V  ->  ( (/) supp  .0.  )  =  (/) )
9838, 97ax-mp 5 . . . . . . . . . . . . . . . . 17  |-  ( (/) supp  .0.  )  =  (/)
9996, 98syl6eq 2486 . . . . . . . . . . . . . . . 16  |-  ( F  =  (/)  ->  ( ( F  o.  H ) supp 
.0.  )  =  (/) )
10032, 99syl5eq 2482 . . . . . . . . . . . . . . 15  |-  ( F  =  (/)  ->  W  =  (/) )
10192, 100syl6bir 232 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( A  =  (/)  ->  W  =  (/) ) )
102101necon3d 2655 . . . . . . . . . . . . 13  |-  ( ph  ->  ( W  =/=  (/)  ->  A  =/=  (/) ) )
103102imp 430 . . . . . . . . . . . 12  |-  ( (
ph  /\  W  =/=  (/) )  ->  A  =/=  (/) )
104103adantr 466 . . . . . . . . . . 11  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  A  =  ( m ... n ) )  ->  A  =/=  (/) )
10585, 104eqnetrrd 2725 . . . . . . . . . 10  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  A  =  ( m ... n ) )  -> 
( m ... n
)  =/=  (/) )
106 fzn0 11811 . . . . . . . . . 10  |-  ( ( m ... n )  =/=  (/)  <->  n  e.  ( ZZ>=
`  m ) )
107105, 106sylib 199 . . . . . . . . 9  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  A  =  ( m ... n ) )  ->  n  e.  ( ZZ>= `  m ) )
1087ad2antrr 730 . . . . . . . . . 10  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  A  =  ( m ... n ) )  ->  F : A --> B )
10985feq2d 5733 . . . . . . . . . 10  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  A  =  ( m ... n ) )  -> 
( F : A --> B 
<->  F : ( m ... n ) --> B ) )
110108, 109mpbid 213 . . . . . . . . 9  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  A  =  ( m ... n ) )  ->  F : ( m ... n ) --> B )
11163, 66, 84, 107, 110gsumval2 16474 . . . . . . . 8  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  A  =  ( m ... n ) )  -> 
( G  gsumg  F )  =  (  seq m (  .+  ,  F ) `  n
) )
112 frn 5752 . . . . . . . . . . . . . . 15  |-  ( H : ( 1 ... M ) --> A  ->  ran  H  C_  A )
11310, 11, 1123syl 18 . . . . . . . . . . . . . 14  |-  ( ph  ->  ran  H  C_  A
)
114113ad2antrr 730 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  A  =  ( m ... n ) )  ->  ran  H  C_  A )
115114, 85sseqtrd 3506 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  A  =  ( m ... n ) )  ->  ran  H  C_  ( m ... n ) )
116 fzssuz 11837 . . . . . . . . . . . . 13  |-  ( m ... n )  C_  ( ZZ>= `  m )
117 uzssz 11178 . . . . . . . . . . . . . 14  |-  ( ZZ>= `  m )  C_  ZZ
118 zssre 10944 . . . . . . . . . . . . . 14  |-  ZZ  C_  RR
119117, 118sstri 3479 . . . . . . . . . . . . 13  |-  ( ZZ>= `  m )  C_  RR
120116, 119sstri 3479 . . . . . . . . . . . 12  |-  ( m ... n )  C_  RR
121115, 120syl6ss 3482 . . . . . . . . . . 11  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  A  =  ( m ... n ) )  ->  ran  H  C_  RR )
122 ltso 9713 . . . . . . . . . . 11  |-  <  Or  RR
123 soss 4793 . . . . . . . . . . 11  |-  ( ran 
H  C_  RR  ->  (  <  Or  RR  ->  < 
Or  ran  H )
)
124121, 122, 123mpisyl 22 . . . . . . . . . 10  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  A  =  ( m ... n ) )  ->  <  Or  ran  H )
125 fzfi 12182 . . . . . . . . . . . 12  |-  ( 1 ... M )  e. 
Fin
126125a1i 11 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( 1 ... M
)  e.  Fin )
127 fex2 6762 . . . . . . . . . . . . . . 15  |-  ( ( H : ( 1 ... M ) --> A  /\  ( 1 ... M )  e.  Fin  /\  A  e.  V )  ->  H  e.  _V )
12812, 126, 2, 127syl3anc 1264 . . . . . . . . . . . . . 14  |-  ( ph  ->  H  e.  _V )
129 f1oen3g 7592 . . . . . . . . . . . . . 14  |-  ( ( H  e.  _V  /\  H : ( 1 ... M ) -1-1-onto-> ran  H )  -> 
( 1 ... M
)  ~~  ran  H )
130128, 15, 129syl2anc 665 . . . . . . . . . . . . 13  |-  ( ph  ->  ( 1 ... M
)  ~~  ran  H )
131 enfi 7794 . . . . . . . . . . . . 13  |-  ( ( 1 ... M ) 
~~  ran  H  ->  ( ( 1 ... M
)  e.  Fin  <->  ran  H  e. 
Fin ) )
132130, 131syl 17 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( 1 ... M )  e.  Fin  <->  ran  H  e.  Fin ) )
133125, 132mpbii 214 . . . . . . . . . . 11  |-  ( ph  ->  ran  H  e.  Fin )
134133ad2antrr 730 . . . . . . . . . 10  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  A  =  ( m ... n ) )  ->  ran  H  e.  Fin )
135 fz1iso 12620 . . . . . . . . . 10  |-  ( (  <  Or  ran  H  /\  ran  H  e.  Fin )  ->  E. f  f  Isom  <  ,  <  ( ( 1 ... ( # `  ran  H ) ) ,  ran  H ) )
136124, 134, 135syl2anc 665 . . . . . . . . 9  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  A  =  ( m ... n ) )  ->  E. f  f  Isom  <  ,  <  ( ( 1 ... ( # `  ran  H ) ) ,  ran  H ) )
13770nnnn0d 10925 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  M  e.  NN0 )
138 hashfz1 12526 . . . . . . . . . . . . . . . 16  |-  ( M  e.  NN0  ->  ( # `  ( 1 ... M
) )  =  M )
139137, 138syl 17 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( # `  (
1 ... M ) )  =  M )
140 hashen 12527 . . . . . . . . . . . . . . . . 17  |-  ( ( ( 1 ... M
)  e.  Fin  /\  ran  H  e.  Fin )  ->  ( ( # `  (
1 ... M ) )  =  ( # `  ran  H )  <->  ( 1 ... M )  ~~  ran  H ) )
141125, 133, 140sylancr 667 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( ( # `  (
1 ... M ) )  =  ( # `  ran  H )  <->  ( 1 ... M )  ~~  ran  H ) )
142130, 141mpbird 235 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( # `  (
1 ... M ) )  =  ( # `  ran  H ) )
143139, 142eqtr3d 2472 . . . . . . . . . . . . . 14  |-  ( ph  ->  M  =  ( # `  ran  H ) )
144143ad2antrr 730 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  ->  M  =  ( # `  ran  H ) )
145144fveq2d 5885 . . . . . . . . . . . 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 730 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  ->  G  e.  Mnd )
14763, 66mndcl 16496 . . . . . . . . . . . . . . 15  |-  ( ( G  e.  Mnd  /\  x  e.  B  /\  y  e.  B )  ->  ( x  .+  y
)  e.  B )
1481473expb 1206 . . . . . . . . . . . . . 14  |-  ( ( G  e.  Mnd  /\  ( x  e.  B  /\  y  e.  B
) )  ->  (
x  .+  y )  e.  B )
149146, 148sylan 473 . . . . . . . . . . . . 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 730 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  ->  ran  F  C_  ( Z `  ran  F ) )
152151sselda 3470 . . . . . . . . . . . . . . 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 16934 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  ( Z `
 ran  F )  /\  y  e.  ran  F )  ->  ( x  .+  y )  =  ( y  .+  x ) )
155152, 154sylan 473 . . . . . . . . . . . . . 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 651 . . . . . . . . . . . . 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 16497 . . . . . . . . . . . . . 14  |-  ( ( G  e.  Mnd  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B
) )  ->  (
( x  .+  y
)  .+  z )  =  ( x  .+  ( y  .+  z
) ) )
158146, 157sylan 473 . . . . . . . . . . . . 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 730 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  ->  M  e.  ( ZZ>= ` 
1 ) )
1607ad2antrr 730 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  ->  F : A --> B )
161 frn 5752 . . . . . . . . . . . . . 14  |-  ( F : A --> B  ->  ran  F  C_  B )
162160, 161syl 17 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  ->  ran  F  C_  B )
163 simprr 764 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  -> 
f  Isom  <  ,  <  ( ( 1 ... ( # `
 ran  H )
) ,  ran  H
) )
164 isof1o 6231 . . . . . . . . . . . . . . . . 17  |-  ( f 
Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
)  ->  f :
( 1 ... ( # `
 ran  H )
)
-1-1-onto-> ran  H )
165163, 164syl 17 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  -> 
f : ( 1 ... ( # `  ran  H ) ) -1-1-onto-> ran  H )
166144oveq2d 6321 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  -> 
( 1 ... M
)  =  ( 1 ... ( # `  ran  H ) ) )
167 f1oeq2 5823 . . . . . . . . . . . . . . . . 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 17 . . . . . . . . . . . . . . . 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 235 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  -> 
f : ( 1 ... M ) -1-1-onto-> ran  H
)
170 f1ocnv 5843 . . . . . . . . . . . . . . 15  |-  ( f : ( 1 ... M ) -1-1-onto-> ran  H  ->  `' f : ran  H -1-1-onto-> ( 1 ... M ) )
171169, 170syl 17 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  ->  `' f : ran  H -1-1-onto-> ( 1 ... M ) )
17215ad2antrr 730 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  ->  H : ( 1 ... M ) -1-1-onto-> ran  H )
173 f1oco 5853 . . . . . . . . . . . . . 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 665 . . . . . . . . . . . . 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 5746 . . . . . . . . . . . . . . . . 17  |-  ( F : A --> B  ->  F  Fn  A )
176 dffn4 5816 . . . . . . . . . . . . . . . . 17  |-  ( F  Fn  A  <->  F : A -onto-> ran  F )
177175, 176sylib 199 . . . . . . . . . . . . . . . 16  |-  ( F : A --> B  ->  F : A -onto-> ran  F
)
178 fof 5810 . . . . . . . . . . . . . . . 16  |-  ( F : A -onto-> ran  F  ->  F : A --> ran  F
)
179160, 177, 1783syl 18 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  ->  F : A --> ran  F
)
180 f1of 5831 . . . . . . . . . . . . . . . . 17  |-  ( f : ( 1 ... M ) -1-1-onto-> ran  H  ->  f : ( 1 ... M ) --> ran  H
)
181169, 180syl 17 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  -> 
f : ( 1 ... M ) --> ran 
H )
182113ad2antrr 730 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  ->  ran  H  C_  A )
183181, 182fssd 5755 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  -> 
f : ( 1 ... M ) --> A )
184 fco 5756 . . . . . . . . . . . . . . 15  |-  ( ( F : A --> ran  F  /\  f : ( 1 ... M ) --> A )  ->  ( F  o.  f ) : ( 1 ... M ) --> ran  F )
185179, 183, 184syl2anc 665 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  -> 
( F  o.  f
) : ( 1 ... M ) --> ran 
F )
186185ffvelrnda 6037 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  /\  x  e.  ( 1 ... M ) )  ->  ( ( F  o.  f ) `  x )  e.  ran  F )
187 f1ococnv2 5857 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( f : ( 1 ... M ) -1-1-onto-> ran  H  ->  (
f  o.  `' f )  =  (  _I  |`  ran  H ) )
188169, 187syl 17 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  -> 
( f  o.  `' f )  =  (  _I  |`  ran  H ) )
189188coeq1d 5016 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  -> 
( ( f  o.  `' f )  o.  H )  =  ( (  _I  |`  ran  H
)  o.  H ) )
190 f1of 5831 . . . . . . . . . . . . . . . . . . . . 21  |-  ( H : ( 1 ... M ) -1-1-onto-> ran  H  ->  H : ( 1 ... M ) --> ran  H
)
191 fcoi2 5775 . . . . . . . . . . . . . . . . . . . . 21  |-  ( H : ( 1 ... M ) --> ran  H  ->  ( (  _I  |`  ran  H
)  o.  H )  =  H )
192172, 190, 1913syl 18 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  -> 
( (  _I  |`  ran  H
)  o.  H )  =  H )
193189, 192eqtr2d 2471 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  ->  H  =  ( (
f  o.  `' f )  o.  H ) )
194 coass 5374 . . . . . . . . . . . . . . . . . . 19  |-  ( ( f  o.  `' f )  o.  H )  =  ( f  o.  ( `' f  o.  H ) )
195193, 194syl6eq 2486 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  ->  H  =  ( f  o.  ( `' f  o.  H ) ) )
196195coeq2d 5017 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  -> 
( F  o.  H
)  =  ( F  o.  ( f  o.  ( `' f  o.  H ) ) ) )
197 coass 5374 . . . . . . . . . . . . . . . . 17  |-  ( ( F  o.  f )  o.  ( `' f  o.  H ) )  =  ( F  o.  ( f  o.  ( `' f  o.  H
) ) )
198196, 197syl6eqr 2488 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  -> 
( F  o.  H
)  =  ( ( F  o.  f )  o.  ( `' f  o.  H ) ) )
199198fveq1d 5883 . . . . . . . . . . . . . . 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 ) )
200199adantr 466 . . . . . . . . . . . . . 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
) )
201 f1of 5831 . . . . . . . . . . . . . . . . 17  |-  ( `' f : ran  H -1-1-onto-> (
1 ... M )  ->  `' f : ran  H --> ( 1 ... M
) )
202169, 170, 2013syl 18 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  ->  `' f : ran  H --> ( 1 ... M
) )
203172, 190syl 17 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  ->  H : ( 1 ... M ) --> ran  H
)
204 fco 5756 . . . . . . . . . . . . . . . 16  |-  ( ( `' f : ran  H --> ( 1 ... M
)  /\  H :
( 1 ... M
) --> ran  H )  ->  ( `' f  o.  H ) : ( 1 ... M ) --> ( 1 ... M
) )
205202, 203, 204syl2anc 665 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  -> 
( `' f  o.  H ) : ( 1 ... M ) --> ( 1 ... M
) )
206 fvco3 5958 . . . . . . . . . . . . . . 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 ) ) )
207205, 206sylan 473 . . . . . . . . . . . . . 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
) ) )
208200, 207eqtrd 2470 . . . . . . . . . . . . 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 )
) )
209149, 156, 158, 159, 162, 174, 186, 208seqf1o 12251 . . . . . . . . . . . 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 ) )
21063, 66, 3mndlid 16508 . . . . . . . . . . . . . 14  |-  ( ( G  e.  Mnd  /\  x  e.  B )  ->  (  .0.  .+  x
)  =  x )
211146, 210sylan 473 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  /\  x  e.  B )  ->  (  .0.  .+  x
)  =  x )
21263, 66, 3mndrid 16509 . . . . . . . . . . . . . 14  |-  ( ( G  e.  Mnd  /\  x  e.  B )  ->  ( x  .+  .0.  )  =  x )
213146, 212sylan 473 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  /\  x  e.  B )  ->  ( x  .+  .0.  )  =  x )
214146, 64syl 17 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  ->  .0.  e.  B )
215 fdm 5750 . . . . . . . . . . . . . . . . 17  |-  ( H : ( 1 ... M ) --> A  ->  dom  H  =  ( 1 ... M ) )
21610, 11, 2153syl 18 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  dom  H  =  ( 1 ... M ) )
217 eluzfz1 11804 . . . . . . . . . . . . . . . . 17  |-  ( M  e.  ( ZZ>= `  1
)  ->  1  e.  ( 1 ... M
) )
218 ne0i 3773 . . . . . . . . . . . . . . . . 17  |-  ( 1  e.  ( 1 ... M )  ->  (
1 ... M )  =/=  (/) )
21972, 217, 2183syl 18 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( 1 ... M
)  =/=  (/) )
220216, 219eqnetrd 2724 . . . . . . . . . . . . . . 15  |-  ( ph  ->  dom  H  =/=  (/) )
221 dm0rn0 5071 . . . . . . . . . . . . . . . 16  |-  ( dom 
H  =  (/)  <->  ran  H  =  (/) )
222221necon3bii 2699 . . . . . . . . . . . . . . 15  |-  ( dom 
H  =/=  (/)  <->  ran  H  =/=  (/) )
223220, 222sylib 199 . . . . . . . . . . . . . 14  |-  ( ph  ->  ran  H  =/=  (/) )
224223ad2antrr 730 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  ->  ran  H  =/=  (/) )
225115adantrr 721 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  ->  ran  H  C_  ( m ... n ) )
226 simprl 762 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  ->  A  =  ( m ... n ) )
227226eleq2d 2499 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  -> 
( x  e.  A  <->  x  e.  ( m ... n ) ) )
228227biimpar 487 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  /\  x  e.  ( m ... n ) )  ->  x  e.  A )
229160ffvelrnda 6037 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  /\  x  e.  A )  ->  ( F `  x
)  e.  B )
230228, 229syldan 472 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  /\  x  e.  ( m ... n ) )  -> 
( F `  x
)  e.  B )
231226difeq1d 3588 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  -> 
( A  \  ran  H )  =  ( ( m ... n ) 
\  ran  H )
)
232231eleq2d 2499 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  -> 
( x  e.  ( A  \  ran  H
)  <->  x  e.  (
( m ... n
)  \  ran  H ) ) )
233232biimpar 487 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  /\  x  e.  ( (
m ... n )  \  ran  H ) )  ->  x  e.  ( A  \  ran  H ) )
234 simpll 758 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  ->  ph )
235234, 53sylan 473 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  /\  x  e.  ( A  \  ran  H ) )  ->  ( F `  x )  =  .0.  )
236233, 235syldan 472 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  /\  x  e.  ( (
m ... n )  \  ran  H ) )  -> 
( F `  x
)  =  .0.  )
237 f1of 5831 . . . . . . . . . . . . . . 15  |-  ( f : ( 1 ... ( # `  ran  H ) ) -1-1-onto-> ran  H  ->  f : ( 1 ... ( # `  ran  H ) ) --> ran  H
)
238163, 164, 2373syl 18 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  -> 
f : ( 1 ... ( # `  ran  H ) ) --> ran  H
)
239 fvco3 5958 . . . . . . . . . . . . . 14  |-  ( ( f : ( 1 ... ( # `  ran  H ) ) --> ran  H  /\  y  e.  (
1 ... ( # `  ran  H ) ) )  -> 
( ( F  o.  f ) `  y
)  =  ( F `
 ( f `  y ) ) )
240238, 239sylan 473 . . . . . . . . . . . . 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 ) ) )
241211, 213, 149, 214, 163, 224, 225, 230, 236, 240seqcoll2 12622 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  -> 
(  seq m (  .+  ,  F ) `  n
)  =  (  seq 1 (  .+  , 
( F  o.  f
) ) `  ( # `
 ran  H )
) )
242145, 209, 2413eqtr4d 2480 . . . . . . . . . . 11  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  -> 
(  seq 1 (  .+  ,  ( F  o.  H ) ) `  M )  =  (  seq m (  .+  ,  F ) `  n
) )
243242expr 618 . . . . . . . . . 10  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  A  =  ( m ... n ) )  -> 
( f  Isom  <  ,  <  ( ( 1 ... ( # `  ran  H ) ) ,  ran  H )  ->  (  seq 1 (  .+  , 
( F  o.  H
) ) `  M
)  =  (  seq m (  .+  ,  F ) `  n
) ) )
244243exlimdv 1771 . . . . . . . . 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
) ) )
245136, 244mpd 15 . . . . . . . 8  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  A  =  ( m ... n ) )  -> 
(  seq 1 (  .+  ,  ( F  o.  H ) ) `  M )  =  (  seq m (  .+  ,  F ) `  n
) )
246111, 245eqtr4d 2473 . . . . . . 7  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  A  =  ( m ... n ) )  -> 
( G  gsumg  F )  =  (  seq 1 (  .+  ,  ( F  o.  H ) ) `  M ) )
247246ex 435 . . . . . 6  |-  ( (
ph  /\  W  =/=  (/) )  ->  ( A  =  ( m ... n )  ->  ( G  gsumg  F )  =  (  seq 1 (  .+  ,  ( F  o.  H ) ) `  M ) ) )
248247rexlimdvw 2927 . . . . 5  |-  ( (
ph  /\  W  =/=  (/) )  ->  ( E. n  e.  ZZ  A  =  ( m ... n )  ->  ( G  gsumg  F )  =  (  seq 1 (  .+  ,  ( F  o.  H ) ) `  M ) ) )
249248rexlimdvw 2927 . . . 4  |-  ( (
ph  /\  W  =/=  (/) )  ->  ( E. m  e.  ZZ  E. n  e.  ZZ  A  =  ( m ... n )  ->  ( G  gsumg  F )  =  (  seq 1
(  .+  ,  ( F  o.  H )
) `  M )
) )
25083, 249syl5bi 220 . . 3  |-  ( (
ph  /\  W  =/=  (/) )  ->  ( A  e.  ran  ...  ->  ( G 
gsumg  F )  =  (  seq 1 (  .+  ,  ( F  o.  H ) ) `  M ) ) )
251 suppssdm 6938 . . . . . . . . . . 11  |-  ( ( F  o.  H ) supp 
.0.  )  C_  dom  ( F  o.  H
)
25232, 251eqsstri 3500 . . . . . . . . . 10  |-  W  C_  dom  ( F  o.  H
)
253 fdm 5750 . . . . . . . . . . 11  |-  ( ( F  o.  H ) : ( 1 ... M ) --> B  ->  dom  ( F  o.  H
)  =  ( 1 ... M ) )
25430, 253syl 17 . . . . . . . . . 10  |-  ( ph  ->  dom  ( F  o.  H )  =  ( 1 ... M ) )
255252, 254syl5sseq 3518 . . . . . . . . 9  |-  ( ph  ->  W  C_  ( 1 ... M ) )
256 fzssuz 11837 . . . . . . . . . . 11  |-  ( 1 ... M )  C_  ( ZZ>= `  1 )
257256, 71sseqtr4i 3503 . . . . . . . . . 10  |-  ( 1 ... M )  C_  NN
258 nnssre 10613 . . . . . . . . . 10  |-  NN  C_  RR
259257, 258sstri 3479 . . . . . . . . 9  |-  ( 1 ... M )  C_  RR
260255, 259syl6ss 3482 . . . . . . . 8  |-  ( ph  ->  W  C_  RR )
261 soss 4793 . . . . . . . 8  |-  ( W 
C_  RR  ->  (  < 
Or  RR  ->  <  Or  W ) )
262260, 122, 261mpisyl 22 . . . . . . 7  |-  ( ph  ->  <  Or  W )
263 ssfi 7798 . . . . . . . 8  |-  ( ( ( 1 ... M
)  e.  Fin  /\  W  C_  ( 1 ... M ) )  ->  W  e.  Fin )
264125, 255, 263sylancr 667 . . . . . . 7  |-  ( ph  ->  W  e.  Fin )
265 fz1iso 12620 . . . . . . 7  |-  ( (  <  Or  W  /\  W  e.  Fin )  ->  E. f  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) )
266262, 264, 265syl2anc 665 . . . . . 6  |-  ( ph  ->  E. f  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) )
267266ad2antrr 730 . . . . 5  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  -.  A  e.  ran  ... )  ->  E. f  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) )
26863, 3, 66, 153, 1, 2, 7, 150, 70, 10, 51, 32gsumval3lem2 17475 . . . . . . . 8  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  -> 
( G  gsumg  F )  =  (  seq 1 (  .+  ,  ( F  o.  ( H  o.  f
) ) ) `  ( # `  W ) ) )
2691ad2antrr 730 . . . . . . . . . 10  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  ->  G  e.  Mnd )
270269, 210sylan 473 . . . . . . . . 9  |-  ( ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  /\  x  e.  B )  ->  (  .0.  .+  x
)  =  x )
271269, 212sylan 473 . . . . . . . . 9  |-  ( ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  /\  x  e.  B )  ->  ( x  .+  .0.  )  =  x )
272269, 148sylan 473 . . . . . . . . 9  |-  ( ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  /\  ( x  e.  B  /\  y  e.  B
) )  ->  (
x  .+  y )  e.  B )
273269, 64syl 17 . . . . . . . . 9  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  ->  .0.  e.  B )
274 simprr 764 . . . . . . . . 9  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  -> 
f  Isom  <  ,  <  ( ( 1 ... ( # `
 W ) ) ,  W ) )
275 simplr 760 . . . . . . . . 9  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  ->  W  =/=  (/) )
276255ad2antrr 730 . . . . . . . . 9  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  ->  W  C_  ( 1 ... M ) )
27730ad2antrr 730 . . . . . . . . . 10  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  -> 
( F  o.  H
) : ( 1 ... M ) --> B )
278277ffvelrnda 6037 . . . . . . . . 9  |-  ( ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  /\  x  e.  ( 1 ... M ) )  ->  ( ( F  o.  H ) `  x )  e.  B
)
27933a1i 11 . . . . . . . . . 10  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  -> 
( ( F  o.  H ) supp  .0.  )  C_  W )
28035a1i 11 . . . . . . . . . 10  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  -> 
( 1 ... M
)  e.  _V )
28138a1i 11 . . . . . . . . . 10  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  ->  .0.  e.  _V )
282277, 279, 280, 281suppssr 6957 . . . . . . . . 9  |-  ( ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  /\  x  e.  ( (
1 ... M )  \  W ) )  -> 
( ( F  o.  H ) `  x
)  =  .0.  )
283 coass 5374 . . . . . . . . . . 11  |-  ( ( F  o.  H )  o.  f )  =  ( F  o.  ( H  o.  f )
)
284283fveq1i 5882 . . . . . . . . . 10  |-  ( ( ( F  o.  H
)  o.  f ) `
 y )  =  ( ( F  o.  ( H  o.  f
) ) `  y
)
285 isof1o 6231 . . . . . . . . . . . 12  |-  ( f 
Isom  <  ,  <  (
( 1 ... ( # `
 W ) ) ,  W )  -> 
f : ( 1 ... ( # `  W
) ) -1-1-onto-> W )
286 f1of 5831 . . . . . . . . . . . 12  |-  ( f : ( 1 ... ( # `  W
) ) -1-1-onto-> W  ->  f :
( 1 ... ( # `
 W ) ) --> W )
287274, 285, 2863syl 18 . . . . . . . . . . 11  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  -> 
f : ( 1 ... ( # `  W
) ) --> W )
288 fvco3 5958 . . . . . . . . . . 11  |-  ( ( f : ( 1 ... ( # `  W
) ) --> W  /\  y  e.  ( 1 ... ( # `  W
) ) )  -> 
( ( ( F  o.  H )  o.  f ) `  y
)  =  ( ( F  o.  H ) `
 ( f `  y ) ) )
289287, 288sylan 473 . . . . . . . . . 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 ) ) )
290284, 289syl5eqr 2484 . . . . . . . . 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 ) ) )
291270, 271, 272, 273, 274, 275, 276, 278, 282, 290seqcoll2 12622 . . . . . . . 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 ) ) )
292268, 291eqtr4d 2473 . . . . . . 7  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  -> 
( G  gsumg  F )  =  (  seq 1 (  .+  ,  ( F  o.  H ) ) `  M ) )
293292expr 618 . . . . . 6  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  -.  A  e.  ran  ... )  ->  ( f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
)  ->  ( G  gsumg  F )  =  (  seq 1 (  .+  , 
( F  o.  H
) ) `  M
) ) )
294293exlimdv 1771 . . . . 5  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  -.  A  e.  ran  ... )  ->  ( E. f  f 
Isom  <  ,  <  (
( 1 ... ( # `
 W ) ) ,  W )  -> 
( G  gsumg  F )  =  (  seq 1 (  .+  ,  ( F  o.  H ) ) `  M ) ) )
295267, 294mpd 15 . . . 4  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  -.  A  e.  ran  ... )  ->  ( G  gsumg  F )  =  (  seq 1 (  .+  ,  ( F  o.  H ) ) `  M ) )
296295ex 435 . . 3  |-  ( (
ph  /\  W  =/=  (/) )  ->  ( -.  A  e.  ran  ...  ->  ( G  gsumg  F )  =  (  seq 1 (  .+  ,  ( F  o.  H ) ) `  M ) ) )
297250, 296pm2.61d 161 . 2  |-  ( (
ph  /\  W  =/=  (/) )  ->  ( G  gsumg  F )  =  (  seq 1 (  .+  , 
( F  o.  H
) ) `  M
) )
29879, 297pm2.61dane 2749 1  |-  ( ph  ->  ( G  gsumg  F )  =  (  seq 1 (  .+  ,  ( F  o.  H ) ) `  M ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437   E.wex 1659    e. wcel 1870    =/= wne 2625   E.wrex 2783   _Vcvv 3087    \ cdif 3439    C_ wss 3442   (/)c0 3767   ~Pcpw 3985   {csn 4002   class class class wbr 4426    |-> cmpt 4484    _I cid 4764    Or wor 4774    X. cxp 4852   `'ccnv 4853   dom cdm 4854   ran crn 4855    |` cres 4856    o. ccom 4858   Rel wrel 4859    Fn wfn 5596   -->wf 5597   -1-1->wf1 5598   -onto->wfo 5599   -1-1-onto->wf1o 5600   ` cfv 5601    Isom wiso 5602  (class class class)co 6305   supp csupp 6925    ~~ cen 7574   Fincfn 7577   RRcr 9537   1c1 9539    < clt 9674   NNcn 10609   NN0cn0 10869   ZZcz 10937   ZZ>=cuz 11159   ...cfz 11782    seqcseq 12210   #chash 12512   Basecbs 15084   +g cplusg 15152   0gc0g 15297    gsumg cgsu 15298   Mndcmnd 16486  Cntzccntz 16920
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-se 4814  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-supp 6926  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-oadd 7194  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-oi 8025  df-card 8372  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11783  df-fzo 11914  df-seq 12211  df-hash 12513  df-0g 15299  df-gsum 15300  df-mgm 16439  df-sgrp 16478  df-mnd 16488  df-cntz 16922
This theorem is referenced by:  gsumzres  17478  gsumzcl2  17479  gsumzf1o  17481  gsumzaddlem  17489  gsumconst  17502  gsumzmhm  17505  gsumzoppg  17512  gsumfsum  18969  wilthlem3  23860
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