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Theorem gsumval3 17619
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 16699 . . . . 5  |-  ( ( G  e.  Mnd  /\  A  e.  V )  ->  ( G  gsumg  ( x  e.  A  |->  .0.  ) )  =  .0.  )
51, 2, 4syl2anc 673 . . . 4  |-  ( ph  ->  ( G  gsumg  ( x  e.  A  |->  .0.  ) )  =  .0.  )
65adantr 472 . . 3  |-  ( (
ph  /\  W  =  (/) )  ->  ( G  gsumg  ( x  e.  A  |->  .0.  ) )  =  .0.  )
7 gsumval3.f . . . . . . 7  |-  ( ph  ->  F : A --> B )
87feqmptd 5932 . . . . . 6  |-  ( ph  ->  F  =  ( x  e.  A  |->  ( F `
 x ) ) )
98adantr 472 . . . . 5  |-  ( (
ph  /\  W  =  (/) )  ->  F  =  ( x  e.  A  |->  ( F `  x
) ) )
10 gsumval3.h . . . . . . . . . . . . . 14  |-  ( ph  ->  H : ( 1 ... M ) -1-1-> A
)
11 f1f 5792 . . . . . . . . . . . . . 14  |-  ( H : ( 1 ... M ) -1-1-> A  ->  H : ( 1 ... M ) --> A )
1210, 11syl 17 . . . . . . . . . . . . 13  |-  ( ph  ->  H : ( 1 ... M ) --> A )
1312ad2antrr 740 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  W  =  (/) )  /\  x  e.  ran  H )  ->  H : ( 1 ... M ) --> A )
14 f1f1orn 5839 . . . . . . . . . . . . . . . 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 472 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  W  =  (/) )  ->  H :
( 1 ... M
)
-1-1-onto-> ran  H )
17 f1ocnv 5840 . . . . . . . . . . . . . 14  |-  ( H : ( 1 ... M ) -1-1-onto-> ran  H  ->  `' H : ran  H -1-1-onto-> ( 1 ... M ) )
18 f1of 5828 . . . . . . . . . . . . . 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 5957 . . . . . . . . . . . 12  |-  ( ( H : ( 1 ... M ) --> A  /\  ( `' H `  x )  e.  ( 1 ... M ) )  ->  ( ( F  o.  H ) `  ( `' H `  x ) )  =  ( F `  ( H `  ( `' H `  x )
) ) )
2213, 20, 21syl2anc 673 . . . . . . . . . . 11  |-  ( ( ( ph  /\  W  =  (/) )  /\  x  e.  ran  H )  -> 
( ( F  o.  H ) `  ( `' H `  x ) )  =  ( F `
 ( H `  ( `' H `  x ) ) ) )
23 simpr 468 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  W  =  (/) )  ->  W  =  (/) )
2423difeq2d 3540 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  W  =  (/) )  ->  ( (
1 ... M )  \  W )  =  ( ( 1 ... M
)  \  (/) ) )
25 dif0 3749 . . . . . . . . . . . . . . 15  |-  ( ( 1 ... M ) 
\  (/) )  =  ( 1 ... M )
2624, 25syl6eq 2521 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  W  =  (/) )  ->  ( (
1 ... M )  \  W )  =  ( 1 ... M ) )
2726adantr 472 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  W  =  (/) )  /\  x  e.  ran  H )  -> 
( ( 1 ... M )  \  W
)  =  ( 1 ... M ) )
2820, 27eleqtrrd 2552 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  W  =  (/) )  /\  x  e.  ran  H )  -> 
( `' H `  x )  e.  ( ( 1 ... M
)  \  W )
)
29 fco 5751 . . . . . . . . . . . . . . 15  |-  ( ( F : A --> B  /\  H : ( 1 ... M ) --> A )  ->  ( F  o.  H ) : ( 1 ... M ) --> B )
307, 12, 29syl2anc 673 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( F  o.  H
) : ( 1 ... M ) --> B )
3130adantr 472 . . . . . . . . . . . . 13  |-  ( (
ph  /\  W  =  (/) )  ->  ( F  o.  H ) : ( 1 ... M ) --> B )
32 gsumval3.w . . . . . . . . . . . . . . 15  |-  W  =  ( ( F  o.  H ) supp  .0.  )
3332eqimss2i 3473 . . . . . . . . . . . . . 14  |-  ( ( F  o.  H ) supp 
.0.  )  C_  W
3433a1i 11 . . . . . . . . . . . . 13  |-  ( (
ph  /\  W  =  (/) )  ->  ( ( F  o.  H ) supp  .0.  )  C_  W )
35 ovex 6336 . . . . . . . . . . . . . 14  |-  ( 1 ... M )  e. 
_V
3635a1i 11 . . . . . . . . . . . . 13  |-  ( (
ph  /\  W  =  (/) )  ->  ( 1 ... M )  e. 
_V )
37 fvex 5889 . . . . . . . . . . . . . . 15  |-  ( 0g
`  G )  e. 
_V
383, 37eqeltri 2545 . . . . . . . . . . . . . 14  |-  .0.  e.  _V
3938a1i 11 . . . . . . . . . . . . 13  |-  ( (
ph  /\  W  =  (/) )  ->  .0.  e.  _V )
4031, 34, 36, 39suppssr 6965 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  W  =  (/) )  /\  ( `' H `  x )  e.  ( ( 1 ... M )  \  W ) )  -> 
( ( F  o.  H ) `  ( `' H `  x ) )  =  .0.  )
4128, 40syldan 478 . . . . . . . . . . 11  |-  ( ( ( ph  /\  W  =  (/) )  /\  x  e.  ran  H )  -> 
( ( F  o.  H ) `  ( `' H `  x ) )  =  .0.  )
42 f1ocnvfv2 6194 . . . . . . . . . . . . 13  |-  ( ( H : ( 1 ... M ) -1-1-onto-> ran  H  /\  x  e.  ran  H )  ->  ( H `  ( `' H `  x ) )  =  x )
4316, 42sylan 479 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  W  =  (/) )  /\  x  e.  ran  H )  -> 
( H `  ( `' H `  x ) )  =  x )
4443fveq2d 5883 . . . . . . . . . . 11  |-  ( ( ( ph  /\  W  =  (/) )  /\  x  e.  ran  H )  -> 
( F `  ( H `  ( `' H `  x )
) )  =  ( F `  x ) )
4522, 41, 443eqtr3rd 2514 . . . . . . . . . 10  |-  ( ( ( ph  /\  W  =  (/) )  /\  x  e.  ran  H )  -> 
( F `  x
)  =  .0.  )
46 fvex 5889 . . . . . . . . . . 11  |-  ( F `
 x )  e. 
_V
4746elsnc 3984 . . . . . . . . . 10  |-  ( ( F `  x )  e.  {  .0.  }  <->  ( F `  x )  =  .0.  )
4845, 47sylibr 217 . . . . . . . . 9  |-  ( ( ( ph  /\  W  =  (/) )  /\  x  e.  ran  H )  -> 
( F `  x
)  e.  {  .0.  } )
4948adantlr 729 . . . . . . . 8  |-  ( ( ( ( ph  /\  W  =  (/) )  /\  x  e.  A )  /\  x  e.  ran  H )  ->  ( F `  x )  e.  {  .0.  } )
50 eldif 3400 . . . . . . . . . . 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 6965 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( A  \  ran  H
) )  ->  ( F `  x )  =  .0.  )
5453, 47sylibr 217 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( A  \  ran  H
) )  ->  ( F `  x )  e.  {  .0.  } )
5550, 54sylan2br 484 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  A  /\  -.  x  e.  ran  H ) )  ->  ( F `  x )  e.  {  .0.  } )
5655adantlr 729 . . . . . . . . 9  |-  ( ( ( ph  /\  W  =  (/) )  /\  (
x  e.  A  /\  -.  x  e.  ran  H ) )  ->  ( F `  x )  e.  {  .0.  } )
5756anassrs 660 . . . . . . . 8  |-  ( ( ( ( ph  /\  W  =  (/) )  /\  x  e.  A )  /\  -.  x  e.  ran  H )  ->  ( F `  x )  e.  {  .0.  } )
5849, 57pm2.61dan 808 . . . . . . 7  |-  ( ( ( ph  /\  W  =  (/) )  /\  x  e.  A )  ->  ( F `  x )  e.  {  .0.  } )
5958, 47sylib 201 . . . . . 6  |-  ( ( ( ph  /\  W  =  (/) )  /\  x  e.  A )  ->  ( F `  x )  =  .0.  )
6059mpteq2dva 4482 . . . . 5  |-  ( (
ph  /\  W  =  (/) )  ->  ( x  e.  A  |->  ( F `
 x ) )  =  ( x  e.  A  |->  .0.  ) )
619, 60eqtrd 2505 . . . 4  |-  ( (
ph  /\  W  =  (/) )  ->  F  =  ( x  e.  A  |->  .0.  ) )
6261oveq2d 6324 . . 3  |-  ( (
ph  /\  W  =  (/) )  ->  ( G  gsumg  F )  =  ( G 
gsumg  ( x  e.  A  |->  .0.  ) ) )
63 gsumval3.b . . . . . . . 8  |-  B  =  ( Base `  G
)
6463, 3mndidcl 16632 . . . . . . 7  |-  ( G  e.  Mnd  ->  .0.  e.  B )
651, 64syl 17 . . . . . 6  |-  ( ph  ->  .0.  e.  B )
66 gsumval3.p . . . . . . 7  |-  .+  =  ( +g  `  G )
6763, 66, 3mndlid 16635 . . . . . 6  |-  ( ( G  e.  Mnd  /\  .0.  e.  B )  -> 
(  .0.  .+  .0.  )  =  .0.  )
681, 65, 67syl2anc 673 . . . . 5  |-  ( ph  ->  (  .0.  .+  .0.  )  =  .0.  )
6968adantr 472 . . . 4  |-  ( (
ph  /\  W  =  (/) )  ->  (  .0.  .+  .0.  )  =  .0.  )
70 gsumval3.m . . . . . 6  |-  ( ph  ->  M  e.  NN )
71 nnuz 11218 . . . . . 6  |-  NN  =  ( ZZ>= `  1 )
7270, 71syl6eleq 2559 . . . . 5  |-  ( ph  ->  M  e.  ( ZZ>= ` 
1 ) )
7372adantr 472 . . . 4  |-  ( (
ph  /\  W  =  (/) )  ->  M  e.  ( ZZ>= `  1 )
)
7426eleq2d 2534 . . . . . 6  |-  ( (
ph  /\  W  =  (/) )  ->  ( x  e.  ( ( 1 ... M )  \  W
)  <->  x  e.  (
1 ... M ) ) )
7574biimpar 493 . . . . 5  |-  ( ( ( ph  /\  W  =  (/) )  /\  x  e.  ( 1 ... M
) )  ->  x  e.  ( ( 1 ... M )  \  W
) )
7631, 34, 36, 39suppssr 6965 . . . . 5  |-  ( ( ( ph  /\  W  =  (/) )  /\  x  e.  ( ( 1 ... M )  \  W
) )  ->  (
( F  o.  H
) `  x )  =  .0.  )
7775, 76syldan 478 . . . 4  |-  ( ( ( ph  /\  W  =  (/) )  /\  x  e.  ( 1 ... M
) )  ->  (
( F  o.  H
) `  x )  =  .0.  )
7869, 73, 77seqid3 12295 . . 3  |-  ( (
ph  /\  W  =  (/) )  ->  (  seq 1 (  .+  , 
( F  o.  H
) ) `  M
)  =  .0.  )
796, 62, 783eqtr4d 2515 . 2  |-  ( (
ph  /\  W  =  (/) )  ->  ( G  gsumg  F )  =  (  seq 1 (  .+  , 
( F  o.  H
) ) `  M
) )
80 fzf 11814 . . . . 5  |-  ... :
( ZZ  X.  ZZ )
--> ~P ZZ
81 ffn 5739 . . . . 5  |-  ( ...
: ( ZZ  X.  ZZ ) --> ~P ZZ  ->  ... 
Fn  ( ZZ  X.  ZZ ) )
82 ovelrn 6464 . . . . 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 740 . . . . . . . . 9  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  A  =  ( m ... n ) )  ->  G  e.  Mnd )
85 simpr 468 . . . . . . . . . . 11  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  A  =  ( m ... n ) )  ->  A  =  ( m ... n ) )
86 frel 5744 . . . . . . . . . . . . . . . . 17  |-  ( F : A --> B  ->  Rel  F )
87 reldm0 5058 . . . . . . . . . . . . . . . . 17  |-  ( Rel 
F  ->  ( F  =  (/)  <->  dom  F  =  (/) ) )
887, 86, 873syl 18 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( F  =  (/)  <->  dom  F  =  (/) ) )
89 fdm 5745 . . . . . . . . . . . . . . . . . 18  |-  ( F : A --> B  ->  dom  F  =  A )
907, 89syl 17 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  dom  F  =  A )
9190eqeq1d 2473 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( dom  F  =  (/) 
<->  A  =  (/) ) )
9288, 91bitrd 261 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( F  =  (/)  <->  A  =  (/) ) )
93 coeq1 4997 . . . . . . . . . . . . . . . . . . 19  |-  ( F  =  (/)  ->  ( F  o.  H )  =  ( (/)  o.  H
) )
94 co01 5357 . . . . . . . . . . . . . . . . . . 19  |-  ( (/)  o.  H )  =  (/)
9593, 94syl6eq 2521 . . . . . . . . . . . . . . . . . 18  |-  ( F  =  (/)  ->  ( F  o.  H )  =  (/) )
9695oveq1d 6323 . . . . . . . . . . . . . . . . 17  |-  ( F  =  (/)  ->  ( ( F  o.  H ) supp 
.0.  )  =  (
(/) supp  .0.  ) )
97 supp0 6938 . . . . . . . . . . . . . . . . . 18  |-  (  .0. 
e.  _V  ->  ( (/) supp  .0.  )  =  (/) )
9838, 97ax-mp 5 . . . . . . . . . . . . . . . . 17  |-  ( (/) supp  .0.  )  =  (/)
9996, 98syl6eq 2521 . . . . . . . . . . . . . . . 16  |-  ( F  =  (/)  ->  ( ( F  o.  H ) supp 
.0.  )  =  (/) )
10032, 99syl5eq 2517 . . . . . . . . . . . . . . 15  |-  ( F  =  (/)  ->  W  =  (/) )
10192, 100syl6bir 237 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( A  =  (/)  ->  W  =  (/) ) )
102101necon3d 2664 . . . . . . . . . . . . 13  |-  ( ph  ->  ( W  =/=  (/)  ->  A  =/=  (/) ) )
103102imp 436 . . . . . . . . . . . 12  |-  ( (
ph  /\  W  =/=  (/) )  ->  A  =/=  (/) )
104103adantr 472 . . . . . . . . . . 11  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  A  =  ( m ... n ) )  ->  A  =/=  (/) )
10585, 104eqnetrrd 2711 . . . . . . . . . 10  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  A  =  ( m ... n ) )  -> 
( m ... n
)  =/=  (/) )
106 fzn0 11839 . . . . . . . . . 10  |-  ( ( m ... n )  =/=  (/)  <->  n  e.  ( ZZ>=
`  m ) )
107105, 106sylib 201 . . . . . . . . 9  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  A  =  ( m ... n ) )  ->  n  e.  ( ZZ>= `  m ) )
1087ad2antrr 740 . . . . . . . . . 10  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  A  =  ( m ... n ) )  ->  F : A --> B )
10985feq2d 5725 . . . . . . . . . 10  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  A  =  ( m ... n ) )  -> 
( F : A --> B 
<->  F : ( m ... n ) --> B ) )
110108, 109mpbid 215 . . . . . . . . 9  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  A  =  ( m ... n ) )  ->  F : ( m ... n ) --> B )
11163, 66, 84, 107, 110gsumval2 16601 . . . . . . . 8  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  A  =  ( m ... n ) )  -> 
( G  gsumg  F )  =  (  seq m (  .+  ,  F ) `  n
) )
112 frn 5747 . . . . . . . . . . . . . . 15  |-  ( H : ( 1 ... M ) --> A  ->  ran  H  C_  A )
11310, 11, 1123syl 18 . . . . . . . . . . . . . 14  |-  ( ph  ->  ran  H  C_  A
)
114113ad2antrr 740 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  A  =  ( m ... n ) )  ->  ran  H  C_  A )
115114, 85sseqtrd 3454 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  A  =  ( m ... n ) )  ->  ran  H  C_  ( m ... n ) )
116 fzssuz 11865 . . . . . . . . . . . . 13  |-  ( m ... n )  C_  ( ZZ>= `  m )
117 uzssz 11202 . . . . . . . . . . . . . 14  |-  ( ZZ>= `  m )  C_  ZZ
118 zssre 10968 . . . . . . . . . . . . . 14  |-  ZZ  C_  RR
119117, 118sstri 3427 . . . . . . . . . . . . 13  |-  ( ZZ>= `  m )  C_  RR
120116, 119sstri 3427 . . . . . . . . . . . 12  |-  ( m ... n )  C_  RR
121115, 120syl6ss 3430 . . . . . . . . . . 11  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  A  =  ( m ... n ) )  ->  ran  H  C_  RR )
122 ltso 9732 . . . . . . . . . . 11  |-  <  Or  RR
123 soss 4778 . . . . . . . . . . 11  |-  ( ran 
H  C_  RR  ->  (  <  Or  RR  ->  < 
Or  ran  H )
)
124121, 122, 123mpisyl 21 . . . . . . . . . 10  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  A  =  ( m ... n ) )  ->  <  Or  ran  H )
125 fzfi 12223 . . . . . . . . . . . 12  |-  ( 1 ... M )  e. 
Fin
126125a1i 11 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( 1 ... M
)  e.  Fin )
127 fex2 6767 . . . . . . . . . . . . . . 15  |-  ( ( H : ( 1 ... M ) --> A  /\  ( 1 ... M )  e.  Fin  /\  A  e.  V )  ->  H  e.  _V )
12812, 126, 2, 127syl3anc 1292 . . . . . . . . . . . . . 14  |-  ( ph  ->  H  e.  _V )
129 f1oen3g 7603 . . . . . . . . . . . . . 14  |-  ( ( H  e.  _V  /\  H : ( 1 ... M ) -1-1-onto-> ran  H )  -> 
( 1 ... M
)  ~~  ran  H )
130128, 15, 129syl2anc 673 . . . . . . . . . . . . 13  |-  ( ph  ->  ( 1 ... M
)  ~~  ran  H )
131 enfi 7806 . . . . . . . . . . . . 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 216 . . . . . . . . . . 11  |-  ( ph  ->  ran  H  e.  Fin )
134133ad2antrr 740 . . . . . . . . . 10  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  A  =  ( m ... n ) )  ->  ran  H  e.  Fin )
135 fz1iso 12666 . . . . . . . . . 10  |-  ( (  <  Or  ran  H  /\  ran  H  e.  Fin )  ->  E. f  f  Isom  <  ,  <  ( ( 1 ... ( # `  ran  H ) ) ,  ran  H ) )
136124, 134, 135syl2anc 673 . . . . . . . . 9  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  A  =  ( m ... n ) )  ->  E. f  f  Isom  <  ,  <  ( ( 1 ... ( # `  ran  H ) ) ,  ran  H ) )
13770nnnn0d 10949 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  M  e.  NN0 )
138 hashfz1 12567 . . . . . . . . . . . . . . . 16  |-  ( M  e.  NN0  ->  ( # `  ( 1 ... M
) )  =  M )
139137, 138syl 17 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( # `  (
1 ... M ) )  =  M )
140 hashen 12568 . . . . . . . . . . . . . . . . 17  |-  ( ( ( 1 ... M
)  e.  Fin  /\  ran  H  e.  Fin )  ->  ( ( # `  (
1 ... M ) )  =  ( # `  ran  H )  <->  ( 1 ... M )  ~~  ran  H ) )
141125, 133, 140sylancr 676 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( ( # `  (
1 ... M ) )  =  ( # `  ran  H )  <->  ( 1 ... M )  ~~  ran  H ) )
142130, 141mpbird 240 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( # `  (
1 ... M ) )  =  ( # `  ran  H ) )
143139, 142eqtr3d 2507 . . . . . . . . . . . . . 14  |-  ( ph  ->  M  =  ( # `  ran  H ) )
144143ad2antrr 740 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  ->  M  =  ( # `  ran  H ) )
145144fveq2d 5883 . . . . . . . . . . . 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 740 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  ->  G  e.  Mnd )
14763, 66mndcl 16623 . . . . . . . . . . . . . . 15  |-  ( ( G  e.  Mnd  /\  x  e.  B  /\  y  e.  B )  ->  ( x  .+  y
)  e.  B )
1481473expb 1232 . . . . . . . . . . . . . 14  |-  ( ( G  e.  Mnd  /\  ( x  e.  B  /\  y  e.  B
) )  ->  (
x  .+  y )  e.  B )
149146, 148sylan 479 . . . . . . . . . . . . 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 740 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  ->  ran  F  C_  ( Z `  ran  F ) )
152151sselda 3418 . . . . . . . . . . . . . . 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 17061 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  ( Z `
 ran  F )  /\  y  e.  ran  F )  ->  ( x  .+  y )  =  ( y  .+  x ) )
155152, 154sylan 479 . . . . . . . . . . . . . 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 659 . . . . . . . . . . . . 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 16624 . . . . . . . . . . . . . 14  |-  ( ( G  e.  Mnd  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B
) )  ->  (
( x  .+  y
)  .+  z )  =  ( x  .+  ( y  .+  z
) ) )
158146, 157sylan 479 . . . . . . . . . . . . 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 740 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  ->  M  e.  ( ZZ>= ` 
1 ) )
1607ad2antrr 740 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  ->  F : A --> B )
161 frn 5747 . . . . . . . . . . . . . 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 774 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  -> 
f  Isom  <  ,  <  ( ( 1 ... ( # `
 ran  H )
) ,  ran  H
) )
164 isof1o 6234 . . . . . . . . . . . . . . . . 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 6324 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  -> 
( 1 ... M
)  =  ( 1 ... ( # `  ran  H ) ) )
167 f1oeq2 5819 . . . . . . . . . . . . . . . . 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 240 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  -> 
f : ( 1 ... M ) -1-1-onto-> ran  H
)
170 f1ocnv 5840 . . . . . . . . . . . . . . 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 740 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  ->  H : ( 1 ... M ) -1-1-onto-> ran  H )
173 f1oco 5850 . . . . . . . . . . . . . 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 673 . . . . . . . . . . . . 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 5739 . . . . . . . . . . . . . . . . 17  |-  ( F : A --> B  ->  F  Fn  A )
176 dffn4 5812 . . . . . . . . . . . . . . . . 17  |-  ( F  Fn  A  <->  F : A -onto-> ran  F )
177175, 176sylib 201 . . . . . . . . . . . . . . . 16  |-  ( F : A --> B  ->  F : A -onto-> ran  F
)
178 fof 5806 . . . . . . . . . . . . . . . 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 5828 . . . . . . . . . . . . . . . . 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 740 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  ->  ran  H  C_  A )
183181, 182fssd 5750 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  -> 
f : ( 1 ... M ) --> A )
184 fco 5751 . . . . . . . . . . . . . . 15  |-  ( ( F : A --> ran  F  /\  f : ( 1 ... M ) --> A )  ->  ( F  o.  f ) : ( 1 ... M ) --> ran  F )
185179, 183, 184syl2anc 673 . . . . . . . . . . . . . 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 5854 . . . . . . . . . . . . . . . . . . . . . 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 5001 . . . . . . . . . . . . . . . . . . . 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 5828 . . . . . . . . . . . . . . . . . . . . 21  |-  ( H : ( 1 ... M ) -1-1-onto-> ran  H  ->  H : ( 1 ... M ) --> ran  H
)
191 fcoi2 5770 . . . . . . . . . . . . . . . . . . . . 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 2506 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  ->  H  =  ( (
f  o.  `' f )  o.  H ) )
194 coass 5361 . . . . . . . . . . . . . . . . . . 19  |-  ( ( f  o.  `' f )  o.  H )  =  ( f  o.  ( `' f  o.  H ) )
195193, 194syl6eq 2521 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  ->  H  =  ( f  o.  ( `' f  o.  H ) ) )
196195coeq2d 5002 . . . . . . . . . . . . . . . . 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 5361 . . . . . . . . . . . . . . . . 17  |-  ( ( F  o.  f )  o.  ( `' f  o.  H ) )  =  ( F  o.  ( f  o.  ( `' f  o.  H
) ) )
198196, 197syl6eqr 2523 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  -> 
( F  o.  H
)  =  ( ( F  o.  f )  o.  ( `' f  o.  H ) ) )
199198fveq1d 5881 . . . . . . . . . . . . . . 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 472 . . . . . . . . . . . . . 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 5828 . . . . . . . . . . . . . . . . 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 5751 . . . . . . . . . . . . . . . 16  |-  ( ( `' f : ran  H --> ( 1 ... M
)  /\  H :
( 1 ... M
) --> ran  H )  ->  ( `' f  o.  H ) : ( 1 ... M ) --> ( 1 ... M
) )
205202, 203, 204syl2anc 673 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  -> 
( `' f  o.  H ) : ( 1 ... M ) --> ( 1 ... M
) )
206 fvco3 5957 . . . . . . . . . . . . . . 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 479 . . . . . . . . . . . . . 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 2505 . . . . . . . . . . . . 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 12292 . . . . . . . . . . . 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 16635 . . . . . . . . . . . . . 14  |-  ( ( G  e.  Mnd  /\  x  e.  B )  ->  (  .0.  .+  x
)  =  x )
211146, 210sylan 479 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  /\  x  e.  B )  ->  (  .0.  .+  x
)  =  x )
21263, 66, 3mndrid 16636 . . . . . . . . . . . . . 14  |-  ( ( G  e.  Mnd  /\  x  e.  B )  ->  ( x  .+  .0.  )  =  x )
213146, 212sylan 479 . . . . . . . . . . . . 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 5745 . . . . . . . . . . . . . . . . 17  |-  ( H : ( 1 ... M ) --> A  ->  dom  H  =  ( 1 ... M ) )
21610, 11, 2153syl 18 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  dom  H  =  ( 1 ... M ) )
217 eluzfz1 11832 . . . . . . . . . . . . . . . . 17  |-  ( M  e.  ( ZZ>= `  1
)  ->  1  e.  ( 1 ... M
) )
218 ne0i 3728 . . . . . . . . . . . . . . . . 17  |-  ( 1  e.  ( 1 ... M )  ->  (
1 ... M )  =/=  (/) )
21972, 217, 2183syl 18 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( 1 ... M
)  =/=  (/) )
220216, 219eqnetrd 2710 . . . . . . . . . . . . . . 15  |-  ( ph  ->  dom  H  =/=  (/) )
221 dm0rn0 5057 . . . . . . . . . . . . . . . 16  |-  ( dom 
H  =  (/)  <->  ran  H  =  (/) )
222221necon3bii 2695 . . . . . . . . . . . . . . 15  |-  ( dom 
H  =/=  (/)  <->  ran  H  =/=  (/) )
223220, 222sylib 201 . . . . . . . . . . . . . 14  |-  ( ph  ->  ran  H  =/=  (/) )
224223ad2antrr 740 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  ->  ran  H  =/=  (/) )
225115adantrr 731 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  ->  ran  H  C_  ( m ... n ) )
226 simprl 772 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  ->  A  =  ( m ... n ) )
227226eleq2d 2534 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  -> 
( x  e.  A  <->  x  e.  ( m ... n ) ) )
228227biimpar 493 . . . . . . . . . . . . . 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 478 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  /\  x  e.  ( m ... n ) )  -> 
( F `  x
)  e.  B )
231226difeq1d 3539 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  -> 
( A  \  ran  H )  =  ( ( m ... n ) 
\  ran  H )
)
232231eleq2d 2534 . . . . . . . . . . . . . . 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 493 . . . . . . . . . . . . . 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 768 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  ->  ph )
235234, 53sylan 479 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  /\  x  e.  ( A  \  ran  H ) )  ->  ( F `  x )  =  .0.  )
236233, 235syldan 478 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  /\  x  e.  ( (
m ... n )  \  ran  H ) )  -> 
( F `  x
)  =  .0.  )
237 f1of 5828 . . . . . . . . . . . . . . 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 5957 . . . . . . . . . . . . . 14  |-  ( ( f : ( 1 ... ( # `  ran  H ) ) --> ran  H  /\  y  e.  (
1 ... ( # `  ran  H ) ) )  -> 
( ( F  o.  f ) `  y
)  =  ( F `
 ( f `  y ) ) )
240238, 239sylan 479 . . . . . . . . . . . . 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 12669 . . . . . . . . . . . 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 2515 . . . . . . . . . . 11  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( A  =  ( m ... n )  /\  f  Isom  <  ,  <  (
( 1 ... ( # `
 ran  H )
) ,  ran  H
) ) )  -> 
(  seq 1 (  .+  ,  ( F  o.  H ) ) `  M )  =  (  seq m (  .+  ,  F ) `  n
) )
243242expr 626 . . . . . . . . . 10  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  A  =  ( m ... n ) )  -> 
( f  Isom  <  ,  <  ( ( 1 ... ( # `  ran  H ) ) ,  ran  H )  ->  (  seq 1 (  .+  , 
( F  o.  H
) ) `  M
)  =  (  seq m (  .+  ,  F ) `  n
) ) )
244243exlimdv 1787 . . . . . . . . 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 2508 . . . . . . 7  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  A  =  ( m ... n ) )  -> 
( G  gsumg  F )  =  (  seq 1 (  .+  ,  ( F  o.  H ) ) `  M ) )
247246ex 441 . . . . . 6  |-  ( (
ph  /\  W  =/=  (/) )  ->  ( A  =  ( m ... n )  ->  ( G  gsumg  F )  =  (  seq 1 (  .+  ,  ( F  o.  H ) ) `  M ) ) )
248247rexlimdvw 2874 . . . . 5  |-  ( (
ph  /\  W  =/=  (/) )  ->  ( E. n  e.  ZZ  A  =  ( m ... n )  ->  ( G  gsumg  F )  =  (  seq 1 (  .+  ,  ( F  o.  H ) ) `  M ) ) )
249248rexlimdvw 2874 . . . 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 225 . . 3  |-  ( (
ph  /\  W  =/=  (/) )  ->  ( A  e.  ran  ...  ->  ( G 
gsumg  F )  =  (  seq 1 (  .+  ,  ( F  o.  H ) ) `  M ) ) )
251 suppssdm 6946 . . . . . . . . . . 11  |-  ( ( F  o.  H ) supp 
.0.  )  C_  dom  ( F  o.  H
)
25232, 251eqsstri 3448 . . . . . . . . . 10  |-  W  C_  dom  ( F  o.  H
)
253 fdm 5745 . . . . . . . . . . 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 3466 . . . . . . . . 9  |-  ( ph  ->  W  C_  ( 1 ... M ) )
256 fzssuz 11865 . . . . . . . . . . 11  |-  ( 1 ... M )  C_  ( ZZ>= `  1 )
257256, 71sseqtr4i 3451 . . . . . . . . . 10  |-  ( 1 ... M )  C_  NN
258 nnssre 10635 . . . . . . . . . 10  |-  NN  C_  RR
259257, 258sstri 3427 . . . . . . . . 9  |-  ( 1 ... M )  C_  RR
260255, 259syl6ss 3430 . . . . . . . 8  |-  ( ph  ->  W  C_  RR )
261 soss 4778 . . . . . . . 8  |-  ( W 
C_  RR  ->  (  < 
Or  RR  ->  <  Or  W ) )
262260, 122, 261mpisyl 21 . . . . . . 7  |-  ( ph  ->  <  Or  W )
263 ssfi 7810 . . . . . . . 8  |-  ( ( ( 1 ... M
)  e.  Fin  /\  W  C_  ( 1 ... M ) )  ->  W  e.  Fin )
264125, 255, 263sylancr 676 . . . . . . 7  |-  ( ph  ->  W  e.  Fin )
265 fz1iso 12666 . . . . . . 7  |-  ( (  <  Or  W  /\  W  e.  Fin )  ->  E. f  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) )
266262, 264, 265syl2anc 673 . . . . . 6  |-  ( ph  ->  E. f  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) )
267266ad2antrr 740 . . . . 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 17618 . . . . . . . 8  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  -> 
( G  gsumg  F )  =  (  seq 1 (  .+  ,  ( F  o.  ( H  o.  f
) ) ) `  ( # `  W ) ) )
2691ad2antrr 740 . . . . . . . . . 10  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  ->  G  e.  Mnd )
270269, 210sylan 479 . . . . . . . . 9  |-  ( ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  /\  x  e.  B )  ->  (  .0.  .+  x
)  =  x )
271269, 212sylan 479 . . . . . . . . 9  |-  ( ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  /\  x  e.  B )  ->  ( x  .+  .0.  )  =  x )
272269, 148sylan 479 . . . . . . . . 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 774 . . . . . . . . 9  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  -> 
f  Isom  <  ,  <  ( ( 1 ... ( # `
 W ) ) ,  W ) )
275 simplr 770 . . . . . . . . 9  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  ->  W  =/=  (/) )
276255ad2antrr 740 . . . . . . . . 9  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  ->  W  C_  ( 1 ... M ) )
27730ad2antrr 740 . . . . . . . . . 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 6965 . . . . . . . . 9  |-  ( ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  /\  x  e.  ( (
1 ... M )  \  W ) )  -> 
( ( F  o.  H ) `  x
)  =  .0.  )
283 coass 5361 . . . . . . . . . . 11  |-  ( ( F  o.  H )  o.  f )  =  ( F  o.  ( H  o.  f )
)
284283fveq1i 5880 . . . . . . . . . 10  |-  ( ( ( F  o.  H
)  o.  f ) `
 y )  =  ( ( F  o.  ( H  o.  f
) ) `  y
)
285 isof1o 6234 . . . . . . . . . . . 12  |-  ( f 
Isom  <  ,  <  (
( 1 ... ( # `
 W ) ) ,  W )  -> 
f : ( 1 ... ( # `  W
) ) -1-1-onto-> W )
286 f1of 5828 . . . . . . . . . . . 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 5957 . . . . . . . . . . 11  |-  ( ( f : ( 1 ... ( # `  W
) ) --> W  /\  y  e.  ( 1 ... ( # `  W
) ) )  -> 
( ( ( F  o.  H )  o.  f ) `  y
)  =  ( ( F  o.  H ) `
 ( f `  y ) ) )
289287, 288sylan 479 . . . . . . . . . 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 2519 . . . . . . . . 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 12669 . . . . . . . 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 2508 . . . . . . 7  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  -> 
( G  gsumg  F )  =  (  seq 1 (  .+  ,  ( F  o.  H ) ) `  M ) )
293292expr 626 . . . . . 6  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  -.  A  e.  ran  ... )  ->  ( f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
)  ->  ( G  gsumg  F )  =  (  seq 1 (  .+  , 
( F  o.  H
) ) `  M
) ) )
294293exlimdv 1787 . . . . 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 441 . . 3  |-  ( (
ph  /\  W  =/=  (/) )  ->  ( -.  A  e.  ran  ...  ->  ( G  gsumg  F )  =  (  seq 1 (  .+  ,  ( F  o.  H ) ) `  M ) ) )
297250, 296pm2.61d 163 . 2  |-  ( (
ph  /\  W  =/=  (/) )  ->  ( G  gsumg  F )  =  (  seq 1 (  .+  , 
( F  o.  H
) ) `  M
) )
29879, 297pm2.61dane 2730 1  |-  ( ph  ->  ( G  gsumg  F )  =  (  seq 1 (  .+  ,  ( F  o.  H ) ) `  M ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    /\ wa 376    /\ w3a 1007    = wceq 1452   E.wex 1671    e. wcel 1904    =/= wne 2641   E.wrex 2757   _Vcvv 3031    \ cdif 3387    C_ wss 3390   (/)c0 3722   ~Pcpw 3942   {csn 3959   class class class wbr 4395    |-> cmpt 4454    _I cid 4749    Or wor 4759    X. cxp 4837   `'ccnv 4838   dom cdm 4839   ran crn 4840    |` cres 4841    o. ccom 4843   Rel wrel 4844    Fn wfn 5584   -->wf 5585   -1-1->wf1 5586   -onto->wfo 5587   -1-1-onto->wf1o 5588   ` cfv 5589    Isom wiso 5590  (class class class)co 6308   supp csupp 6933    ~~ cen 7584   Fincfn 7587   RRcr 9556   1c1 9558    < clt 9693   NNcn 10631   NN0cn0 10893   ZZcz 10961   ZZ>=cuz 11182   ...cfz 11810    seqcseq 12251   #chash 12553   Basecbs 15199   +g cplusg 15268   0gc0g 15416    gsumg cgsu 15417   Mndcmnd 16613  Cntzccntz 17047
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-oi 8043  df-card 8391  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-fzo 11943  df-seq 12252  df-hash 12554  df-0g 15418  df-gsum 15419  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-cntz 17049
This theorem is referenced by:  gsumzres  17621  gsumzcl2  17622  gsumzf1o  17624  gsumzaddlem  17632  gsumconst  17645  gsumzmhm  17648  gsumzoppg  17655  gsumfsum  19111  wilthlem3  24074
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