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Theorem gsumval3lem2 17475
Description: Lemma 2 for gsumval3 17476. (Contributed 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
gsumval3lem2  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  -> 
( G  gsumg  F )  =  (  seq 1 (  .+  ,  ( F  o.  ( H  o.  f
) ) ) `  ( # `  W ) ) )
Distinct variable groups:    .+ , f    A, f    ph, f    f, G   
f, M    B, f    f, F    f, H    f, W
Allowed substitution hints:    V( f)    .0. ( f)    Z( f)

Proof of Theorem gsumval3lem2
Dummy variables  g  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 gsumval3.h . . . . . . 7  |-  ( ph  ->  H : ( 1 ... M ) -1-1-> A
)
2 f1f 5796 . . . . . . 7  |-  ( H : ( 1 ... M ) -1-1-> A  ->  H : ( 1 ... M ) --> A )
31, 2syl 17 . . . . . 6  |-  ( ph  ->  H : ( 1 ... M ) --> A )
4 fzfid 12183 . . . . . 6  |-  ( ph  ->  ( 1 ... M
)  e.  Fin )
5 gsumval3.a . . . . . 6  |-  ( ph  ->  A  e.  V )
6 fex2 6762 . . . . . 6  |-  ( ( H : ( 1 ... M ) --> A  /\  ( 1 ... M )  e.  Fin  /\  A  e.  V )  ->  H  e.  _V )
73, 4, 5, 6syl3anc 1264 . . . . 5  |-  ( ph  ->  H  e.  _V )
8 vex 3090 . . . . 5  |-  f  e. 
_V
9 coexg 6758 . . . . 5  |-  ( ( H  e.  _V  /\  f  e.  _V )  ->  ( H  o.  f
)  e.  _V )
107, 8, 9sylancl 666 . . . 4  |-  ( ph  ->  ( H  o.  f
)  e.  _V )
1110ad2antrr 730 . . 3  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  -> 
( H  o.  f
)  e.  _V )
12 gsumval3.b . . . . 5  |-  B  =  ( Base `  G
)
13 gsumval3.0 . . . . 5  |-  .0.  =  ( 0g `  G )
14 gsumval3.p . . . . 5  |-  .+  =  ( +g  `  G )
15 gsumval3.z . . . . 5  |-  Z  =  (Cntz `  G )
16 gsumval3.g . . . . 5  |-  ( ph  ->  G  e.  Mnd )
17 gsumval3.f . . . . 5  |-  ( ph  ->  F : A --> B )
18 gsumval3.c . . . . 5  |-  ( ph  ->  ran  F  C_  ( Z `  ran  F ) )
19 gsumval3.m . . . . 5  |-  ( ph  ->  M  e.  NN )
20 gsumval3.n . . . . 5  |-  ( ph  ->  ( F supp  .0.  )  C_ 
ran  H )
21 gsumval3.w . . . . 5  |-  W  =  ( ( F  o.  H ) supp  .0.  )
2212, 13, 14, 15, 16, 5, 17, 18, 19, 1, 20, 21gsumval3lem1 17474 . . . 4  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  -> 
( H  o.  f
) : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
23 resexg 5167 . . . . . . . . 9  |-  ( H  e.  _V  ->  ( H  |`  W )  e. 
_V )
247, 23syl 17 . . . . . . . 8  |-  ( ph  ->  ( H  |`  W )  e.  _V )
2524ad2antrr 730 . . . . . . 7  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  -> 
( H  |`  W )  e.  _V )
261ad2antrr 730 . . . . . . . . 9  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  ->  H : ( 1 ... M ) -1-1-> A )
27 suppssdm 6938 . . . . . . . . . . . 12  |-  ( ( F  o.  H ) supp 
.0.  )  C_  dom  ( F  o.  H
)
2821, 27eqsstri 3500 . . . . . . . . . . 11  |-  W  C_  dom  ( F  o.  H
)
29 fco 5756 . . . . . . . . . . . . 13  |-  ( ( F : A --> B  /\  H : ( 1 ... M ) --> A )  ->  ( F  o.  H ) : ( 1 ... M ) --> B )
3017, 3, 29syl2anc 665 . . . . . . . . . . . 12  |-  ( ph  ->  ( F  o.  H
) : ( 1 ... M ) --> B )
31 fdm 5750 . . . . . . . . . . . 12  |-  ( ( F  o.  H ) : ( 1 ... M ) --> B  ->  dom  ( F  o.  H
)  =  ( 1 ... M ) )
3230, 31syl 17 . . . . . . . . . . 11  |-  ( ph  ->  dom  ( F  o.  H )  =  ( 1 ... M ) )
3328, 32syl5sseq 3518 . . . . . . . . . 10  |-  ( ph  ->  W  C_  ( 1 ... M ) )
3433ad2antrr 730 . . . . . . . . 9  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  ->  W  C_  ( 1 ... M ) )
35 f1ores 5845 . . . . . . . . 9  |-  ( ( H : ( 1 ... M ) -1-1-> A  /\  W  C_  ( 1 ... M ) )  ->  ( H  |`  W ) : W -1-1-onto-> ( H " W ) )
3626, 34, 35syl2anc 665 . . . . . . . 8  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  -> 
( H  |`  W ) : W -1-1-onto-> ( H " W
) )
3721imaeq2i 5186 . . . . . . . . . . 11  |-  ( H
" W )  =  ( H " (
( F  o.  H
) supp  .0.  ) )
38 fex 6153 . . . . . . . . . . . . . . 15  |-  ( ( F : A --> B  /\  A  e.  V )  ->  F  e.  _V )
3917, 5, 38syl2anc 665 . . . . . . . . . . . . . 14  |-  ( ph  ->  F  e.  _V )
40 ovex 6333 . . . . . . . . . . . . . . 15  |-  ( 1 ... M )  e. 
_V
41 fex 6153 . . . . . . . . . . . . . . 15  |-  ( ( H : ( 1 ... M ) --> A  /\  ( 1 ... M )  e.  _V )  ->  H  e.  _V )
423, 40, 41sylancl 666 . . . . . . . . . . . . . 14  |-  ( ph  ->  H  e.  _V )
4339, 42jca 534 . . . . . . . . . . . . 13  |-  ( ph  ->  ( F  e.  _V  /\  H  e.  _V )
)
44 f1fun 5798 . . . . . . . . . . . . . . 15  |-  ( H : ( 1 ... M ) -1-1-> A  ->  Fun  H )
451, 44syl 17 . . . . . . . . . . . . . 14  |-  ( ph  ->  Fun  H )
4645, 20jca 534 . . . . . . . . . . . . 13  |-  ( ph  ->  ( Fun  H  /\  ( F supp  .0.  )  C_  ran  H ) )
47 imacosupp 6966 . . . . . . . . . . . . 13  |-  ( ( F  e.  _V  /\  H  e.  _V )  ->  ( ( Fun  H  /\  ( F supp  .0.  )  C_ 
ran  H )  -> 
( H " (
( F  o.  H
) supp  .0.  ) )  =  ( F supp  .0.  ) ) )
4843, 46, 47sylc 62 . . . . . . . . . . . 12  |-  ( ph  ->  ( H " (
( F  o.  H
) supp  .0.  ) )  =  ( F supp  .0.  ) )
4948adantr 466 . . . . . . . . . . 11  |-  ( (
ph  /\  W  =/=  (/) )  ->  ( H " ( ( F  o.  H ) supp  .0.  )
)  =  ( F supp 
.0.  ) )
5037, 49syl5eq 2482 . . . . . . . . . 10  |-  ( (
ph  /\  W  =/=  (/) )  ->  ( H " W )  =  ( F supp  .0.  ) )
5150adantr 466 . . . . . . . . 9  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  -> 
( H " W
)  =  ( F supp 
.0.  ) )
52 f1oeq3 5824 . . . . . . . . 9  |-  ( ( H " W )  =  ( F supp  .0.  )  ->  ( ( H  |`  W ) : W -1-1-onto-> ( H " W )  <->  ( H  |`  W ) : W -1-1-onto-> ( F supp  .0.  ) ) )
5351, 52syl 17 . . . . . . . 8  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  -> 
( ( H  |`  W ) : W -1-1-onto-> ( H " W )  <->  ( H  |`  W ) : W -1-1-onto-> ( F supp  .0.  ) ) )
5436, 53mpbid 213 . . . . . . 7  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  -> 
( H  |`  W ) : W -1-1-onto-> ( F supp  .0.  )
)
55 f1oen3g 7592 . . . . . . 7  |-  ( ( ( H  |`  W )  e.  _V  /\  ( H  |`  W ) : W -1-1-onto-> ( F supp  .0.  )
)  ->  W  ~~  ( F supp  .0.  ) )
5625, 54, 55syl2anc 665 . . . . . 6  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  ->  W  ~~  ( F supp  .0.  ) )
57 fzfi 12182 . . . . . . . . 9  |-  ( 1 ... M )  e. 
Fin
58 ssfi 7798 . . . . . . . . 9  |-  ( ( ( 1 ... M
)  e.  Fin  /\  W  C_  ( 1 ... M ) )  ->  W  e.  Fin )
5957, 33, 58sylancr 667 . . . . . . . 8  |-  ( ph  ->  W  e.  Fin )
6059ad2antrr 730 . . . . . . 7  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  ->  W  e.  Fin )
61 f1f1orn 5842 . . . . . . . . . . . . 13  |-  ( H : ( 1 ... M ) -1-1-> A  ->  H : ( 1 ... M ) -1-1-onto-> ran  H )
621, 61syl 17 . . . . . . . . . . . 12  |-  ( ph  ->  H : ( 1 ... M ) -1-1-onto-> ran  H
)
63 f1oen3g 7592 . . . . . . . . . . . 12  |-  ( ( H  e.  _V  /\  H : ( 1 ... M ) -1-1-onto-> ran  H )  -> 
( 1 ... M
)  ~~  ran  H )
647, 62, 63syl2anc 665 . . . . . . . . . . 11  |-  ( ph  ->  ( 1 ... M
)  ~~  ran  H )
65 enfi 7794 . . . . . . . . . . 11  |-  ( ( 1 ... M ) 
~~  ran  H  ->  ( ( 1 ... M
)  e.  Fin  <->  ran  H  e. 
Fin ) )
6664, 65syl 17 . . . . . . . . . 10  |-  ( ph  ->  ( ( 1 ... M )  e.  Fin  <->  ran  H  e.  Fin ) )
6757, 66mpbii 214 . . . . . . . . 9  |-  ( ph  ->  ran  H  e.  Fin )
68 ssfi 7798 . . . . . . . . 9  |-  ( ( ran  H  e.  Fin  /\  ( F supp  .0.  )  C_ 
ran  H )  -> 
( F supp  .0.  )  e.  Fin )
6967, 20, 68syl2anc 665 . . . . . . . 8  |-  ( ph  ->  ( F supp  .0.  )  e.  Fin )
7069ad2antrr 730 . . . . . . 7  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  -> 
( F supp  .0.  )  e.  Fin )
71 hashen 12527 . . . . . . 7  |-  ( ( W  e.  Fin  /\  ( F supp  .0.  )  e. 
Fin )  ->  (
( # `  W )  =  ( # `  ( F supp  .0.  ) )  <->  W  ~~  ( F supp  .0.  ) ) )
7260, 70, 71syl2anc 665 . . . . . 6  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  -> 
( ( # `  W
)  =  ( # `  ( F supp  .0.  )
)  <->  W  ~~  ( F supp 
.0.  ) ) )
7356, 72mpbird 235 . . . . 5  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  -> 
( # `  W )  =  ( # `  ( F supp  .0.  ) ) )
7473fveq2d 5885 . . . 4  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  -> 
(  seq 1 (  .+  ,  ( F  o.  ( H  o.  f
) ) ) `  ( # `  W ) )  =  (  seq 1 (  .+  , 
( F  o.  ( H  o.  f )
) ) `  ( # `
 ( F supp  .0.  ) ) ) )
7522, 74jca 534 . . 3  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  -> 
( ( H  o.  f ) : ( 1 ... ( # `  ( F supp  .0.  )
) ) -1-1-onto-> ( F supp  .0.  )  /\  (  seq 1
(  .+  ,  ( F  o.  ( H  o.  f ) ) ) `
 ( # `  W
) )  =  (  seq 1 (  .+  ,  ( F  o.  ( H  o.  f
) ) ) `  ( # `  ( F supp 
.0.  ) ) ) ) )
76 f1oeq1 5822 . . . . 5  |-  ( g  =  ( H  o.  f )  ->  (
g : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  )  <->  ( H  o.  f ) : ( 1 ... ( # `  ( F supp  .0.  )
) ) -1-1-onto-> ( F supp  .0.  )
) )
77 coeq2 5013 . . . . . . . 8  |-  ( g  =  ( H  o.  f )  ->  ( F  o.  g )  =  ( F  o.  ( H  o.  f
) ) )
7877seqeq3d 12218 . . . . . . 7  |-  ( g  =  ( H  o.  f )  ->  seq 1 (  .+  , 
( F  o.  g
) )  =  seq 1 (  .+  , 
( F  o.  ( H  o.  f )
) ) )
7978fveq1d 5883 . . . . . 6  |-  ( g  =  ( H  o.  f )  ->  (  seq 1 (  .+  , 
( F  o.  g
) ) `  ( # `
 ( F supp  .0.  ) ) )  =  (  seq 1 ( 
.+  ,  ( F  o.  ( H  o.  f ) ) ) `
 ( # `  ( F supp  .0.  ) ) ) )
8079eqeq2d 2443 . . . . 5  |-  ( g  =  ( H  o.  f )  ->  (
(  seq 1 (  .+  ,  ( F  o.  ( H  o.  f
) ) ) `  ( # `  W ) )  =  (  seq 1 (  .+  , 
( F  o.  g
) ) `  ( # `
 ( F supp  .0.  ) ) )  <->  (  seq 1 (  .+  , 
( F  o.  ( H  o.  f )
) ) `  ( # `
 W ) )  =  (  seq 1
(  .+  ,  ( F  o.  ( H  o.  f ) ) ) `
 ( # `  ( F supp  .0.  ) ) ) ) )
8176, 80anbi12d 715 . . . 4  |-  ( g  =  ( H  o.  f )  ->  (
( g : ( 1 ... ( # `  ( F supp  .0.  )
) ) -1-1-onto-> ( F supp  .0.  )  /\  (  seq 1
(  .+  ,  ( F  o.  ( H  o.  f ) ) ) `
 ( # `  W
) )  =  (  seq 1 (  .+  ,  ( F  o.  g ) ) `  ( # `  ( F supp 
.0.  ) ) ) )  <->  ( ( H  o.  f ) : ( 1 ... ( # `
 ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp 
.0.  )  /\  (  seq 1 (  .+  , 
( F  o.  ( H  o.  f )
) ) `  ( # `
 W ) )  =  (  seq 1
(  .+  ,  ( F  o.  ( H  o.  f ) ) ) `
 ( # `  ( F supp  .0.  ) ) ) ) ) )
8281spcegv 3173 . . 3  |-  ( ( H  o.  f )  e.  _V  ->  (
( ( H  o.  f ) : ( 1 ... ( # `  ( F supp  .0.  )
) ) -1-1-onto-> ( F supp  .0.  )  /\  (  seq 1
(  .+  ,  ( F  o.  ( H  o.  f ) ) ) `
 ( # `  W
) )  =  (  seq 1 (  .+  ,  ( F  o.  ( H  o.  f
) ) ) `  ( # `  ( F supp 
.0.  ) ) ) )  ->  E. g
( g : ( 1 ... ( # `  ( F supp  .0.  )
) ) -1-1-onto-> ( F supp  .0.  )  /\  (  seq 1
(  .+  ,  ( F  o.  ( H  o.  f ) ) ) `
 ( # `  W
) )  =  (  seq 1 (  .+  ,  ( F  o.  g ) ) `  ( # `  ( F supp 
.0.  ) ) ) ) ) )
8311, 75, 82sylc 62 . 2  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  ->  E. g ( g : ( 1 ... ( # `
 ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp 
.0.  )  /\  (  seq 1 (  .+  , 
( F  o.  ( H  o.  f )
) ) `  ( # `
 W ) )  =  (  seq 1
(  .+  ,  ( F  o.  g )
) `  ( # `  ( F supp  .0.  ) ) ) ) )
8416ad2antrr 730 . . . . . . 7  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  ->  G  e.  Mnd )
855ad2antrr 730 . . . . . . 7  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  ->  A  e.  V )
8617ad2antrr 730 . . . . . . 7  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  ->  F : A --> B )
8718ad2antrr 730 . . . . . . 7  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  ->  ran  F  C_  ( Z `  ran  F ) )
8821neeq1i 2716 . . . . . . . . . 10  |-  ( W  =/=  (/)  <->  ( ( F  o.  H ) supp  .0.  )  =/=  (/) )
89 supp0cosupp0 6965 . . . . . . . . . . . 12  |-  ( ( F  e.  _V  /\  H  e.  _V )  ->  ( ( F supp  .0.  )  =  (/)  ->  (
( F  o.  H
) supp  .0.  )  =  (/) ) )
9089necon3d 2655 . . . . . . . . . . 11  |-  ( ( F  e.  _V  /\  H  e.  _V )  ->  ( ( ( F  o.  H ) supp  .0.  )  =/=  (/)  ->  ( F supp  .0.  )  =/=  (/) ) )
9139, 42, 90syl2anc 665 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( F  o.  H ) supp  .0.  )  =/=  (/)  ->  ( F supp  .0.  )  =/=  (/) ) )
9288, 91syl5bi 220 . . . . . . . . 9  |-  ( ph  ->  ( W  =/=  (/)  ->  ( F supp  .0.  )  =/=  (/) ) )
9392imp 430 . . . . . . . 8  |-  ( (
ph  /\  W  =/=  (/) )  ->  ( F supp  .0.  )  =/=  (/) )
9493adantr 466 . . . . . . 7  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  -> 
( F supp  .0.  )  =/=  (/) )
9520ad2antrr 730 . . . . . . . 8  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  -> 
( F supp  .0.  )  C_ 
ran  H )
96 frn 5752 . . . . . . . . . 10  |-  ( H : ( 1 ... M ) --> A  ->  ran  H  C_  A )
973, 96syl 17 . . . . . . . . 9  |-  ( ph  ->  ran  H  C_  A
)
9897ad2antrr 730 . . . . . . . 8  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  ->  ran  H  C_  A )
9995, 98sstrd 3480 . . . . . . 7  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  -> 
( F supp  .0.  )  C_  A )
10012, 13, 14, 15, 84, 85, 86, 87, 70, 94, 99gsumval3eu 17473 . . . . . 6  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  ->  E! x E. g ( g : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  )  /\  x  =  (  seq 1 (  .+  , 
( F  o.  g
) ) `  ( # `
 ( F supp  .0.  ) ) ) ) )
101 iota1 5579 . . . . . 6  |-  ( E! x E. g ( g : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  )  /\  x  =  (  seq 1 (  .+  , 
( F  o.  g
) ) `  ( # `
 ( F supp  .0.  ) ) ) )  ->  ( E. g
( g : ( 1 ... ( # `  ( F supp  .0.  )
) ) -1-1-onto-> ( F supp  .0.  )  /\  x  =  (  seq 1 (  .+  , 
( F  o.  g
) ) `  ( # `
 ( F supp  .0.  ) ) ) )  <-> 
( iota x E. g
( g : ( 1 ... ( # `  ( F supp  .0.  )
) ) -1-1-onto-> ( F supp  .0.  )  /\  x  =  (  seq 1 (  .+  , 
( F  o.  g
) ) `  ( # `
 ( F supp  .0.  ) ) ) ) )  =  x ) )
102100, 101syl 17 . . . . 5  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  -> 
( E. g ( g : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  )  /\  x  =  (  seq 1 (  .+  , 
( F  o.  g
) ) `  ( # `
 ( F supp  .0.  ) ) ) )  <-> 
( iota x E. g
( g : ( 1 ... ( # `  ( F supp  .0.  )
) ) -1-1-onto-> ( F supp  .0.  )  /\  x  =  (  seq 1 (  .+  , 
( F  o.  g
) ) `  ( # `
 ( F supp  .0.  ) ) ) ) )  =  x ) )
103 eqid 2429 . . . . . . 7  |-  ( F supp 
.0.  )  =  ( F supp  .0.  )
104 simprl 762 . . . . . . 7  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  ->  -.  A  e.  ran  ... )
10512, 13, 14, 15, 84, 85, 86, 87, 70, 94, 103, 104gsumval3a 17472 . . . . . 6  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  -> 
( G  gsumg  F )  =  ( iota x E. g
( g : ( 1 ... ( # `  ( F supp  .0.  )
) ) -1-1-onto-> ( F supp  .0.  )  /\  x  =  (  seq 1 (  .+  , 
( F  o.  g
) ) `  ( # `
 ( F supp  .0.  ) ) ) ) ) )
106105eqeq1d 2431 . . . . 5  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  -> 
( ( G  gsumg  F )  =  x  <->  ( iota x E. g ( g : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  )  /\  x  =  (  seq 1 (  .+  , 
( F  o.  g
) ) `  ( # `
 ( F supp  .0.  ) ) ) ) )  =  x ) )
107102, 106bitr4d 259 . . . 4  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  -> 
( E. g ( g : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  )  /\  x  =  (  seq 1 (  .+  , 
( F  o.  g
) ) `  ( # `
 ( F supp  .0.  ) ) ) )  <-> 
( G  gsumg  F )  =  x ) )
108107alrimiv 1766 . . 3  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  ->  A. x ( E. g
( g : ( 1 ... ( # `  ( F supp  .0.  )
) ) -1-1-onto-> ( F supp  .0.  )  /\  x  =  (  seq 1 (  .+  , 
( F  o.  g
) ) `  ( # `
 ( F supp  .0.  ) ) ) )  <-> 
( G  gsumg  F )  =  x ) )
109 fvex 5891 . . . 4  |-  (  seq 1 (  .+  , 
( F  o.  ( H  o.  f )
) ) `  ( # `
 W ) )  e.  _V
110 eqeq1 2433 . . . . . . 7  |-  ( x  =  (  seq 1
(  .+  ,  ( F  o.  ( H  o.  f ) ) ) `
 ( # `  W
) )  ->  (
x  =  (  seq 1 (  .+  , 
( F  o.  g
) ) `  ( # `
 ( F supp  .0.  ) ) )  <->  (  seq 1 (  .+  , 
( F  o.  ( H  o.  f )
) ) `  ( # `
 W ) )  =  (  seq 1
(  .+  ,  ( F  o.  g )
) `  ( # `  ( F supp  .0.  ) ) ) ) )
111110anbi2d 708 . . . . . 6  |-  ( x  =  (  seq 1
(  .+  ,  ( F  o.  ( H  o.  f ) ) ) `
 ( # `  W
) )  ->  (
( g : ( 1 ... ( # `  ( F supp  .0.  )
) ) -1-1-onto-> ( F supp  .0.  )  /\  x  =  (  seq 1 (  .+  , 
( F  o.  g
) ) `  ( # `
 ( F supp  .0.  ) ) ) )  <-> 
( g : ( 1 ... ( # `  ( F supp  .0.  )
) ) -1-1-onto-> ( F supp  .0.  )  /\  (  seq 1
(  .+  ,  ( F  o.  ( H  o.  f ) ) ) `
 ( # `  W
) )  =  (  seq 1 (  .+  ,  ( F  o.  g ) ) `  ( # `  ( F supp 
.0.  ) ) ) ) ) )
112111exbidv 1761 . . . . 5  |-  ( x  =  (  seq 1
(  .+  ,  ( F  o.  ( H  o.  f ) ) ) `
 ( # `  W
) )  ->  ( E. g ( g : ( 1 ... ( # `
 ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp 
.0.  )  /\  x  =  (  seq 1
(  .+  ,  ( F  o.  g )
) `  ( # `  ( F supp  .0.  ) ) ) )  <->  E. g ( g : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  )  /\  (  seq 1 (  .+  ,  ( F  o.  ( H  o.  f
) ) ) `  ( # `  W ) )  =  (  seq 1 (  .+  , 
( F  o.  g
) ) `  ( # `
 ( F supp  .0.  ) ) ) ) ) )
113 eqeq2 2444 . . . . 5  |-  ( x  =  (  seq 1
(  .+  ,  ( F  o.  ( H  o.  f ) ) ) `
 ( # `  W
) )  ->  (
( G  gsumg  F )  =  x  <-> 
( G  gsumg  F )  =  (  seq 1 (  .+  ,  ( F  o.  ( H  o.  f
) ) ) `  ( # `  W ) ) ) )
114112, 113bibi12d 322 . . . 4  |-  ( x  =  (  seq 1
(  .+  ,  ( F  o.  ( H  o.  f ) ) ) `
 ( # `  W
) )  ->  (
( E. g ( g : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  )  /\  x  =  (  seq 1 (  .+  , 
( F  o.  g
) ) `  ( # `
 ( F supp  .0.  ) ) ) )  <-> 
( G  gsumg  F )  =  x )  <->  ( E. g
( g : ( 1 ... ( # `  ( F supp  .0.  )
) ) -1-1-onto-> ( F supp  .0.  )  /\  (  seq 1
(  .+  ,  ( F  o.  ( H  o.  f ) ) ) `
 ( # `  W
) )  =  (  seq 1 (  .+  ,  ( F  o.  g ) ) `  ( # `  ( F supp 
.0.  ) ) ) )  <->  ( G  gsumg  F )  =  (  seq 1
(  .+  ,  ( F  o.  ( H  o.  f ) ) ) `
 ( # `  W
) ) ) ) )
115109, 114spcv 3178 . . 3  |-  ( A. x ( E. g
( g : ( 1 ... ( # `  ( F supp  .0.  )
) ) -1-1-onto-> ( F supp  .0.  )  /\  x  =  (  seq 1 (  .+  , 
( F  o.  g
) ) `  ( # `
 ( F supp  .0.  ) ) ) )  <-> 
( G  gsumg  F )  =  x )  ->  ( E. g ( g : ( 1 ... ( # `
 ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp 
.0.  )  /\  (  seq 1 (  .+  , 
( F  o.  ( H  o.  f )
) ) `  ( # `
 W ) )  =  (  seq 1
(  .+  ,  ( F  o.  g )
) `  ( # `  ( F supp  .0.  ) ) ) )  <->  ( G  gsumg  F )  =  (  seq 1
(  .+  ,  ( F  o.  ( H  o.  f ) ) ) `
 ( # `  W
) ) ) )
116108, 115syl 17 . 2  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  -> 
( E. g ( g : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  )  /\  (  seq 1 (  .+  ,  ( F  o.  ( H  o.  f
) ) ) `  ( # `  W ) )  =  (  seq 1 (  .+  , 
( F  o.  g
) ) `  ( # `
 ( F supp  .0.  ) ) ) )  <-> 
( G  gsumg  F )  =  (  seq 1 (  .+  ,  ( F  o.  ( H  o.  f
) ) ) `  ( # `  W ) ) ) )
11783, 116mpbid 213 1  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  -> 
( G  gsumg  F )  =  (  seq 1 (  .+  ,  ( F  o.  ( H  o.  f
) ) ) `  ( # `  W ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370   A.wal 1435    = wceq 1437   E.wex 1659    e. wcel 1870   E!weu 2266    =/= wne 2625   _Vcvv 3087    C_ wss 3442   (/)c0 3767   class class class wbr 4426   dom cdm 4854   ran crn 4855    |` cres 4856   "cima 4857    o. ccom 4858   iotacio 5563   Fun wfun 5595   -->wf 5597   -1-1->wf1 5598   -1-1-onto->wf1o 5600   ` cfv 5601    Isom wiso 5602  (class class class)co 6305   supp csupp 6925    ~~ cen 7574   Fincfn 7577   1c1 9539    < clt 9674   NNcn 10609   ...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-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-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:  gsumval3  17476
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