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Theorem gsumval3lem2 16713
Description: Lemma 2 for gsumval3 16714. (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 5781 . . . . . . 7  |-  ( H : ( 1 ... M ) -1-1-> A  ->  H : ( 1 ... M ) --> A )
31, 2syl 16 . . . . . 6  |-  ( ph  ->  H : ( 1 ... M ) --> A )
4 fzfid 12051 . . . . . 6  |-  ( ph  ->  ( 1 ... M
)  e.  Fin )
5 gsumval3.a . . . . . 6  |-  ( ph  ->  A  e.  V )
6 fex2 6739 . . . . . 6  |-  ( ( H : ( 1 ... M ) --> A  /\  ( 1 ... M )  e.  Fin  /\  A  e.  V )  ->  H  e.  _V )
73, 4, 5, 6syl3anc 1228 . . . . 5  |-  ( ph  ->  H  e.  _V )
8 vex 3116 . . . . 5  |-  f  e. 
_V
9 coexg 6735 . . . . 5  |-  ( ( H  e.  _V  /\  f  e.  _V )  ->  ( H  o.  f
)  e.  _V )
107, 8, 9sylancl 662 . . . 4  |-  ( ph  ->  ( H  o.  f
)  e.  _V )
1110ad2antrr 725 . . 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 16712 . . . 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 5316 . . . . . . . . 9  |-  ( H  e.  _V  ->  ( H  |`  W )  e. 
_V )
247, 23syl 16 . . . . . . . 8  |-  ( ph  ->  ( H  |`  W )  e.  _V )
2524ad2antrr 725 . . . . . . 7  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  -> 
( H  |`  W )  e.  _V )
261ad2antrr 725 . . . . . . . . 9  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  ->  H : ( 1 ... M ) -1-1-> A )
27 suppssdm 6914 . . . . . . . . . . . 12  |-  ( ( F  o.  H ) supp 
.0.  )  C_  dom  ( F  o.  H
)
2821, 27eqsstri 3534 . . . . . . . . . . 11  |-  W  C_  dom  ( F  o.  H
)
29 fco 5741 . . . . . . . . . . . . 13  |-  ( ( F : A --> B  /\  H : ( 1 ... M ) --> A )  ->  ( F  o.  H ) : ( 1 ... M ) --> B )
3017, 3, 29syl2anc 661 . . . . . . . . . . . 12  |-  ( ph  ->  ( F  o.  H
) : ( 1 ... M ) --> B )
31 fdm 5735 . . . . . . . . . . . 12  |-  ( ( F  o.  H ) : ( 1 ... M ) --> B  ->  dom  ( F  o.  H
)  =  ( 1 ... M ) )
3230, 31syl 16 . . . . . . . . . . 11  |-  ( ph  ->  dom  ( F  o.  H )  =  ( 1 ... M ) )
3328, 32syl5sseq 3552 . . . . . . . . . 10  |-  ( ph  ->  W  C_  ( 1 ... M ) )
3433ad2antrr 725 . . . . . . . . 9  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  ->  W  C_  ( 1 ... M ) )
35 f1ores 5830 . . . . . . . . 9  |-  ( ( H : ( 1 ... M ) -1-1-> A  /\  W  C_  ( 1 ... M ) )  ->  ( H  |`  W ) : W -1-1-onto-> ( H " W ) )
3626, 34, 35syl2anc 661 . . . . . . . 8  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  -> 
( H  |`  W ) : W -1-1-onto-> ( H " W
) )
3721imaeq2i 5335 . . . . . . . . . . 11  |-  ( H
" W )  =  ( H " (
( F  o.  H
) supp  .0.  ) )
38 fex 6133 . . . . . . . . . . . . . . 15  |-  ( ( F : A --> B  /\  A  e.  V )  ->  F  e.  _V )
3917, 5, 38syl2anc 661 . . . . . . . . . . . . . 14  |-  ( ph  ->  F  e.  _V )
40 ovex 6309 . . . . . . . . . . . . . . 15  |-  ( 1 ... M )  e. 
_V
41 fex 6133 . . . . . . . . . . . . . . 15  |-  ( ( H : ( 1 ... M ) --> A  /\  ( 1 ... M )  e.  _V )  ->  H  e.  _V )
423, 40, 41sylancl 662 . . . . . . . . . . . . . 14  |-  ( ph  ->  H  e.  _V )
4339, 42jca 532 . . . . . . . . . . . . 13  |-  ( ph  ->  ( F  e.  _V  /\  H  e.  _V )
)
44 f1fun 5783 . . . . . . . . . . . . . . 15  |-  ( H : ( 1 ... M ) -1-1-> A  ->  Fun  H )
451, 44syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  Fun  H )
4645, 20jca 532 . . . . . . . . . . . . 13  |-  ( ph  ->  ( Fun  H  /\  ( F supp  .0.  )  C_  ran  H ) )
47 imacosupp 6940 . . . . . . . . . . . . 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 60 . . . . . . . . . . . 12  |-  ( ph  ->  ( H " (
( F  o.  H
) supp  .0.  ) )  =  ( F supp  .0.  ) )
4948adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  W  =/=  (/) )  ->  ( H " ( ( F  o.  H ) supp  .0.  )
)  =  ( F supp 
.0.  ) )
5037, 49syl5eq 2520 . . . . . . . . . 10  |-  ( (
ph  /\  W  =/=  (/) )  ->  ( H " W )  =  ( F supp  .0.  ) )
5150adantr 465 . . . . . . . . 9  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  -> 
( H " W
)  =  ( F supp 
.0.  ) )
52 f1oeq3 5809 . . . . . . . . 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 16 . . . . . . . 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 210 . . . . . . 7  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  -> 
( H  |`  W ) : W -1-1-onto-> ( F supp  .0.  )
)
55 f1oen3g 7531 . . . . . . 7  |-  ( ( ( H  |`  W )  e.  _V  /\  ( H  |`  W ) : W -1-1-onto-> ( F supp  .0.  )
)  ->  W  ~~  ( F supp  .0.  ) )
5625, 54, 55syl2anc 661 . . . . . 6  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  ->  W  ~~  ( F supp  .0.  ) )
57 fzfi 12050 . . . . . . . . 9  |-  ( 1 ... M )  e. 
Fin
58 ssfi 7740 . . . . . . . . 9  |-  ( ( ( 1 ... M
)  e.  Fin  /\  W  C_  ( 1 ... M ) )  ->  W  e.  Fin )
5957, 33, 58sylancr 663 . . . . . . . 8  |-  ( ph  ->  W  e.  Fin )
6059ad2antrr 725 . . . . . . 7  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  ->  W  e.  Fin )
61 f1f1orn 5827 . . . . . . . . . . . . 13  |-  ( H : ( 1 ... M ) -1-1-> A  ->  H : ( 1 ... M ) -1-1-onto-> ran  H )
621, 61syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  H : ( 1 ... M ) -1-1-onto-> ran  H
)
63 f1oen3g 7531 . . . . . . . . . . . 12  |-  ( ( H  e.  _V  /\  H : ( 1 ... M ) -1-1-onto-> ran  H )  -> 
( 1 ... M
)  ~~  ran  H )
647, 62, 63syl2anc 661 . . . . . . . . . . 11  |-  ( ph  ->  ( 1 ... M
)  ~~  ran  H )
65 enfi 7736 . . . . . . . . . . 11  |-  ( ( 1 ... M ) 
~~  ran  H  ->  ( ( 1 ... M
)  e.  Fin  <->  ran  H  e. 
Fin ) )
6664, 65syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( ( 1 ... M )  e.  Fin  <->  ran  H  e.  Fin ) )
6757, 66mpbii 211 . . . . . . . . 9  |-  ( ph  ->  ran  H  e.  Fin )
68 ssfi 7740 . . . . . . . . 9  |-  ( ( ran  H  e.  Fin  /\  ( F supp  .0.  )  C_ 
ran  H )  -> 
( F supp  .0.  )  e.  Fin )
6967, 20, 68syl2anc 661 . . . . . . . 8  |-  ( ph  ->  ( F supp  .0.  )  e.  Fin )
7069ad2antrr 725 . . . . . . 7  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  -> 
( F supp  .0.  )  e.  Fin )
71 hashen 12388 . . . . . . 7  |-  ( ( W  e.  Fin  /\  ( F supp  .0.  )  e. 
Fin )  ->  (
( # `  W )  =  ( # `  ( F supp  .0.  ) )  <->  W  ~~  ( F supp  .0.  ) ) )
7260, 70, 71syl2anc 661 . . . . . 6  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  -> 
( ( # `  W
)  =  ( # `  ( F supp  .0.  )
)  <->  W  ~~  ( F supp 
.0.  ) ) )
7356, 72mpbird 232 . . . . 5  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  -> 
( # `  W )  =  ( # `  ( F supp  .0.  ) ) )
7473fveq2d 5870 . . . 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 532 . . 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 5807 . . . . 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 5161 . . . . . . . 8  |-  ( g  =  ( H  o.  f )  ->  ( F  o.  g )  =  ( F  o.  ( H  o.  f
) ) )
7877seqeq3d 12083 . . . . . . 7  |-  ( g  =  ( H  o.  f )  ->  seq 1 (  .+  , 
( F  o.  g
) )  =  seq 1 (  .+  , 
( F  o.  ( H  o.  f )
) ) )
7978fveq1d 5868 . . . . . 6  |-  ( g  =  ( H  o.  f )  ->  (  seq 1 (  .+  , 
( F  o.  g
) ) `  ( # `
 ( F supp  .0.  ) ) )  =  (  seq 1 ( 
.+  ,  ( F  o.  ( H  o.  f ) ) ) `
 ( # `  ( F supp  .0.  ) ) ) )
8079eqeq2d 2481 . . . . 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 710 . . . 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 3199 . . 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 60 . 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 725 . . . . . . 7  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  ->  G  e.  Mnd )
855ad2antrr 725 . . . . . . 7  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  ->  A  e.  V )
8617ad2antrr 725 . . . . . . 7  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  ->  F : A --> B )
8718ad2antrr 725 . . . . . . 7  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  ->  ran  F  C_  ( Z `  ran  F ) )
8821neeq1i 2752 . . . . . . . . . 10  |-  ( W  =/=  (/)  <->  ( ( F  o.  H ) supp  .0.  )  =/=  (/) )
89 supp0cosupp0 6939 . . . . . . . . . . . 12  |-  ( ( F  e.  _V  /\  H  e.  _V )  ->  ( ( F supp  .0.  )  =  (/)  ->  (
( F  o.  H
) supp  .0.  )  =  (/) ) )
9089necon3d 2691 . . . . . . . . . . 11  |-  ( ( F  e.  _V  /\  H  e.  _V )  ->  ( ( ( F  o.  H ) supp  .0.  )  =/=  (/)  ->  ( F supp  .0.  )  =/=  (/) ) )
9139, 42, 90syl2anc 661 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( F  o.  H ) supp  .0.  )  =/=  (/)  ->  ( F supp  .0.  )  =/=  (/) ) )
9288, 91syl5bi 217 . . . . . . . . 9  |-  ( ph  ->  ( W  =/=  (/)  ->  ( F supp  .0.  )  =/=  (/) ) )
9392imp 429 . . . . . . . 8  |-  ( (
ph  /\  W  =/=  (/) )  ->  ( F supp  .0.  )  =/=  (/) )
9493adantr 465 . . . . . . 7  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  -> 
( F supp  .0.  )  =/=  (/) )
9520ad2antrr 725 . . . . . . . 8  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  -> 
( F supp  .0.  )  C_ 
ran  H )
96 frn 5737 . . . . . . . . . 10  |-  ( H : ( 1 ... M ) --> A  ->  ran  H  C_  A )
973, 96syl 16 . . . . . . . . 9  |-  ( ph  ->  ran  H  C_  A
)
9897ad2antrr 725 . . . . . . . 8  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  ->  ran  H  C_  A )
9995, 98sstrd 3514 . . . . . . 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 16710 . . . . . 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 5565 . . . . . 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 16 . . . . 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 2467 . . . . . . 7  |-  ( F supp 
.0.  )  =  ( F supp  .0.  )
104 simprl 755 . . . . . . 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 16708 . . . . . 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 2469 . . . . 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 256 . . . 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 1695 . . 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 5876 . . . 4  |-  (  seq 1 (  .+  , 
( F  o.  ( H  o.  f )
) ) `  ( # `
 W ) )  e.  _V
110 eqeq1 2471 . . . . . . 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 703 . . . . . 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 1690 . . . . 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 2482 . . . . 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 321 . . . 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 3204 . . 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 16 . 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 210 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 184    /\ wa 369   A.wal 1377    = wceq 1379   E.wex 1596    e. wcel 1767   E!weu 2275    =/= wne 2662   _Vcvv 3113    C_ wss 3476   (/)c0 3785   class class class wbr 4447   dom cdm 4999   ran crn 5000    |` cres 5001   "cima 5002    o. ccom 5003   iotacio 5549   Fun wfun 5582   -->wf 5584   -1-1->wf1 5585   -1-1-onto->wf1o 5587   ` cfv 5588    Isom wiso 5589  (class class class)co 6284   supp csupp 6901    ~~ cen 7513   Fincfn 7516   1c1 9493    < clt 9628   NNcn 10536   ...cfz 11672    seqcseq 12075   #chash 12373   Basecbs 14490   +g cplusg 14555   0gc0g 14695    gsumg cgsu 14696   Mndcmnd 15726  Cntzccntz 16158
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-supp 6902  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-card 8320  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-nn 10537  df-n0 10796  df-z 10865  df-uz 11083  df-fz 11673  df-fzo 11793  df-seq 12076  df-hash 12374  df-0g 14697  df-gsum 14698  df-mnd 15732  df-cntz 16160
This theorem is referenced by:  gsumval3  16714
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