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Theorem gsumval3lem2 16382
Description: Lemma 2 for gsumval3 16383. (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 5604 . . . . . . 7  |-  ( H : ( 1 ... M ) -1-1-> A  ->  H : ( 1 ... M ) --> A )
31, 2syl 16 . . . . . 6  |-  ( ph  ->  H : ( 1 ... M ) --> A )
4 fzfid 11793 . . . . . 6  |-  ( ph  ->  ( 1 ... M
)  e.  Fin )
5 gsumval3.a . . . . . 6  |-  ( ph  ->  A  e.  V )
6 fex2 6530 . . . . . 6  |-  ( ( H : ( 1 ... M ) --> A  /\  ( 1 ... M )  e.  Fin  /\  A  e.  V )  ->  H  e.  _V )
73, 4, 5, 6syl3anc 1218 . . . . 5  |-  ( ph  ->  H  e.  _V )
8 vex 2973 . . . . 5  |-  f  e. 
_V
9 coexg 6526 . . . . 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 16381 . . . 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 5147 . . . . . . . . 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 6701 . . . . . . . . . . . 12  |-  ( ( F  o.  H ) supp 
.0.  )  C_  dom  ( F  o.  H
)
2821, 27eqsstri 3384 . . . . . . . . . . 11  |-  W  C_  dom  ( F  o.  H
)
29 fco 5566 . . . . . . . . . . . . 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 5561 . . . . . . . . . . . 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 3402 . . . . . . . . . 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 5653 . . . . . . . . 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 5165 . . . . . . . . . . 11  |-  ( H
" W )  =  ( H " (
( F  o.  H
) supp  .0.  ) )
38 fex 5948 . . . . . . . . . . . . . . 15  |-  ( ( F : A --> B  /\  A  e.  V )  ->  F  e.  _V )
3917, 5, 38syl2anc 661 . . . . . . . . . . . . . 14  |-  ( ph  ->  F  e.  _V )
40 ovex 6114 . . . . . . . . . . . . . . 15  |-  ( 1 ... M )  e. 
_V
41 fex 5948 . . . . . . . . . . . . . . 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 5606 . . . . . . . . . . . . . . 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 6727 . . . . . . . . . . . . 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 2485 . . . . . . . . . 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 5632 . . . . . . . . 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 7323 . . . . . . 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 11792 . . . . . . . . 9  |-  ( 1 ... M )  e. 
Fin
58 ssfi 7531 . . . . . . . . 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 5650 . . . . . . . . . . . . 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 7323 . . . . . . . . . . . 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 7527 . . . . . . . . . . 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 7531 . . . . . . . . 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 12116 . . . . . . 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 5693 . . . 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 5630 . . . . 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 4996 . . . . . . . 8  |-  ( g  =  ( H  o.  f )  ->  ( F  o.  g )  =  ( F  o.  ( H  o.  f
) ) )
7877seqeq3d 11812 . . . . . . 7  |-  ( g  =  ( H  o.  f )  ->  seq 1 (  .+  , 
( F  o.  g
) )  =  seq 1 (  .+  , 
( F  o.  ( H  o.  f )
) ) )
7978fveq1d 5691 . . . . . 6  |-  ( g  =  ( H  o.  f )  ->  (  seq 1 (  .+  , 
( F  o.  g
) ) `  ( # `
 ( F supp  .0.  ) ) )  =  (  seq 1 ( 
.+  ,  ( F  o.  ( H  o.  f ) ) ) `
 ( # `  ( F supp  .0.  ) ) ) )
8079eqeq2d 2452 . . . . 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 3056 . . 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 2616 . . . . . . . . . 10  |-  ( W  =/=  (/)  <->  ( ( F  o.  H ) supp  .0.  )  =/=  (/) )
89 supp0cosupp0 6726 . . . . . . . . . . . 12  |-  ( ( F  e.  _V  /\  H  e.  _V )  ->  ( ( F supp  .0.  )  =  (/)  ->  (
( F  o.  H
) supp  .0.  )  =  (/) ) )
9089necon3d 2644 . . . . . . . . . . 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 5563 . . . . . . . . . 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 3364 . . . . . . 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 16379 . . . . . 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 5393 . . . . . 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 2441 . . . . . . 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 16377 . . . . . 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 2449 . . . . 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 1685 . . 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 5699 . . . 4  |-  (  seq 1 (  .+  , 
( F  o.  ( H  o.  f )
) ) `  ( # `
 W ) )  e.  _V
110 eqeq1 2447 . . . . . . 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 1680 . . . . 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 2450 . . . . 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 3061 . . 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 1367    = wceq 1369   E.wex 1586    e. wcel 1756   E!weu 2253    =/= wne 2604   _Vcvv 2970    C_ wss 3326   (/)c0 3635   class class class wbr 4290   dom cdm 4838   ran crn 4839    |` cres 4840   "cima 4841    o. ccom 4842   iotacio 5377   Fun wfun 5410   -->wf 5412   -1-1->wf1 5413   -1-1-onto->wf1o 5415   ` cfv 5416    Isom wiso 5417  (class class class)co 6089   supp csupp 6688    ~~ cen 7305   Fincfn 7308   1c1 9281    < clt 9416   NNcn 10320   ...cfz 11435    seqcseq 11804   #chash 12101   Basecbs 14172   +g cplusg 14236   0gc0g 14376    gsumg cgsu 14377   Mndcmnd 15407  Cntzccntz 15831
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-cnex 9336  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356  ax-pre-mulgt0 9357
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-int 4127  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-isom 5425  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-om 6475  df-1st 6575  df-2nd 6576  df-supp 6689  df-recs 6830  df-rdg 6864  df-1o 6918  df-oadd 6922  df-er 7099  df-en 7309  df-dom 7310  df-sdom 7311  df-fin 7312  df-card 8107  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-sub 9595  df-neg 9596  df-nn 10321  df-n0 10578  df-z 10645  df-uz 10860  df-fz 11436  df-fzo 11547  df-seq 11805  df-hash 12102  df-0g 14378  df-gsum 14379  df-mnd 15413  df-cntz 15833
This theorem is referenced by:  gsumval3  16383
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