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Theorem gsumval3lem1 16496
Description: Lemma 1 for gsumval3 16498. (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
gsumval3lem1  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  -> 
( H  o.  f
) : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
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 gsumval3lem1
StepHypRef Expression
1 gsumval3.h . . . . . . 7  |-  ( ph  ->  H : ( 1 ... M ) -1-1-> A
)
21ad2antrr 725 . . . . . 6  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  ->  H : ( 1 ... M ) -1-1-> A )
3 gsumval3.w . . . . . . . . 9  |-  W  =  ( ( F  o.  H ) supp  .0.  )
4 suppssdm 6806 . . . . . . . . 9  |-  ( ( F  o.  H ) supp 
.0.  )  C_  dom  ( F  o.  H
)
53, 4eqsstri 3487 . . . . . . . 8  |-  W  C_  dom  ( F  o.  H
)
6 gsumval3.f . . . . . . . . . 10  |-  ( ph  ->  F : A --> B )
7 f1f 5707 . . . . . . . . . . 11  |-  ( H : ( 1 ... M ) -1-1-> A  ->  H : ( 1 ... M ) --> A )
81, 7syl 16 . . . . . . . . . 10  |-  ( ph  ->  H : ( 1 ... M ) --> A )
9 fco 5669 . . . . . . . . . 10  |-  ( ( F : A --> B  /\  H : ( 1 ... M ) --> A )  ->  ( F  o.  H ) : ( 1 ... M ) --> B )
106, 8, 9syl2anc 661 . . . . . . . . 9  |-  ( ph  ->  ( F  o.  H
) : ( 1 ... M ) --> B )
11 fdm 5664 . . . . . . . . 9  |-  ( ( F  o.  H ) : ( 1 ... M ) --> B  ->  dom  ( F  o.  H
)  =  ( 1 ... M ) )
1210, 11syl 16 . . . . . . . 8  |-  ( ph  ->  dom  ( F  o.  H )  =  ( 1 ... M ) )
135, 12syl5sseq 3505 . . . . . . 7  |-  ( ph  ->  W  C_  ( 1 ... M ) )
1413ad2antrr 725 . . . . . 6  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  ->  W  C_  ( 1 ... M ) )
15 f1ores 5756 . . . . . 6  |-  ( ( H : ( 1 ... M ) -1-1-> A  /\  W  C_  ( 1 ... M ) )  ->  ( H  |`  W ) : W -1-1-onto-> ( H " W ) )
162, 14, 15syl2anc 661 . . . . 5  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  -> 
( H  |`  W ) : W -1-1-onto-> ( H " W
) )
173imaeq2i 5268 . . . . . . 7  |-  ( H
" W )  =  ( H " (
( F  o.  H
) supp  .0.  ) )
18 gsumval3.a . . . . . . . . . . 11  |-  ( ph  ->  A  e.  V )
19 fex 6052 . . . . . . . . . . 11  |-  ( ( F : A --> B  /\  A  e.  V )  ->  F  e.  _V )
206, 18, 19syl2anc 661 . . . . . . . . . 10  |-  ( ph  ->  F  e.  _V )
21 ovex 6218 . . . . . . . . . . . 12  |-  ( 1 ... M )  e. 
_V
22 fex 6052 . . . . . . . . . . . 12  |-  ( ( H : ( 1 ... M ) --> A  /\  ( 1 ... M )  e.  _V )  ->  H  e.  _V )
237, 21, 22sylancl 662 . . . . . . . . . . 11  |-  ( H : ( 1 ... M ) -1-1-> A  ->  H  e.  _V )
241, 23syl 16 . . . . . . . . . 10  |-  ( ph  ->  H  e.  _V )
25 f1fun 5709 . . . . . . . . . . . 12  |-  ( H : ( 1 ... M ) -1-1-> A  ->  Fun  H )
261, 25syl 16 . . . . . . . . . . 11  |-  ( ph  ->  Fun  H )
27 gsumval3.n . . . . . . . . . . 11  |-  ( ph  ->  ( F supp  .0.  )  C_ 
ran  H )
2826, 27jca 532 . . . . . . . . . 10  |-  ( ph  ->  ( Fun  H  /\  ( F supp  .0.  )  C_  ran  H ) )
2920, 24, 28jca31 534 . . . . . . . . 9  |-  ( ph  ->  ( ( F  e. 
_V  /\  H  e.  _V )  /\  ( Fun  H  /\  ( F supp 
.0.  )  C_  ran  H ) ) )
3029ad2antrr 725 . . . . . . . 8  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  -> 
( ( F  e. 
_V  /\  H  e.  _V )  /\  ( Fun  H  /\  ( F supp 
.0.  )  C_  ran  H ) ) )
31 imacosupp 6832 . . . . . . . . 9  |-  ( ( F  e.  _V  /\  H  e.  _V )  ->  ( ( Fun  H  /\  ( F supp  .0.  )  C_ 
ran  H )  -> 
( H " (
( F  o.  H
) supp  .0.  ) )  =  ( F supp  .0.  ) ) )
3231imp 429 . . . . . . . 8  |-  ( ( ( F  e.  _V  /\  H  e.  _V )  /\  ( Fun  H  /\  ( F supp  .0.  )  C_  ran  H ) )  -> 
( H " (
( F  o.  H
) supp  .0.  ) )  =  ( F supp  .0.  ) )
3330, 32syl 16 . . . . . . 7  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  -> 
( H " (
( F  o.  H
) supp  .0.  ) )  =  ( F supp  .0.  ) )
3417, 33syl5eq 2504 . . . . . 6  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  -> 
( H " W
)  =  ( F supp 
.0.  ) )
35 f1oeq3 5735 . . . . . 6  |-  ( ( H " W )  =  ( F supp  .0.  )  ->  ( ( H  |`  W ) : W -1-1-onto-> ( H " W )  <->  ( H  |`  W ) : W -1-1-onto-> ( F supp  .0.  ) ) )
3634, 35syl 16 . . . . 5  |-  ( ( ( 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.  ) ) )
3716, 36mpbid 210 . . . 4  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  -> 
( H  |`  W ) : W -1-1-onto-> ( F supp  .0.  )
)
38 isof1o 6118 . . . . 5  |-  ( f 
Isom  <  ,  <  (
( 1 ... ( # `
 W ) ) ,  W )  -> 
f : ( 1 ... ( # `  W
) ) -1-1-onto-> W )
3938ad2antll 728 . . . 4  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  -> 
f : ( 1 ... ( # `  W
) ) -1-1-onto-> W )
40 f1oco 5764 . . . 4  |-  ( ( ( H  |`  W ) : W -1-1-onto-> ( F supp  .0.  )  /\  f : ( 1 ... ( # `  W
) ) -1-1-onto-> W )  ->  (
( H  |`  W )  o.  f ) : ( 1 ... ( # `
 W ) ) -1-1-onto-> ( F supp  .0.  ) )
4137, 39, 40syl2anc 661 . . 3  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  -> 
( ( H  |`  W )  o.  f
) : ( 1 ... ( # `  W
) ) -1-1-onto-> ( F supp  .0.  )
)
42 f1of 5742 . . . . 5  |-  ( f : ( 1 ... ( # `  W
) ) -1-1-onto-> W  ->  f :
( 1 ... ( # `
 W ) ) --> W )
43 frn 5666 . . . . 5  |-  ( f : ( 1 ... ( # `  W
) ) --> W  ->  ran  f  C_  W )
4439, 42, 433syl 20 . . . 4  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  ->  ran  f  C_  W )
45 cores 5442 . . . 4  |-  ( ran  f  C_  W  ->  ( ( H  |`  W )  o.  f )  =  ( H  o.  f
) )
46 f1oeq1 5733 . . . 4  |-  ( ( ( H  |`  W )  o.  f )  =  ( H  o.  f
)  ->  ( (
( H  |`  W )  o.  f ) : ( 1 ... ( # `
 W ) ) -1-1-onto-> ( F supp  .0.  )  <->  ( H  o.  f ) : ( 1 ... ( # `  W ) ) -1-1-onto-> ( F supp 
.0.  ) ) )
4744, 45, 463syl 20 . . 3  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  -> 
( ( ( H  |`  W )  o.  f
) : ( 1 ... ( # `  W
) ) -1-1-onto-> ( F supp  .0.  )  <->  ( H  o.  f ) : ( 1 ... ( # `  W
) ) -1-1-onto-> ( F supp  .0.  )
) )
4841, 47mpbid 210 . 2  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  -> 
( H  o.  f
) : ( 1 ... ( # `  W
) ) -1-1-onto-> ( F supp  .0.  )
)
49 fzfi 11904 . . . . . . . . . 10  |-  ( 1 ... M )  e. 
Fin
5049a1i 11 . . . . . . . . 9  |-  ( ph  ->  ( 1 ... M
)  e.  Fin )
51 fex2 6635 . . . . . . . . 9  |-  ( ( H : ( 1 ... M ) --> A  /\  ( 1 ... M )  e.  Fin  /\  A  e.  V )  ->  H  e.  _V )
528, 50, 18, 51syl3anc 1219 . . . . . . . 8  |-  ( ph  ->  H  e.  _V )
53 resexg 5250 . . . . . . . 8  |-  ( H  e.  _V  ->  ( H  |`  W )  e. 
_V )
5452, 53syl 16 . . . . . . 7  |-  ( ph  ->  ( H  |`  W )  e.  _V )
5554ad2antrr 725 . . . . . 6  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  -> 
( H  |`  W )  e.  _V )
563a1i 11 . . . . . . . . . 10  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  ->  W  =  ( ( F  o.  H ) supp  .0.  ) )
5756imaeq2d 5270 . . . . . . . . 9  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  -> 
( H " W
)  =  ( H
" ( ( F  o.  H ) supp  .0.  ) ) )
5820, 52, 28jca31 534 . . . . . . . . . . 11  |-  ( ph  ->  ( ( F  e. 
_V  /\  H  e.  _V )  /\  ( Fun  H  /\  ( F supp 
.0.  )  C_  ran  H ) ) )
5958ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  -> 
( ( F  e. 
_V  /\  H  e.  _V )  /\  ( Fun  H  /\  ( F supp 
.0.  )  C_  ran  H ) ) )
6059, 32syl 16 . . . . . . . . 9  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  -> 
( H " (
( F  o.  H
) supp  .0.  ) )  =  ( F supp  .0.  ) )
6157, 60eqtrd 2492 . . . . . . . 8  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  -> 
( H " W
)  =  ( F supp 
.0.  ) )
6261, 35syl 16 . . . . . . 7  |-  ( ( ( 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.  ) ) )
6316, 62mpbid 210 . . . . . 6  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  -> 
( H  |`  W ) : W -1-1-onto-> ( F supp  .0.  )
)
64 f1oen3g 7428 . . . . . 6  |-  ( ( ( H  |`  W )  e.  _V  /\  ( H  |`  W ) : W -1-1-onto-> ( F supp  .0.  )
)  ->  W  ~~  ( F supp  .0.  ) )
6555, 63, 64syl2anc 661 . . . . 5  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  ->  W  ~~  ( F supp  .0.  ) )
66 ssfi 7637 . . . . . . . 8  |-  ( ( ( 1 ... M
)  e.  Fin  /\  W  C_  ( 1 ... M ) )  ->  W  e.  Fin )
6749, 13, 66sylancr 663 . . . . . . 7  |-  ( ph  ->  W  e.  Fin )
6867ad2antrr 725 . . . . . 6  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  ->  W  e.  Fin )
69 f1f1orn 5753 . . . . . . . . . . . 12  |-  ( H : ( 1 ... M ) -1-1-> A  ->  H : ( 1 ... M ) -1-1-onto-> ran  H )
701, 69syl 16 . . . . . . . . . . 11  |-  ( ph  ->  H : ( 1 ... M ) -1-1-onto-> ran  H
)
71 f1oen3g 7428 . . . . . . . . . . 11  |-  ( ( H  e.  _V  /\  H : ( 1 ... M ) -1-1-onto-> ran  H )  -> 
( 1 ... M
)  ~~  ran  H )
7252, 70, 71syl2anc 661 . . . . . . . . . 10  |-  ( ph  ->  ( 1 ... M
)  ~~  ran  H )
73 enfi 7633 . . . . . . . . . 10  |-  ( ( 1 ... M ) 
~~  ran  H  ->  ( ( 1 ... M
)  e.  Fin  <->  ran  H  e. 
Fin ) )
7472, 73syl 16 . . . . . . . . 9  |-  ( ph  ->  ( ( 1 ... M )  e.  Fin  <->  ran  H  e.  Fin ) )
7549, 74mpbii 211 . . . . . . . 8  |-  ( ph  ->  ran  H  e.  Fin )
76 ssfi 7637 . . . . . . . 8  |-  ( ( ran  H  e.  Fin  /\  ( F supp  .0.  )  C_ 
ran  H )  -> 
( F supp  .0.  )  e.  Fin )
7775, 27, 76syl2anc 661 . . . . . . 7  |-  ( ph  ->  ( F supp  .0.  )  e.  Fin )
7877ad2antrr 725 . . . . . 6  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  -> 
( F supp  .0.  )  e.  Fin )
79 hashen 12228 . . . . . 6  |-  ( ( W  e.  Fin  /\  ( F supp  .0.  )  e. 
Fin )  ->  (
( # `  W )  =  ( # `  ( F supp  .0.  ) )  <->  W  ~~  ( F supp  .0.  ) ) )
8068, 78, 79syl2anc 661 . . . . 5  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  -> 
( ( # `  W
)  =  ( # `  ( F supp  .0.  )
)  <->  W  ~~  ( F supp 
.0.  ) ) )
8165, 80mpbird 232 . . . 4  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  -> 
( # `  W )  =  ( # `  ( F supp  .0.  ) ) )
8281oveq2d 6209 . . 3  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  -> 
( 1 ... ( # `
 W ) )  =  ( 1 ... ( # `  ( F supp  .0.  ) ) ) )
83 f1oeq2 5734 . . 3  |-  ( ( 1 ... ( # `  W ) )  =  ( 1 ... ( # `
 ( F supp  .0.  ) ) )  -> 
( ( H  o.  f ) : ( 1 ... ( # `  W ) ) -1-1-onto-> ( F supp 
.0.  )  <->  ( H  o.  f ) : ( 1 ... ( # `  ( F supp  .0.  )
) ) -1-1-onto-> ( F supp  .0.  )
) )
8482, 83syl 16 . 2  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  -> 
( ( H  o.  f ) : ( 1 ... ( # `  W ) ) -1-1-onto-> ( F supp 
.0.  )  <->  ( H  o.  f ) : ( 1 ... ( # `  ( F supp  .0.  )
) ) -1-1-onto-> ( F supp  .0.  )
) )
8548, 84mpbid 210 1  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  -> 
( H  o.  f
) : ( 1 ... ( # `  ( F supp  .0.  ) ) ) -1-1-onto-> ( F supp  .0.  ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2644   _Vcvv 3071    C_ wss 3429   (/)c0 3738   class class class wbr 4393   dom cdm 4941   ran crn 4942    |` cres 4943   "cima 4944    o. ccom 4945   Fun wfun 5513   -->wf 5515   -1-1->wf1 5516   -1-1-onto->wf1o 5518   ` cfv 5519    Isom wiso 5520  (class class class)co 6193   supp csupp 6793    ~~ cen 7410   Fincfn 7413   1c1 9387    < clt 9522   NNcn 10426   ...cfz 11547   #chash 12213   Basecbs 14285   +g cplusg 14349   0gc0g 14489   Mndcmnd 15520  Cntzccntz 15944
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-int 4230  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-isom 5528  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-om 6580  df-1st 6680  df-2nd 6681  df-supp 6794  df-recs 6935  df-rdg 6969  df-1o 7023  df-er 7204  df-en 7414  df-dom 7415  df-sdom 7416  df-fin 7417  df-card 8213  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-nn 10427  df-n0 10684  df-z 10751  df-uz 10966  df-fz 11548  df-hash 12214
This theorem is referenced by:  gsumval3lem2  16497
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