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Theorem gsumval3lem1 16374
Description: Lemma 1 for gsumval3 16376. (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 6698 . . . . . . . . 9  |-  ( ( F  o.  H ) supp 
.0.  )  C_  dom  ( F  o.  H
)
53, 4eqsstri 3381 . . . . . . . 8  |-  W  C_  dom  ( F  o.  H
)
6 gsumval3.f . . . . . . . . . 10  |-  ( ph  ->  F : A --> B )
7 f1f 5601 . . . . . . . . . . 11  |-  ( H : ( 1 ... M ) -1-1-> A  ->  H : ( 1 ... M ) --> A )
81, 7syl 16 . . . . . . . . . 10  |-  ( ph  ->  H : ( 1 ... M ) --> A )
9 fco 5563 . . . . . . . . . 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 5558 . . . . . . . . 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 3399 . . . . . . 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 5650 . . . . . 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 5162 . . . . . . 7  |-  ( H
" W )  =  ( H " (
( F  o.  H
) supp  .0.  ) )
18 gsumval3.a . . . . . . . . . . 11  |-  ( ph  ->  A  e.  V )
19 fex 5945 . . . . . . . . . . 11  |-  ( ( F : A --> B  /\  A  e.  V )  ->  F  e.  _V )
206, 18, 19syl2anc 661 . . . . . . . . . 10  |-  ( ph  ->  F  e.  _V )
21 ovex 6111 . . . . . . . . . . . 12  |-  ( 1 ... M )  e. 
_V
22 fex 5945 . . . . . . . . . . . 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 5603 . . . . . . . . . . . 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 6724 . . . . . . . . 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 2482 . . . . . 6  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  -> 
( H " W
)  =  ( F supp 
.0.  ) )
35 f1oeq3 5629 . . . . . 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 6011 . . . . 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 5658 . . . 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 5636 . . . . 5  |-  ( f : ( 1 ... ( # `  W
) ) -1-1-onto-> W  ->  f :
( 1 ... ( # `
 W ) ) --> W )
43 frn 5560 . . . . 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 5336 . . . 4  |-  ( ran  f  C_  W  ->  ( ( H  |`  W )  o.  f )  =  ( H  o.  f
) )
46 f1oeq1 5627 . . . 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 11786 . . . . . . . . . 10  |-  ( 1 ... M )  e. 
Fin
5049a1i 11 . . . . . . . . 9  |-  ( ph  ->  ( 1 ... M
)  e.  Fin )
51 fex2 6527 . . . . . . . . 9  |-  ( ( H : ( 1 ... M ) --> A  /\  ( 1 ... M )  e.  Fin  /\  A  e.  V )  ->  H  e.  _V )
528, 50, 18, 51syl3anc 1218 . . . . . . . 8  |-  ( ph  ->  H  e.  _V )
53 resexg 5144 . . . . . . . 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 5164 . . . . . . . . 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 2470 . . . . . . . 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 7317 . . . . . 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 7525 . . . . . . . 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 5647 . . . . . . . . . . . 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 7317 . . . . . . . . . . 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 7521 . . . . . . . . . 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 7525 . . . . . . . 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 12110 . . . . . 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 6102 . . 3  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  -> 
( 1 ... ( # `
 W ) )  =  ( 1 ... ( # `  ( F supp  .0.  ) ) ) )
83 f1oeq2 5628 . . 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 1369    e. wcel 1756    =/= wne 2601   _Vcvv 2967    C_ wss 3323   (/)c0 3632   class class class wbr 4287   dom cdm 4835   ran crn 4836    |` cres 4837   "cima 4838    o. ccom 4839   Fun wfun 5407   -->wf 5409   -1-1->wf1 5410   -1-1-onto->wf1o 5412   ` cfv 5413    Isom wiso 5414  (class class class)co 6086   supp csupp 6685    ~~ cen 7299   Fincfn 7302   1c1 9275    < clt 9410   NNcn 10314   ...cfz 11429   #chash 12095   Basecbs 14166   +g cplusg 14230   0gc0g 14370   Mndcmnd 15401  Cntzccntz 15824
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 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351
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 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-supp 6686  df-recs 6824  df-rdg 6858  df-1o 6912  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-card 8101  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-nn 10315  df-n0 10572  df-z 10639  df-uz 10854  df-fz 11430  df-hash 12096
This theorem is referenced by:  gsumval3lem2  16375
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