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Theorem gsumval3lem1 17035
Description: Lemma 1 for gsumval3 17037. (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 6930 . . . . . . . . 9  |-  ( ( F  o.  H ) supp 
.0.  )  C_  dom  ( F  o.  H
)
53, 4eqsstri 3529 . . . . . . . 8  |-  W  C_  dom  ( F  o.  H
)
6 gsumval3.f . . . . . . . . . 10  |-  ( ph  ->  F : A --> B )
7 f1f 5787 . . . . . . . . . . 11  |-  ( H : ( 1 ... M ) -1-1-> A  ->  H : ( 1 ... M ) --> A )
81, 7syl 16 . . . . . . . . . 10  |-  ( ph  ->  H : ( 1 ... M ) --> A )
9 fco 5747 . . . . . . . . . 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 5741 . . . . . . . . 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 3547 . . . . . . 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 5836 . . . . . 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 5345 . . . . . . 7  |-  ( H
" W )  =  ( H " (
( F  o.  H
) supp  .0.  ) )
18 gsumval3.a . . . . . . . . . . 11  |-  ( ph  ->  A  e.  V )
19 fex 6146 . . . . . . . . . . 11  |-  ( ( F : A --> B  /\  A  e.  V )  ->  F  e.  _V )
206, 18, 19syl2anc 661 . . . . . . . . . 10  |-  ( ph  ->  F  e.  _V )
21 ovex 6324 . . . . . . . . . . . 12  |-  ( 1 ... M )  e. 
_V
22 fex 6146 . . . . . . . . . . . 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 5789 . . . . . . . . . . . 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 6958 . . . . . . . . 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 2510 . . . . . 6  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  -> 
( H " W
)  =  ( F supp 
.0.  ) )
35 f1oeq3 5815 . . . . . 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 6222 . . . . 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 5844 . . . 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 5822 . . . . 5  |-  ( f : ( 1 ... ( # `  W
) ) -1-1-onto-> W  ->  f :
( 1 ... ( # `
 W ) ) --> W )
43 frn 5743 . . . . 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 5516 . . . 4  |-  ( ran  f  C_  W  ->  ( ( H  |`  W )  o.  f )  =  ( H  o.  f
) )
46 f1oeq1 5813 . . . 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 12084 . . . . . . . . . 10  |-  ( 1 ... M )  e. 
Fin
5049a1i 11 . . . . . . . . 9  |-  ( ph  ->  ( 1 ... M
)  e.  Fin )
51 fex2 6754 . . . . . . . . 9  |-  ( ( H : ( 1 ... M ) --> A  /\  ( 1 ... M )  e.  Fin  /\  A  e.  V )  ->  H  e.  _V )
528, 50, 18, 51syl3anc 1228 . . . . . . . 8  |-  ( ph  ->  H  e.  _V )
53 resexg 5326 . . . . . . . 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 5347 . . . . . . . . 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 2498 . . . . . . . 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 7550 . . . . . 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 7759 . . . . . . . 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 5833 . . . . . . . . . . . 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 7550 . . . . . . . . . . 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 7755 . . . . . . . . . 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 7759 . . . . . . . 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 12422 . . . . . 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 6312 . . 3  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  -> 
( 1 ... ( # `
 W ) )  =  ( 1 ... ( # `  ( F supp  .0.  ) ) ) )
83 f1oeq2 5814 . . 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 1395    e. wcel 1819    =/= wne 2652   _Vcvv 3109    C_ wss 3471   (/)c0 3793   class class class wbr 4456   dom cdm 5008   ran crn 5009    |` cres 5010   "cima 5011    o. ccom 5012   Fun wfun 5588   -->wf 5590   -1-1->wf1 5591   -1-1-onto->wf1o 5593   ` cfv 5594    Isom wiso 5595  (class class class)co 6296   supp csupp 6917    ~~ cen 7532   Fincfn 7535   1c1 9510    < clt 9645   NNcn 10556   ...cfz 11697   #chash 12407   Basecbs 14643   +g cplusg 14711   0gc0g 14856   Mndcmnd 16045  Cntzccntz 16479
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-supp 6918  df-recs 7060  df-rdg 7094  df-1o 7148  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-card 8337  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-hash 12408
This theorem is referenced by:  gsumval3lem2  17036
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