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Theorem gsumval3lem1 16700
Description: Lemma 1 for gsumval3 16702. (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 6911 . . . . . . . . 9  |-  ( ( F  o.  H ) supp 
.0.  )  C_  dom  ( F  o.  H
)
53, 4eqsstri 3534 . . . . . . . 8  |-  W  C_  dom  ( F  o.  H
)
6 gsumval3.f . . . . . . . . . 10  |-  ( ph  ->  F : A --> B )
7 f1f 5779 . . . . . . . . . . 11  |-  ( H : ( 1 ... M ) -1-1-> A  ->  H : ( 1 ... M ) --> A )
81, 7syl 16 . . . . . . . . . 10  |-  ( ph  ->  H : ( 1 ... M ) --> A )
9 fco 5739 . . . . . . . . . 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 5733 . . . . . . . . 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 3552 . . . . . . 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 5828 . . . . . 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 5333 . . . . . . 7  |-  ( H
" W )  =  ( H " (
( F  o.  H
) supp  .0.  ) )
18 gsumval3.a . . . . . . . . . . 11  |-  ( ph  ->  A  e.  V )
19 fex 6131 . . . . . . . . . . 11  |-  ( ( F : A --> B  /\  A  e.  V )  ->  F  e.  _V )
206, 18, 19syl2anc 661 . . . . . . . . . 10  |-  ( ph  ->  F  e.  _V )
21 ovex 6307 . . . . . . . . . . . 12  |-  ( 1 ... M )  e. 
_V
22 fex 6131 . . . . . . . . . . . 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 5781 . . . . . . . . . . . 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 6937 . . . . . . . . 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 2520 . . . . . 6  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  -> 
( H " W
)  =  ( F supp 
.0.  ) )
35 f1oeq3 5807 . . . . . 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 6207 . . . . 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 5836 . . . 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 5814 . . . . 5  |-  ( f : ( 1 ... ( # `  W
) ) -1-1-onto-> W  ->  f :
( 1 ... ( # `
 W ) ) --> W )
43 frn 5735 . . . . 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 5508 . . . 4  |-  ( ran  f  C_  W  ->  ( ( H  |`  W )  o.  f )  =  ( H  o.  f
) )
46 f1oeq1 5805 . . . 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 12046 . . . . . . . . . 10  |-  ( 1 ... M )  e. 
Fin
5049a1i 11 . . . . . . . . 9  |-  ( ph  ->  ( 1 ... M
)  e.  Fin )
51 fex2 6736 . . . . . . . . 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 5314 . . . . . . . 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 5335 . . . . . . . . 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 2508 . . . . . . . 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 7528 . . . . . 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 7737 . . . . . . . 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 5825 . . . . . . . . . . . 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 7528 . . . . . . . . . . 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 7733 . . . . . . . . . 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 7737 . . . . . . . 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 12384 . . . . . 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 6298 . . 3  |-  ( ( ( ph  /\  W  =/=  (/) )  /\  ( -.  A  e.  ran  ... 
/\  f  Isom  <  ,  <  ( ( 1 ... ( # `  W
) ) ,  W
) ) )  -> 
( 1 ... ( # `
 W ) )  =  ( 1 ... ( # `  ( F supp  .0.  ) ) ) )
83 f1oeq2 5806 . . 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 1379    e. wcel 1767    =/= 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   Fun wfun 5580   -->wf 5582   -1-1->wf1 5583   -1-1-onto->wf1o 5585   ` cfv 5586    Isom wiso 5587  (class class class)co 6282   supp csupp 6898    ~~ cen 7510   Fincfn 7513   1c1 9489    < clt 9624   NNcn 10532   ...cfz 11668   #chash 12369   Basecbs 14486   +g cplusg 14551   0gc0g 14691   Mndcmnd 15722  Cntzccntz 16148
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 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
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-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 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-supp 6899  df-recs 7039  df-rdg 7073  df-1o 7127  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-card 8316  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-n0 10792  df-z 10861  df-uz 11079  df-fz 11669  df-hash 12370
This theorem is referenced by:  gsumval3lem2  16701
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