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Theorem dchrfi 20600
Description: The group of Dirichlet characters is a finite group. (Contributed by Mario Carneiro, 19-Apr-2016.)
Hypotheses
Ref Expression
dchrabl.g  |-  G  =  (DChr `  N )
dchrfi.b  |-  D  =  ( Base `  G
)
Assertion
Ref Expression
dchrfi  |-  ( N  e.  NN  ->  D  e.  Fin )

Proof of Theorem dchrfi
Dummy variables  x  f  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 snfi 7026 . . . 4  |-  { 0 }  e.  Fin
2 cnex 8905 . . . . . . . . 9  |-  CC  e.  _V
32a1i 10 . . . . . . . 8  |-  ( N  e.  NN  ->  CC  e.  _V )
4 ovex 5967 . . . . . . . . 9  |-  ( z ^ ( phi `  N ) )  e. 
_V
54a1i 10 . . . . . . . 8  |-  ( ( N  e.  NN  /\  z  e.  CC )  ->  ( z ^ ( phi `  N ) )  e.  _V )
6 ax-1cn 8882 . . . . . . . . 9  |-  1  e.  CC
76a1i 10 . . . . . . . 8  |-  ( ( N  e.  NN  /\  z  e.  CC )  ->  1  e.  CC )
8 eqidd 2359 . . . . . . . 8  |-  ( N  e.  NN  ->  (
z  e.  CC  |->  ( z ^ ( phi `  N ) ) )  =  ( z  e.  CC  |->  ( z ^
( phi `  N
) ) ) )
9 fconstmpt 4811 . . . . . . . . 9  |-  ( CC 
X.  { 1 } )  =  ( z  e.  CC  |->  1 )
109a1i 10 . . . . . . . 8  |-  ( N  e.  NN  ->  ( CC  X.  { 1 } )  =  ( z  e.  CC  |->  1 ) )
113, 5, 7, 8, 10offval2 6179 . . . . . . 7  |-  ( N  e.  NN  ->  (
( z  e.  CC  |->  ( z ^ ( phi `  N ) ) )  o F  -  ( CC  X.  { 1 } ) )  =  ( z  e.  CC  |->  ( ( z ^
( phi `  N
) )  -  1 ) ) )
12 ssid 3273 . . . . . . . . . 10  |-  CC  C_  CC
1312a1i 10 . . . . . . . . 9  |-  ( N  e.  NN  ->  CC  C_  CC )
146a1i 10 . . . . . . . . 9  |-  ( N  e.  NN  ->  1  e.  CC )
15 phicl 12928 . . . . . . . . . 10  |-  ( N  e.  NN  ->  ( phi `  N )  e.  NN )
1615nnnn0d 10107 . . . . . . . . 9  |-  ( N  e.  NN  ->  ( phi `  N )  e. 
NN0 )
17 plypow 19685 . . . . . . . . 9  |-  ( ( CC  C_  CC  /\  1  e.  CC  /\  ( phi `  N )  e.  NN0 )  ->  ( z  e.  CC  |->  ( z ^
( phi `  N
) ) )  e.  (Poly `  CC )
)
1813, 14, 16, 17syl3anc 1182 . . . . . . . 8  |-  ( N  e.  NN  ->  (
z  e.  CC  |->  ( z ^ ( phi `  N ) ) )  e.  (Poly `  CC ) )
19 plyconst 19686 . . . . . . . . 9  |-  ( ( CC  C_  CC  /\  1  e.  CC )  ->  ( CC  X.  { 1 } )  e.  (Poly `  CC ) )
2012, 6, 19mp2an 653 . . . . . . . 8  |-  ( CC 
X.  { 1 } )  e.  (Poly `  CC )
21 plysubcl 19702 . . . . . . . 8  |-  ( ( ( z  e.  CC  |->  ( z ^ ( phi `  N ) ) )  e.  (Poly `  CC )  /\  ( CC  X.  { 1 } )  e.  (Poly `  CC ) )  ->  (
( z  e.  CC  |->  ( z ^ ( phi `  N ) ) )  o F  -  ( CC  X.  { 1 } ) )  e.  (Poly `  CC )
)
2218, 20, 21sylancl 643 . . . . . . 7  |-  ( N  e.  NN  ->  (
( z  e.  CC  |->  ( z ^ ( phi `  N ) ) )  o F  -  ( CC  X.  { 1 } ) )  e.  (Poly `  CC )
)
2311, 22eqeltrrd 2433 . . . . . 6  |-  ( N  e.  NN  ->  (
z  e.  CC  |->  ( ( z ^ ( phi `  N ) )  -  1 ) )  e.  (Poly `  CC ) )
24 0cn 8918 . . . . . . 7  |-  0  e.  CC
25 ax-1ne0 8893 . . . . . . . . 9  |-  1  =/=  0
266, 25negne0i 9208 . . . . . . . 8  |-  -u 1  =/=  0
27150expd 11351 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  (
0 ^ ( phi `  N ) )  =  0 )
2827oveq1d 5957 . . . . . . . . . 10  |-  ( N  e.  NN  ->  (
( 0 ^ ( phi `  N ) )  -  1 )  =  ( 0  -  1 ) )
29 oveq1 5949 . . . . . . . . . . . . 13  |-  ( z  =  0  ->  (
z ^ ( phi `  N ) )  =  ( 0 ^ ( phi `  N ) ) )
3029oveq1d 5957 . . . . . . . . . . . 12  |-  ( z  =  0  ->  (
( z ^ ( phi `  N ) )  -  1 )  =  ( ( 0 ^ ( phi `  N
) )  -  1 ) )
31 eqid 2358 . . . . . . . . . . . 12  |-  ( z  e.  CC  |->  ( ( z ^ ( phi `  N ) )  - 
1 ) )  =  ( z  e.  CC  |->  ( ( z ^
( phi `  N
) )  -  1 ) )
32 ovex 5967 . . . . . . . . . . . 12  |-  ( ( 0 ^ ( phi `  N ) )  - 
1 )  e.  _V
3330, 31, 32fvmpt 5682 . . . . . . . . . . 11  |-  ( 0  e.  CC  ->  (
( z  e.  CC  |->  ( ( z ^
( phi `  N
) )  -  1 ) ) `  0
)  =  ( ( 0 ^ ( phi `  N ) )  - 
1 ) )
3424, 33ax-mp 8 . . . . . . . . . 10  |-  ( ( z  e.  CC  |->  ( ( z ^ ( phi `  N ) )  -  1 ) ) `
 0 )  =  ( ( 0 ^ ( phi `  N
) )  -  1 )
35 df-neg 9127 . . . . . . . . . 10  |-  -u 1  =  ( 0  -  1 )
3628, 34, 353eqtr4g 2415 . . . . . . . . 9  |-  ( N  e.  NN  ->  (
( z  e.  CC  |->  ( ( z ^
( phi `  N
) )  -  1 ) ) `  0
)  =  -u 1
)
3736neeq1d 2534 . . . . . . . 8  |-  ( N  e.  NN  ->  (
( ( z  e.  CC  |->  ( ( z ^ ( phi `  N ) )  - 
1 ) ) ` 
0 )  =/=  0  <->  -u 1  =/=  0 ) )
3826, 37mpbiri 224 . . . . . . 7  |-  ( N  e.  NN  ->  (
( z  e.  CC  |->  ( ( z ^
( phi `  N
) )  -  1 ) ) `  0
)  =/=  0 )
39 ne0p 19687 . . . . . . 7  |-  ( ( 0  e.  CC  /\  ( ( z  e.  CC  |->  ( ( z ^ ( phi `  N ) )  - 
1 ) ) ` 
0 )  =/=  0
)  ->  ( z  e.  CC  |->  ( ( z ^ ( phi `  N ) )  - 
1 ) )  =/=  0 p )
4024, 38, 39sylancr 644 . . . . . 6  |-  ( N  e.  NN  ->  (
z  e.  CC  |->  ( ( z ^ ( phi `  N ) )  -  1 ) )  =/=  0 p )
4131mptiniseg 5246 . . . . . . . . 9  |-  ( 0  e.  CC  ->  ( `' ( z  e.  CC  |->  ( ( z ^ ( phi `  N ) )  - 
1 ) ) " { 0 } )  =  { z  e.  CC  |  ( ( z ^ ( phi `  N ) )  - 
1 )  =  0 } )
4224, 41ax-mp 8 . . . . . . . 8  |-  ( `' ( z  e.  CC  |->  ( ( z ^
( phi `  N
) )  -  1 ) ) " {
0 } )  =  { z  e.  CC  |  ( ( z ^ ( phi `  N ) )  - 
1 )  =  0 }
4342eqcomi 2362 . . . . . . 7  |-  { z  e.  CC  |  ( ( z ^ ( phi `  N ) )  -  1 )  =  0 }  =  ( `' ( z  e.  CC  |->  ( ( z ^ ( phi `  N ) )  - 
1 ) ) " { 0 } )
4443fta1 19786 . . . . . 6  |-  ( ( ( z  e.  CC  |->  ( ( z ^
( phi `  N
) )  -  1 ) )  e.  (Poly `  CC )  /\  (
z  e.  CC  |->  ( ( z ^ ( phi `  N ) )  -  1 ) )  =/=  0 p )  ->  ( { z  e.  CC  |  ( ( z ^ ( phi `  N ) )  -  1 )  =  0 }  e.  Fin  /\  ( # `  {
z  e.  CC  | 
( ( z ^
( phi `  N
) )  -  1 )  =  0 } )  <_  (deg `  (
z  e.  CC  |->  ( ( z ^ ( phi `  N ) )  -  1 ) ) ) ) )
4523, 40, 44syl2anc 642 . . . . 5  |-  ( N  e.  NN  ->  ( { z  e.  CC  |  ( ( z ^ ( phi `  N ) )  - 
1 )  =  0 }  e.  Fin  /\  ( # `  { z  e.  CC  |  ( ( z ^ ( phi `  N ) )  -  1 )  =  0 } )  <_ 
(deg `  ( z  e.  CC  |->  ( ( z ^ ( phi `  N ) )  - 
1 ) ) ) ) )
4645simpld 445 . . . 4  |-  ( N  e.  NN  ->  { z  e.  CC  |  ( ( z ^ ( phi `  N ) )  -  1 )  =  0 }  e.  Fin )
47 unfi 7211 . . . 4  |-  ( ( { 0 }  e.  Fin  /\  { z  e.  CC  |  ( ( z ^ ( phi `  N ) )  - 
1 )  =  0 }  e.  Fin )  ->  ( { 0 }  u.  { z  e.  CC  |  ( ( z ^ ( phi `  N ) )  - 
1 )  =  0 } )  e.  Fin )
481, 46, 47sylancr 644 . . 3  |-  ( N  e.  NN  ->  ( { 0 }  u.  { z  e.  CC  | 
( ( z ^
( phi `  N
) )  -  1 )  =  0 } )  e.  Fin )
49 eqid 2358 . . . 4  |-  (ℤ/n `  N
)  =  (ℤ/n `  N
)
50 eqid 2358 . . . 4  |-  ( Base `  (ℤ/n `  N ) )  =  ( Base `  (ℤ/n `  N
) )
5149, 50znfi 16613 . . 3  |-  ( N  e.  NN  ->  ( Base `  (ℤ/n `  N ) )  e. 
Fin )
52 mapfi 7239 . . 3  |-  ( ( ( { 0 }  u.  { z  e.  CC  |  ( ( z ^ ( phi `  N ) )  - 
1 )  =  0 } )  e.  Fin  /\  ( Base `  (ℤ/n `  N
) )  e.  Fin )  ->  ( ( { 0 }  u.  {
z  e.  CC  | 
( ( z ^
( phi `  N
) )  -  1 )  =  0 } )  ^m  ( Base `  (ℤ/n `  N ) ) )  e.  Fin )
5348, 51, 52syl2anc 642 . 2  |-  ( N  e.  NN  ->  (
( { 0 }  u.  { z  e.  CC  |  ( ( z ^ ( phi `  N ) )  - 
1 )  =  0 } )  ^m  ( Base `  (ℤ/n `  N ) ) )  e.  Fin )
54 dchrabl.g . . . . . . . 8  |-  G  =  (DChr `  N )
55 dchrfi.b . . . . . . . 8  |-  D  =  ( Base `  G
)
56 simpr 447 . . . . . . . 8  |-  ( ( N  e.  NN  /\  f  e.  D )  ->  f  e.  D )
5754, 49, 55, 50, 56dchrf 20587 . . . . . . 7  |-  ( ( N  e.  NN  /\  f  e.  D )  ->  f : ( Base `  (ℤ/n `  N ) ) --> CC )
58 ffn 5469 . . . . . . 7  |-  ( f : ( Base `  (ℤ/n `  N
) ) --> CC  ->  f  Fn  ( Base `  (ℤ/n `  N
) ) )
5957, 58syl 15 . . . . . 6  |-  ( ( N  e.  NN  /\  f  e.  D )  ->  f  Fn  ( Base `  (ℤ/n `  N ) ) )
60 df-ne 2523 . . . . . . . . . . 11  |-  ( ( f `  x )  =/=  0  <->  -.  (
f `  x )  =  0 )
61 fvex 5619 . . . . . . . . . . . 12  |-  ( f `
 x )  e. 
_V
6261elsnc 3739 . . . . . . . . . . 11  |-  ( ( f `  x )  e.  { 0 }  <-> 
( f `  x
)  =  0 )
6360, 62xchbinxr 302 . . . . . . . . . 10  |-  ( ( f `  x )  =/=  0  <->  -.  (
f `  x )  e.  { 0 } )
64 simpl 443 . . . . . . . . . . . . 13  |-  ( ( x  e.  ( Base `  (ℤ/n `  N ) )  /\  ( f `  x
)  =/=  0 )  ->  x  e.  (
Base `  (ℤ/n `  N ) ) )
65 ffvelrn 5743 . . . . . . . . . . . . 13  |-  ( ( f : ( Base `  (ℤ/n `  N ) ) --> CC 
/\  x  e.  (
Base `  (ℤ/n `  N ) ) )  ->  ( f `  x )  e.  CC )
6657, 64, 65syl2an 463 . . . . . . . . . . . 12  |-  ( ( ( N  e.  NN  /\  f  e.  D )  /\  ( x  e.  ( Base `  (ℤ/n `  N
) )  /\  (
f `  x )  =/=  0 ) )  -> 
( f `  x
)  e.  CC )
6754, 49, 55dchrmhm 20586 . . . . . . . . . . . . . . . . . 18  |-  D  C_  ( (mulGrp `  (ℤ/n `  N ) ) MndHom  (mulGrp ` fld ) )
68 simplr 731 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( N  e.  NN  /\  f  e.  D )  /\  ( x  e.  ( Base `  (ℤ/n `  N
) )  /\  (
f `  x )  =/=  0 ) )  -> 
f  e.  D )
6967, 68sseldi 3254 . . . . . . . . . . . . . . . . 17  |-  ( ( ( N  e.  NN  /\  f  e.  D )  /\  ( x  e.  ( Base `  (ℤ/n `  N
) )  /\  (
f `  x )  =/=  0 ) )  -> 
f  e.  ( (mulGrp `  (ℤ/n `  N ) ) MndHom  (mulGrp ` fld ) ) )
7016ad2antrr 706 . . . . . . . . . . . . . . . . 17  |-  ( ( ( N  e.  NN  /\  f  e.  D )  /\  ( x  e.  ( Base `  (ℤ/n `  N
) )  /\  (
f `  x )  =/=  0 ) )  -> 
( phi `  N
)  e.  NN0 )
71 simprl 732 . . . . . . . . . . . . . . . . 17  |-  ( ( ( N  e.  NN  /\  f  e.  D )  /\  ( x  e.  ( Base `  (ℤ/n `  N
) )  /\  (
f `  x )  =/=  0 ) )  ->  x  e.  ( Base `  (ℤ/n `  N ) ) )
72 eqid 2358 . . . . . . . . . . . . . . . . . . 19  |-  (mulGrp `  (ℤ/n `  N ) )  =  (mulGrp `  (ℤ/n `  N ) )
7372, 50mgpbas 15424 . . . . . . . . . . . . . . . . . 18  |-  ( Base `  (ℤ/n `  N ) )  =  ( Base `  (mulGrp `  (ℤ/n `  N ) ) )
74 eqid 2358 . . . . . . . . . . . . . . . . . 18  |-  (.g `  (mulGrp `  (ℤ/n `  N ) ) )  =  (.g `  (mulGrp `  (ℤ/n `  N
) ) )
75 eqid 2358 . . . . . . . . . . . . . . . . . 18  |-  (.g `  (mulGrp ` fld ) )  =  (.g `  (mulGrp ` fld ) )
7673, 74, 75mhmmulg 14692 . . . . . . . . . . . . . . . . 17  |-  ( ( f  e.  ( (mulGrp `  (ℤ/n `  N ) ) MndHom  (mulGrp ` fld ) )  /\  ( phi `  N )  e.  NN0  /\  x  e.  ( Base `  (ℤ/n `  N ) ) )  ->  ( f `  ( ( phi `  N ) (.g `  (mulGrp `  (ℤ/n `  N ) ) ) x ) )  =  ( ( phi `  N ) (.g `  (mulGrp ` fld ) ) ( f `  x ) ) )
7769, 70, 71, 76syl3anc 1182 . . . . . . . . . . . . . . . 16  |-  ( ( ( N  e.  NN  /\  f  e.  D )  /\  ( x  e.  ( Base `  (ℤ/n `  N
) )  /\  (
f `  x )  =/=  0 ) )  -> 
( f `  (
( phi `  N
) (.g `  (mulGrp `  (ℤ/n `  N
) ) ) x ) )  =  ( ( phi `  N
) (.g `  (mulGrp ` fld ) ) ( f `
 x ) ) )
78 nnnn0 10061 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( N  e.  NN  ->  N  e.  NN0 )
7949zncrng 16598 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( N  e.  NN0  ->  (ℤ/n `  N
)  e.  CRing )
8078, 79syl 15 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( N  e.  NN  ->  (ℤ/n `  N
)  e.  CRing )
81 crngrng 15444 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( (ℤ/n `  N )  e.  CRing  -> 
(ℤ/n `  N )  e.  Ring )
8280, 81syl 15 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( N  e.  NN  ->  (ℤ/n `  N
)  e.  Ring )
8382ad2antrr 706 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( N  e.  NN  /\  f  e.  D )  /\  ( x  e.  ( Base `  (ℤ/n `  N
) )  /\  (
f `  x )  =/=  0 ) )  -> 
(ℤ/n `  N )  e.  Ring )
84 eqid 2358 . . . . . . . . . . . . . . . . . . . . . . 23  |-  (Unit `  (ℤ/n `  N ) )  =  (Unit `  (ℤ/n `  N ) )
85 eqid 2358 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( (mulGrp `  (ℤ/n `  N ) )s  (Unit `  (ℤ/n `  N ) ) )  =  ( (mulGrp `  (ℤ/n `  N ) )s  (Unit `  (ℤ/n `  N ) ) )
8684, 85unitgrp 15542 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( (ℤ/n `  N )  e.  Ring  -> 
( (mulGrp `  (ℤ/n `  N
) )s  (Unit `  (ℤ/n `  N ) ) )  e.  Grp )
8783, 86syl 15 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( N  e.  NN  /\  f  e.  D )  /\  ( x  e.  ( Base `  (ℤ/n `  N
) )  /\  (
f `  x )  =/=  0 ) )  -> 
( (mulGrp `  (ℤ/n `  N
) )s  (Unit `  (ℤ/n `  N ) ) )  e.  Grp )
8849, 84znunithash 16618 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( N  e.  NN  ->  ( # `
 (Unit `  (ℤ/n `  N
) ) )  =  ( phi `  N
) )
8988, 16eqeltrd 2432 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( N  e.  NN  ->  ( # `
 (Unit `  (ℤ/n `  N
) ) )  e. 
NN0 )
90 fvex 5619 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  (Unit `  (ℤ/n `  N ) )  e. 
_V
91 hashclb 11442 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( (Unit `  (ℤ/n `  N ) )  e. 
_V  ->  ( (Unit `  (ℤ/n `  N ) )  e. 
Fin 
<->  ( # `  (Unit `  (ℤ/n `  N ) ) )  e.  NN0 ) )
9290, 91ax-mp 8 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( (Unit `  (ℤ/n `  N ) )  e. 
Fin 
<->  ( # `  (Unit `  (ℤ/n `  N ) ) )  e.  NN0 )
9389, 92sylibr 203 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( N  e.  NN  ->  (Unit `  (ℤ/n `  N ) )  e. 
Fin )
9493ad2antrr 706 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( N  e.  NN  /\  f  e.  D )  /\  ( x  e.  ( Base `  (ℤ/n `  N
) )  /\  (
f `  x )  =/=  0 ) )  -> 
(Unit `  (ℤ/n `  N ) )  e. 
Fin )
95 simprr 733 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( N  e.  NN  /\  f  e.  D )  /\  ( x  e.  ( Base `  (ℤ/n `  N
) )  /\  (
f `  x )  =/=  0 ) )  -> 
( f `  x
)  =/=  0 )
9654, 49, 55, 50, 84, 68, 71dchrn0 20595 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( N  e.  NN  /\  f  e.  D )  /\  ( x  e.  ( Base `  (ℤ/n `  N
) )  /\  (
f `  x )  =/=  0 ) )  -> 
( ( f `  x )  =/=  0  <->  x  e.  (Unit `  (ℤ/n `  N
) ) ) )
9795, 96mpbid 201 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( N  e.  NN  /\  f  e.  D )  /\  ( x  e.  ( Base `  (ℤ/n `  N
) )  /\  (
f `  x )  =/=  0 ) )  ->  x  e.  (Unit `  (ℤ/n `  N
) ) )
9884, 85unitgrpbas 15541 . . . . . . . . . . . . . . . . . . . . . 22  |-  (Unit `  (ℤ/n `  N ) )  =  ( Base `  (
(mulGrp `  (ℤ/n `  N ) )s  (Unit `  (ℤ/n `  N ) ) ) )
99 eqid 2358 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( od
`  ( (mulGrp `  (ℤ/n `  N ) )s  (Unit `  (ℤ/n `  N ) ) ) )  =  ( od
`  ( (mulGrp `  (ℤ/n `  N ) )s  (Unit `  (ℤ/n `  N ) ) ) )
10098, 99oddvds2 14972 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( (mulGrp `  (ℤ/n `  N
) )s  (Unit `  (ℤ/n `  N ) ) )  e.  Grp  /\  (Unit `  (ℤ/n `  N ) )  e. 
Fin  /\  x  e.  (Unit `  (ℤ/n `  N ) ) )  ->  ( ( od
`  ( (mulGrp `  (ℤ/n `  N ) )s  (Unit `  (ℤ/n `  N ) ) ) ) `  x ) 
||  ( # `  (Unit `  (ℤ/n `  N ) ) ) )
10187, 94, 97, 100syl3anc 1182 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( N  e.  NN  /\  f  e.  D )  /\  ( x  e.  ( Base `  (ℤ/n `  N
) )  /\  (
f `  x )  =/=  0 ) )  -> 
( ( od `  ( (mulGrp `  (ℤ/n `  N ) )s  (Unit `  (ℤ/n `  N ) ) ) ) `  x ) 
||  ( # `  (Unit `  (ℤ/n `  N ) ) ) )
10288ad2antrr 706 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( N  e.  NN  /\  f  e.  D )  /\  ( x  e.  ( Base `  (ℤ/n `  N
) )  /\  (
f `  x )  =/=  0 ) )  -> 
( # `  (Unit `  (ℤ/n `  N ) ) )  =  ( phi `  N ) )
103101, 102breqtrd 4126 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( N  e.  NN  /\  f  e.  D )  /\  ( x  e.  ( Base `  (ℤ/n `  N
) )  /\  (
f `  x )  =/=  0 ) )  -> 
( ( od `  ( (mulGrp `  (ℤ/n `  N ) )s  (Unit `  (ℤ/n `  N ) ) ) ) `  x ) 
||  ( phi `  N ) )
10415ad2antrr 706 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( N  e.  NN  /\  f  e.  D )  /\  ( x  e.  ( Base `  (ℤ/n `  N
) )  /\  (
f `  x )  =/=  0 ) )  -> 
( phi `  N
)  e.  NN )
105104nnzd 10205 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( N  e.  NN  /\  f  e.  D )  /\  ( x  e.  ( Base `  (ℤ/n `  N
) )  /\  (
f `  x )  =/=  0 ) )  -> 
( phi `  N
)  e.  ZZ )
106 eqid 2358 . . . . . . . . . . . . . . . . . . . . 21  |-  (.g `  (
(mulGrp `  (ℤ/n `  N ) )s  (Unit `  (ℤ/n `  N ) ) ) )  =  (.g `  (
(mulGrp `  (ℤ/n `  N ) )s  (Unit `  (ℤ/n `  N ) ) ) )
107 eqid 2358 . . . . . . . . . . . . . . . . . . . . 21  |-  ( 0g
`  ( (mulGrp `  (ℤ/n `  N ) )s  (Unit `  (ℤ/n `  N ) ) ) )  =  ( 0g
`  ( (mulGrp `  (ℤ/n `  N ) )s  (Unit `  (ℤ/n `  N ) ) ) )
10898, 99, 106, 107oddvds 14955 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( (mulGrp `  (ℤ/n `  N
) )s  (Unit `  (ℤ/n `  N ) ) )  e.  Grp  /\  x  e.  (Unit `  (ℤ/n `  N ) )  /\  ( phi `  N )  e.  ZZ )  -> 
( ( ( od
`  ( (mulGrp `  (ℤ/n `  N ) )s  (Unit `  (ℤ/n `  N ) ) ) ) `  x ) 
||  ( phi `  N )  <->  ( ( phi `  N ) (.g `  ( (mulGrp `  (ℤ/n `  N
) )s  (Unit `  (ℤ/n `  N ) ) ) ) x )  =  ( 0g `  (
(mulGrp `  (ℤ/n `  N ) )s  (Unit `  (ℤ/n `  N ) ) ) ) ) )
10987, 97, 105, 108syl3anc 1182 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( N  e.  NN  /\  f  e.  D )  /\  ( x  e.  ( Base `  (ℤ/n `  N
) )  /\  (
f `  x )  =/=  0 ) )  -> 
( ( ( od
`  ( (mulGrp `  (ℤ/n `  N ) )s  (Unit `  (ℤ/n `  N ) ) ) ) `  x ) 
||  ( phi `  N )  <->  ( ( phi `  N ) (.g `  ( (mulGrp `  (ℤ/n `  N
) )s  (Unit `  (ℤ/n `  N ) ) ) ) x )  =  ( 0g `  (
(mulGrp `  (ℤ/n `  N ) )s  (Unit `  (ℤ/n `  N ) ) ) ) ) )
110103, 109mpbid 201 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( N  e.  NN  /\  f  e.  D )  /\  ( x  e.  ( Base `  (ℤ/n `  N
) )  /\  (
f `  x )  =/=  0 ) )  -> 
( ( phi `  N ) (.g `  (
(mulGrp `  (ℤ/n `  N ) )s  (Unit `  (ℤ/n `  N ) ) ) ) x )  =  ( 0g `  (
(mulGrp `  (ℤ/n `  N ) )s  (Unit `  (ℤ/n `  N ) ) ) ) )
11184, 72unitsubm 15545 . . . . . . . . . . . . . . . . . . . 20  |-  ( (ℤ/n `  N )  e.  Ring  -> 
(Unit `  (ℤ/n `  N ) )  e.  (SubMnd `  (mulGrp `  (ℤ/n `  N
) ) ) )
11283, 111syl 15 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( N  e.  NN  /\  f  e.  D )  /\  ( x  e.  ( Base `  (ℤ/n `  N
) )  /\  (
f `  x )  =/=  0 ) )  -> 
(Unit `  (ℤ/n `  N ) )  e.  (SubMnd `  (mulGrp `  (ℤ/n `  N
) ) ) )
11374, 85, 106submmulg 14695 . . . . . . . . . . . . . . . . . . 19  |-  ( ( (Unit `  (ℤ/n `  N ) )  e.  (SubMnd `  (mulGrp `  (ℤ/n `  N
) ) )  /\  ( phi `  N )  e.  NN0  /\  x  e.  (Unit `  (ℤ/n `  N ) ) )  ->  ( ( phi `  N ) (.g `  (mulGrp `  (ℤ/n `  N ) ) ) x )  =  ( ( phi `  N
) (.g `  ( (mulGrp `  (ℤ/n `  N ) )s  (Unit `  (ℤ/n `  N ) ) ) ) x ) )
114112, 70, 97, 113syl3anc 1182 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( N  e.  NN  /\  f  e.  D )  /\  ( x  e.  ( Base `  (ℤ/n `  N
) )  /\  (
f `  x )  =/=  0 ) )  -> 
( ( phi `  N ) (.g `  (mulGrp `  (ℤ/n `  N ) ) ) x )  =  ( ( phi `  N
) (.g `  ( (mulGrp `  (ℤ/n `  N ) )s  (Unit `  (ℤ/n `  N ) ) ) ) x ) )
115 eqid 2358 . . . . . . . . . . . . . . . . . . . . 21  |-  ( 1r
`  (ℤ/n `  N ) )  =  ( 1r `  (ℤ/n `  N
) )
11672, 115rngidval 15436 . . . . . . . . . . . . . . . . . . . 20  |-  ( 1r
`  (ℤ/n `  N ) )  =  ( 0g `  (mulGrp `  (ℤ/n `  N ) ) )
11785, 116subm0 14526 . . . . . . . . . . . . . . . . . . 19  |-  ( (Unit `  (ℤ/n `  N ) )  e.  (SubMnd `  (mulGrp `  (ℤ/n `  N
) ) )  -> 
( 1r `  (ℤ/n `  N
) )  =  ( 0g `  ( (mulGrp `  (ℤ/n `  N ) )s  (Unit `  (ℤ/n `  N ) ) ) ) )
118112, 117syl 15 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( N  e.  NN  /\  f  e.  D )  /\  ( x  e.  ( Base `  (ℤ/n `  N
) )  /\  (
f `  x )  =/=  0 ) )  -> 
( 1r `  (ℤ/n `  N
) )  =  ( 0g `  ( (mulGrp `  (ℤ/n `  N ) )s  (Unit `  (ℤ/n `  N ) ) ) ) )
119110, 114, 1183eqtr4d 2400 . . . . . . . . . . . . . . . . 17  |-  ( ( ( N  e.  NN  /\  f  e.  D )  /\  ( x  e.  ( Base `  (ℤ/n `  N
) )  /\  (
f `  x )  =/=  0 ) )  -> 
( ( phi `  N ) (.g `  (mulGrp `  (ℤ/n `  N ) ) ) x )  =  ( 1r `  (ℤ/n `  N
) ) )
120119fveq2d 5609 . . . . . . . . . . . . . . . 16  |-  ( ( ( N  e.  NN  /\  f  e.  D )  /\  ( x  e.  ( Base `  (ℤ/n `  N
) )  /\  (
f `  x )  =/=  0 ) )  -> 
( f `  (
( phi `  N
) (.g `  (mulGrp `  (ℤ/n `  N
) ) ) x ) )  =  ( f `  ( 1r
`  (ℤ/n `  N ) ) ) )
12177, 120eqtr3d 2392 . . . . . . . . . . . . . . 15  |-  ( ( ( N  e.  NN  /\  f  e.  D )  /\  ( x  e.  ( Base `  (ℤ/n `  N
) )  /\  (
f `  x )  =/=  0 ) )  -> 
( ( phi `  N ) (.g `  (mulGrp ` fld ) ) ( f `  x ) )  =  ( f `  ( 1r `  (ℤ/n `  N ) ) ) )
122 cnfldexp 16507 . . . . . . . . . . . . . . . 16  |-  ( ( ( f `  x
)  e.  CC  /\  ( phi `  N )  e.  NN0 )  -> 
( ( phi `  N ) (.g `  (mulGrp ` fld ) ) ( f `  x ) )  =  ( ( f `  x ) ^ ( phi `  N ) ) )
12366, 70, 122syl2anc 642 . . . . . . . . . . . . . . 15  |-  ( ( ( N  e.  NN  /\  f  e.  D )  /\  ( x  e.  ( Base `  (ℤ/n `  N
) )  /\  (
f `  x )  =/=  0 ) )  -> 
( ( phi `  N ) (.g `  (mulGrp ` fld ) ) ( f `  x ) )  =  ( ( f `  x ) ^ ( phi `  N ) ) )
124 eqid 2358 . . . . . . . . . . . . . . . . . 18  |-  (mulGrp ` fld )  =  (mulGrp ` fld )
125 cnfld1 16499 . . . . . . . . . . . . . . . . . 18  |-  1  =  ( 1r ` fld )
126124, 125rngidval 15436 . . . . . . . . . . . . . . . . 17  |-  1  =  ( 0g `  (mulGrp ` fld ) )
127116, 126mhm0 14516 . . . . . . . . . . . . . . . 16  |-  ( f  e.  ( (mulGrp `  (ℤ/n `  N ) ) MndHom  (mulGrp ` fld ) )  ->  ( f `  ( 1r `  (ℤ/n `  N
) ) )  =  1 )
12869, 127syl 15 . . . . . . . . . . . . . . 15  |-  ( ( ( N  e.  NN  /\  f  e.  D )  /\  ( x  e.  ( Base `  (ℤ/n `  N
) )  /\  (
f `  x )  =/=  0 ) )  -> 
( f `  ( 1r `  (ℤ/n `  N ) ) )  =  1 )
129121, 123, 1283eqtr3d 2398 . . . . . . . . . . . . . 14  |-  ( ( ( N  e.  NN  /\  f  e.  D )  /\  ( x  e.  ( Base `  (ℤ/n `  N
) )  /\  (
f `  x )  =/=  0 ) )  -> 
( ( f `  x ) ^ ( phi `  N ) )  =  1 )
130129oveq1d 5957 . . . . . . . . . . . . 13  |-  ( ( ( N  e.  NN  /\  f  e.  D )  /\  ( x  e.  ( Base `  (ℤ/n `  N
) )  /\  (
f `  x )  =/=  0 ) )  -> 
( ( ( f `
 x ) ^
( phi `  N
) )  -  1 )  =  ( 1  -  1 ) )
131 1m1e0 9901 . . . . . . . . . . . . 13  |-  ( 1  -  1 )  =  0
132130, 131syl6eq 2406 . . . . . . . . . . . 12  |-  ( ( ( N  e.  NN  /\  f  e.  D )  /\  ( x  e.  ( Base `  (ℤ/n `  N
) )  /\  (
f `  x )  =/=  0 ) )  -> 
( ( ( f `
 x ) ^
( phi `  N
) )  -  1 )  =  0 )
133 oveq1 5949 . . . . . . . . . . . . . . 15  |-  ( z  =  ( f `  x )  ->  (
z ^ ( phi `  N ) )  =  ( ( f `  x ) ^ ( phi `  N ) ) )
134133oveq1d 5957 . . . . . . . . . . . . . 14  |-  ( z  =  ( f `  x )  ->  (
( z ^ ( phi `  N ) )  -  1 )  =  ( ( ( f `
 x ) ^
( phi `  N
) )  -  1 ) )
135134eqeq1d 2366 . . . . . . . . . . . . 13  |-  ( z  =  ( f `  x )  ->  (
( ( z ^
( phi `  N
) )  -  1 )  =  0  <->  (
( ( f `  x ) ^ ( phi `  N ) )  -  1 )  =  0 ) )
136135elrab 2999 . . . . . . . . . . . 12  |-  ( ( f `  x )  e.  { z  e.  CC  |  ( ( z ^ ( phi `  N ) )  - 
1 )  =  0 }  <->  ( ( f `
 x )  e.  CC  /\  ( ( ( f `  x
) ^ ( phi `  N ) )  - 
1 )  =  0 ) )
13766, 132, 136sylanbrc 645 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN  /\  f  e.  D )  /\  ( x  e.  ( Base `  (ℤ/n `  N
) )  /\  (
f `  x )  =/=  0 ) )  -> 
( f `  x
)  e.  { z  e.  CC  |  ( ( z ^ ( phi `  N ) )  -  1 )  =  0 } )
138137expr 598 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  f  e.  D )  /\  x  e.  (
Base `  (ℤ/n `  N ) ) )  ->  ( ( f `
 x )  =/=  0  ->  ( f `  x )  e.  {
z  e.  CC  | 
( ( z ^
( phi `  N
) )  -  1 )  =  0 } ) )
13963, 138syl5bir 209 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  f  e.  D )  /\  x  e.  (
Base `  (ℤ/n `  N ) ) )  ->  ( -.  (
f `  x )  e.  { 0 }  ->  ( f `  x )  e.  { z  e.  CC  |  ( ( z ^ ( phi `  N ) )  - 
1 )  =  0 } ) )
140139orrd 367 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  f  e.  D )  /\  x  e.  (
Base `  (ℤ/n `  N ) ) )  ->  ( ( f `
 x )  e. 
{ 0 }  \/  ( f `  x
)  e.  { z  e.  CC  |  ( ( z ^ ( phi `  N ) )  -  1 )  =  0 } ) )
141 elun 3392 . . . . . . . 8  |-  ( ( f `  x )  e.  ( { 0 }  u.  { z  e.  CC  |  ( ( z ^ ( phi `  N ) )  -  1 )  =  0 } )  <->  ( (
f `  x )  e.  { 0 }  \/  ( f `  x
)  e.  { z  e.  CC  |  ( ( z ^ ( phi `  N ) )  -  1 )  =  0 } ) )
142140, 141sylibr 203 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  f  e.  D )  /\  x  e.  (
Base `  (ℤ/n `  N ) ) )  ->  ( f `  x )  e.  ( { 0 }  u.  { z  e.  CC  | 
( ( z ^
( phi `  N
) )  -  1 )  =  0 } ) )
143142ralrimiva 2702 . . . . . 6  |-  ( ( N  e.  NN  /\  f  e.  D )  ->  A. x  e.  (
Base `  (ℤ/n `  N ) ) ( f `  x )  e.  ( { 0 }  u.  { z  e.  CC  |  ( ( z ^ ( phi `  N ) )  -  1 )  =  0 } ) )
144 ffnfv 5765 . . . . . 6  |-  ( f : ( Base `  (ℤ/n `  N
) ) --> ( { 0 }  u.  {
z  e.  CC  | 
( ( z ^
( phi `  N
) )  -  1 )  =  0 } )  <->  ( f  Fn  ( Base `  (ℤ/n `  N
) )  /\  A. x  e.  ( Base `  (ℤ/n `  N ) ) ( f `  x )  e.  ( { 0 }  u.  { z  e.  CC  |  ( ( z ^ ( phi `  N ) )  -  1 )  =  0 } ) ) )
14559, 143, 144sylanbrc 645 . . . . 5  |-  ( ( N  e.  NN  /\  f  e.  D )  ->  f : ( Base `  (ℤ/n `  N ) ) --> ( { 0 }  u.  { z  e.  CC  | 
( ( z ^
( phi `  N
) )  -  1 )  =  0 } ) )
146145ex 423 . . . 4  |-  ( N  e.  NN  ->  (
f  e.  D  -> 
f : ( Base `  (ℤ/n `  N ) ) --> ( { 0 }  u.  { z  e.  CC  | 
( ( z ^
( phi `  N
) )  -  1 )  =  0 } ) ) )
147 elmapg 6870 . . . . 5  |-  ( ( ( { 0 }  u.  { z  e.  CC  |  ( ( z ^ ( phi `  N ) )  - 
1 )  =  0 } )  e.  Fin  /\  ( Base `  (ℤ/n `  N
) )  e.  Fin )  ->  ( f  e.  ( ( { 0 }  u.  { z  e.  CC  |  ( ( z ^ ( phi `  N ) )  -  1 )  =  0 } )  ^m  ( Base `  (ℤ/n `  N ) ) )  <-> 
f : ( Base `  (ℤ/n `  N ) ) --> ( { 0 }  u.  { z  e.  CC  | 
( ( z ^
( phi `  N
) )  -  1 )  =  0 } ) ) )
14848, 51, 147syl2anc 642 . . . 4  |-  ( N  e.  NN  ->  (
f  e.  ( ( { 0 }  u.  { z  e.  CC  | 
( ( z ^
( phi `  N
) )  -  1 )  =  0 } )  ^m  ( Base `  (ℤ/n `  N ) ) )  <-> 
f : ( Base `  (ℤ/n `  N ) ) --> ( { 0 }  u.  { z  e.  CC  | 
( ( z ^
( phi `  N
) )  -  1 )  =  0 } ) ) )
149146, 148sylibrd 225 . . 3  |-  ( N  e.  NN  ->  (
f  e.  D  -> 
f  e.  ( ( { 0 }  u.  { z  e.  CC  | 
( ( z ^
( phi `  N
) )  -  1 )  =  0 } )  ^m  ( Base `  (ℤ/n `  N ) ) ) ) )
150149ssrdv 3261 . 2  |-  ( N  e.  NN  ->  D  C_  ( ( { 0 }  u.  { z  e.  CC  |  ( ( z ^ ( phi `  N ) )  -  1 )  =  0 } )  ^m  ( Base `  (ℤ/n `  N ) ) ) )
151 ssfi 7168 . 2  |-  ( ( ( ( { 0 }  u.  { z  e.  CC  |  ( ( z ^ ( phi `  N ) )  -  1 )  =  0 } )  ^m  ( Base `  (ℤ/n `  N ) ) )  e.  Fin  /\  D  C_  ( ( { 0 }  u.  { z  e.  CC  |  ( ( z ^ ( phi `  N ) )  -  1 )  =  0 } )  ^m  ( Base `  (ℤ/n `  N ) ) ) )  ->  D  e.  Fin )
15253, 150, 151syl2anc 642 1  |-  ( N  e.  NN  ->  D  e.  Fin )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1642    e. wcel 1710    =/= wne 2521   A.wral 2619   {crab 2623   _Vcvv 2864    u. cun 3226    C_ wss 3228   {csn 3716   class class class wbr 4102    e. cmpt 4156    X. cxp 4766   `'ccnv 4767   "cima 4771    Fn wfn 5329   -->wf 5330   ` cfv 5334  (class class class)co 5942    o Fcof 6160    ^m cmap 6857   Fincfn 6948   CCcc 8822   0cc0 8824   1c1 8825    <_ cle 8955    - cmin 9124   -ucneg 9125   NNcn 9833   NN0cn0 10054   ZZcz 10113   ^cexp 11194   #chash 11427    || cdivides 12622   phicphi 12923   Basecbs 13239   ↾s cress 13240   0gc0g 13493   Grpcgrp 14455  .gcmg 14459   MndHom cmhm 14506  SubMndcsubmnd 14507   odcod 14933  mulGrpcmgp 15418   Ringcrg 15430   CRingccrg 15431   1rcur 15432  Unitcui 15514  ℂfldccnfld 16476  ℤ/nczn 16554   0 pc0p 19122  Polycply 19664  degcdgr 19667  DChrcdchr 20577
This theorem is referenced by:  sumdchr2  20615  dchrhash  20616  rpvmasum2  20767  dchrisum0re  20768
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-inf2 7429  ax-cnex 8880  ax-resscn 8881  ax-1cn 8882  ax-icn 8883  ax-addcl 8884  ax-addrcl 8885  ax-mulcl 8886  ax-mulrcl 8887  ax-mulcom 8888  ax-addass 8889  ax-mulass 8890  ax-distr 8891  ax-i2m1 8892  ax-1ne0 8893  ax-1rid 8894  ax-rnegex 8895  ax-rrecex 8896  ax-cnre 8897  ax-pre-lttri 8898  ax-pre-lttrn 8899  ax-pre-ltadd 8900  ax-pre-mulgt0 8901  ax-pre-sup 8902  ax-addf 8903  ax-mulf 8904
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-int 3942  df-iun 3986  df-disj 4073  df-br 4103  df-opab 4157  df-mpt 4158  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-se 4432  df-we 4433  df-ord 4474  df-on 4475  df-lim 4476  df-suc 4477  df-om 4736  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-isom 5343  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-of 6162  df-1st 6206  df-2nd 6207  df-tpos 6318  df-riota 6388  df-recs 6472  df-rdg 6507  df-1o 6563  df-2o 6564  df-oadd 6567  df-omul 6568  df-er 6744  df-ec 6746  df-qs 6750  df-map 6859  df-pm 6860  df-en 6949  df-dom 6950  df-sdom 6951  df-fin 6952  df-sup 7281  df-oi 7312  df-card 7659  df-acn 7662  df-cda 7881  df-pnf 8956  df-mnf 8957  df-xr 8958  df-ltxr 8959  df-le 8960  df-sub 9126  df-neg 9127  df-div 9511  df-nn 9834  df-2 9891  df-3 9892  df-4 9893  df-5 9894  df-6 9895  df-7 9896  df-8 9897  df-9 9898  df-10 9899  df-n0 10055  df-z 10114  df-dec 10214  df-uz 10320  df-rp 10444  df-fz 10872  df-fzo 10960  df-fl 11014  df-mod 11063  df-seq 11136  df-exp 11195  df-hash 11428  df-cj 11674  df-re 11675  df-im 11676  df-sqr 11810  df-abs 11811  df-clim 12052  df-rlim 12053  df-sum 12250  df-dvds 12623  df-gcd 12777  df-phi 12925  df-struct 13241  df-ndx 13242  df-slot 13243  df-base 13244  df-sets 13245  df-ress 13246  df-plusg 13312  df-mulr 13313  df-starv 13314  df-sca 13315  df-vsca 13316  df-tset 13318  df-ple 13319  df-ds 13321  df-unif 13322  df-0g 13497  df-imas 13504  df-divs 13505  df-mnd 14460  df-mhm 14508  df-submnd 14509  df-grp 14582  df-minusg 14583  df-sbg 14584  df-mulg 14585  df-subg 14711  df-nsg 14712  df-eqg 14713  df-ghm 14774  df-od 14937  df-cmn 15184  df-abl 15185  df-mgp 15419  df-rng 15433  df-cring 15434  df-ur 15435  df-oppr 15498  df-dvdsr 15516  df-unit 15517  df-invr 15547  df-rnghom 15589  df-subrg 15636  df-lmod 15722  df-lss 15783  df-lsp 15822  df-sra 16018  df-rgmod 16019  df-lidl 16020  df-rsp 16021  df-2idl 16077  df-cnfld 16477  df-zrh 16555  df-zn 16558  df-0p 19123  df-ply 19668  df-idp 19669  df-coe 19670  df-dgr 19671  df-quot 19769  df-dchr 20578
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