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Theorem dchrfi 21000
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 7154 . . . 4  |-  { 0 }  e.  Fin
2 cnex 9035 . . . . . . . . 9  |-  CC  e.  _V
32a1i 11 . . . . . . . 8  |-  ( N  e.  NN  ->  CC  e.  _V )
4 ovex 6073 . . . . . . . . 9  |-  ( z ^ ( phi `  N ) )  e. 
_V
54a1i 11 . . . . . . . 8  |-  ( ( N  e.  NN  /\  z  e.  CC )  ->  ( z ^ ( phi `  N ) )  e.  _V )
6 ax-1cn 9012 . . . . . . . . 9  |-  1  e.  CC
76a1i 11 . . . . . . . 8  |-  ( ( N  e.  NN  /\  z  e.  CC )  ->  1  e.  CC )
8 eqidd 2413 . . . . . . . 8  |-  ( N  e.  NN  ->  (
z  e.  CC  |->  ( z ^ ( phi `  N ) ) )  =  ( z  e.  CC  |->  ( z ^
( phi `  N
) ) ) )
9 fconstmpt 4888 . . . . . . . . 9  |-  ( CC 
X.  { 1 } )  =  ( z  e.  CC  |->  1 )
109a1i 11 . . . . . . . 8  |-  ( N  e.  NN  ->  ( CC  X.  { 1 } )  =  ( z  e.  CC  |->  1 ) )
113, 5, 7, 8, 10offval2 6289 . . . . . . 7  |-  ( N  e.  NN  ->  (
( z  e.  CC  |->  ( z ^ ( phi `  N ) ) )  o F  -  ( CC  X.  { 1 } ) )  =  ( z  e.  CC  |->  ( ( z ^
( phi `  N
) )  -  1 ) ) )
12 ssid 3335 . . . . . . . . . 10  |-  CC  C_  CC
1312a1i 11 . . . . . . . . 9  |-  ( N  e.  NN  ->  CC  C_  CC )
146a1i 11 . . . . . . . . 9  |-  ( N  e.  NN  ->  1  e.  CC )
15 phicl 13121 . . . . . . . . . 10  |-  ( N  e.  NN  ->  ( phi `  N )  e.  NN )
1615nnnn0d 10238 . . . . . . . . 9  |-  ( N  e.  NN  ->  ( phi `  N )  e. 
NN0 )
17 plypow 20085 . . . . . . . . 9  |-  ( ( CC  C_  CC  /\  1  e.  CC  /\  ( phi `  N )  e.  NN0 )  ->  ( z  e.  CC  |->  ( z ^
( phi `  N
) ) )  e.  (Poly `  CC )
)
1813, 14, 16, 17syl3anc 1184 . . . . . . . 8  |-  ( N  e.  NN  ->  (
z  e.  CC  |->  ( z ^ ( phi `  N ) ) )  e.  (Poly `  CC ) )
19 plyconst 20086 . . . . . . . . 9  |-  ( ( CC  C_  CC  /\  1  e.  CC )  ->  ( CC  X.  { 1 } )  e.  (Poly `  CC ) )
2012, 6, 19mp2an 654 . . . . . . . 8  |-  ( CC 
X.  { 1 } )  e.  (Poly `  CC )
21 plysubcl 20102 . . . . . . . 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 644 . . . . . . 7  |-  ( N  e.  NN  ->  (
( z  e.  CC  |->  ( z ^ ( phi `  N ) ) )  o F  -  ( CC  X.  { 1 } ) )  e.  (Poly `  CC )
)
2311, 22eqeltrrd 2487 . . . . . 6  |-  ( N  e.  NN  ->  (
z  e.  CC  |->  ( ( z ^ ( phi `  N ) )  -  1 ) )  e.  (Poly `  CC ) )
24 0cn 9048 . . . . . . 7  |-  0  e.  CC
25 ax-1ne0 9023 . . . . . . . . 9  |-  1  =/=  0
266, 25negne0i 9339 . . . . . . . 8  |-  -u 1  =/=  0
27150expd 11502 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  (
0 ^ ( phi `  N ) )  =  0 )
2827oveq1d 6063 . . . . . . . . . 10  |-  ( N  e.  NN  ->  (
( 0 ^ ( phi `  N ) )  -  1 )  =  ( 0  -  1 ) )
29 oveq1 6055 . . . . . . . . . . . . 13  |-  ( z  =  0  ->  (
z ^ ( phi `  N ) )  =  ( 0 ^ ( phi `  N ) ) )
3029oveq1d 6063 . . . . . . . . . . . 12  |-  ( z  =  0  ->  (
( z ^ ( phi `  N ) )  -  1 )  =  ( ( 0 ^ ( phi `  N
) )  -  1 ) )
31 eqid 2412 . . . . . . . . . . . 12  |-  ( z  e.  CC  |->  ( ( z ^ ( phi `  N ) )  - 
1 ) )  =  ( z  e.  CC  |->  ( ( z ^
( phi `  N
) )  -  1 ) )
32 ovex 6073 . . . . . . . . . . . 12  |-  ( ( 0 ^ ( phi `  N ) )  - 
1 )  e.  _V
3330, 31, 32fvmpt 5773 . . . . . . . . . . 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 9258 . . . . . . . . . 10  |-  -u 1  =  ( 0  -  1 )
3628, 34, 353eqtr4g 2469 . . . . . . . . 9  |-  ( N  e.  NN  ->  (
( z  e.  CC  |->  ( ( z ^
( phi `  N
) )  -  1 ) ) `  0
)  =  -u 1
)
3736neeq1d 2588 . . . . . . . 8  |-  ( N  e.  NN  ->  (
( ( z  e.  CC  |->  ( ( z ^ ( phi `  N ) )  - 
1 ) ) ` 
0 )  =/=  0  <->  -u 1  =/=  0 ) )
3826, 37mpbiri 225 . . . . . . 7  |-  ( N  e.  NN  ->  (
( z  e.  CC  |->  ( ( z ^
( phi `  N
) )  -  1 ) ) `  0
)  =/=  0 )
39 ne0p 20087 . . . . . . 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 645 . . . . . 6  |-  ( N  e.  NN  ->  (
z  e.  CC  |->  ( ( z ^ ( phi `  N ) )  -  1 ) )  =/=  0 p )
4131mptiniseg 5331 . . . . . . . . 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 2416 . . . . . . 7  |-  { z  e.  CC  |  ( ( z ^ ( phi `  N ) )  -  1 )  =  0 }  =  ( `' ( z  e.  CC  |->  ( ( z ^ ( phi `  N ) )  - 
1 ) ) " { 0 } )
4443fta1 20186 . . . . . 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 643 . . . . 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 446 . . . 4  |-  ( N  e.  NN  ->  { z  e.  CC  |  ( ( z ^ ( phi `  N ) )  -  1 )  =  0 }  e.  Fin )
47 unfi 7341 . . . 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 645 . . 3  |-  ( N  e.  NN  ->  ( { 0 }  u.  { z  e.  CC  | 
( ( z ^
( phi `  N
) )  -  1 )  =  0 } )  e.  Fin )
49 eqid 2412 . . . 4  |-  (ℤ/n `  N
)  =  (ℤ/n `  N
)
50 eqid 2412 . . . 4  |-  ( Base `  (ℤ/n `  N ) )  =  ( Base `  (ℤ/n `  N
) )
5149, 50znfi 16803 . . 3  |-  ( N  e.  NN  ->  ( Base `  (ℤ/n `  N ) )  e. 
Fin )
52 mapfi 7369 . . 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 643 . 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 448 . . . . . . . 8  |-  ( ( N  e.  NN  /\  f  e.  D )  ->  f  e.  D )
5754, 49, 55, 50, 56dchrf 20987 . . . . . . 7  |-  ( ( N  e.  NN  /\  f  e.  D )  ->  f : ( Base `  (ℤ/n `  N ) ) --> CC )
58 ffn 5558 . . . . . . 7  |-  ( f : ( Base `  (ℤ/n `  N
) ) --> CC  ->  f  Fn  ( Base `  (ℤ/n `  N
) ) )
5957, 58syl 16 . . . . . 6  |-  ( ( N  e.  NN  /\  f  e.  D )  ->  f  Fn  ( Base `  (ℤ/n `  N ) ) )
60 df-ne 2577 . . . . . . . . . . 11  |-  ( ( f `  x )  =/=  0  <->  -.  (
f `  x )  =  0 )
61 fvex 5709 . . . . . . . . . . . 12  |-  ( f `
 x )  e. 
_V
6261elsnc 3805 . . . . . . . . . . 11  |-  ( ( f `  x )  e.  { 0 }  <-> 
( f `  x
)  =  0 )
6360, 62xchbinxr 303 . . . . . . . . . 10  |-  ( ( f `  x )  =/=  0  <->  -.  (
f `  x )  e.  { 0 } )
64 simpl 444 . . . . . . . . . . . . 13  |-  ( ( x  e.  ( Base `  (ℤ/n `  N ) )  /\  ( f `  x
)  =/=  0 )  ->  x  e.  (
Base `  (ℤ/n `  N ) ) )
65 ffvelrn 5835 . . . . . . . . . . . . 13  |-  ( ( f : ( Base `  (ℤ/n `  N ) ) --> CC 
/\  x  e.  (
Base `  (ℤ/n `  N ) ) )  ->  ( f `  x )  e.  CC )
6657, 64, 65syl2an 464 . . . . . . . . . . . 12  |-  ( ( ( N  e.  NN  /\  f  e.  D )  /\  ( x  e.  ( Base `  (ℤ/n `  N
) )  /\  (
f `  x )  =/=  0 ) )  -> 
( f `  x
)  e.  CC )
6754, 49, 55dchrmhm 20986 . . . . . . . . . . . . . . . . . 18  |-  D  C_  ( (mulGrp `  (ℤ/n `  N ) ) MndHom  (mulGrp ` fld ) )
68 simplr 732 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( N  e.  NN  /\  f  e.  D )  /\  ( x  e.  ( Base `  (ℤ/n `  N
) )  /\  (
f `  x )  =/=  0 ) )  -> 
f  e.  D )
6967, 68sseldi 3314 . . . . . . . . . . . . . . . . 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 707 . . . . . . . . . . . . . . . . 17  |-  ( ( ( N  e.  NN  /\  f  e.  D )  /\  ( x  e.  ( Base `  (ℤ/n `  N
) )  /\  (
f `  x )  =/=  0 ) )  -> 
( phi `  N
)  e.  NN0 )
71 simprl 733 . . . . . . . . . . . . . . . . 17  |-  ( ( ( N  e.  NN  /\  f  e.  D )  /\  ( x  e.  ( Base `  (ℤ/n `  N
) )  /\  (
f `  x )  =/=  0 ) )  ->  x  e.  ( Base `  (ℤ/n `  N ) ) )
72 eqid 2412 . . . . . . . . . . . . . . . . . . 19  |-  (mulGrp `  (ℤ/n `  N ) )  =  (mulGrp `  (ℤ/n `  N ) )
7372, 50mgpbas 15617 . . . . . . . . . . . . . . . . . 18  |-  ( Base `  (ℤ/n `  N ) )  =  ( Base `  (mulGrp `  (ℤ/n `  N ) ) )
74 eqid 2412 . . . . . . . . . . . . . . . . . 18  |-  (.g `  (mulGrp `  (ℤ/n `  N ) ) )  =  (.g `  (mulGrp `  (ℤ/n `  N
) ) )
75 eqid 2412 . . . . . . . . . . . . . . . . . 18  |-  (.g `  (mulGrp ` fld ) )  =  (.g `  (mulGrp ` fld ) )
7673, 74, 75mhmmulg 14885 . . . . . . . . . . . . . . . . 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 1184 . . . . . . . . . . . . . . . 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 10192 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( N  e.  NN  ->  N  e.  NN0 )
7949zncrng 16788 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( N  e.  NN0  ->  (ℤ/n `  N
)  e.  CRing )
8078, 79syl 16 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( N  e.  NN  ->  (ℤ/n `  N
)  e.  CRing )
81 crngrng 15637 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( (ℤ/n `  N )  e.  CRing  -> 
(ℤ/n `  N )  e.  Ring )
8280, 81syl 16 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( N  e.  NN  ->  (ℤ/n `  N
)  e.  Ring )
8382ad2antrr 707 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( N  e.  NN  /\  f  e.  D )  /\  ( x  e.  ( Base `  (ℤ/n `  N
) )  /\  (
f `  x )  =/=  0 ) )  -> 
(ℤ/n `  N )  e.  Ring )
84 eqid 2412 . . . . . . . . . . . . . . . . . . . . . . 23  |-  (Unit `  (ℤ/n `  N ) )  =  (Unit `  (ℤ/n `  N ) )
85 eqid 2412 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( (mulGrp `  (ℤ/n `  N ) )s  (Unit `  (ℤ/n `  N ) ) )  =  ( (mulGrp `  (ℤ/n `  N ) )s  (Unit `  (ℤ/n `  N ) ) )
8684, 85unitgrp 15735 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( (ℤ/n `  N )  e.  Ring  -> 
( (mulGrp `  (ℤ/n `  N
) )s  (Unit `  (ℤ/n `  N ) ) )  e.  Grp )
8783, 86syl 16 . . . . . . . . . . . . . . . . . . . . 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 16808 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( N  e.  NN  ->  ( # `
 (Unit `  (ℤ/n `  N
) ) )  =  ( phi `  N
) )
8988, 16eqeltrd 2486 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( N  e.  NN  ->  ( # `
 (Unit `  (ℤ/n `  N
) ) )  e. 
NN0 )
90 fvex 5709 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  (Unit `  (ℤ/n `  N ) )  e. 
_V
91 hashclb 11604 . . . . . . . . . . . . . . . . . . . . . . . 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 204 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( N  e.  NN  ->  (Unit `  (ℤ/n `  N ) )  e. 
Fin )
9493ad2antrr 707 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( N  e.  NN  /\  f  e.  D )  /\  ( x  e.  ( Base `  (ℤ/n `  N
) )  /\  (
f `  x )  =/=  0 ) )  -> 
(Unit `  (ℤ/n `  N ) )  e. 
Fin )
95 simprr 734 . . . . . . . . . . . . . . . . . . . . . 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 20995 . . . . . . . . . . . . . . . . . . . . . 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 202 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( N  e.  NN  /\  f  e.  D )  /\  ( x  e.  ( Base `  (ℤ/n `  N
) )  /\  (
f `  x )  =/=  0 ) )  ->  x  e.  (Unit `  (ℤ/n `  N
) ) )
9884, 85unitgrpbas 15734 . . . . . . . . . . . . . . . . . . . . . 22  |-  (Unit `  (ℤ/n `  N ) )  =  ( Base `  (
(mulGrp `  (ℤ/n `  N ) )s  (Unit `  (ℤ/n `  N ) ) ) )
99 eqid 2412 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( od
`  ( (mulGrp `  (ℤ/n `  N ) )s  (Unit `  (ℤ/n `  N ) ) ) )  =  ( od
`  ( (mulGrp `  (ℤ/n `  N ) )s  (Unit `  (ℤ/n `  N ) ) ) )
10098, 99oddvds2 15165 . . . . . . . . . . . . . . . . . . . . 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 1184 . . . . . . . . . . . . . . . . . . . 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 707 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( N  e.  NN  /\  f  e.  D )  /\  ( x  e.  ( Base `  (ℤ/n `  N
) )  /\  (
f `  x )  =/=  0 ) )  -> 
( # `  (Unit `  (ℤ/n `  N ) ) )  =  ( phi `  N ) )
103101, 102breqtrd 4204 . . . . . . . . . . . . . . . . . . 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 707 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( N  e.  NN  /\  f  e.  D )  /\  ( x  e.  ( Base `  (ℤ/n `  N
) )  /\  (
f `  x )  =/=  0 ) )  -> 
( phi `  N
)  e.  NN )
105104nnzd 10338 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( N  e.  NN  /\  f  e.  D )  /\  ( x  e.  ( Base `  (ℤ/n `  N
) )  /\  (
f `  x )  =/=  0 ) )  -> 
( phi `  N
)  e.  ZZ )
106 eqid 2412 . . . . . . . . . . . . . . . . . . . . 21  |-  (.g `  (
(mulGrp `  (ℤ/n `  N ) )s  (Unit `  (ℤ/n `  N ) ) ) )  =  (.g `  (
(mulGrp `  (ℤ/n `  N ) )s  (Unit `  (ℤ/n `  N ) ) ) )
107 eqid 2412 . . . . . . . . . . . . . . . . . . . . 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 15148 . . . . . . . . . . . . . . . . . . . 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 1184 . . . . . . . . . . . . . . . . . . 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 202 . . . . . . . . . . . . . . . . . 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 15738 . . . . . . . . . . . . . . . . . . . 20  |-  ( (ℤ/n `  N )  e.  Ring  -> 
(Unit `  (ℤ/n `  N ) )  e.  (SubMnd `  (mulGrp `  (ℤ/n `  N
) ) ) )
11283, 111syl 16 . . . . . . . . . . . . . . . . . . 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 14888 . . . . . . . . . . . . . . . . . . 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 1184 . . . . . . . . . . . . . . . . . 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 2412 . . . . . . . . . . . . . . . . . . . . 21  |-  ( 1r
`  (ℤ/n `  N ) )  =  ( 1r `  (ℤ/n `  N
) )
11672, 115rngidval 15629 . . . . . . . . . . . . . . . . . . . 20  |-  ( 1r
`  (ℤ/n `  N ) )  =  ( 0g `  (mulGrp `  (ℤ/n `  N ) ) )
11785, 116subm0 14719 . . . . . . . . . . . . . . . . . . 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 16 . . . . . . . . . . . . . . . . . 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 2454 . . . . . . . . . . . . . . . . 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 5699 . . . . . . . . . . . . . . . 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 2446 . . . . . . . . . . . . . . 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 16697 . . . . . . . . . . . . . . . 16  |-  ( ( ( f `  x
)  e.  CC  /\  ( phi `  N )  e.  NN0 )  -> 
( ( phi `  N ) (.g `  (mulGrp ` fld ) ) ( f `  x ) )  =  ( ( f `  x ) ^ ( phi `  N ) ) )
12366, 70, 122syl2anc 643 . . . . . . . . . . . . . . 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 2412 . . . . . . . . . . . . . . . . . 18  |-  (mulGrp ` fld )  =  (mulGrp ` fld )
125 cnfld1 16689 . . . . . . . . . . . . . . . . . 18  |-  1  =  ( 1r ` fld )
126124, 125rngidval 15629 . . . . . . . . . . . . . . . . 17  |-  1  =  ( 0g `  (mulGrp ` fld ) )
127116, 126mhm0 14709 . . . . . . . . . . . . . . . 16  |-  ( f  e.  ( (mulGrp `  (ℤ/n `  N ) ) MndHom  (mulGrp ` fld ) )  ->  ( f `  ( 1r `  (ℤ/n `  N
) ) )  =  1 )
12869, 127syl 16 . . . . . . . . . . . . . . 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 2452 . . . . . . . . . . . . . 14  |-  ( ( ( N  e.  NN  /\  f  e.  D )  /\  ( x  e.  ( Base `  (ℤ/n `  N
) )  /\  (
f `  x )  =/=  0 ) )  -> 
( ( f `  x ) ^ ( phi `  N ) )  =  1 )
130129oveq1d 6063 . . . . . . . . . . . . 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 10032 . . . . . . . . . . . . 13  |-  ( 1  -  1 )  =  0
132130, 131syl6eq 2460 . . . . . . . . . . . 12  |-  ( ( ( N  e.  NN  /\  f  e.  D )  /\  ( x  e.  ( Base `  (ℤ/n `  N
) )  /\  (
f `  x )  =/=  0 ) )  -> 
( ( ( f `
 x ) ^
( phi `  N
) )  -  1 )  =  0 )
133 oveq1 6055 . . . . . . . . . . . . . . 15  |-  ( z  =  ( f `  x )  ->  (
z ^ ( phi `  N ) )  =  ( ( f `  x ) ^ ( phi `  N ) ) )
134133oveq1d 6063 . . . . . . . . . . . . . 14  |-  ( z  =  ( f `  x )  ->  (
( z ^ ( phi `  N ) )  -  1 )  =  ( ( ( f `
 x ) ^
( phi `  N
) )  -  1 ) )
135134eqeq1d 2420 . . . . . . . . . . . . 13  |-  ( z  =  ( f `  x )  ->  (
( ( z ^
( phi `  N
) )  -  1 )  =  0  <->  (
( ( f `  x ) ^ ( phi `  N ) )  -  1 )  =  0 ) )
136135elrab 3060 . . . . . . . . . . . 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 646 . . . . . . . . . . 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 599 . . . . . . . . . 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 210 . . . . . . . . 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 368 . . . . . . . 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 3456 . . . . . . . 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 204 . . . . . . 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 2757 . . . . . 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 5861 . . . . . 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 646 . . . . 5  |-  ( ( N  e.  NN  /\  f  e.  D )  ->  f : ( Base `  (ℤ/n `  N ) ) --> ( { 0 }  u.  { z  e.  CC  | 
( ( z ^
( phi `  N
) )  -  1 )  =  0 } ) )
146145ex 424 . . . 4  |-  ( N  e.  NN  ->  (
f  e.  D  -> 
f : ( Base `  (ℤ/n `  N ) ) --> ( { 0 }  u.  { z  e.  CC  | 
( ( z ^
( phi `  N
) )  -  1 )  =  0 } ) ) )
147 elmapg 6998 . . . . 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 643 . . . 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 226 . . 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 3322 . 2  |-  ( N  e.  NN  ->  D  C_  ( ( { 0 }  u.  { z  e.  CC  |  ( ( z ^ ( phi `  N ) )  -  1 )  =  0 } )  ^m  ( Base `  (ℤ/n `  N ) ) ) )
151 ssfi 7296 . 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 643 1  |-  ( N  e.  NN  ->  D  e.  Fin )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2575   A.wral 2674   {crab 2678   _Vcvv 2924    u. cun 3286    C_ wss 3288   {csn 3782   class class class wbr 4180    e. cmpt 4234    X. cxp 4843   `'ccnv 4844   "cima 4848    Fn wfn 5416   -->wf 5417   ` cfv 5421  (class class class)co 6048    o Fcof 6270    ^m cmap 6985   Fincfn 7076   CCcc 8952   0cc0 8954   1c1 8955    <_ cle 9085    - cmin 9255   -ucneg 9256   NNcn 9964   NN0cn0 10185   ZZcz 10246   ^cexp 11345   #chash 11581    || cdivides 12815   phicphi 13116   Basecbs 13432   ↾s cress 13433   0gc0g 13686   Grpcgrp 14648  .gcmg 14652   MndHom cmhm 14699  SubMndcsubmnd 14700   odcod 15126  mulGrpcmgp 15611   Ringcrg 15623   CRingccrg 15624   1rcur 15625  Unitcui 15707  ℂfldccnfld 16666  ℤ/nczn 16744   0 pc0p 19522  Polycply 20064  degcdgr 20067  DChrcdchr 20977
This theorem is referenced by:  sumdchr2  21015  dchrhash  21016  rpvmasum2  21167  dchrisum0re  21168
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-inf2 7560  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031  ax-pre-sup 9032  ax-addf 9033  ax-mulf 9034
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-disj 4151  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-se 4510  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-isom 5430  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-of 6272  df-1st 6316  df-2nd 6317  df-tpos 6446  df-riota 6516  df-recs 6600  df-rdg 6635  df-1o 6691  df-2o 6692  df-oadd 6695  df-omul 6696  df-er 6872  df-ec 6874  df-qs 6878  df-map 6987  df-pm 6988  df-en 7077  df-dom 7078  df-sdom 7079  df-fin 7080  df-sup 7412  df-oi 7443  df-card 7790  df-acn 7793  df-cda 8012  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-div 9642  df-nn 9965  df-2 10022  df-3 10023  df-4 10024  df-5 10025  df-6 10026  df-7 10027  df-8 10028  df-9 10029  df-10 10030  df-n0 10186  df-z 10247  df-dec 10347  df-uz 10453  df-rp 10577  df-fz 11008  df-fzo 11099  df-fl 11165  df-mod 11214  df-seq 11287  df-exp 11346  df-hash 11582  df-cj 11867  df-re 11868  df-im 11869  df-sqr 12003  df-abs 12004  df-clim 12245  df-rlim 12246  df-sum 12443  df-dvds 12816  df-gcd 12970  df-phi 13118  df-struct 13434  df-ndx 13435  df-slot 13436  df-base 13437  df-sets 13438  df-ress 13439  df-plusg 13505  df-mulr 13506  df-starv 13507  df-sca 13508  df-vsca 13509  df-tset 13511  df-ple 13512  df-ds 13514  df-unif 13515  df-0g 13690  df-imas 13697  df-divs 13698  df-mnd 14653  df-mhm 14701  df-submnd 14702  df-grp 14775  df-minusg 14776  df-sbg 14777  df-mulg 14778  df-subg 14904  df-nsg 14905  df-eqg 14906  df-ghm 14967  df-od 15130  df-cmn 15377  df-abl 15378  df-mgp 15612  df-rng 15626  df-cring 15627  df-ur 15628  df-oppr 15691  df-dvdsr 15709  df-unit 15710  df-invr 15740  df-rnghom 15782  df-subrg 15829  df-lmod 15915  df-lss 15972  df-lsp 16011  df-sra 16207  df-rgmod 16208  df-lidl 16209  df-rsp 16210  df-2idl 16266  df-cnfld 16667  df-zrh 16745  df-zn 16748  df-0p 19523  df-ply 20068  df-idp 20069  df-coe 20070  df-dgr 20071  df-quot 20169  df-dchr 20978
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