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Theorem dchrfi 23395
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 7594 . . . 4  |-  { 0 }  e.  Fin
2 cnex 9571 . . . . . . . . 9  |-  CC  e.  _V
32a1i 11 . . . . . . . 8  |-  ( N  e.  NN  ->  CC  e.  _V )
4 ovex 6305 . . . . . . . . 9  |-  ( z ^ ( phi `  N ) )  e. 
_V
54a1i 11 . . . . . . . 8  |-  ( ( N  e.  NN  /\  z  e.  CC )  ->  ( z ^ ( phi `  N ) )  e.  _V )
6 1cnd 9610 . . . . . . . 8  |-  ( ( N  e.  NN  /\  z  e.  CC )  ->  1  e.  CC )
7 eqidd 2442 . . . . . . . 8  |-  ( N  e.  NN  ->  (
z  e.  CC  |->  ( z ^ ( phi `  N ) ) )  =  ( z  e.  CC  |->  ( z ^
( phi `  N
) ) ) )
8 fconstmpt 5029 . . . . . . . . 9  |-  ( CC 
X.  { 1 } )  =  ( z  e.  CC  |->  1 )
98a1i 11 . . . . . . . 8  |-  ( N  e.  NN  ->  ( CC  X.  { 1 } )  =  ( z  e.  CC  |->  1 ) )
103, 5, 6, 7, 9offval2 6537 . . . . . . 7  |-  ( N  e.  NN  ->  (
( z  e.  CC  |->  ( z ^ ( phi `  N ) ) )  oF  -  ( CC  X.  { 1 } ) )  =  ( z  e.  CC  |->  ( ( z ^
( phi `  N
) )  -  1 ) ) )
11 ssid 3505 . . . . . . . . . 10  |-  CC  C_  CC
1211a1i 11 . . . . . . . . 9  |-  ( N  e.  NN  ->  CC  C_  CC )
13 1cnd 9610 . . . . . . . . 9  |-  ( N  e.  NN  ->  1  e.  CC )
14 phicl 14171 . . . . . . . . . 10  |-  ( N  e.  NN  ->  ( phi `  N )  e.  NN )
1514nnnn0d 10853 . . . . . . . . 9  |-  ( N  e.  NN  ->  ( phi `  N )  e. 
NN0 )
16 plypow 22468 . . . . . . . . 9  |-  ( ( CC  C_  CC  /\  1  e.  CC  /\  ( phi `  N )  e.  NN0 )  ->  ( z  e.  CC  |->  ( z ^
( phi `  N
) ) )  e.  (Poly `  CC )
)
1712, 13, 15, 16syl3anc 1227 . . . . . . . 8  |-  ( N  e.  NN  ->  (
z  e.  CC  |->  ( z ^ ( phi `  N ) ) )  e.  (Poly `  CC ) )
18 ax-1cn 9548 . . . . . . . . 9  |-  1  e.  CC
19 plyconst 22469 . . . . . . . . 9  |-  ( ( CC  C_  CC  /\  1  e.  CC )  ->  ( CC  X.  { 1 } )  e.  (Poly `  CC ) )
2011, 18, 19mp2an 672 . . . . . . . 8  |-  ( CC 
X.  { 1 } )  e.  (Poly `  CC )
21 plysubcl 22485 . . . . . . . 8  |-  ( ( ( z  e.  CC  |->  ( z ^ ( phi `  N ) ) )  e.  (Poly `  CC )  /\  ( CC  X.  { 1 } )  e.  (Poly `  CC ) )  ->  (
( z  e.  CC  |->  ( z ^ ( phi `  N ) ) )  oF  -  ( CC  X.  { 1 } ) )  e.  (Poly `  CC )
)
2217, 20, 21sylancl 662 . . . . . . 7  |-  ( N  e.  NN  ->  (
( z  e.  CC  |->  ( z ^ ( phi `  N ) ) )  oF  -  ( CC  X.  { 1 } ) )  e.  (Poly `  CC )
)
2310, 22eqeltrrd 2530 . . . . . 6  |-  ( N  e.  NN  ->  (
z  e.  CC  |->  ( ( z ^ ( phi `  N ) )  -  1 ) )  e.  (Poly `  CC ) )
24 0cn 9586 . . . . . . 7  |-  0  e.  CC
25 neg1ne0 10642 . . . . . . . 8  |-  -u 1  =/=  0
26140expd 12300 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  (
0 ^ ( phi `  N ) )  =  0 )
2726oveq1d 6292 . . . . . . . . . 10  |-  ( N  e.  NN  ->  (
( 0 ^ ( phi `  N ) )  -  1 )  =  ( 0  -  1 ) )
28 oveq1 6284 . . . . . . . . . . . . 13  |-  ( z  =  0  ->  (
z ^ ( phi `  N ) )  =  ( 0 ^ ( phi `  N ) ) )
2928oveq1d 6292 . . . . . . . . . . . 12  |-  ( z  =  0  ->  (
( z ^ ( phi `  N ) )  -  1 )  =  ( ( 0 ^ ( phi `  N
) )  -  1 ) )
30 eqid 2441 . . . . . . . . . . . 12  |-  ( z  e.  CC  |->  ( ( z ^ ( phi `  N ) )  - 
1 ) )  =  ( z  e.  CC  |->  ( ( z ^
( phi `  N
) )  -  1 ) )
31 ovex 6305 . . . . . . . . . . . 12  |-  ( ( 0 ^ ( phi `  N ) )  - 
1 )  e.  _V
3229, 30, 31fvmpt 5937 . . . . . . . . . . 11  |-  ( 0  e.  CC  ->  (
( z  e.  CC  |->  ( ( z ^
( phi `  N
) )  -  1 ) ) `  0
)  =  ( ( 0 ^ ( phi `  N ) )  - 
1 ) )
3324, 32ax-mp 5 . . . . . . . . . 10  |-  ( ( z  e.  CC  |->  ( ( z ^ ( phi `  N ) )  -  1 ) ) `
 0 )  =  ( ( 0 ^ ( phi `  N
) )  -  1 )
34 df-neg 9808 . . . . . . . . . 10  |-  -u 1  =  ( 0  -  1 )
3527, 33, 343eqtr4g 2507 . . . . . . . . 9  |-  ( N  e.  NN  ->  (
( z  e.  CC  |->  ( ( z ^
( phi `  N
) )  -  1 ) ) `  0
)  =  -u 1
)
3635neeq1d 2718 . . . . . . . 8  |-  ( N  e.  NN  ->  (
( ( z  e.  CC  |->  ( ( z ^ ( phi `  N ) )  - 
1 ) ) ` 
0 )  =/=  0  <->  -u 1  =/=  0 ) )
3725, 36mpbiri 233 . . . . . . 7  |-  ( N  e.  NN  ->  (
( z  e.  CC  |->  ( ( z ^
( phi `  N
) )  -  1 ) ) `  0
)  =/=  0 )
38 ne0p 22470 . . . . . . 7  |-  ( ( 0  e.  CC  /\  ( ( z  e.  CC  |->  ( ( z ^ ( phi `  N ) )  - 
1 ) ) ` 
0 )  =/=  0
)  ->  ( z  e.  CC  |->  ( ( z ^ ( phi `  N ) )  - 
1 ) )  =/=  0p )
3924, 37, 38sylancr 663 . . . . . 6  |-  ( N  e.  NN  ->  (
z  e.  CC  |->  ( ( z ^ ( phi `  N ) )  -  1 ) )  =/=  0p )
4030mptiniseg 5487 . . . . . . . . 9  |-  ( 0  e.  CC  ->  ( `' ( z  e.  CC  |->  ( ( z ^ ( phi `  N ) )  - 
1 ) ) " { 0 } )  =  { z  e.  CC  |  ( ( z ^ ( phi `  N ) )  - 
1 )  =  0 } )
4124, 40ax-mp 5 . . . . . . . 8  |-  ( `' ( z  e.  CC  |->  ( ( z ^
( phi `  N
) )  -  1 ) ) " {
0 } )  =  { z  e.  CC  |  ( ( z ^ ( phi `  N ) )  - 
1 )  =  0 }
4241eqcomi 2454 . . . . . . 7  |-  { z  e.  CC  |  ( ( z ^ ( phi `  N ) )  -  1 )  =  0 }  =  ( `' ( z  e.  CC  |->  ( ( z ^ ( phi `  N ) )  - 
1 ) ) " { 0 } )
4342fta1 22569 . . . . . 6  |-  ( ( ( z  e.  CC  |->  ( ( z ^
( phi `  N
) )  -  1 ) )  e.  (Poly `  CC )  /\  (
z  e.  CC  |->  ( ( z ^ ( phi `  N ) )  -  1 ) )  =/=  0p )  ->  ( { 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 ) ) ) ) )
4423, 39, 43syl2anc 661 . . . . 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 ) ) ) ) )
4544simpld 459 . . . 4  |-  ( N  e.  NN  ->  { z  e.  CC  |  ( ( z ^ ( phi `  N ) )  -  1 )  =  0 }  e.  Fin )
46 unfi 7785 . . . 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 )
471, 45, 46sylancr 663 . . 3  |-  ( N  e.  NN  ->  ( { 0 }  u.  { z  e.  CC  | 
( ( z ^
( phi `  N
) )  -  1 )  =  0 } )  e.  Fin )
48 eqid 2441 . . . 4  |-  (ℤ/n `  N
)  =  (ℤ/n `  N
)
49 eqid 2441 . . . 4  |-  ( Base `  (ℤ/n `  N ) )  =  ( Base `  (ℤ/n `  N
) )
5048, 49znfi 18465 . . 3  |-  ( N  e.  NN  ->  ( Base `  (ℤ/n `  N ) )  e. 
Fin )
51 mapfi 7814 . . 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 )
5247, 50, 51syl2anc 661 . 2  |-  ( N  e.  NN  ->  (
( { 0 }  u.  { z  e.  CC  |  ( ( z ^ ( phi `  N ) )  - 
1 )  =  0 } )  ^m  ( Base `  (ℤ/n `  N ) ) )  e.  Fin )
53 dchrabl.g . . . . . . . 8  |-  G  =  (DChr `  N )
54 dchrfi.b . . . . . . . 8  |-  D  =  ( Base `  G
)
55 simpr 461 . . . . . . . 8  |-  ( ( N  e.  NN  /\  f  e.  D )  ->  f  e.  D )
5653, 48, 54, 49, 55dchrf 23382 . . . . . . 7  |-  ( ( N  e.  NN  /\  f  e.  D )  ->  f : ( Base `  (ℤ/n `  N ) ) --> CC )
57 ffn 5717 . . . . . . 7  |-  ( f : ( Base `  (ℤ/n `  N
) ) --> CC  ->  f  Fn  ( Base `  (ℤ/n `  N
) ) )
5856, 57syl 16 . . . . . 6  |-  ( ( N  e.  NN  /\  f  e.  D )  ->  f  Fn  ( Base `  (ℤ/n `  N ) ) )
59 df-ne 2638 . . . . . . . . . . 11  |-  ( ( f `  x )  =/=  0  <->  -.  (
f `  x )  =  0 )
60 fvex 5862 . . . . . . . . . . . 12  |-  ( f `
 x )  e. 
_V
6160elsnc 4034 . . . . . . . . . . 11  |-  ( ( f `  x )  e.  { 0 }  <-> 
( f `  x
)  =  0 )
6259, 61xchbinxr 311 . . . . . . . . . 10  |-  ( ( f `  x )  =/=  0  <->  -.  (
f `  x )  e.  { 0 } )
63 simpl 457 . . . . . . . . . . . . 13  |-  ( ( x  e.  ( Base `  (ℤ/n `  N ) )  /\  ( f `  x
)  =/=  0 )  ->  x  e.  (
Base `  (ℤ/n `  N ) ) )
64 ffvelrn 6010 . . . . . . . . . . . . 13  |-  ( ( f : ( Base `  (ℤ/n `  N ) ) --> CC 
/\  x  e.  (
Base `  (ℤ/n `  N ) ) )  ->  ( f `  x )  e.  CC )
6556, 63, 64syl2an 477 . . . . . . . . . . . 12  |-  ( ( ( N  e.  NN  /\  f  e.  D )  /\  ( x  e.  ( Base `  (ℤ/n `  N
) )  /\  (
f `  x )  =/=  0 ) )  -> 
( f `  x
)  e.  CC )
6653, 48, 54dchrmhm 23381 . . . . . . . . . . . . . . . . . 18  |-  D  C_  ( (mulGrp `  (ℤ/n `  N ) ) MndHom  (mulGrp ` fld ) )
67 simplr 754 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( N  e.  NN  /\  f  e.  D )  /\  ( x  e.  ( Base `  (ℤ/n `  N
) )  /\  (
f `  x )  =/=  0 ) )  -> 
f  e.  D )
6866, 67sseldi 3484 . . . . . . . . . . . . . . . . 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 ) ) )
6915ad2antrr 725 . . . . . . . . . . . . . . . . 17  |-  ( ( ( N  e.  NN  /\  f  e.  D )  /\  ( x  e.  ( Base `  (ℤ/n `  N
) )  /\  (
f `  x )  =/=  0 ) )  -> 
( phi `  N
)  e.  NN0 )
70 simprl 755 . . . . . . . . . . . . . . . . 17  |-  ( ( ( N  e.  NN  /\  f  e.  D )  /\  ( x  e.  ( Base `  (ℤ/n `  N
) )  /\  (
f `  x )  =/=  0 ) )  ->  x  e.  ( Base `  (ℤ/n `  N ) ) )
71 eqid 2441 . . . . . . . . . . . . . . . . . . 19  |-  (mulGrp `  (ℤ/n `  N ) )  =  (mulGrp `  (ℤ/n `  N ) )
7271, 49mgpbas 17015 . . . . . . . . . . . . . . . . . 18  |-  ( Base `  (ℤ/n `  N ) )  =  ( Base `  (mulGrp `  (ℤ/n `  N ) ) )
73 eqid 2441 . . . . . . . . . . . . . . . . . 18  |-  (.g `  (mulGrp `  (ℤ/n `  N ) ) )  =  (.g `  (mulGrp `  (ℤ/n `  N
) ) )
74 eqid 2441 . . . . . . . . . . . . . . . . . 18  |-  (.g `  (mulGrp ` fld ) )  =  (.g `  (mulGrp ` fld ) )
7572, 73, 74mhmmulg 16043 . . . . . . . . . . . . . . . . 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 ) ) )
7668, 69, 70, 75syl3anc 1227 . . . . . . . . . . . . . . . 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 ) ) )
77 nnnn0 10803 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( N  e.  NN  ->  N  e.  NN0 )
7848zncrng 18450 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( N  e.  NN0  ->  (ℤ/n `  N
)  e.  CRing )
7977, 78syl 16 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( N  e.  NN  ->  (ℤ/n `  N
)  e.  CRing )
80 crngring 17077 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( (ℤ/n `  N )  e.  CRing  -> 
(ℤ/n `  N )  e.  Ring )
8179, 80syl 16 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( N  e.  NN  ->  (ℤ/n `  N
)  e.  Ring )
8281ad2antrr 725 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( N  e.  NN  /\  f  e.  D )  /\  ( x  e.  ( Base `  (ℤ/n `  N
) )  /\  (
f `  x )  =/=  0 ) )  -> 
(ℤ/n `  N )  e.  Ring )
83 eqid 2441 . . . . . . . . . . . . . . . . . . . . . . 23  |-  (Unit `  (ℤ/n `  N ) )  =  (Unit `  (ℤ/n `  N ) )
84 eqid 2441 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( (mulGrp `  (ℤ/n `  N ) )s  (Unit `  (ℤ/n `  N ) ) )  =  ( (mulGrp `  (ℤ/n `  N ) )s  (Unit `  (ℤ/n `  N ) ) )
8583, 84unitgrp 17184 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( (ℤ/n `  N )  e.  Ring  -> 
( (mulGrp `  (ℤ/n `  N
) )s  (Unit `  (ℤ/n `  N ) ) )  e.  Grp )
8682, 85syl 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 )
8748, 83znunithash 18470 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( N  e.  NN  ->  ( # `
 (Unit `  (ℤ/n `  N
) ) )  =  ( phi `  N
) )
8887, 15eqeltrd 2529 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( N  e.  NN  ->  ( # `
 (Unit `  (ℤ/n `  N
) ) )  e. 
NN0 )
89 fvex 5862 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  (Unit `  (ℤ/n `  N ) )  e. 
_V
90 hashclb 12404 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( (Unit `  (ℤ/n `  N ) )  e. 
_V  ->  ( (Unit `  (ℤ/n `  N ) )  e. 
Fin 
<->  ( # `  (Unit `  (ℤ/n `  N ) ) )  e.  NN0 ) )
9189, 90ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( (Unit `  (ℤ/n `  N ) )  e. 
Fin 
<->  ( # `  (Unit `  (ℤ/n `  N ) ) )  e.  NN0 )
9288, 91sylibr 212 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( N  e.  NN  ->  (Unit `  (ℤ/n `  N ) )  e. 
Fin )
9392ad2antrr 725 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( N  e.  NN  /\  f  e.  D )  /\  ( x  e.  ( Base `  (ℤ/n `  N
) )  /\  (
f `  x )  =/=  0 ) )  -> 
(Unit `  (ℤ/n `  N ) )  e. 
Fin )
94 simprr 756 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( N  e.  NN  /\  f  e.  D )  /\  ( x  e.  ( Base `  (ℤ/n `  N
) )  /\  (
f `  x )  =/=  0 ) )  -> 
( f `  x
)  =/=  0 )
9553, 48, 54, 49, 83, 67, 70dchrn0 23390 . . . . . . . . . . . . . . . . . . . . . 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
) ) ) )
9694, 95mpbid 210 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( N  e.  NN  /\  f  e.  D )  /\  ( x  e.  ( Base `  (ℤ/n `  N
) )  /\  (
f `  x )  =/=  0 ) )  ->  x  e.  (Unit `  (ℤ/n `  N
) ) )
9783, 84unitgrpbas 17183 . . . . . . . . . . . . . . . . . . . . . 22  |-  (Unit `  (ℤ/n `  N ) )  =  ( Base `  (
(mulGrp `  (ℤ/n `  N ) )s  (Unit `  (ℤ/n `  N ) ) ) )
98 eqid 2441 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( od
`  ( (mulGrp `  (ℤ/n `  N ) )s  (Unit `  (ℤ/n `  N ) ) ) )  =  ( od
`  ( (mulGrp `  (ℤ/n `  N ) )s  (Unit `  (ℤ/n `  N ) ) ) )
9997, 98oddvds2 16457 . . . . . . . . . . . . . . . . . . . . 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 ) ) ) )
10086, 93, 96, 99syl3anc 1227 . . . . . . . . . . . . . . . . . . . 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 ) ) ) )
10187ad2antrr 725 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( N  e.  NN  /\  f  e.  D )  /\  ( x  e.  ( Base `  (ℤ/n `  N
) )  /\  (
f `  x )  =/=  0 ) )  -> 
( # `  (Unit `  (ℤ/n `  N ) ) )  =  ( phi `  N ) )
102100, 101breqtrd 4457 . . . . . . . . . . . . . . . . . . 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 ) )
10314ad2antrr 725 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( N  e.  NN  /\  f  e.  D )  /\  ( x  e.  ( Base `  (ℤ/n `  N
) )  /\  (
f `  x )  =/=  0 ) )  -> 
( phi `  N
)  e.  NN )
104103nnzd 10968 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( N  e.  NN  /\  f  e.  D )  /\  ( x  e.  ( Base `  (ℤ/n `  N
) )  /\  (
f `  x )  =/=  0 ) )  -> 
( phi `  N
)  e.  ZZ )
105 eqid 2441 . . . . . . . . . . . . . . . . . . . . 21  |-  (.g `  (
(mulGrp `  (ℤ/n `  N ) )s  (Unit `  (ℤ/n `  N ) ) ) )  =  (.g `  (
(mulGrp `  (ℤ/n `  N ) )s  (Unit `  (ℤ/n `  N ) ) ) )
106 eqid 2441 . . . . . . . . . . . . . . . . . . . . 21  |-  ( 0g
`  ( (mulGrp `  (ℤ/n `  N ) )s  (Unit `  (ℤ/n `  N ) ) ) )  =  ( 0g
`  ( (mulGrp `  (ℤ/n `  N ) )s  (Unit `  (ℤ/n `  N ) ) ) )
10797, 98, 105, 106oddvds 16440 . . . . . . . . . . . . . . . . . . . 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 ) ) ) ) ) )
10886, 96, 104, 107syl3anc 1227 . . . . . . . . . . . . . . . . . . 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 ) ) ) ) ) )
109102, 108mpbid 210 . . . . . . . . . . . . . . . . . 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 ) ) ) ) )
11083, 71unitsubm 17187 . . . . . . . . . . . . . . . . . . . 20  |-  ( (ℤ/n `  N )  e.  Ring  -> 
(Unit `  (ℤ/n `  N ) )  e.  (SubMnd `  (mulGrp `  (ℤ/n `  N
) ) ) )
11182, 110syl 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
) ) ) )
11273, 84, 105submmulg 16046 . . . . . . . . . . . . . . . . . . 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 ) )
113111, 69, 96, 112syl3anc 1227 . . . . . . . . . . . . . . . . . 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 ) )
114 eqid 2441 . . . . . . . . . . . . . . . . . . . . 21  |-  ( 1r
`  (ℤ/n `  N ) )  =  ( 1r `  (ℤ/n `  N
) )
11571, 114ringidval 17023 . . . . . . . . . . . . . . . . . . . 20  |-  ( 1r
`  (ℤ/n `  N ) )  =  ( 0g `  (mulGrp `  (ℤ/n `  N ) ) )
11684, 115subm0 15856 . . . . . . . . . . . . . . . . . . 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 ) ) ) ) )
117111, 116syl 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 ) ) ) ) )
118109, 113, 1173eqtr4d 2492 . . . . . . . . . . . . . . . . 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
) ) )
119118fveq2d 5856 . . . . . . . . . . . . . . . 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 ) ) ) )
12076, 119eqtr3d 2484 . . . . . . . . . . . . . . 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 ) ) ) )
121 cnfldexp 18319 . . . . . . . . . . . . . . . 16  |-  ( ( ( f `  x
)  e.  CC  /\  ( phi `  N )  e.  NN0 )  -> 
( ( phi `  N ) (.g `  (mulGrp ` fld ) ) ( f `  x ) )  =  ( ( f `  x ) ^ ( phi `  N ) ) )
12265, 69, 121syl2anc 661 . . . . . . . . . . . . . . 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 ) ) )
123 eqid 2441 . . . . . . . . . . . . . . . . . 18  |-  (mulGrp ` fld )  =  (mulGrp ` fld )
124 cnfld1 18311 . . . . . . . . . . . . . . . . . 18  |-  1  =  ( 1r ` fld )
125123, 124ringidval 17023 . . . . . . . . . . . . . . . . 17  |-  1  =  ( 0g `  (mulGrp ` fld ) )
126115, 125mhm0 15843 . . . . . . . . . . . . . . . 16  |-  ( f  e.  ( (mulGrp `  (ℤ/n `  N ) ) MndHom  (mulGrp ` fld ) )  ->  ( f `  ( 1r `  (ℤ/n `  N
) ) )  =  1 )
12768, 126syl 16 . . . . . . . . . . . . . . 15  |-  ( ( ( N  e.  NN  /\  f  e.  D )  /\  ( x  e.  ( Base `  (ℤ/n `  N
) )  /\  (
f `  x )  =/=  0 ) )  -> 
( f `  ( 1r `  (ℤ/n `  N ) ) )  =  1 )
128120, 122, 1273eqtr3d 2490 . . . . . . . . . . . . . 14  |-  ( ( ( N  e.  NN  /\  f  e.  D )  /\  ( x  e.  ( Base `  (ℤ/n `  N
) )  /\  (
f `  x )  =/=  0 ) )  -> 
( ( f `  x ) ^ ( phi `  N ) )  =  1 )
129128oveq1d 6292 . . . . . . . . . . . . 13  |-  ( ( ( N  e.  NN  /\  f  e.  D )  /\  ( x  e.  ( Base `  (ℤ/n `  N
) )  /\  (
f `  x )  =/=  0 ) )  -> 
( ( ( f `
 x ) ^
( phi `  N
) )  -  1 )  =  ( 1  -  1 ) )
130 1m1e0 10605 . . . . . . . . . . . . 13  |-  ( 1  -  1 )  =  0
131129, 130syl6eq 2498 . . . . . . . . . . . 12  |-  ( ( ( N  e.  NN  /\  f  e.  D )  /\  ( x  e.  ( Base `  (ℤ/n `  N
) )  /\  (
f `  x )  =/=  0 ) )  -> 
( ( ( f `
 x ) ^
( phi `  N
) )  -  1 )  =  0 )
132 oveq1 6284 . . . . . . . . . . . . . . 15  |-  ( z  =  ( f `  x )  ->  (
z ^ ( phi `  N ) )  =  ( ( f `  x ) ^ ( phi `  N ) ) )
133132oveq1d 6292 . . . . . . . . . . . . . 14  |-  ( z  =  ( f `  x )  ->  (
( z ^ ( phi `  N ) )  -  1 )  =  ( ( ( f `
 x ) ^
( phi `  N
) )  -  1 ) )
134133eqeq1d 2443 . . . . . . . . . . . . 13  |-  ( z  =  ( f `  x )  ->  (
( ( z ^
( phi `  N
) )  -  1 )  =  0  <->  (
( ( f `  x ) ^ ( phi `  N ) )  -  1 )  =  0 ) )
135134elrab 3241 . . . . . . . . . . . 12  |-  ( ( f `  x )  e.  { z  e.  CC  |  ( ( z ^ ( phi `  N ) )  - 
1 )  =  0 }  <->  ( ( f `
 x )  e.  CC  /\  ( ( ( f `  x
) ^ ( phi `  N ) )  - 
1 )  =  0 ) )
13665, 131, 135sylanbrc 664 . . . . . . . . . . 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 } )
137136expr 615 . . . . . . . . . 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 } ) )
13862, 137syl5bir 218 . . . . . . . . 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 } ) )
139138orrd 378 . . . . . . . 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 } ) )
140 elun 3627 . . . . . . . 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 } ) )
141139, 140sylibr 212 . . . . . . 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 } ) )
142141ralrimiva 2855 . . . . . 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 } ) )
143 ffnfv 6038 . . . . . 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 } ) ) )
14458, 142, 143sylanbrc 664 . . . . 5  |-  ( ( N  e.  NN  /\  f  e.  D )  ->  f : ( Base `  (ℤ/n `  N ) ) --> ( { 0 }  u.  { z  e.  CC  | 
( ( z ^
( phi `  N
) )  -  1 )  =  0 } ) )
145144ex 434 . . . 4  |-  ( N  e.  NN  ->  (
f  e.  D  -> 
f : ( Base `  (ℤ/n `  N ) ) --> ( { 0 }  u.  { z  e.  CC  | 
( ( z ^
( phi `  N
) )  -  1 )  =  0 } ) ) )
146 elmapg 7431 . . . . 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 } ) ) )
14747, 50, 146syl2anc 661 . . . 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 } ) ) )
148145, 147sylibrd 234 . . 3  |-  ( N  e.  NN  ->  (
f  e.  D  -> 
f  e.  ( ( { 0 }  u.  { z  e.  CC  | 
( ( z ^
( phi `  N
) )  -  1 )  =  0 } )  ^m  ( Base `  (ℤ/n `  N ) ) ) ) )
149148ssrdv 3492 . 2  |-  ( N  e.  NN  ->  D  C_  ( ( { 0 }  u.  { z  e.  CC  |  ( ( z ^ ( phi `  N ) )  -  1 )  =  0 } )  ^m  ( Base `  (ℤ/n `  N ) ) ) )
150 ssfi 7738 . 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 )
15152, 149, 150syl2anc 661 1  |-  ( N  e.  NN  ->  D  e.  Fin )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1381    e. wcel 1802    =/= wne 2636   A.wral 2791   {crab 2795   _Vcvv 3093    u. cun 3456    C_ wss 3458   {csn 4010   class class class wbr 4433    |-> cmpt 4491    X. cxp 4983   `'ccnv 4984   "cima 4988    Fn wfn 5569   -->wf 5570   ` cfv 5574  (class class class)co 6277    oFcof 6519    ^m cmap 7418   Fincfn 7514   CCcc 9488   0cc0 9490   1c1 9491    <_ cle 9627    - cmin 9805   -ucneg 9806   NNcn 10537   NN0cn0 10796   ZZcz 10865   ^cexp 12140   #chash 12379    || cdvds 13858   phicphi 14166   Basecbs 14504   ↾s cress 14505   0gc0g 14709   MndHom cmhm 15833  SubMndcsubmnd 15834   Grpcgrp 15922  .gcmg 15925   odcod 16418  mulGrpcmgp 17009   1rcur 17021   Ringcrg 17066   CRingccrg 17067  Unitcui 17156  ℂfldccnfld 18288  ℤ/nczn 18407   0pc0p 21942  Polycply 22447  degcdgr 22450  DChrcdchr 23372
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-inf2 8056  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567  ax-pre-sup 9568  ax-addf 9569  ax-mulf 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-fal 1387  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-int 4268  df-iun 4313  df-disj 4404  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-se 4825  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-isom 5583  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-of 6521  df-om 6682  df-1st 6781  df-2nd 6782  df-tpos 6953  df-recs 7040  df-rdg 7074  df-1o 7128  df-2o 7129  df-oadd 7132  df-omul 7133  df-er 7309  df-ec 7311  df-qs 7315  df-map 7420  df-pm 7421  df-en 7515  df-dom 7516  df-sdom 7517  df-fin 7518  df-sup 7899  df-oi 7933  df-card 8318  df-acn 8321  df-cda 8546  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-div 10208  df-nn 10538  df-2 10595  df-3 10596  df-4 10597  df-5 10598  df-6 10599  df-7 10600  df-8 10601  df-9 10602  df-10 10603  df-n0 10797  df-z 10866  df-dec 10980  df-uz 11086  df-rp 11225  df-fz 11677  df-fzo 11799  df-fl 11903  df-mod 11971  df-seq 12082  df-exp 12141  df-hash 12380  df-cj 12906  df-re 12907  df-im 12908  df-sqrt 13042  df-abs 13043  df-clim 13285  df-rlim 13286  df-sum 13483  df-dvds 13859  df-gcd 14017  df-phi 14168  df-struct 14506  df-ndx 14507  df-slot 14508  df-base 14509  df-sets 14510  df-ress 14511  df-plusg 14582  df-mulr 14583  df-starv 14584  df-sca 14585  df-vsca 14586  df-ip 14587  df-tset 14588  df-ple 14589  df-ds 14591  df-unif 14592  df-0g 14711  df-imas 14777  df-qus 14778  df-mgm 15741  df-sgrp 15780  df-mnd 15790  df-mhm 15835  df-submnd 15836  df-grp 15926  df-minusg 15927  df-sbg 15928  df-mulg 15929  df-subg 16067  df-nsg 16068  df-eqg 16069  df-ghm 16134  df-od 16422  df-cmn 16669  df-abl 16670  df-mgp 17010  df-ur 17022  df-ring 17068  df-cring 17069  df-oppr 17140  df-dvdsr 17158  df-unit 17159  df-invr 17189  df-rnghom 17232  df-subrg 17295  df-lmod 17382  df-lss 17447  df-lsp 17486  df-sra 17686  df-rgmod 17687  df-lidl 17688  df-rsp 17689  df-2idl 17748  df-cnfld 18289  df-zring 18357  df-zrh 18408  df-zn 18411  df-0p 21943  df-ply 22451  df-idp 22452  df-coe 22453  df-dgr 22454  df-quot 22552  df-dchr 23373
This theorem is referenced by:  sumdchr2  23410  dchrhash  23411  rpvmasum2  23562  dchrisum0re  23563
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