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Theorem dchrfi 24167
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 7653 . . . 4  |-  { 0 }  e.  Fin
2 cnex 9620 . . . . . . . . 9  |-  CC  e.  _V
32a1i 11 . . . . . . . 8  |-  ( N  e.  NN  ->  CC  e.  _V )
4 ovex 6329 . . . . . . . . 9  |-  ( z ^ ( phi `  N ) )  e. 
_V
54a1i 11 . . . . . . . 8  |-  ( ( N  e.  NN  /\  z  e.  CC )  ->  ( z ^ ( phi `  N ) )  e.  _V )
6 1cnd 9659 . . . . . . . 8  |-  ( ( N  e.  NN  /\  z  e.  CC )  ->  1  e.  CC )
7 eqidd 2423 . . . . . . . 8  |-  ( N  e.  NN  ->  (
z  e.  CC  |->  ( z ^ ( phi `  N ) ) )  =  ( z  e.  CC  |->  ( z ^
( phi `  N
) ) ) )
8 fconstmpt 4893 . . . . . . . . 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 6558 . . . . . . 7  |-  ( N  e.  NN  ->  (
( z  e.  CC  |->  ( z ^ ( phi `  N ) ) )  oF  -  ( CC  X.  { 1 } ) )  =  ( z  e.  CC  |->  ( ( z ^
( phi `  N
) )  -  1 ) ) )
11 ssid 3483 . . . . . . . . . 10  |-  CC  C_  CC
1211a1i 11 . . . . . . . . 9  |-  ( N  e.  NN  ->  CC  C_  CC )
13 1cnd 9659 . . . . . . . . 9  |-  ( N  e.  NN  ->  1  e.  CC )
14 phicl 14702 . . . . . . . . . 10  |-  ( N  e.  NN  ->  ( phi `  N )  e.  NN )
1514nnnn0d 10925 . . . . . . . . 9  |-  ( N  e.  NN  ->  ( phi `  N )  e. 
NN0 )
16 plypow 23143 . . . . . . . . 9  |-  ( ( CC  C_  CC  /\  1  e.  CC  /\  ( phi `  N )  e.  NN0 )  ->  ( z  e.  CC  |->  ( z ^
( phi `  N
) ) )  e.  (Poly `  CC )
)
1712, 13, 15, 16syl3anc 1264 . . . . . . . 8  |-  ( N  e.  NN  ->  (
z  e.  CC  |->  ( z ^ ( phi `  N ) ) )  e.  (Poly `  CC ) )
18 ax-1cn 9597 . . . . . . . . 9  |-  1  e.  CC
19 plyconst 23144 . . . . . . . . 9  |-  ( ( CC  C_  CC  /\  1  e.  CC )  ->  ( CC  X.  { 1 } )  e.  (Poly `  CC ) )
2011, 18, 19mp2an 676 . . . . . . . 8  |-  ( CC 
X.  { 1 } )  e.  (Poly `  CC )
21 plysubcl 23160 . . . . . . . 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 666 . . . . . . 7  |-  ( N  e.  NN  ->  (
( z  e.  CC  |->  ( z ^ ( phi `  N ) ) )  oF  -  ( CC  X.  { 1 } ) )  e.  (Poly `  CC )
)
2310, 22eqeltrrd 2511 . . . . . 6  |-  ( N  e.  NN  ->  (
z  e.  CC  |->  ( ( z ^ ( phi `  N ) )  -  1 ) )  e.  (Poly `  CC ) )
24 0cn 9635 . . . . . . 7  |-  0  e.  CC
25 neg1ne0 10715 . . . . . . . 8  |-  -u 1  =/=  0
26140expd 12431 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  (
0 ^ ( phi `  N ) )  =  0 )
2726oveq1d 6316 . . . . . . . . . 10  |-  ( N  e.  NN  ->  (
( 0 ^ ( phi `  N ) )  -  1 )  =  ( 0  -  1 ) )
28 oveq1 6308 . . . . . . . . . . . . 13  |-  ( z  =  0  ->  (
z ^ ( phi `  N ) )  =  ( 0 ^ ( phi `  N ) ) )
2928oveq1d 6316 . . . . . . . . . . . 12  |-  ( z  =  0  ->  (
( z ^ ( phi `  N ) )  -  1 )  =  ( ( 0 ^ ( phi `  N
) )  -  1 ) )
30 eqid 2422 . . . . . . . . . . . 12  |-  ( z  e.  CC  |->  ( ( z ^ ( phi `  N ) )  - 
1 ) )  =  ( z  e.  CC  |->  ( ( z ^
( phi `  N
) )  -  1 ) )
31 ovex 6329 . . . . . . . . . . . 12  |-  ( ( 0 ^ ( phi `  N ) )  - 
1 )  e.  _V
3229, 30, 31fvmpt 5960 . . . . . . . . . . 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 9863 . . . . . . . . . 10  |-  -u 1  =  ( 0  -  1 )
3527, 33, 343eqtr4g 2488 . . . . . . . . 9  |-  ( N  e.  NN  ->  (
( z  e.  CC  |->  ( ( z ^
( phi `  N
) )  -  1 ) ) `  0
)  =  -u 1
)
3635neeq1d 2701 . . . . . . . 8  |-  ( N  e.  NN  ->  (
( ( z  e.  CC  |->  ( ( z ^ ( phi `  N ) )  - 
1 ) ) ` 
0 )  =/=  0  <->  -u 1  =/=  0 ) )
3725, 36mpbiri 236 . . . . . . 7  |-  ( N  e.  NN  ->  (
( z  e.  CC  |->  ( ( z ^
( phi `  N
) )  -  1 ) ) `  0
)  =/=  0 )
38 ne0p 23145 . . . . . . 7  |-  ( ( 0  e.  CC  /\  ( ( z  e.  CC  |->  ( ( z ^ ( phi `  N ) )  - 
1 ) ) ` 
0 )  =/=  0
)  ->  ( z  e.  CC  |->  ( ( z ^ ( phi `  N ) )  - 
1 ) )  =/=  0p )
3924, 37, 38sylancr 667 . . . . . 6  |-  ( N  e.  NN  ->  (
z  e.  CC  |->  ( ( z ^ ( phi `  N ) )  -  1 ) )  =/=  0p )
4030mptiniseg 5344 . . . . . . . . 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 2435 . . . . . . 7  |-  { z  e.  CC  |  ( ( z ^ ( phi `  N ) )  -  1 )  =  0 }  =  ( `' ( z  e.  CC  |->  ( ( z ^ ( phi `  N ) )  - 
1 ) ) " { 0 } )
4342fta1 23245 . . . . . 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 665 . . . . 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 460 . . . 4  |-  ( N  e.  NN  ->  { z  e.  CC  |  ( ( z ^ ( phi `  N ) )  -  1 )  =  0 }  e.  Fin )
46 unfi 7840 . . . 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 667 . . 3  |-  ( N  e.  NN  ->  ( { 0 }  u.  { z  e.  CC  | 
( ( z ^
( phi `  N
) )  -  1 )  =  0 } )  e.  Fin )
48 eqid 2422 . . . 4  |-  (ℤ/n `  N
)  =  (ℤ/n `  N
)
49 eqid 2422 . . . 4  |-  ( Base `  (ℤ/n `  N ) )  =  ( Base `  (ℤ/n `  N
) )
5048, 49znfi 19114 . . 3  |-  ( N  e.  NN  ->  ( Base `  (ℤ/n `  N ) )  e. 
Fin )
51 mapfi 7872 . . 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 665 . 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 462 . . . . . . . 8  |-  ( ( N  e.  NN  /\  f  e.  D )  ->  f  e.  D )
5653, 48, 54, 49, 55dchrf 24154 . . . . . . 7  |-  ( ( N  e.  NN  /\  f  e.  D )  ->  f : ( Base `  (ℤ/n `  N ) ) --> CC )
57 ffn 5742 . . . . . . 7  |-  ( f : ( Base `  (ℤ/n `  N
) ) --> CC  ->  f  Fn  ( Base `  (ℤ/n `  N
) ) )
5856, 57syl 17 . . . . . 6  |-  ( ( N  e.  NN  /\  f  e.  D )  ->  f  Fn  ( Base `  (ℤ/n `  N ) ) )
59 df-ne 2620 . . . . . . . . . . 11  |-  ( ( f `  x )  =/=  0  <->  -.  (
f `  x )  =  0 )
60 fvex 5887 . . . . . . . . . . . 12  |-  ( f `
 x )  e. 
_V
6160elsnc 4020 . . . . . . . . . . 11  |-  ( ( f `  x )  e.  { 0 }  <-> 
( f `  x
)  =  0 )
6259, 61xchbinxr 312 . . . . . . . . . 10  |-  ( ( f `  x )  =/=  0  <->  -.  (
f `  x )  e.  { 0 } )
63 simpl 458 . . . . . . . . . . . . 13  |-  ( ( x  e.  ( Base `  (ℤ/n `  N ) )  /\  ( f `  x
)  =/=  0 )  ->  x  e.  (
Base `  (ℤ/n `  N ) ) )
64 ffvelrn 6031 . . . . . . . . . . . . 13  |-  ( ( f : ( Base `  (ℤ/n `  N ) ) --> CC 
/\  x  e.  (
Base `  (ℤ/n `  N ) ) )  ->  ( f `  x )  e.  CC )
6556, 63, 64syl2an 479 . . . . . . . . . . . 12  |-  ( ( ( N  e.  NN  /\  f  e.  D )  /\  ( x  e.  ( Base `  (ℤ/n `  N
) )  /\  (
f `  x )  =/=  0 ) )  -> 
( f `  x
)  e.  CC )
6653, 48, 54dchrmhm 24153 . . . . . . . . . . . . . . . . . 18  |-  D  C_  ( (mulGrp `  (ℤ/n `  N ) ) MndHom  (mulGrp ` fld ) )
67 simplr 760 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( N  e.  NN  /\  f  e.  D )  /\  ( x  e.  ( Base `  (ℤ/n `  N
) )  /\  (
f `  x )  =/=  0 ) )  -> 
f  e.  D )
6866, 67sseldi 3462 . . . . . . . . . . . . . . . . 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 730 . . . . . . . . . . . . . . . . 17  |-  ( ( ( N  e.  NN  /\  f  e.  D )  /\  ( x  e.  ( Base `  (ℤ/n `  N
) )  /\  (
f `  x )  =/=  0 ) )  -> 
( phi `  N
)  e.  NN0 )
70 simprl 762 . . . . . . . . . . . . . . . . 17  |-  ( ( ( N  e.  NN  /\  f  e.  D )  /\  ( x  e.  ( Base `  (ℤ/n `  N
) )  /\  (
f `  x )  =/=  0 ) )  ->  x  e.  ( Base `  (ℤ/n `  N ) ) )
71 eqid 2422 . . . . . . . . . . . . . . . . . . 19  |-  (mulGrp `  (ℤ/n `  N ) )  =  (mulGrp `  (ℤ/n `  N ) )
7271, 49mgpbas 17714 . . . . . . . . . . . . . . . . . 18  |-  ( Base `  (ℤ/n `  N ) )  =  ( Base `  (mulGrp `  (ℤ/n `  N ) ) )
73 eqid 2422 . . . . . . . . . . . . . . . . . 18  |-  (.g `  (mulGrp `  (ℤ/n `  N ) ) )  =  (.g `  (mulGrp `  (ℤ/n `  N
) ) )
74 eqid 2422 . . . . . . . . . . . . . . . . . 18  |-  (.g `  (mulGrp ` fld ) )  =  (.g `  (mulGrp ` fld ) )
7572, 73, 74mhmmulg 16775 . . . . . . . . . . . . . . . . 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 1264 . . . . . . . . . . . . . . . 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 10876 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( N  e.  NN  ->  N  e.  NN0 )
7848zncrng 19099 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( N  e.  NN0  ->  (ℤ/n `  N
)  e.  CRing )
7977, 78syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( N  e.  NN  ->  (ℤ/n `  N
)  e.  CRing )
80 crngring 17776 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( (ℤ/n `  N )  e.  CRing  -> 
(ℤ/n `  N )  e.  Ring )
8179, 80syl 17 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( N  e.  NN  ->  (ℤ/n `  N
)  e.  Ring )
8281ad2antrr 730 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( N  e.  NN  /\  f  e.  D )  /\  ( x  e.  ( Base `  (ℤ/n `  N
) )  /\  (
f `  x )  =/=  0 ) )  -> 
(ℤ/n `  N )  e.  Ring )
83 eqid 2422 . . . . . . . . . . . . . . . . . . . . . . 23  |-  (Unit `  (ℤ/n `  N ) )  =  (Unit `  (ℤ/n `  N ) )
84 eqid 2422 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( (mulGrp `  (ℤ/n `  N ) )s  (Unit `  (ℤ/n `  N ) ) )  =  ( (mulGrp `  (ℤ/n `  N ) )s  (Unit `  (ℤ/n `  N ) ) )
8583, 84unitgrp 17880 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( (ℤ/n `  N )  e.  Ring  -> 
( (mulGrp `  (ℤ/n `  N
) )s  (Unit `  (ℤ/n `  N ) ) )  e.  Grp )
8682, 85syl 17 . . . . . . . . . . . . . . . . . . . . 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 19119 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( N  e.  NN  ->  ( # `
 (Unit `  (ℤ/n `  N
) ) )  =  ( phi `  N
) )
8887, 15eqeltrd 2510 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( N  e.  NN  ->  ( # `
 (Unit `  (ℤ/n `  N
) ) )  e. 
NN0 )
89 fvex 5887 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  (Unit `  (ℤ/n `  N ) )  e. 
_V
90 hashclb 12539 . . . . . . . . . . . . . . . . . . . . . . . 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 215 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( N  e.  NN  ->  (Unit `  (ℤ/n `  N ) )  e. 
Fin )
9392ad2antrr 730 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( N  e.  NN  /\  f  e.  D )  /\  ( x  e.  ( Base `  (ℤ/n `  N
) )  /\  (
f `  x )  =/=  0 ) )  -> 
(Unit `  (ℤ/n `  N ) )  e. 
Fin )
94 simprr 764 . . . . . . . . . . . . . . . . . . . . . 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 24162 . . . . . . . . . . . . . . . . . . . . . 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 213 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( N  e.  NN  /\  f  e.  D )  /\  ( x  e.  ( Base `  (ℤ/n `  N
) )  /\  (
f `  x )  =/=  0 ) )  ->  x  e.  (Unit `  (ℤ/n `  N
) ) )
9783, 84unitgrpbas 17879 . . . . . . . . . . . . . . . . . . . . . 22  |-  (Unit `  (ℤ/n `  N ) )  =  ( Base `  (
(mulGrp `  (ℤ/n `  N ) )s  (Unit `  (ℤ/n `  N ) ) ) )
98 eqid 2422 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( od
`  ( (mulGrp `  (ℤ/n `  N ) )s  (Unit `  (ℤ/n `  N ) ) ) )  =  ( od
`  ( (mulGrp `  (ℤ/n `  N ) )s  (Unit `  (ℤ/n `  N ) ) ) )
9997, 98oddvds2 17202 . . . . . . . . . . . . . . . . . . . . 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 1264 . . . . . . . . . . . . . . . . . . . 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 730 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( N  e.  NN  /\  f  e.  D )  /\  ( x  e.  ( Base `  (ℤ/n `  N
) )  /\  (
f `  x )  =/=  0 ) )  -> 
( # `  (Unit `  (ℤ/n `  N ) ) )  =  ( phi `  N ) )
102100, 101breqtrd 4445 . . . . . . . . . . . . . . . . . . 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 730 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( N  e.  NN  /\  f  e.  D )  /\  ( x  e.  ( Base `  (ℤ/n `  N
) )  /\  (
f `  x )  =/=  0 ) )  -> 
( phi `  N
)  e.  NN )
104103nnzd 11039 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( N  e.  NN  /\  f  e.  D )  /\  ( x  e.  ( Base `  (ℤ/n `  N
) )  /\  (
f `  x )  =/=  0 ) )  -> 
( phi `  N
)  e.  ZZ )
105 eqid 2422 . . . . . . . . . . . . . . . . . . . . 21  |-  (.g `  (
(mulGrp `  (ℤ/n `  N ) )s  (Unit `  (ℤ/n `  N ) ) ) )  =  (.g `  (
(mulGrp `  (ℤ/n `  N ) )s  (Unit `  (ℤ/n `  N ) ) ) )
106 eqid 2422 . . . . . . . . . . . . . . . . . . . . 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 17181 . . . . . . . . . . . . . . . . . . . 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 1264 . . . . . . . . . . . . . . . . . . 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 213 . . . . . . . . . . . . . . . . . 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 17883 . . . . . . . . . . . . . . . . . . . 20  |-  ( (ℤ/n `  N )  e.  Ring  -> 
(Unit `  (ℤ/n `  N ) )  e.  (SubMnd `  (mulGrp `  (ℤ/n `  N
) ) ) )
11182, 110syl 17 . . . . . . . . . . . . . . . . . . 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 16778 . . . . . . . . . . . . . . . . . . 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 1264 . . . . . . . . . . . . . . . . . 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 2422 . . . . . . . . . . . . . . . . . . . . 21  |-  ( 1r
`  (ℤ/n `  N ) )  =  ( 1r `  (ℤ/n `  N
) )
11571, 114ringidval 17722 . . . . . . . . . . . . . . . . . . . 20  |-  ( 1r
`  (ℤ/n `  N ) )  =  ( 0g `  (mulGrp `  (ℤ/n `  N ) ) )
11684, 115subm0 16588 . . . . . . . . . . . . . . . . . . 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 17 . . . . . . . . . . . . . . . . . 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 2473 . . . . . . . . . . . . . . . . 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 5881 . . . . . . . . . . . . . . . 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 2465 . . . . . . . . . . . . . . 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 18986 . . . . . . . . . . . . . . . 16  |-  ( ( ( f `  x
)  e.  CC  /\  ( phi `  N )  e.  NN0 )  -> 
( ( phi `  N ) (.g `  (mulGrp ` fld ) ) ( f `  x ) )  =  ( ( f `  x ) ^ ( phi `  N ) ) )
12265, 69, 121syl2anc 665 . . . . . . . . . . . . . . 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 2422 . . . . . . . . . . . . . . . . . 18  |-  (mulGrp ` fld )  =  (mulGrp ` fld )
124 cnfld1 18978 . . . . . . . . . . . . . . . . . 18  |-  1  =  ( 1r ` fld )
125123, 124ringidval 17722 . . . . . . . . . . . . . . . . 17  |-  1  =  ( 0g `  (mulGrp ` fld ) )
126115, 125mhm0 16575 . . . . . . . . . . . . . . . 16  |-  ( f  e.  ( (mulGrp `  (ℤ/n `  N ) ) MndHom  (mulGrp ` fld ) )  ->  ( f `  ( 1r `  (ℤ/n `  N
) ) )  =  1 )
12768, 126syl 17 . . . . . . . . . . . . . . 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 2471 . . . . . . . . . . . . . 14  |-  ( ( ( N  e.  NN  /\  f  e.  D )  /\  ( x  e.  ( Base `  (ℤ/n `  N
) )  /\  (
f `  x )  =/=  0 ) )  -> 
( ( f `  x ) ^ ( phi `  N ) )  =  1 )
129128oveq1d 6316 . . . . . . . . . . . . 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 10678 . . . . . . . . . . . . 13  |-  ( 1  -  1 )  =  0
131129, 130syl6eq 2479 . . . . . . . . . . . 12  |-  ( ( ( N  e.  NN  /\  f  e.  D )  /\  ( x  e.  ( Base `  (ℤ/n `  N
) )  /\  (
f `  x )  =/=  0 ) )  -> 
( ( ( f `
 x ) ^
( phi `  N
) )  -  1 )  =  0 )
132 oveq1 6308 . . . . . . . . . . . . . . 15  |-  ( z  =  ( f `  x )  ->  (
z ^ ( phi `  N ) )  =  ( ( f `  x ) ^ ( phi `  N ) ) )
133132oveq1d 6316 . . . . . . . . . . . . . 14  |-  ( z  =  ( f `  x )  ->  (
( z ^ ( phi `  N ) )  -  1 )  =  ( ( ( f `
 x ) ^
( phi `  N
) )  -  1 ) )
134133eqeq1d 2424 . . . . . . . . . . . . 13  |-  ( z  =  ( f `  x )  ->  (
( ( z ^
( phi `  N
) )  -  1 )  =  0  <->  (
( ( f `  x ) ^ ( phi `  N ) )  -  1 )  =  0 ) )
135134elrab 3229 . . . . . . . . . . . 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 668 . . . . . . . . . . 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 618 . . . . . . . . . 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 221 . . . . . . . . 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 379 . . . . . . . 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 3606 . . . . . . . 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 215 . . . . . . 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 2839 . . . . . 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 6060 . . . . . 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 668 . . . . 5  |-  ( ( N  e.  NN  /\  f  e.  D )  ->  f : ( Base `  (ℤ/n `  N ) ) --> ( { 0 }  u.  { z  e.  CC  | 
( ( z ^
( phi `  N
) )  -  1 )  =  0 } ) )
145144ex 435 . . . 4  |-  ( N  e.  NN  ->  (
f  e.  D  -> 
f : ( Base `  (ℤ/n `  N ) ) --> ( { 0 }  u.  { z  e.  CC  | 
( ( z ^
( phi `  N
) )  -  1 )  =  0 } ) ) )
14647, 50elmapd 7490 . . . 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 } ) ) )
147145, 146sylibrd 237 . . 3  |-  ( N  e.  NN  ->  (
f  e.  D  -> 
f  e.  ( ( { 0 }  u.  { z  e.  CC  | 
( ( z ^
( phi `  N
) )  -  1 )  =  0 } )  ^m  ( Base `  (ℤ/n `  N ) ) ) ) )
148147ssrdv 3470 . 2  |-  ( N  e.  NN  ->  D  C_  ( ( { 0 }  u.  { z  e.  CC  |  ( ( z ^ ( phi `  N ) )  -  1 )  =  0 } )  ^m  ( Base `  (ℤ/n `  N ) ) ) )
149 ssfi 7794 . 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 )
15052, 148, 149syl2anc 665 1  |-  ( N  e.  NN  ->  D  e.  Fin )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    \/ wo 369    /\ wa 370    = wceq 1437    e. wcel 1868    =/= wne 2618   A.wral 2775   {crab 2779   _Vcvv 3081    u. cun 3434    C_ wss 3436   {csn 3996   class class class wbr 4420    |-> cmpt 4479    X. cxp 4847   `'ccnv 4848   "cima 4852    Fn wfn 5592   -->wf 5593   ` cfv 5597  (class class class)co 6301    oFcof 6539    ^m cmap 7476   Fincfn 7573   CCcc 9537   0cc0 9539   1c1 9540    <_ cle 9676    - cmin 9860   -ucneg 9861   NNcn 10609   NN0cn0 10869   ZZcz 10937   ^cexp 12271   #chash 12514    || cdvds 14290   phicphi 14696   Basecbs 15106   ↾s cress 15107   0gc0g 15323   MndHom cmhm 16565  SubMndcsubmnd 16566   Grpcgrp 16654  .gcmg 16657   odcod 17150  mulGrpcmgp 17708   1rcur 17720   Ringcrg 17765   CRingccrg 17766  Unitcui 17852  ℂfldccnfld 18955  ℤ/nczn 19058   0pc0p 22611  Polycply 23122  degcdgr 23125  DChrcdchr 24144
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656  ax-un 6593  ax-inf2 8148  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617  ax-addf 9618  ax-mulf 9619
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-int 4253  df-iun 4298  df-disj 4392  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4760  df-id 4764  df-po 4770  df-so 4771  df-fr 4808  df-se 4809  df-we 4810  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-res 4861  df-ima 4862  df-pred 5395  df-ord 5441  df-on 5442  df-lim 5443  df-suc 5444  df-iota 5561  df-fun 5599  df-fn 5600  df-f 5601  df-f1 5602  df-fo 5603  df-f1o 5604  df-fv 5605  df-isom 5606  df-riota 6263  df-ov 6304  df-oprab 6305  df-mpt2 6306  df-of 6541  df-om 6703  df-1st 6803  df-2nd 6804  df-tpos 6977  df-wrecs 7032  df-recs 7094  df-rdg 7132  df-1o 7186  df-2o 7187  df-oadd 7190  df-omul 7191  df-er 7367  df-ec 7369  df-qs 7373  df-map 7478  df-pm 7479  df-en 7574  df-dom 7575  df-sdom 7576  df-fin 7577  df-sup 7958  df-inf 7959  df-oi 8027  df-card 8374  df-acn 8377  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-rp 11303  df-fz 11785  df-fzo 11916  df-fl 12027  df-mod 12096  df-seq 12213  df-exp 12272  df-hash 12515  df-cj 13148  df-re 13149  df-im 13150  df-sqrt 13284  df-abs 13285  df-clim 13537  df-rlim 13538  df-sum 13738  df-dvds 14291  df-gcd 14454  df-phi 14699  df-struct 15108  df-ndx 15109  df-slot 15110  df-base 15111  df-sets 15112  df-ress 15113  df-plusg 15188  df-mulr 15189  df-starv 15190  df-sca 15191  df-vsca 15192  df-ip 15193  df-tset 15194  df-ple 15195  df-ds 15197  df-unif 15198  df-0g 15325  df-imas 15392  df-qus 15394  df-mgm 16473  df-sgrp 16512  df-mnd 16522  df-mhm 16567  df-submnd 16568  df-grp 16658  df-minusg 16659  df-sbg 16660  df-mulg 16661  df-subg 16799  df-nsg 16800  df-eqg 16801  df-ghm 16866  df-od 17157  df-cmn 17417  df-abl 17418  df-mgp 17709  df-ur 17721  df-ring 17767  df-cring 17768  df-oppr 17836  df-dvdsr 17854  df-unit 17855  df-invr 17885  df-rnghom 17928  df-subrg 17991  df-lmod 18078  df-lss 18141  df-lsp 18180  df-sra 18380  df-rgmod 18381  df-lidl 18382  df-rsp 18383  df-2idl 18441  df-cnfld 18956  df-zring 19024  df-zrh 19059  df-zn 19062  df-0p 22612  df-ply 23126  df-idp 23127  df-coe 23128  df-dgr 23129  df-quot 23228  df-dchr 24145
This theorem is referenced by:  sumdchr2  24182  dchrhash  24183  rpvmasum2  24334  dchrisum0re  24335
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