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Theorem dchrfi 21044
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 7190 . . . 4  |-  { 0 }  e.  Fin
2 cnex 9076 . . . . . . . . 9  |-  CC  e.  _V
32a1i 11 . . . . . . . 8  |-  ( N  e.  NN  ->  CC  e.  _V )
4 ovex 6109 . . . . . . . . 9  |-  ( z ^ ( phi `  N ) )  e. 
_V
54a1i 11 . . . . . . . 8  |-  ( ( N  e.  NN  /\  z  e.  CC )  ->  ( z ^ ( phi `  N ) )  e.  _V )
6 ax-1cn 9053 . . . . . . . . 9  |-  1  e.  CC
76a1i 11 . . . . . . . 8  |-  ( ( N  e.  NN  /\  z  e.  CC )  ->  1  e.  CC )
8 eqidd 2439 . . . . . . . 8  |-  ( N  e.  NN  ->  (
z  e.  CC  |->  ( z ^ ( phi `  N ) ) )  =  ( z  e.  CC  |->  ( z ^
( phi `  N
) ) ) )
9 fconstmpt 4924 . . . . . . . . 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 6325 . . . . . . 7  |-  ( N  e.  NN  ->  (
( z  e.  CC  |->  ( z ^ ( phi `  N ) ) )  o F  -  ( CC  X.  { 1 } ) )  =  ( z  e.  CC  |->  ( ( z ^
( phi `  N
) )  -  1 ) ) )
12 ssid 3369 . . . . . . . . . 10  |-  CC  C_  CC
1312a1i 11 . . . . . . . . 9  |-  ( N  e.  NN  ->  CC  C_  CC )
146a1i 11 . . . . . . . . 9  |-  ( N  e.  NN  ->  1  e.  CC )
15 phicl 13163 . . . . . . . . . 10  |-  ( N  e.  NN  ->  ( phi `  N )  e.  NN )
1615nnnn0d 10279 . . . . . . . . 9  |-  ( N  e.  NN  ->  ( phi `  N )  e. 
NN0 )
17 plypow 20129 . . . . . . . . 9  |-  ( ( CC  C_  CC  /\  1  e.  CC  /\  ( phi `  N )  e.  NN0 )  ->  ( z  e.  CC  |->  ( z ^
( phi `  N
) ) )  e.  (Poly `  CC )
)
1813, 14, 16, 17syl3anc 1185 . . . . . . . 8  |-  ( N  e.  NN  ->  (
z  e.  CC  |->  ( z ^ ( phi `  N ) ) )  e.  (Poly `  CC ) )
19 plyconst 20130 . . . . . . . . 9  |-  ( ( CC  C_  CC  /\  1  e.  CC )  ->  ( CC  X.  { 1 } )  e.  (Poly `  CC ) )
2012, 6, 19mp2an 655 . . . . . . . 8  |-  ( CC 
X.  { 1 } )  e.  (Poly `  CC )
21 plysubcl 20146 . . . . . . . 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 645 . . . . . . 7  |-  ( N  e.  NN  ->  (
( z  e.  CC  |->  ( z ^ ( phi `  N ) ) )  o F  -  ( CC  X.  { 1 } ) )  e.  (Poly `  CC )
)
2311, 22eqeltrrd 2513 . . . . . 6  |-  ( N  e.  NN  ->  (
z  e.  CC  |->  ( ( z ^ ( phi `  N ) )  -  1 ) )  e.  (Poly `  CC ) )
24 0cn 9089 . . . . . . 7  |-  0  e.  CC
25 ax-1ne0 9064 . . . . . . . . 9  |-  1  =/=  0
266, 25negne0i 9380 . . . . . . . 8  |-  -u 1  =/=  0
27150expd 11544 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  (
0 ^ ( phi `  N ) )  =  0 )
2827oveq1d 6099 . . . . . . . . . 10  |-  ( N  e.  NN  ->  (
( 0 ^ ( phi `  N ) )  -  1 )  =  ( 0  -  1 ) )
29 oveq1 6091 . . . . . . . . . . . . 13  |-  ( z  =  0  ->  (
z ^ ( phi `  N ) )  =  ( 0 ^ ( phi `  N ) ) )
3029oveq1d 6099 . . . . . . . . . . . 12  |-  ( z  =  0  ->  (
( z ^ ( phi `  N ) )  -  1 )  =  ( ( 0 ^ ( phi `  N
) )  -  1 ) )
31 eqid 2438 . . . . . . . . . . . 12  |-  ( z  e.  CC  |->  ( ( z ^ ( phi `  N ) )  - 
1 ) )  =  ( z  e.  CC  |->  ( ( z ^
( phi `  N
) )  -  1 ) )
32 ovex 6109 . . . . . . . . . . . 12  |-  ( ( 0 ^ ( phi `  N ) )  - 
1 )  e.  _V
3330, 31, 32fvmpt 5809 . . . . . . . . . . 11  |-  ( 0  e.  CC  ->  (
( z  e.  CC  |->  ( ( z ^
( phi `  N
) )  -  1 ) ) `  0
)  =  ( ( 0 ^ ( phi `  N ) )  - 
1 ) )
3424, 33ax-mp 5 . . . . . . . . . 10  |-  ( ( z  e.  CC  |->  ( ( z ^ ( phi `  N ) )  -  1 ) ) `
 0 )  =  ( ( 0 ^ ( phi `  N
) )  -  1 )
35 df-neg 9299 . . . . . . . . . 10  |-  -u 1  =  ( 0  -  1 )
3628, 34, 353eqtr4g 2495 . . . . . . . . 9  |-  ( N  e.  NN  ->  (
( z  e.  CC  |->  ( ( z ^
( phi `  N
) )  -  1 ) ) `  0
)  =  -u 1
)
3736neeq1d 2616 . . . . . . . 8  |-  ( N  e.  NN  ->  (
( ( z  e.  CC  |->  ( ( z ^ ( phi `  N ) )  - 
1 ) ) ` 
0 )  =/=  0  <->  -u 1  =/=  0 ) )
3826, 37mpbiri 226 . . . . . . 7  |-  ( N  e.  NN  ->  (
( z  e.  CC  |->  ( ( z ^
( phi `  N
) )  -  1 ) ) `  0
)  =/=  0 )
39 ne0p 20131 . . . . . . 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 646 . . . . . 6  |-  ( N  e.  NN  ->  (
z  e.  CC  |->  ( ( z ^ ( phi `  N ) )  -  1 ) )  =/=  0 p )
4131mptiniseg 5367 . . . . . . . . 9  |-  ( 0  e.  CC  ->  ( `' ( z  e.  CC  |->  ( ( z ^ ( phi `  N ) )  - 
1 ) ) " { 0 } )  =  { z  e.  CC  |  ( ( z ^ ( phi `  N ) )  - 
1 )  =  0 } )
4224, 41ax-mp 5 . . . . . . . 8  |-  ( `' ( z  e.  CC  |->  ( ( z ^
( phi `  N
) )  -  1 ) ) " {
0 } )  =  { z  e.  CC  |  ( ( z ^ ( phi `  N ) )  - 
1 )  =  0 }
4342eqcomi 2442 . . . . . . 7  |-  { z  e.  CC  |  ( ( z ^ ( phi `  N ) )  -  1 )  =  0 }  =  ( `' ( z  e.  CC  |->  ( ( z ^ ( phi `  N ) )  - 
1 ) ) " { 0 } )
4443fta1 20230 . . . . . 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 644 . . . . 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 447 . . . 4  |-  ( N  e.  NN  ->  { z  e.  CC  |  ( ( z ^ ( phi `  N ) )  -  1 )  =  0 }  e.  Fin )
47 unfi 7377 . . . 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 646 . . 3  |-  ( N  e.  NN  ->  ( { 0 }  u.  { z  e.  CC  | 
( ( z ^
( phi `  N
) )  -  1 )  =  0 } )  e.  Fin )
49 eqid 2438 . . . 4  |-  (ℤ/n `  N
)  =  (ℤ/n `  N
)
50 eqid 2438 . . . 4  |-  ( Base `  (ℤ/n `  N ) )  =  ( Base `  (ℤ/n `  N
) )
5149, 50znfi 16845 . . 3  |-  ( N  e.  NN  ->  ( Base `  (ℤ/n `  N ) )  e. 
Fin )
52 mapfi 7406 . . 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 644 . 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 449 . . . . . . . 8  |-  ( ( N  e.  NN  /\  f  e.  D )  ->  f  e.  D )
5754, 49, 55, 50, 56dchrf 21031 . . . . . . 7  |-  ( ( N  e.  NN  /\  f  e.  D )  ->  f : ( Base `  (ℤ/n `  N ) ) --> CC )
58 ffn 5594 . . . . . . 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 2603 . . . . . . . . . . 11  |-  ( ( f `  x )  =/=  0  <->  -.  (
f `  x )  =  0 )
61 fvex 5745 . . . . . . . . . . . 12  |-  ( f `
 x )  e. 
_V
6261elsnc 3839 . . . . . . . . . . 11  |-  ( ( f `  x )  e.  { 0 }  <-> 
( f `  x
)  =  0 )
6360, 62xchbinxr 304 . . . . . . . . . 10  |-  ( ( f `  x )  =/=  0  <->  -.  (
f `  x )  e.  { 0 } )
64 simpl 445 . . . . . . . . . . . . 13  |-  ( ( x  e.  ( Base `  (ℤ/n `  N ) )  /\  ( f `  x
)  =/=  0 )  ->  x  e.  (
Base `  (ℤ/n `  N ) ) )
65 ffvelrn 5871 . . . . . . . . . . . . 13  |-  ( ( f : ( Base `  (ℤ/n `  N ) ) --> CC 
/\  x  e.  (
Base `  (ℤ/n `  N ) ) )  ->  ( f `  x )  e.  CC )
6657, 64, 65syl2an 465 . . . . . . . . . . . 12  |-  ( ( ( N  e.  NN  /\  f  e.  D )  /\  ( x  e.  ( Base `  (ℤ/n `  N
) )  /\  (
f `  x )  =/=  0 ) )  -> 
( f `  x
)  e.  CC )
6754, 49, 55dchrmhm 21030 . . . . . . . . . . . . . . . . . 18  |-  D  C_  ( (mulGrp `  (ℤ/n `  N ) ) MndHom  (mulGrp ` fld ) )
68 simplr 733 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( N  e.  NN  /\  f  e.  D )  /\  ( x  e.  ( Base `  (ℤ/n `  N
) )  /\  (
f `  x )  =/=  0 ) )  -> 
f  e.  D )
6967, 68sseldi 3348 . . . . . . . . . . . . . . . . 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 708 . . . . . . . . . . . . . . . . 17  |-  ( ( ( N  e.  NN  /\  f  e.  D )  /\  ( x  e.  ( Base `  (ℤ/n `  N
) )  /\  (
f `  x )  =/=  0 ) )  -> 
( phi `  N
)  e.  NN0 )
71 simprl 734 . . . . . . . . . . . . . . . . 17  |-  ( ( ( N  e.  NN  /\  f  e.  D )  /\  ( x  e.  ( Base `  (ℤ/n `  N
) )  /\  (
f `  x )  =/=  0 ) )  ->  x  e.  ( Base `  (ℤ/n `  N ) ) )
72 eqid 2438 . . . . . . . . . . . . . . . . . . 19  |-  (mulGrp `  (ℤ/n `  N ) )  =  (mulGrp `  (ℤ/n `  N ) )
7372, 50mgpbas 15659 . . . . . . . . . . . . . . . . . 18  |-  ( Base `  (ℤ/n `  N ) )  =  ( Base `  (mulGrp `  (ℤ/n `  N ) ) )
74 eqid 2438 . . . . . . . . . . . . . . . . . 18  |-  (.g `  (mulGrp `  (ℤ/n `  N ) ) )  =  (.g `  (mulGrp `  (ℤ/n `  N
) ) )
75 eqid 2438 . . . . . . . . . . . . . . . . . 18  |-  (.g `  (mulGrp ` fld ) )  =  (.g `  (mulGrp ` fld ) )
7673, 74, 75mhmmulg 14927 . . . . . . . . . . . . . . . . 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 1185 . . . . . . . . . . . . . . . 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 10233 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( N  e.  NN  ->  N  e.  NN0 )
7949zncrng 16830 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( N  e.  NN0  ->  (ℤ/n `  N
)  e.  CRing )
8078, 79syl 16 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( N  e.  NN  ->  (ℤ/n `  N
)  e.  CRing )
81 crngrng 15679 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( (ℤ/n `  N )  e.  CRing  -> 
(ℤ/n `  N )  e.  Ring )
8280, 81syl 16 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( N  e.  NN  ->  (ℤ/n `  N
)  e.  Ring )
8382ad2antrr 708 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( N  e.  NN  /\  f  e.  D )  /\  ( x  e.  ( Base `  (ℤ/n `  N
) )  /\  (
f `  x )  =/=  0 ) )  -> 
(ℤ/n `  N )  e.  Ring )
84 eqid 2438 . . . . . . . . . . . . . . . . . . . . . . 23  |-  (Unit `  (ℤ/n `  N ) )  =  (Unit `  (ℤ/n `  N ) )
85 eqid 2438 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( (mulGrp `  (ℤ/n `  N ) )s  (Unit `  (ℤ/n `  N ) ) )  =  ( (mulGrp `  (ℤ/n `  N ) )s  (Unit `  (ℤ/n `  N ) ) )
8684, 85unitgrp 15777 . . . . . . . . . . . . . . . . . . . . . 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 16850 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( N  e.  NN  ->  ( # `
 (Unit `  (ℤ/n `  N
) ) )  =  ( phi `  N
) )
8988, 16eqeltrd 2512 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( N  e.  NN  ->  ( # `
 (Unit `  (ℤ/n `  N
) ) )  e. 
NN0 )
90 fvex 5745 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  (Unit `  (ℤ/n `  N ) )  e. 
_V
91 hashclb 11646 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( (Unit `  (ℤ/n `  N ) )  e. 
_V  ->  ( (Unit `  (ℤ/n `  N ) )  e. 
Fin 
<->  ( # `  (Unit `  (ℤ/n `  N ) ) )  e.  NN0 ) )
9290, 91ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( (Unit `  (ℤ/n `  N ) )  e. 
Fin 
<->  ( # `  (Unit `  (ℤ/n `  N ) ) )  e.  NN0 )
9389, 92sylibr 205 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( N  e.  NN  ->  (Unit `  (ℤ/n `  N ) )  e. 
Fin )
9493ad2antrr 708 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( N  e.  NN  /\  f  e.  D )  /\  ( x  e.  ( Base `  (ℤ/n `  N
) )  /\  (
f `  x )  =/=  0 ) )  -> 
(Unit `  (ℤ/n `  N ) )  e. 
Fin )
95 simprr 735 . . . . . . . . . . . . . . . . . . . . . 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 21039 . . . . . . . . . . . . . . . . . . . . . 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 203 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( N  e.  NN  /\  f  e.  D )  /\  ( x  e.  ( Base `  (ℤ/n `  N
) )  /\  (
f `  x )  =/=  0 ) )  ->  x  e.  (Unit `  (ℤ/n `  N
) ) )
9884, 85unitgrpbas 15776 . . . . . . . . . . . . . . . . . . . . . 22  |-  (Unit `  (ℤ/n `  N ) )  =  ( Base `  (
(mulGrp `  (ℤ/n `  N ) )s  (Unit `  (ℤ/n `  N ) ) ) )
99 eqid 2438 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( od
`  ( (mulGrp `  (ℤ/n `  N ) )s  (Unit `  (ℤ/n `  N ) ) ) )  =  ( od
`  ( (mulGrp `  (ℤ/n `  N ) )s  (Unit `  (ℤ/n `  N ) ) ) )
10098, 99oddvds2 15207 . . . . . . . . . . . . . . . . . . . . 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 1185 . . . . . . . . . . . . . . . . . . . 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 708 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( N  e.  NN  /\  f  e.  D )  /\  ( x  e.  ( Base `  (ℤ/n `  N
) )  /\  (
f `  x )  =/=  0 ) )  -> 
( # `  (Unit `  (ℤ/n `  N ) ) )  =  ( phi `  N ) )
103101, 102breqtrd 4239 . . . . . . . . . . . . . . . . . . 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 708 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( N  e.  NN  /\  f  e.  D )  /\  ( x  e.  ( Base `  (ℤ/n `  N
) )  /\  (
f `  x )  =/=  0 ) )  -> 
( phi `  N
)  e.  NN )
105104nnzd 10379 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( N  e.  NN  /\  f  e.  D )  /\  ( x  e.  ( Base `  (ℤ/n `  N
) )  /\  (
f `  x )  =/=  0 ) )  -> 
( phi `  N
)  e.  ZZ )
106 eqid 2438 . . . . . . . . . . . . . . . . . . . . 21  |-  (.g `  (
(mulGrp `  (ℤ/n `  N ) )s  (Unit `  (ℤ/n `  N ) ) ) )  =  (.g `  (
(mulGrp `  (ℤ/n `  N ) )s  (Unit `  (ℤ/n `  N ) ) ) )
107 eqid 2438 . . . . . . . . . . . . . . . . . . . . 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 15190 . . . . . . . . . . . . . . . . . . . 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 1185 . . . . . . . . . . . . . . . . . . 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 203 . . . . . . . . . . . . . . . . . 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 15780 . . . . . . . . . . . . . . . . . . . 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 14930 . . . . . . . . . . . . . . . . . . 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 1185 . . . . . . . . . . . . . . . . . 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 2438 . . . . . . . . . . . . . . . . . . . . 21  |-  ( 1r
`  (ℤ/n `  N ) )  =  ( 1r `  (ℤ/n `  N
) )
11672, 115rngidval 15671 . . . . . . . . . . . . . . . . . . . 20  |-  ( 1r
`  (ℤ/n `  N ) )  =  ( 0g `  (mulGrp `  (ℤ/n `  N ) ) )
11785, 116subm0 14761 . . . . . . . . . . . . . . . . . . 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 2480 . . . . . . . . . . . . . . . . 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 5735 . . . . . . . . . . . . . . . 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 2472 . . . . . . . . . . . . . . 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 16739 . . . . . . . . . . . . . . . 16  |-  ( ( ( f `  x
)  e.  CC  /\  ( phi `  N )  e.  NN0 )  -> 
( ( phi `  N ) (.g `  (mulGrp ` fld ) ) ( f `  x ) )  =  ( ( f `  x ) ^ ( phi `  N ) ) )
12366, 70, 122syl2anc 644 . . . . . . . . . . . . . . 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 2438 . . . . . . . . . . . . . . . . . 18  |-  (mulGrp ` fld )  =  (mulGrp ` fld )
125 cnfld1 16731 . . . . . . . . . . . . . . . . . 18  |-  1  =  ( 1r ` fld )
126124, 125rngidval 15671 . . . . . . . . . . . . . . . . 17  |-  1  =  ( 0g `  (mulGrp ` fld ) )
127116, 126mhm0 14751 . . . . . . . . . . . . . . . 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 2478 . . . . . . . . . . . . . 14  |-  ( ( ( N  e.  NN  /\  f  e.  D )  /\  ( x  e.  ( Base `  (ℤ/n `  N
) )  /\  (
f `  x )  =/=  0 ) )  -> 
( ( f `  x ) ^ ( phi `  N ) )  =  1 )
130129oveq1d 6099 . . . . . . . . . . . . 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 10073 . . . . . . . . . . . . 13  |-  ( 1  -  1 )  =  0
132130, 131syl6eq 2486 . . . . . . . . . . . 12  |-  ( ( ( N  e.  NN  /\  f  e.  D )  /\  ( x  e.  ( Base `  (ℤ/n `  N
) )  /\  (
f `  x )  =/=  0 ) )  -> 
( ( ( f `
 x ) ^
( phi `  N
) )  -  1 )  =  0 )
133 oveq1 6091 . . . . . . . . . . . . . . 15  |-  ( z  =  ( f `  x )  ->  (
z ^ ( phi `  N ) )  =  ( ( f `  x ) ^ ( phi `  N ) ) )
134133oveq1d 6099 . . . . . . . . . . . . . 14  |-  ( z  =  ( f `  x )  ->  (
( z ^ ( phi `  N ) )  -  1 )  =  ( ( ( f `
 x ) ^
( phi `  N
) )  -  1 ) )
135134eqeq1d 2446 . . . . . . . . . . . . 13  |-  ( z  =  ( f `  x )  ->  (
( ( z ^
( phi `  N
) )  -  1 )  =  0  <->  (
( ( f `  x ) ^ ( phi `  N ) )  -  1 )  =  0 ) )
136135elrab 3094 . . . . . . . . . . . 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 647 . . . . . . . . . . 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 600 . . . . . . . . . 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 211 . . . . . . . . 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 369 . . . . . . . 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 3490 . . . . . . . 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 205 . . . . . . 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 2791 . . . . . 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 5897 . . . . . 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 647 . . . . 5  |-  ( ( N  e.  NN  /\  f  e.  D )  ->  f : ( Base `  (ℤ/n `  N ) ) --> ( { 0 }  u.  { z  e.  CC  | 
( ( z ^
( phi `  N
) )  -  1 )  =  0 } ) )
146145ex 425 . . . 4  |-  ( N  e.  NN  ->  (
f  e.  D  -> 
f : ( Base `  (ℤ/n `  N ) ) --> ( { 0 }  u.  { z  e.  CC  | 
( ( z ^
( phi `  N
) )  -  1 )  =  0 } ) ) )
147 elmapg 7034 . . . . 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 644 . . . 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 227 . . 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 3356 . 2  |-  ( N  e.  NN  ->  D  C_  ( ( { 0 }  u.  { z  e.  CC  |  ( ( z ^ ( phi `  N ) )  -  1 )  =  0 } )  ^m  ( Base `  (ℤ/n `  N ) ) ) )
151 ssfi 7332 . 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 644 1  |-  ( N  e.  NN  ->  D  e.  Fin )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    \/ wo 359    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601   A.wral 2707   {crab 2711   _Vcvv 2958    u. cun 3320    C_ wss 3322   {csn 3816   class class class wbr 4215    e. cmpt 4269    X. cxp 4879   `'ccnv 4880   "cima 4884    Fn wfn 5452   -->wf 5453   ` cfv 5457  (class class class)co 6084    o Fcof 6306    ^m cmap 7021   Fincfn 7112   CCcc 8993   0cc0 8995   1c1 8996    <_ cle 9126    - cmin 9296   -ucneg 9297   NNcn 10005   NN0cn0 10226   ZZcz 10287   ^cexp 11387   #chash 11623    || cdivides 12857   phicphi 13158   Basecbs 13474   ↾s cress 13475   0gc0g 13728   Grpcgrp 14690  .gcmg 14694   MndHom cmhm 14741  SubMndcsubmnd 14742   odcod 15168  mulGrpcmgp 15653   Ringcrg 15665   CRingccrg 15666   1rcur 15667  Unitcui 15749  ℂfldccnfld 16708  ℤ/nczn 16786   0 pc0p 19564  Polycply 20108  degcdgr 20111  DChrcdchr 21021
This theorem is referenced by:  sumdchr2  21059  dchrhash  21060  rpvmasum2  21211  dchrisum0re  21212
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-inf2 7599  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072  ax-pre-sup 9073  ax-addf 9074  ax-mulf 9075
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-disj 4186  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-se 4545  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-isom 5466  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-of 6308  df-1st 6352  df-2nd 6353  df-tpos 6482  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-2o 6728  df-oadd 6731  df-omul 6732  df-er 6908  df-ec 6910  df-qs 6914  df-map 7023  df-pm 7024  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-sup 7449  df-oi 7482  df-card 7831  df-acn 7834  df-cda 8053  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683  df-nn 10006  df-2 10063  df-3 10064  df-4 10065  df-5 10066  df-6 10067  df-7 10068  df-8 10069  df-9 10070  df-10 10071  df-n0 10227  df-z 10288  df-dec 10388  df-uz 10494  df-rp 10618  df-fz 11049  df-fzo 11141  df-fl 11207  df-mod 11256  df-seq 11329  df-exp 11388  df-hash 11624  df-cj 11909  df-re 11910  df-im 11911  df-sqr 12045  df-abs 12046  df-clim 12287  df-rlim 12288  df-sum 12485  df-dvds 12858  df-gcd 13012  df-phi 13160  df-struct 13476  df-ndx 13477  df-slot 13478  df-base 13479  df-sets 13480  df-ress 13481  df-plusg 13547  df-mulr 13548  df-starv 13549  df-sca 13550  df-vsca 13551  df-tset 13553  df-ple 13554  df-ds 13556  df-unif 13557  df-0g 13732  df-imas 13739  df-divs 13740  df-mnd 14695  df-mhm 14743  df-submnd 14744  df-grp 14817  df-minusg 14818  df-sbg 14819  df-mulg 14820  df-subg 14946  df-nsg 14947  df-eqg 14948  df-ghm 15009  df-od 15172  df-cmn 15419  df-abl 15420  df-mgp 15654  df-rng 15668  df-cring 15669  df-ur 15670  df-oppr 15733  df-dvdsr 15751  df-unit 15752  df-invr 15782  df-rnghom 15824  df-subrg 15871  df-lmod 15957  df-lss 16014  df-lsp 16053  df-sra 16249  df-rgmod 16250  df-lidl 16251  df-rsp 16252  df-2idl 16308  df-cnfld 16709  df-zrh 16787  df-zn 16790  df-0p 19565  df-ply 20112  df-idp 20113  df-coe 20114  df-dgr 20115  df-quot 20213  df-dchr 21022
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