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Theorem dchrinv 20500
Description: The inverse of a Dirichlet character is the conjugate (which is also the multiplicative inverse, because the values of  X are unimodular). (Contributed by Mario Carneiro, 28-Apr-2016.)
Hypotheses
Ref Expression
dchrabs.g  |-  G  =  (DChr `  N )
dchrabs.d  |-  D  =  ( Base `  G
)
dchrabs.x  |-  ( ph  ->  X  e.  D )
dchrinv.i  |-  I  =  ( inv g `  G )
Assertion
Ref Expression
dchrinv  |-  ( ph  ->  ( I `  X
)  =  ( *  o.  X ) )

Proof of Theorem dchrinv
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dchrabs.g . . . . . . . 8  |-  G  =  (DChr `  N )
2 eqid 2283 . . . . . . . 8  |-  (ℤ/n `  N
)  =  (ℤ/n `  N
)
3 dchrabs.d . . . . . . . 8  |-  D  =  ( Base `  G
)
4 eqid 2283 . . . . . . . 8  |-  ( +g  `  G )  =  ( +g  `  G )
5 dchrabs.x . . . . . . . 8  |-  ( ph  ->  X  e.  D )
6 cjf 11589 . . . . . . . . . 10  |-  * : CC --> CC
7 eqid 2283 . . . . . . . . . . 11  |-  ( Base `  (ℤ/n `  N ) )  =  ( Base `  (ℤ/n `  N
) )
81, 2, 3, 7, 5dchrf 20481 . . . . . . . . . 10  |-  ( ph  ->  X : ( Base `  (ℤ/n `  N ) ) --> CC )
9 fco 5398 . . . . . . . . . 10  |-  ( ( * : CC --> CC  /\  X : ( Base `  (ℤ/n `  N
) ) --> CC )  ->  ( *  o.  X ) : (
Base `  (ℤ/n `  N ) ) --> CC )
106, 8, 9sylancr 644 . . . . . . . . 9  |-  ( ph  ->  ( *  o.  X
) : ( Base `  (ℤ/n `  N ) ) --> CC )
11 eqid 2283 . . . . . . . . . . . . . . . . . . . . 21  |-  (Unit `  (ℤ/n `  N ) )  =  (Unit `  (ℤ/n `  N ) )
121, 3dchrrcl 20479 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( X  e.  D  ->  N  e.  NN )
135, 12syl 15 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  N  e.  NN )
141, 2, 7, 11, 13, 3dchrelbas3 20477 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ( X  e.  D  <->  ( X : ( Base `  (ℤ/n `  N ) ) --> CC 
/\  ( A. x  e.  (Unit `  (ℤ/n `  N ) ) A. y  e.  (Unit `  (ℤ/n `  N
) ) ( X `
 ( x ( .r `  (ℤ/n `  N
) ) y ) )  =  ( ( X `  x )  x.  ( X `  y ) )  /\  ( X `  ( 1r
`  (ℤ/n `  N ) ) )  =  1  /\  A. x  e.  ( Base `  (ℤ/n `  N ) ) ( ( X `  x
)  =/=  0  ->  x  e.  (Unit `  (ℤ/n `  N
) ) ) ) ) ) )
155, 14mpbid 201 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( X : (
Base `  (ℤ/n `  N ) ) --> CC 
/\  ( A. x  e.  (Unit `  (ℤ/n `  N ) ) A. y  e.  (Unit `  (ℤ/n `  N
) ) ( X `
 ( x ( .r `  (ℤ/n `  N
) ) y ) )  =  ( ( X `  x )  x.  ( X `  y ) )  /\  ( X `  ( 1r
`  (ℤ/n `  N ) ) )  =  1  /\  A. x  e.  ( Base `  (ℤ/n `  N ) ) ( ( X `  x
)  =/=  0  ->  x  e.  (Unit `  (ℤ/n `  N
) ) ) ) ) )
1615simprd 449 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( A. x  e.  (Unit `  (ℤ/n `  N ) ) A. y  e.  (Unit `  (ℤ/n `  N
) ) ( X `
 ( x ( .r `  (ℤ/n `  N
) ) y ) )  =  ( ( X `  x )  x.  ( X `  y ) )  /\  ( X `  ( 1r
`  (ℤ/n `  N ) ) )  =  1  /\  A. x  e.  ( Base `  (ℤ/n `  N ) ) ( ( X `  x
)  =/=  0  ->  x  e.  (Unit `  (ℤ/n `  N
) ) ) ) )
1716simp1d 967 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  A. x  e.  (Unit `  (ℤ/n `  N ) ) A. y  e.  (Unit `  (ℤ/n `  N
) ) ( X `
 ( x ( .r `  (ℤ/n `  N
) ) y ) )  =  ( ( X `  x )  x.  ( X `  y ) ) )
1817r19.21bi 2641 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  (Unit `  (ℤ/n `  N ) ) )  ->  A. y  e.  (Unit `  (ℤ/n `  N ) ) ( X `  ( x ( .r `  (ℤ/n `  N
) ) y ) )  =  ( ( X `  x )  x.  ( X `  y ) ) )
1918r19.21bi 2641 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  e.  (Unit `  (ℤ/n `  N ) ) )  /\  y  e.  (Unit `  (ℤ/n `  N ) ) )  ->  ( X `  ( x ( .r
`  (ℤ/n `  N ) ) y ) )  =  ( ( X `  x
)  x.  ( X `
 y ) ) )
2019anasss 628 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( x  e.  (Unit `  (ℤ/n `  N ) )  /\  y  e.  (Unit `  (ℤ/n `  N
) ) ) )  ->  ( X `  ( x ( .r
`  (ℤ/n `  N ) ) y ) )  =  ( ( X `  x
)  x.  ( X `
 y ) ) )
2120fveq2d 5529 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( x  e.  (Unit `  (ℤ/n `  N ) )  /\  y  e.  (Unit `  (ℤ/n `  N
) ) ) )  ->  ( * `  ( X `  ( x ( .r `  (ℤ/n `  N
) ) y ) ) )  =  ( * `  ( ( X `  x )  x.  ( X `  y ) ) ) )
228adantr 451 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( x  e.  (Unit `  (ℤ/n `  N ) )  /\  y  e.  (Unit `  (ℤ/n `  N
) ) ) )  ->  X : (
Base `  (ℤ/n `  N ) ) --> CC )
237, 11unitss 15442 . . . . . . . . . . . . . . . 16  |-  (Unit `  (ℤ/n `  N ) )  C_  ( Base `  (ℤ/n `  N ) )
24 simprl 732 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( x  e.  (Unit `  (ℤ/n `  N ) )  /\  y  e.  (Unit `  (ℤ/n `  N
) ) ) )  ->  x  e.  (Unit `  (ℤ/n `  N ) ) )
2523, 24sseldi 3178 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( x  e.  (Unit `  (ℤ/n `  N ) )  /\  y  e.  (Unit `  (ℤ/n `  N
) ) ) )  ->  x  e.  (
Base `  (ℤ/n `  N ) ) )
26 ffvelrn 5663 . . . . . . . . . . . . . . 15  |-  ( ( X : ( Base `  (ℤ/n `  N ) ) --> CC 
/\  x  e.  (
Base `  (ℤ/n `  N ) ) )  ->  ( X `  x )  e.  CC )
2722, 25, 26syl2anc 642 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( x  e.  (Unit `  (ℤ/n `  N ) )  /\  y  e.  (Unit `  (ℤ/n `  N
) ) ) )  ->  ( X `  x )  e.  CC )
28 simprr 733 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( x  e.  (Unit `  (ℤ/n `  N ) )  /\  y  e.  (Unit `  (ℤ/n `  N
) ) ) )  ->  y  e.  (Unit `  (ℤ/n `  N ) ) )
2923, 28sseldi 3178 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( x  e.  (Unit `  (ℤ/n `  N ) )  /\  y  e.  (Unit `  (ℤ/n `  N
) ) ) )  ->  y  e.  (
Base `  (ℤ/n `  N ) ) )
30 ffvelrn 5663 . . . . . . . . . . . . . . 15  |-  ( ( X : ( Base `  (ℤ/n `  N ) ) --> CC 
/\  y  e.  (
Base `  (ℤ/n `  N ) ) )  ->  ( X `  y )  e.  CC )
3122, 29, 30syl2anc 642 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( x  e.  (Unit `  (ℤ/n `  N ) )  /\  y  e.  (Unit `  (ℤ/n `  N
) ) ) )  ->  ( X `  y )  e.  CC )
3227, 31cjmuld 11706 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( x  e.  (Unit `  (ℤ/n `  N ) )  /\  y  e.  (Unit `  (ℤ/n `  N
) ) ) )  ->  ( * `  ( ( X `  x )  x.  ( X `  y )
) )  =  ( ( * `  ( X `  x )
)  x.  ( * `
 ( X `  y ) ) ) )
3321, 32eqtrd 2315 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( x  e.  (Unit `  (ℤ/n `  N ) )  /\  y  e.  (Unit `  (ℤ/n `  N
) ) ) )  ->  ( * `  ( X `  ( x ( .r `  (ℤ/n `  N
) ) y ) ) )  =  ( ( * `  ( X `  x )
)  x.  ( * `
 ( X `  y ) ) ) )
3413nnnn0d 10018 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  N  e.  NN0 )
352zncrng 16498 . . . . . . . . . . . . . . . 16  |-  ( N  e.  NN0  ->  (ℤ/n `  N
)  e.  CRing )
36 crngrng 15351 . . . . . . . . . . . . . . . 16  |-  ( (ℤ/n `  N )  e.  CRing  -> 
(ℤ/n `  N )  e.  Ring )
3734, 35, 363syl 18 . . . . . . . . . . . . . . 15  |-  ( ph  ->  (ℤ/n `  N )  e.  Ring )
3837adantr 451 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( x  e.  (Unit `  (ℤ/n `  N ) )  /\  y  e.  (Unit `  (ℤ/n `  N
) ) ) )  ->  (ℤ/n `  N )  e.  Ring )
39 eqid 2283 . . . . . . . . . . . . . . 15  |-  ( .r
`  (ℤ/n `  N ) )  =  ( .r `  (ℤ/n `  N
) )
407, 39rngcl 15354 . . . . . . . . . . . . . 14  |-  ( ( (ℤ/n `  N )  e.  Ring  /\  x  e.  ( Base `  (ℤ/n `  N ) )  /\  y  e.  ( Base `  (ℤ/n `  N ) ) )  ->  ( x ( .r `  (ℤ/n `  N
) ) y )  e.  ( Base `  (ℤ/n `  N
) ) )
4138, 25, 29, 40syl3anc 1182 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( x  e.  (Unit `  (ℤ/n `  N ) )  /\  y  e.  (Unit `  (ℤ/n `  N
) ) ) )  ->  ( x ( .r `  (ℤ/n `  N
) ) y )  e.  ( Base `  (ℤ/n `  N
) ) )
42 fvco3 5596 . . . . . . . . . . . . 13  |-  ( ( X : ( Base `  (ℤ/n `  N ) ) --> CC 
/\  ( x ( .r `  (ℤ/n `  N
) ) y )  e.  ( Base `  (ℤ/n `  N
) ) )  -> 
( ( *  o.  X ) `  (
x ( .r `  (ℤ/n `  N ) ) y ) )  =  ( * `  ( X `
 ( x ( .r `  (ℤ/n `  N
) ) y ) ) ) )
4322, 41, 42syl2anc 642 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( x  e.  (Unit `  (ℤ/n `  N ) )  /\  y  e.  (Unit `  (ℤ/n `  N
) ) ) )  ->  ( ( *  o.  X ) `  ( x ( .r
`  (ℤ/n `  N ) ) y ) )  =  ( * `  ( X `
 ( x ( .r `  (ℤ/n `  N
) ) y ) ) ) )
44 fvco3 5596 . . . . . . . . . . . . . 14  |-  ( ( X : ( Base `  (ℤ/n `  N ) ) --> CC 
/\  x  e.  (
Base `  (ℤ/n `  N ) ) )  ->  ( ( *  o.  X ) `  x )  =  ( * `  ( X `
 x ) ) )
4522, 25, 44syl2anc 642 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( x  e.  (Unit `  (ℤ/n `  N ) )  /\  y  e.  (Unit `  (ℤ/n `  N
) ) ) )  ->  ( ( *  o.  X ) `  x )  =  ( * `  ( X `
 x ) ) )
46 fvco3 5596 . . . . . . . . . . . . . 14  |-  ( ( X : ( Base `  (ℤ/n `  N ) ) --> CC 
/\  y  e.  (
Base `  (ℤ/n `  N ) ) )  ->  ( ( *  o.  X ) `  y )  =  ( * `  ( X `
 y ) ) )
4722, 29, 46syl2anc 642 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( x  e.  (Unit `  (ℤ/n `  N ) )  /\  y  e.  (Unit `  (ℤ/n `  N
) ) ) )  ->  ( ( *  o.  X ) `  y )  =  ( * `  ( X `
 y ) ) )
4845, 47oveq12d 5876 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( x  e.  (Unit `  (ℤ/n `  N ) )  /\  y  e.  (Unit `  (ℤ/n `  N
) ) ) )  ->  ( ( ( *  o.  X ) `
 x )  x.  ( ( *  o.  X ) `  y
) )  =  ( ( * `  ( X `  x )
)  x.  ( * `
 ( X `  y ) ) ) )
4933, 43, 483eqtr4d 2325 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  e.  (Unit `  (ℤ/n `  N ) )  /\  y  e.  (Unit `  (ℤ/n `  N
) ) ) )  ->  ( ( *  o.  X ) `  ( x ( .r
`  (ℤ/n `  N ) ) y ) )  =  ( ( ( *  o.  X ) `  x
)  x.  ( ( *  o.  X ) `
 y ) ) )
5049ralrimivva 2635 . . . . . . . . . 10  |-  ( ph  ->  A. x  e.  (Unit `  (ℤ/n `  N ) ) A. y  e.  (Unit `  (ℤ/n `  N
) ) ( ( *  o.  X ) `
 ( x ( .r `  (ℤ/n `  N
) ) y ) )  =  ( ( ( *  o.  X
) `  x )  x.  ( ( *  o.  X ) `  y
) ) )
51 eqid 2283 . . . . . . . . . . . . . 14  |-  ( 1r
`  (ℤ/n `  N ) )  =  ( 1r `  (ℤ/n `  N
) )
527, 51rngidcl 15361 . . . . . . . . . . . . 13  |-  ( (ℤ/n `  N )  e.  Ring  -> 
( 1r `  (ℤ/n `  N
) )  e.  (
Base `  (ℤ/n `  N ) ) )
5337, 52syl 15 . . . . . . . . . . . 12  |-  ( ph  ->  ( 1r `  (ℤ/n `  N
) )  e.  (
Base `  (ℤ/n `  N ) ) )
54 fvco3 5596 . . . . . . . . . . . 12  |-  ( ( X : ( Base `  (ℤ/n `  N ) ) --> CC 
/\  ( 1r `  (ℤ/n `  N ) )  e.  ( Base `  (ℤ/n `  N
) ) )  -> 
( ( *  o.  X ) `  ( 1r `  (ℤ/n `  N ) ) )  =  ( * `  ( X `  ( 1r
`  (ℤ/n `  N ) ) ) ) )
558, 53, 54syl2anc 642 . . . . . . . . . . 11  |-  ( ph  ->  ( ( *  o.  X ) `  ( 1r `  (ℤ/n `  N ) ) )  =  ( * `  ( X `  ( 1r
`  (ℤ/n `  N ) ) ) ) )
5616simp2d 968 . . . . . . . . . . . . 13  |-  ( ph  ->  ( X `  ( 1r `  (ℤ/n `  N ) ) )  =  1 )
5756fveq2d 5529 . . . . . . . . . . . 12  |-  ( ph  ->  ( * `  ( X `  ( 1r `  (ℤ/n `  N ) ) ) )  =  ( * `
 1 ) )
58 1re 8837 . . . . . . . . . . . . 13  |-  1  e.  RR
59 cjre 11624 . . . . . . . . . . . . 13  |-  ( 1  e.  RR  ->  (
* `  1 )  =  1 )
6058, 59ax-mp 8 . . . . . . . . . . . 12  |-  ( * `
 1 )  =  1
6157, 60syl6eq 2331 . . . . . . . . . . 11  |-  ( ph  ->  ( * `  ( X `  ( 1r `  (ℤ/n `  N ) ) ) )  =  1 )
6255, 61eqtrd 2315 . . . . . . . . . 10  |-  ( ph  ->  ( ( *  o.  X ) `  ( 1r `  (ℤ/n `  N ) ) )  =  1 )
6316simp3d 969 . . . . . . . . . . 11  |-  ( ph  ->  A. x  e.  (
Base `  (ℤ/n `  N ) ) ( ( X `  x
)  =/=  0  ->  x  e.  (Unit `  (ℤ/n `  N
) ) ) )
648, 44sylan 457 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  ( Base `  (ℤ/n `  N ) ) )  ->  ( ( *  o.  X ) `  x )  =  ( * `  ( X `
 x ) ) )
65 cj0 11643 . . . . . . . . . . . . . . . . . 18  |-  ( * `
 0 )  =  0
6665eqcomi 2287 . . . . . . . . . . . . . . . . 17  |-  0  =  ( * ` 
0 )
6766a1i 10 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  ( Base `  (ℤ/n `  N ) ) )  ->  0  =  ( * `  0 ) )
6864, 67eqeq12d 2297 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  ( Base `  (ℤ/n `  N ) ) )  ->  ( ( ( *  o.  X ) `
 x )  =  0  <->  ( * `  ( X `  x ) )  =  ( * `
 0 ) ) )
698, 26sylan 457 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  ( Base `  (ℤ/n `  N ) ) )  ->  ( X `  x )  e.  CC )
70 0cn 8831 . . . . . . . . . . . . . . . 16  |-  0  e.  CC
71 cj11 11647 . . . . . . . . . . . . . . . 16  |-  ( ( ( X `  x
)  e.  CC  /\  0  e.  CC )  ->  ( ( * `  ( X `  x ) )  =  ( * `
 0 )  <->  ( X `  x )  =  0 ) )
7269, 70, 71sylancl 643 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  ( Base `  (ℤ/n `  N ) ) )  ->  ( ( * `
 ( X `  x ) )  =  ( * `  0
)  <->  ( X `  x )  =  0 ) )
7368, 72bitrd 244 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  ( Base `  (ℤ/n `  N ) ) )  ->  ( ( ( *  o.  X ) `
 x )  =  0  <->  ( X `  x )  =  0 ) )
7473necon3bid 2481 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ( Base `  (ℤ/n `  N ) ) )  ->  ( ( ( *  o.  X ) `
 x )  =/=  0  <->  ( X `  x )  =/=  0
) )
7574imbi1d 308 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( Base `  (ℤ/n `  N ) ) )  ->  ( ( ( ( *  o.  X
) `  x )  =/=  0  ->  x  e.  (Unit `  (ℤ/n `  N ) ) )  <-> 
( ( X `  x )  =/=  0  ->  x  e.  (Unit `  (ℤ/n `  N ) ) ) ) )
7675ralbidva 2559 . . . . . . . . . . 11  |-  ( ph  ->  ( A. x  e.  ( Base `  (ℤ/n `  N
) ) ( ( ( *  o.  X
) `  x )  =/=  0  ->  x  e.  (Unit `  (ℤ/n `  N ) ) )  <->  A. x  e.  ( Base `  (ℤ/n `  N ) ) ( ( X `  x
)  =/=  0  ->  x  e.  (Unit `  (ℤ/n `  N
) ) ) ) )
7763, 76mpbird 223 . . . . . . . . . 10  |-  ( ph  ->  A. x  e.  (
Base `  (ℤ/n `  N ) ) ( ( ( *  o.  X ) `  x
)  =/=  0  ->  x  e.  (Unit `  (ℤ/n `  N
) ) ) )
7850, 62, 773jca 1132 . . . . . . . . 9  |-  ( ph  ->  ( A. x  e.  (Unit `  (ℤ/n `  N ) ) A. y  e.  (Unit `  (ℤ/n `  N
) ) ( ( *  o.  X ) `
 ( x ( .r `  (ℤ/n `  N
) ) y ) )  =  ( ( ( *  o.  X
) `  x )  x.  ( ( *  o.  X ) `  y
) )  /\  (
( *  o.  X
) `  ( 1r `  (ℤ/n `  N ) ) )  =  1  /\  A. x  e.  ( Base `  (ℤ/n `  N ) ) ( ( ( *  o.  X ) `  x
)  =/=  0  ->  x  e.  (Unit `  (ℤ/n `  N
) ) ) ) )
791, 2, 7, 11, 13, 3dchrelbas3 20477 . . . . . . . . 9  |-  ( ph  ->  ( ( *  o.  X )  e.  D  <->  ( ( *  o.  X
) : ( Base `  (ℤ/n `  N ) ) --> CC 
/\  ( A. x  e.  (Unit `  (ℤ/n `  N ) ) A. y  e.  (Unit `  (ℤ/n `  N
) ) ( ( *  o.  X ) `
 ( x ( .r `  (ℤ/n `  N
) ) y ) )  =  ( ( ( *  o.  X
) `  x )  x.  ( ( *  o.  X ) `  y
) )  /\  (
( *  o.  X
) `  ( 1r `  (ℤ/n `  N ) ) )  =  1  /\  A. x  e.  ( Base `  (ℤ/n `  N ) ) ( ( ( *  o.  X ) `  x
)  =/=  0  ->  x  e.  (Unit `  (ℤ/n `  N
) ) ) ) ) ) )
8010, 78, 79mpbir2and 888 . . . . . . . 8  |-  ( ph  ->  ( *  o.  X
)  e.  D )
811, 2, 3, 4, 5, 80dchrmul 20487 . . . . . . 7  |-  ( ph  ->  ( X ( +g  `  G ) ( *  o.  X ) )  =  ( X  o F  x.  ( *  o.  X ) ) )
8281adantr 451 . . . . . 6  |-  ( (
ph  /\  x  e.  (Unit `  (ℤ/n `  N ) ) )  ->  ( X ( +g  `  G ) ( *  o.  X
) )  =  ( X  o F  x.  ( *  o.  X
) ) )
8382fveq1d 5527 . . . . 5  |-  ( (
ph  /\  x  e.  (Unit `  (ℤ/n `  N ) ) )  ->  ( ( X ( +g  `  G
) ( *  o.  X ) ) `  x )  =  ( ( X  o F  x.  ( *  o.  X ) ) `  x ) )
8423sseli 3176 . . . . . . . . 9  |-  ( x  e.  (Unit `  (ℤ/n `  N
) )  ->  x  e.  ( Base `  (ℤ/n `  N
) ) )
8584, 64sylan2 460 . . . . . . . 8  |-  ( (
ph  /\  x  e.  (Unit `  (ℤ/n `  N ) ) )  ->  ( ( *  o.  X ) `  x )  =  ( * `  ( X `
 x ) ) )
8685oveq2d 5874 . . . . . . 7  |-  ( (
ph  /\  x  e.  (Unit `  (ℤ/n `  N ) ) )  ->  ( ( X `
 x )  x.  ( ( *  o.  X ) `  x
) )  =  ( ( X `  x
)  x.  ( * `
 ( X `  x ) ) ) )
8784, 69sylan2 460 . . . . . . . 8  |-  ( (
ph  /\  x  e.  (Unit `  (ℤ/n `  N ) ) )  ->  ( X `  x )  e.  CC )
8887absvalsqd 11924 . . . . . . 7  |-  ( (
ph  /\  x  e.  (Unit `  (ℤ/n `  N ) ) )  ->  ( ( abs `  ( X `  x
) ) ^ 2 )  =  ( ( X `  x )  x.  ( * `  ( X `  x ) ) ) )
895adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  (Unit `  (ℤ/n `  N ) ) )  ->  X  e.  D
)
90 simpr 447 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  (Unit `  (ℤ/n `  N ) ) )  ->  x  e.  (Unit `  (ℤ/n `  N ) ) )
911, 3, 89, 2, 11, 90dchrabs 20499 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  (Unit `  (ℤ/n `  N ) ) )  ->  ( abs `  ( X `  x )
)  =  1 )
9291oveq1d 5873 . . . . . . . 8  |-  ( (
ph  /\  x  e.  (Unit `  (ℤ/n `  N ) ) )  ->  ( ( abs `  ( X `  x
) ) ^ 2 )  =  ( 1 ^ 2 ) )
93 sq1 11198 . . . . . . . 8  |-  ( 1 ^ 2 )  =  1
9492, 93syl6eq 2331 . . . . . . 7  |-  ( (
ph  /\  x  e.  (Unit `  (ℤ/n `  N ) ) )  ->  ( ( abs `  ( X `  x
) ) ^ 2 )  =  1 )
9586, 88, 943eqtr2d 2321 . . . . . 6  |-  ( (
ph  /\  x  e.  (Unit `  (ℤ/n `  N ) ) )  ->  ( ( X `
 x )  x.  ( ( *  o.  X ) `  x
) )  =  1 )
968adantr 451 . . . . . . . 8  |-  ( (
ph  /\  x  e.  (Unit `  (ℤ/n `  N ) ) )  ->  X : (
Base `  (ℤ/n `  N ) ) --> CC )
97 ffn 5389 . . . . . . . 8  |-  ( X : ( Base `  (ℤ/n `  N
) ) --> CC  ->  X  Fn  ( Base `  (ℤ/n `  N
) ) )
9896, 97syl 15 . . . . . . 7  |-  ( (
ph  /\  x  e.  (Unit `  (ℤ/n `  N ) ) )  ->  X  Fn  ( Base `  (ℤ/n `  N ) ) )
99 ffn 5389 . . . . . . . . 9  |-  ( ( *  o.  X ) : ( Base `  (ℤ/n `  N
) ) --> CC  ->  ( *  o.  X )  Fn  ( Base `  (ℤ/n `  N
) ) )
10010, 99syl 15 . . . . . . . 8  |-  ( ph  ->  ( *  o.  X
)  Fn  ( Base `  (ℤ/n `  N ) ) )
101100adantr 451 . . . . . . 7  |-  ( (
ph  /\  x  e.  (Unit `  (ℤ/n `  N ) ) )  ->  ( *  o.  X )  Fn  ( Base `  (ℤ/n `  N ) ) )
102 fvex 5539 . . . . . . . 8  |-  ( Base `  (ℤ/n `  N ) )  e. 
_V
103102a1i 10 . . . . . . 7  |-  ( (
ph  /\  x  e.  (Unit `  (ℤ/n `  N ) ) )  ->  ( Base `  (ℤ/n `  N
) )  e.  _V )
10484adantl 452 . . . . . . 7  |-  ( (
ph  /\  x  e.  (Unit `  (ℤ/n `  N ) ) )  ->  x  e.  (
Base `  (ℤ/n `  N ) ) )
105 fnfvof 6090 . . . . . . 7  |-  ( ( ( X  Fn  ( Base `  (ℤ/n `  N ) )  /\  ( *  o.  X
)  Fn  ( Base `  (ℤ/n `  N ) ) )  /\  ( ( Base `  (ℤ/n `  N ) )  e. 
_V  /\  x  e.  ( Base `  (ℤ/n `  N ) ) ) )  ->  ( ( X  o F  x.  (
*  o.  X ) ) `  x )  =  ( ( X `
 x )  x.  ( ( *  o.  X ) `  x
) ) )
10698, 101, 103, 104, 105syl22anc 1183 . . . . . 6  |-  ( (
ph  /\  x  e.  (Unit `  (ℤ/n `  N ) ) )  ->  ( ( X  o F  x.  (
*  o.  X ) ) `  x )  =  ( ( X `
 x )  x.  ( ( *  o.  X ) `  x
) ) )
107 eqid 2283 . . . . . . 7  |-  ( 0g
`  G )  =  ( 0g `  G
)
10813adantr 451 . . . . . . 7  |-  ( (
ph  /\  x  e.  (Unit `  (ℤ/n `  N ) ) )  ->  N  e.  NN )
1091, 2, 107, 11, 108, 90dchr1 20496 . . . . . 6  |-  ( (
ph  /\  x  e.  (Unit `  (ℤ/n `  N ) ) )  ->  ( ( 0g
`  G ) `  x )  =  1 )
11095, 106, 1093eqtr4d 2325 . . . . 5  |-  ( (
ph  /\  x  e.  (Unit `  (ℤ/n `  N ) ) )  ->  ( ( X  o F  x.  (
*  o.  X ) ) `  x )  =  ( ( 0g
`  G ) `  x ) )
11183, 110eqtrd 2315 . . . 4  |-  ( (
ph  /\  x  e.  (Unit `  (ℤ/n `  N ) ) )  ->  ( ( X ( +g  `  G
) ( *  o.  X ) ) `  x )  =  ( ( 0g `  G
) `  x )
)
112111ralrimiva 2626 . . 3  |-  ( ph  ->  A. x  e.  (Unit `  (ℤ/n `  N ) ) ( ( X ( +g  `  G ) ( *  o.  X ) ) `
 x )  =  ( ( 0g `  G ) `  x
) )
1131, 2, 3, 4, 5, 80dchrmulcl 20488 . . . 4  |-  ( ph  ->  ( X ( +g  `  G ) ( *  o.  X ) )  e.  D )
1141dchrabl 20493 . . . . . 6  |-  ( N  e.  NN  ->  G  e.  Abel )
115 ablgrp 15094 . . . . . 6  |-  ( G  e.  Abel  ->  G  e. 
Grp )
11613, 114, 1153syl 18 . . . . 5  |-  ( ph  ->  G  e.  Grp )
1173, 107grpidcl 14510 . . . . 5  |-  ( G  e.  Grp  ->  ( 0g `  G )  e.  D )
118116, 117syl 15 . . . 4  |-  ( ph  ->  ( 0g `  G
)  e.  D )
1191, 2, 3, 11, 113, 118dchreq 20497 . . 3  |-  ( ph  ->  ( ( X ( +g  `  G ) ( *  o.  X
) )  =  ( 0g `  G )  <->  A. x  e.  (Unit `  (ℤ/n `  N ) ) ( ( X ( +g  `  G ) ( *  o.  X ) ) `
 x )  =  ( ( 0g `  G ) `  x
) ) )
120112, 119mpbird 223 . 2  |-  ( ph  ->  ( X ( +g  `  G ) ( *  o.  X ) )  =  ( 0g `  G ) )
121 dchrinv.i . . . 4  |-  I  =  ( inv g `  G )
1223, 4, 107, 121grpinvid1 14530 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  D  /\  ( *  o.  X
)  e.  D )  ->  ( ( I `
 X )  =  ( *  o.  X
)  <->  ( X ( +g  `  G ) ( *  o.  X
) )  =  ( 0g `  G ) ) )
123116, 5, 80, 122syl3anc 1182 . 2  |-  ( ph  ->  ( ( I `  X )  =  ( *  o.  X )  <-> 
( X ( +g  `  G ) ( *  o.  X ) )  =  ( 0g `  G ) ) )
124120, 123mpbird 223 1  |-  ( ph  ->  ( I `  X
)  =  ( *  o.  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   _Vcvv 2788    o. ccom 4693    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858    o Fcof 6076   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    x. cmul 8742   NNcn 9746   2c2 9795   NN0cn0 9965   ^cexp 11104   *ccj 11581   abscabs 11719   Basecbs 13148   +g cplusg 13208   .rcmulr 13209   0gc0g 13400   Grpcgrp 14362   inv gcminusg 14363   Abelcabel 15090   Ringcrg 15337   CRingccrg 15338   1rcur 15339  Unitcui 15421  ℤ/nczn 16454  DChrcdchr 20471
This theorem is referenced by:  dchr2sum  20512  dchrisum0re  20662
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-disj 3994  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-tpos 6234  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-omul 6484  df-er 6660  df-ec 6662  df-qs 6666  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-acn 7575  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-ioc 10661  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-fac 11289  df-bc 11316  df-hash 11338  df-shft 11562  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-limsup 11945  df-clim 11962  df-rlim 11963  df-sum 12159  df-ef 12349  df-sin 12351  df-cos 12352  df-pi 12354  df-dvds 12532  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-rest 13327  df-topn 13328  df-topgen 13344  df-pt 13345  df-prds 13348  df-xrs 13403  df-0g 13404  df-gsum 13405  df-qtop 13410  df-imas 13411  df-divs 13412  df-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-mhm 14415  df-submnd 14416  df-grp 14489  df-minusg 14490  df-sbg 14491  df-mulg 14492  df-subg 14618  df-nsg 14619  df-eqg 14620  df-ghm 14681  df-cntz 14793  df-od 14844  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-cring 15341  df-ur 15342  df-oppr 15405  df-dvdsr 15423  df-unit 15424  df-invr 15454  df-dvr 15465  df-rnghom 15496  df-drng 15514  df-subrg 15543  df-lmod 15629  df-lss 15690  df-lsp 15729  df-sra 15925  df-rgmod 15926  df-lidl 15927  df-rsp 15928  df-2idl 15984  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-zrh 16455  df-zn 16458  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cld 16756  df-ntr 16757  df-cls 16758  df-nei 16835  df-lp 16868  df-perf 16869  df-cn 16957  df-cnp 16958  df-haus 17043  df-tx 17257  df-hmeo 17446  df-fbas 17520  df-fg 17521  df-fil 17541  df-fm 17633  df-flim 17634  df-flf 17635  df-xms 17885  df-ms 17886  df-tms 17887  df-cncf 18382  df-limc 19216  df-dv 19217  df-log 19914  df-cxp 19915  df-dchr 20472
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