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Theorem dchrinv 23734
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  =  ( invg `  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 2454 . . . . . . . 8  |-  (ℤ/n `  N
)  =  (ℤ/n `  N
)
3 dchrabs.d . . . . . . . 8  |-  D  =  ( Base `  G
)
4 eqid 2454 . . . . . . . 8  |-  ( +g  `  G )  =  ( +g  `  G )
5 dchrabs.x . . . . . . . 8  |-  ( ph  ->  X  e.  D )
6 cjf 13019 . . . . . . . . . 10  |-  * : CC --> CC
7 eqid 2454 . . . . . . . . . . 11  |-  ( Base `  (ℤ/n `  N ) )  =  ( Base `  (ℤ/n `  N
) )
81, 2, 3, 7, 5dchrf 23715 . . . . . . . . . 10  |-  ( ph  ->  X : ( Base `  (ℤ/n `  N ) ) --> CC )
9 fco 5723 . . . . . . . . . 10  |-  ( ( * : CC --> CC  /\  X : ( Base `  (ℤ/n `  N
) ) --> CC )  ->  ( *  o.  X ) : (
Base `  (ℤ/n `  N ) ) --> CC )
106, 8, 9sylancr 661 . . . . . . . . 9  |-  ( ph  ->  ( *  o.  X
) : ( Base `  (ℤ/n `  N ) ) --> CC )
11 eqid 2454 . . . . . . . . . . . . . . . . . . . . 21  |-  (Unit `  (ℤ/n `  N ) )  =  (Unit `  (ℤ/n `  N ) )
121, 3dchrrcl 23713 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( X  e.  D  ->  N  e.  NN )
135, 12syl 16 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  N  e.  NN )
141, 2, 7, 11, 13, 3dchrelbas3 23711 . . . . . . . . . . . . . . . . . . . 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 210 . . . . . . . . . . . . . . . . . . 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 461 . . . . . . . . . . . . . . . . . 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 1006 . . . . . . . . . . . . . . . . 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 2823 . . . . . . . . . . . . . . . 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 2823 . . . . . . . . . . . . . . 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 645 . . . . . . . . . . . . . 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 5852 . . . . . . . . . . . . 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 463 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( x  e.  (Unit `  (ℤ/n `  N ) )  /\  y  e.  (Unit `  (ℤ/n `  N
) ) ) )  ->  X : (
Base `  (ℤ/n `  N ) ) --> CC )
237, 11unitss 17504 . . . . . . . . . . . . . . . 16  |-  (Unit `  (ℤ/n `  N ) )  C_  ( Base `  (ℤ/n `  N ) )
24 simprl 754 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( x  e.  (Unit `  (ℤ/n `  N ) )  /\  y  e.  (Unit `  (ℤ/n `  N
) ) ) )  ->  x  e.  (Unit `  (ℤ/n `  N ) ) )
2523, 24sseldi 3487 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( x  e.  (Unit `  (ℤ/n `  N ) )  /\  y  e.  (Unit `  (ℤ/n `  N
) ) ) )  ->  x  e.  (
Base `  (ℤ/n `  N ) ) )
2622, 25ffvelrnd 6008 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( x  e.  (Unit `  (ℤ/n `  N ) )  /\  y  e.  (Unit `  (ℤ/n `  N
) ) ) )  ->  ( X `  x )  e.  CC )
27 simprr 755 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( x  e.  (Unit `  (ℤ/n `  N ) )  /\  y  e.  (Unit `  (ℤ/n `  N
) ) ) )  ->  y  e.  (Unit `  (ℤ/n `  N ) ) )
2823, 27sseldi 3487 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( x  e.  (Unit `  (ℤ/n `  N ) )  /\  y  e.  (Unit `  (ℤ/n `  N
) ) ) )  ->  y  e.  (
Base `  (ℤ/n `  N ) ) )
2922, 28ffvelrnd 6008 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( x  e.  (Unit `  (ℤ/n `  N ) )  /\  y  e.  (Unit `  (ℤ/n `  N
) ) ) )  ->  ( X `  y )  e.  CC )
3026, 29cjmuld 13136 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( x  e.  (Unit `  (ℤ/n `  N ) )  /\  y  e.  (Unit `  (ℤ/n `  N
) ) ) )  ->  ( * `  ( ( X `  x )  x.  ( X `  y )
) )  =  ( ( * `  ( X `  x )
)  x.  ( * `
 ( X `  y ) ) ) )
3121, 30eqtrd 2495 . . . . . . . . . . . 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 ) ) ) )
3213nnnn0d 10848 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  N  e.  NN0 )
332zncrng 18756 . . . . . . . . . . . . . . . 16  |-  ( N  e.  NN0  ->  (ℤ/n `  N
)  e.  CRing )
34 crngring 17404 . . . . . . . . . . . . . . . 16  |-  ( (ℤ/n `  N )  e.  CRing  -> 
(ℤ/n `  N )  e.  Ring )
3532, 33, 343syl 20 . . . . . . . . . . . . . . 15  |-  ( ph  ->  (ℤ/n `  N )  e.  Ring )
3635adantr 463 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( x  e.  (Unit `  (ℤ/n `  N ) )  /\  y  e.  (Unit `  (ℤ/n `  N
) ) ) )  ->  (ℤ/n `  N )  e.  Ring )
37 eqid 2454 . . . . . . . . . . . . . . 15  |-  ( .r
`  (ℤ/n `  N ) )  =  ( .r `  (ℤ/n `  N
) )
387, 37ringcl 17407 . . . . . . . . . . . . . 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
) ) )
3936, 25, 28, 38syl3anc 1226 . . . . . . . . . . . . 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
) ) )
40 fvco3 5925 . . . . . . . . . . . . 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 ) ) ) )
4122, 39, 40syl2anc 659 . . . . . . . . . . . 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 ) ) ) )
42 fvco3 5925 . . . . . . . . . . . . . 14  |-  ( ( X : ( Base `  (ℤ/n `  N ) ) --> CC 
/\  x  e.  (
Base `  (ℤ/n `  N ) ) )  ->  ( ( *  o.  X ) `  x )  =  ( * `  ( X `
 x ) ) )
4322, 25, 42syl2anc 659 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( x  e.  (Unit `  (ℤ/n `  N ) )  /\  y  e.  (Unit `  (ℤ/n `  N
) ) ) )  ->  ( ( *  o.  X ) `  x )  =  ( * `  ( X `
 x ) ) )
44 fvco3 5925 . . . . . . . . . . . . . 14  |-  ( ( X : ( Base `  (ℤ/n `  N ) ) --> CC 
/\  y  e.  (
Base `  (ℤ/n `  N ) ) )  ->  ( ( *  o.  X ) `  y )  =  ( * `  ( X `
 y ) ) )
4522, 28, 44syl2anc 659 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( x  e.  (Unit `  (ℤ/n `  N ) )  /\  y  e.  (Unit `  (ℤ/n `  N
) ) ) )  ->  ( ( *  o.  X ) `  y )  =  ( * `  ( X `
 y ) ) )
4643, 45oveq12d 6288 . . . . . . . . . . . 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 ) ) ) )
4731, 41, 463eqtr4d 2505 . . . . . . . . . . 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 ) ) )
4847ralrimivva 2875 . . . . . . . . . 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
) ) )
49 eqid 2454 . . . . . . . . . . . . . 14  |-  ( 1r
`  (ℤ/n `  N ) )  =  ( 1r `  (ℤ/n `  N
) )
507, 49ringidcl 17414 . . . . . . . . . . . . 13  |-  ( (ℤ/n `  N )  e.  Ring  -> 
( 1r `  (ℤ/n `  N
) )  e.  (
Base `  (ℤ/n `  N ) ) )
5135, 50syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  ( 1r `  (ℤ/n `  N
) )  e.  (
Base `  (ℤ/n `  N ) ) )
52 fvco3 5925 . . . . . . . . . . . 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 ) ) ) ) )
538, 51, 52syl2anc 659 . . . . . . . . . . 11  |-  ( ph  ->  ( ( *  o.  X ) `  ( 1r `  (ℤ/n `  N ) ) )  =  ( * `  ( X `  ( 1r
`  (ℤ/n `  N ) ) ) ) )
5416simp2d 1007 . . . . . . . . . . . . 13  |-  ( ph  ->  ( X `  ( 1r `  (ℤ/n `  N ) ) )  =  1 )
5554fveq2d 5852 . . . . . . . . . . . 12  |-  ( ph  ->  ( * `  ( X `  ( 1r `  (ℤ/n `  N ) ) ) )  =  ( * `
 1 ) )
56 1re 9584 . . . . . . . . . . . . 13  |-  1  e.  RR
57 cjre 13054 . . . . . . . . . . . . 13  |-  ( 1  e.  RR  ->  (
* `  1 )  =  1 )
5856, 57ax-mp 5 . . . . . . . . . . . 12  |-  ( * `
 1 )  =  1
5955, 58syl6eq 2511 . . . . . . . . . . 11  |-  ( ph  ->  ( * `  ( X `  ( 1r `  (ℤ/n `  N ) ) ) )  =  1 )
6053, 59eqtrd 2495 . . . . . . . . . 10  |-  ( ph  ->  ( ( *  o.  X ) `  ( 1r `  (ℤ/n `  N ) ) )  =  1 )
6116simp3d 1008 . . . . . . . . . . 11  |-  ( ph  ->  A. x  e.  (
Base `  (ℤ/n `  N ) ) ( ( X `  x
)  =/=  0  ->  x  e.  (Unit `  (ℤ/n `  N
) ) ) )
628, 42sylan 469 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  ( Base `  (ℤ/n `  N ) ) )  ->  ( ( *  o.  X ) `  x )  =  ( * `  ( X `
 x ) ) )
63 cj0 13073 . . . . . . . . . . . . . . . . . 18  |-  ( * `
 0 )  =  0
6463eqcomi 2467 . . . . . . . . . . . . . . . . 17  |-  0  =  ( * ` 
0 )
6564a1i 11 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  ( Base `  (ℤ/n `  N ) ) )  ->  0  =  ( * `  0 ) )
6662, 65eqeq12d 2476 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  ( Base `  (ℤ/n `  N ) ) )  ->  ( ( ( *  o.  X ) `
 x )  =  0  <->  ( * `  ( X `  x ) )  =  ( * `
 0 ) ) )
678ffvelrnda 6007 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  ( Base `  (ℤ/n `  N ) ) )  ->  ( X `  x )  e.  CC )
68 0cn 9577 . . . . . . . . . . . . . . . 16  |-  0  e.  CC
69 cj11 13077 . . . . . . . . . . . . . . . 16  |-  ( ( ( X `  x
)  e.  CC  /\  0  e.  CC )  ->  ( ( * `  ( X `  x ) )  =  ( * `
 0 )  <->  ( X `  x )  =  0 ) )
7067, 68, 69sylancl 660 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  ( Base `  (ℤ/n `  N ) ) )  ->  ( ( * `
 ( X `  x ) )  =  ( * `  0
)  <->  ( X `  x )  =  0 ) )
7166, 70bitrd 253 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  ( Base `  (ℤ/n `  N ) ) )  ->  ( ( ( *  o.  X ) `
 x )  =  0  <->  ( X `  x )  =  0 ) )
7271necon3bid 2712 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ( Base `  (ℤ/n `  N ) ) )  ->  ( ( ( *  o.  X ) `
 x )  =/=  0  <->  ( X `  x )  =/=  0
) )
7372imbi1d 315 . . . . . . . . . . . 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 ) ) ) ) )
7473ralbidva 2890 . . . . . . . . . . 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
) ) ) ) )
7561, 74mpbird 232 . . . . . . . . . 10  |-  ( ph  ->  A. x  e.  (
Base `  (ℤ/n `  N ) ) ( ( ( *  o.  X ) `  x
)  =/=  0  ->  x  e.  (Unit `  (ℤ/n `  N
) ) ) )
7648, 60, 753jca 1174 . . . . . . . . 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
) ) ) ) )
771, 2, 7, 11, 13, 3dchrelbas3 23711 . . . . . . . . 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
) ) ) ) ) ) )
7810, 76, 77mpbir2and 920 . . . . . . . 8  |-  ( ph  ->  ( *  o.  X
)  e.  D )
791, 2, 3, 4, 5, 78dchrmul 23721 . . . . . . 7  |-  ( ph  ->  ( X ( +g  `  G ) ( *  o.  X ) )  =  ( X  oF  x.  ( *  o.  X ) ) )
8079adantr 463 . . . . . 6  |-  ( (
ph  /\  x  e.  (Unit `  (ℤ/n `  N ) ) )  ->  ( X ( +g  `  G ) ( *  o.  X
) )  =  ( X  oF  x.  ( *  o.  X
) ) )
8180fveq1d 5850 . . . . 5  |-  ( (
ph  /\  x  e.  (Unit `  (ℤ/n `  N ) ) )  ->  ( ( X ( +g  `  G
) ( *  o.  X ) ) `  x )  =  ( ( X  oF  x.  ( *  o.  X ) ) `  x ) )
8223sseli 3485 . . . . . . . . 9  |-  ( x  e.  (Unit `  (ℤ/n `  N
) )  ->  x  e.  ( Base `  (ℤ/n `  N
) ) )
8382, 62sylan2 472 . . . . . . . 8  |-  ( (
ph  /\  x  e.  (Unit `  (ℤ/n `  N ) ) )  ->  ( ( *  o.  X ) `  x )  =  ( * `  ( X `
 x ) ) )
8483oveq2d 6286 . . . . . . 7  |-  ( (
ph  /\  x  e.  (Unit `  (ℤ/n `  N ) ) )  ->  ( ( X `
 x )  x.  ( ( *  o.  X ) `  x
) )  =  ( ( X `  x
)  x.  ( * `
 ( X `  x ) ) ) )
8582, 67sylan2 472 . . . . . . . 8  |-  ( (
ph  /\  x  e.  (Unit `  (ℤ/n `  N ) ) )  ->  ( X `  x )  e.  CC )
8685absvalsqd 13355 . . . . . . 7  |-  ( (
ph  /\  x  e.  (Unit `  (ℤ/n `  N ) ) )  ->  ( ( abs `  ( X `  x
) ) ^ 2 )  =  ( ( X `  x )  x.  ( * `  ( X `  x ) ) ) )
875adantr 463 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  (Unit `  (ℤ/n `  N ) ) )  ->  X  e.  D
)
88 simpr 459 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  (Unit `  (ℤ/n `  N ) ) )  ->  x  e.  (Unit `  (ℤ/n `  N ) ) )
891, 3, 87, 2, 11, 88dchrabs 23733 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  (Unit `  (ℤ/n `  N ) ) )  ->  ( abs `  ( X `  x )
)  =  1 )
9089oveq1d 6285 . . . . . . . 8  |-  ( (
ph  /\  x  e.  (Unit `  (ℤ/n `  N ) ) )  ->  ( ( abs `  ( X `  x
) ) ^ 2 )  =  ( 1 ^ 2 ) )
91 sq1 12244 . . . . . . . 8  |-  ( 1 ^ 2 )  =  1
9290, 91syl6eq 2511 . . . . . . 7  |-  ( (
ph  /\  x  e.  (Unit `  (ℤ/n `  N ) ) )  ->  ( ( abs `  ( X `  x
) ) ^ 2 )  =  1 )
9384, 86, 923eqtr2d 2501 . . . . . 6  |-  ( (
ph  /\  x  e.  (Unit `  (ℤ/n `  N ) ) )  ->  ( ( X `
 x )  x.  ( ( *  o.  X ) `  x
) )  =  1 )
948adantr 463 . . . . . . . 8  |-  ( (
ph  /\  x  e.  (Unit `  (ℤ/n `  N ) ) )  ->  X : (
Base `  (ℤ/n `  N ) ) --> CC )
95 ffn 5713 . . . . . . . 8  |-  ( X : ( Base `  (ℤ/n `  N
) ) --> CC  ->  X  Fn  ( Base `  (ℤ/n `  N
) ) )
9694, 95syl 16 . . . . . . 7  |-  ( (
ph  /\  x  e.  (Unit `  (ℤ/n `  N ) ) )  ->  X  Fn  ( Base `  (ℤ/n `  N ) ) )
97 ffn 5713 . . . . . . . . 9  |-  ( ( *  o.  X ) : ( Base `  (ℤ/n `  N
) ) --> CC  ->  ( *  o.  X )  Fn  ( Base `  (ℤ/n `  N
) ) )
9810, 97syl 16 . . . . . . . 8  |-  ( ph  ->  ( *  o.  X
)  Fn  ( Base `  (ℤ/n `  N ) ) )
9998adantr 463 . . . . . . 7  |-  ( (
ph  /\  x  e.  (Unit `  (ℤ/n `  N ) ) )  ->  ( *  o.  X )  Fn  ( Base `  (ℤ/n `  N ) ) )
100 fvex 5858 . . . . . . . 8  |-  ( Base `  (ℤ/n `  N ) )  e. 
_V
101100a1i 11 . . . . . . 7  |-  ( (
ph  /\  x  e.  (Unit `  (ℤ/n `  N ) ) )  ->  ( Base `  (ℤ/n `  N
) )  e.  _V )
10282adantl 464 . . . . . . 7  |-  ( (
ph  /\  x  e.  (Unit `  (ℤ/n `  N ) ) )  ->  x  e.  (
Base `  (ℤ/n `  N ) ) )
103 fnfvof 6526 . . . . . . 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  oF  x.  (
*  o.  X ) ) `  x )  =  ( ( X `
 x )  x.  ( ( *  o.  X ) `  x
) ) )
10496, 99, 101, 102, 103syl22anc 1227 . . . . . 6  |-  ( (
ph  /\  x  e.  (Unit `  (ℤ/n `  N ) ) )  ->  ( ( X  oF  x.  (
*  o.  X ) ) `  x )  =  ( ( X `
 x )  x.  ( ( *  o.  X ) `  x
) ) )
105 eqid 2454 . . . . . . 7  |-  ( 0g
`  G )  =  ( 0g `  G
)
10613adantr 463 . . . . . . 7  |-  ( (
ph  /\  x  e.  (Unit `  (ℤ/n `  N ) ) )  ->  N  e.  NN )
1071, 2, 105, 11, 106, 88dchr1 23730 . . . . . 6  |-  ( (
ph  /\  x  e.  (Unit `  (ℤ/n `  N ) ) )  ->  ( ( 0g
`  G ) `  x )  =  1 )
10893, 104, 1073eqtr4d 2505 . . . . 5  |-  ( (
ph  /\  x  e.  (Unit `  (ℤ/n `  N ) ) )  ->  ( ( X  oF  x.  (
*  o.  X ) ) `  x )  =  ( ( 0g
`  G ) `  x ) )
10981, 108eqtrd 2495 . . . 4  |-  ( (
ph  /\  x  e.  (Unit `  (ℤ/n `  N ) ) )  ->  ( ( X ( +g  `  G
) ( *  o.  X ) ) `  x )  =  ( ( 0g `  G
) `  x )
)
110109ralrimiva 2868 . . 3  |-  ( ph  ->  A. x  e.  (Unit `  (ℤ/n `  N ) ) ( ( X ( +g  `  G ) ( *  o.  X ) ) `
 x )  =  ( ( 0g `  G ) `  x
) )
1111, 2, 3, 4, 5, 78dchrmulcl 23722 . . . 4  |-  ( ph  ->  ( X ( +g  `  G ) ( *  o.  X ) )  e.  D )
1121dchrabl 23727 . . . . . 6  |-  ( N  e.  NN  ->  G  e.  Abel )
113 ablgrp 17002 . . . . . 6  |-  ( G  e.  Abel  ->  G  e. 
Grp )
11413, 112, 1133syl 20 . . . . 5  |-  ( ph  ->  G  e.  Grp )
1153, 105grpidcl 16277 . . . . 5  |-  ( G  e.  Grp  ->  ( 0g `  G )  e.  D )
116114, 115syl 16 . . . 4  |-  ( ph  ->  ( 0g `  G
)  e.  D )
1171, 2, 3, 11, 111, 116dchreq 23731 . . 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
) ) )
118110, 117mpbird 232 . 2  |-  ( ph  ->  ( X ( +g  `  G ) ( *  o.  X ) )  =  ( 0g `  G ) )
119 dchrinv.i . . . 4  |-  I  =  ( invg `  G )
1203, 4, 105, 119grpinvid1 16297 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  D  /\  ( *  o.  X
)  e.  D )  ->  ( ( I `
 X )  =  ( *  o.  X
)  <->  ( X ( +g  `  G ) ( *  o.  X
) )  =  ( 0g `  G ) ) )
121114, 5, 78, 120syl3anc 1226 . 2  |-  ( ph  ->  ( ( I `  X )  =  ( *  o.  X )  <-> 
( X ( +g  `  G ) ( *  o.  X ) )  =  ( 0g `  G ) ) )
122118, 121mpbird 232 1  |-  ( ph  ->  ( I `  X
)  =  ( *  o.  X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   A.wral 2804   _Vcvv 3106    o. ccom 4992    Fn wfn 5565   -->wf 5566   ` cfv 5570  (class class class)co 6270    oFcof 6511   CCcc 9479   RRcr 9480   0cc0 9481   1c1 9482    x. cmul 9486   NNcn 10531   2c2 10581   NN0cn0 10791   ^cexp 12148   *ccj 13011   abscabs 13149   Basecbs 14716   +g cplusg 14784   .rcmulr 14785   0gc0g 14929   Grpcgrp 16252   invgcminusg 16253   Abelcabl 16998   1rcur 17348   Ringcrg 17393   CRingccrg 17394  Unitcui 17483  ℤ/nczn 18715  DChrcdchr 23705
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559  ax-addf 9560  ax-mulf 9561
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-iin 4318  df-disj 4411  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-of 6513  df-om 6674  df-1st 6773  df-2nd 6774  df-supp 6892  df-tpos 6947  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-oadd 7126  df-omul 7127  df-er 7303  df-ec 7305  df-qs 7309  df-map 7414  df-pm 7415  df-ixp 7463  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-fsupp 7822  df-fi 7863  df-sup 7893  df-oi 7927  df-card 8311  df-acn 8314  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10977  df-uz 11083  df-q 11184  df-rp 11222  df-xneg 11321  df-xadd 11322  df-xmul 11323  df-ioo 11536  df-ioc 11537  df-ico 11538  df-icc 11539  df-fz 11676  df-fzo 11800  df-fl 11910  df-mod 11979  df-seq 12090  df-exp 12149  df-fac 12336  df-bc 12363  df-hash 12388  df-shft 12982  df-cj 13014  df-re 13015  df-im 13016  df-sqrt 13150  df-abs 13151  df-limsup 13376  df-clim 13393  df-rlim 13394  df-sum 13591  df-ef 13885  df-sin 13887  df-cos 13888  df-pi 13890  df-dvds 14071  df-struct 14718  df-ndx 14719  df-slot 14720  df-base 14721  df-sets 14722  df-ress 14723  df-plusg 14797  df-mulr 14798  df-starv 14799  df-sca 14800  df-vsca 14801  df-ip 14802  df-tset 14803  df-ple 14804  df-ds 14806  df-unif 14807  df-hom 14808  df-cco 14809  df-rest 14912  df-topn 14913  df-0g 14931  df-gsum 14932  df-topgen 14933  df-pt 14934  df-prds 14937  df-xrs 14991  df-qtop 14996  df-imas 14997  df-qus 14998  df-xps 14999  df-mre 15075  df-mrc 15076  df-acs 15078  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-mhm 16165  df-submnd 16166  df-grp 16256  df-minusg 16257  df-sbg 16258  df-mulg 16259  df-subg 16397  df-nsg 16398  df-eqg 16399  df-ghm 16464  df-cntz 16554  df-od 16752  df-cmn 16999  df-abl 17000  df-mgp 17337  df-ur 17349  df-ring 17395  df-cring 17396  df-oppr 17467  df-dvdsr 17485  df-unit 17486  df-invr 17516  df-dvr 17527  df-rnghom 17559  df-drng 17593  df-subrg 17622  df-lmod 17709  df-lss 17774  df-lsp 17813  df-sra 18013  df-rgmod 18014  df-lidl 18015  df-rsp 18016  df-2idl 18075  df-psmet 18606  df-xmet 18607  df-met 18608  df-bl 18609  df-mopn 18610  df-fbas 18611  df-fg 18612  df-cnfld 18616  df-zring 18684  df-zrh 18716  df-zn 18719  df-top 19566  df-bases 19568  df-topon 19569  df-topsp 19570  df-cld 19687  df-ntr 19688  df-cls 19689  df-nei 19766  df-lp 19804  df-perf 19805  df-cn 19895  df-cnp 19896  df-haus 19983  df-tx 20229  df-hmeo 20422  df-fil 20513  df-fm 20605  df-flim 20606  df-flf 20607  df-xms 20989  df-ms 20990  df-tms 20991  df-cncf 21548  df-limc 22436  df-dv 22437  df-log 23110  df-cxp 23111  df-dchr 23706
This theorem is referenced by:  dchr2sum  23746  dchrisum0re  23896
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