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Theorem dchrinv 22619
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 2443 . . . . . . . 8  |-  (ℤ/n `  N
)  =  (ℤ/n `  N
)
3 dchrabs.d . . . . . . . 8  |-  D  =  ( Base `  G
)
4 eqid 2443 . . . . . . . 8  |-  ( +g  `  G )  =  ( +g  `  G )
5 dchrabs.x . . . . . . . 8  |-  ( ph  ->  X  e.  D )
6 cjf 12612 . . . . . . . . . 10  |-  * : CC --> CC
7 eqid 2443 . . . . . . . . . . 11  |-  ( Base `  (ℤ/n `  N ) )  =  ( Base `  (ℤ/n `  N
) )
81, 2, 3, 7, 5dchrf 22600 . . . . . . . . . 10  |-  ( ph  ->  X : ( Base `  (ℤ/n `  N ) ) --> CC )
9 fco 5587 . . . . . . . . . 10  |-  ( ( * : CC --> CC  /\  X : ( Base `  (ℤ/n `  N
) ) --> CC )  ->  ( *  o.  X ) : (
Base `  (ℤ/n `  N ) ) --> CC )
106, 8, 9sylancr 663 . . . . . . . . 9  |-  ( ph  ->  ( *  o.  X
) : ( Base `  (ℤ/n `  N ) ) --> CC )
11 eqid 2443 . . . . . . . . . . . . . . . . . . . . 21  |-  (Unit `  (ℤ/n `  N ) )  =  (Unit `  (ℤ/n `  N ) )
121, 3dchrrcl 22598 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( X  e.  D  ->  N  e.  NN )
135, 12syl 16 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  N  e.  NN )
141, 2, 7, 11, 13, 3dchrelbas3 22596 . . . . . . . . . . . . . . . . . . . 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 463 . . . . . . . . . . . . . . . . . 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 1000 . . . . . . . . . . . . . . . . 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 2833 . . . . . . . . . . . . . . . 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 2833 . . . . . . . . . . . . . . 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 647 . . . . . . . . . . . . . 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 5714 . . . . . . . . . . . . 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 465 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( x  e.  (Unit `  (ℤ/n `  N ) )  /\  y  e.  (Unit `  (ℤ/n `  N
) ) ) )  ->  X : (
Base `  (ℤ/n `  N ) ) --> CC )
237, 11unitss 16771 . . . . . . . . . . . . . . . 16  |-  (Unit `  (ℤ/n `  N ) )  C_  ( Base `  (ℤ/n `  N ) )
24 simprl 755 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( x  e.  (Unit `  (ℤ/n `  N ) )  /\  y  e.  (Unit `  (ℤ/n `  N
) ) ) )  ->  x  e.  (Unit `  (ℤ/n `  N ) ) )
2523, 24sseldi 3373 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( x  e.  (Unit `  (ℤ/n `  N ) )  /\  y  e.  (Unit `  (ℤ/n `  N
) ) ) )  ->  x  e.  (
Base `  (ℤ/n `  N ) ) )
2622, 25ffvelrnd 5863 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( x  e.  (Unit `  (ℤ/n `  N ) )  /\  y  e.  (Unit `  (ℤ/n `  N
) ) ) )  ->  ( X `  x )  e.  CC )
27 simprr 756 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( x  e.  (Unit `  (ℤ/n `  N ) )  /\  y  e.  (Unit `  (ℤ/n `  N
) ) ) )  ->  y  e.  (Unit `  (ℤ/n `  N ) ) )
2823, 27sseldi 3373 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( x  e.  (Unit `  (ℤ/n `  N ) )  /\  y  e.  (Unit `  (ℤ/n `  N
) ) ) )  ->  y  e.  (
Base `  (ℤ/n `  N ) ) )
2922, 28ffvelrnd 5863 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( x  e.  (Unit `  (ℤ/n `  N ) )  /\  y  e.  (Unit `  (ℤ/n `  N
) ) ) )  ->  ( X `  y )  e.  CC )
3026, 29cjmuld 12729 . . . . . . . . . . . . 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 2475 . . . . . . . . . . . 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 10655 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  N  e.  NN0 )
332zncrng 17996 . . . . . . . . . . . . . . . 16  |-  ( N  e.  NN0  ->  (ℤ/n `  N
)  e.  CRing )
34 crngrng 16674 . . . . . . . . . . . . . . . 16  |-  ( (ℤ/n `  N )  e.  CRing  -> 
(ℤ/n `  N )  e.  Ring )
3532, 33, 343syl 20 . . . . . . . . . . . . . . 15  |-  ( ph  ->  (ℤ/n `  N )  e.  Ring )
3635adantr 465 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( x  e.  (Unit `  (ℤ/n `  N ) )  /\  y  e.  (Unit `  (ℤ/n `  N
) ) ) )  ->  (ℤ/n `  N )  e.  Ring )
37 eqid 2443 . . . . . . . . . . . . . . 15  |-  ( .r
`  (ℤ/n `  N ) )  =  ( .r `  (ℤ/n `  N
) )
387, 37rngcl 16677 . . . . . . . . . . . . . 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 1218 . . . . . . . . . . . . 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 5787 . . . . . . . . . . . . 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 661 . . . . . . . . . . . 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 5787 . . . . . . . . . . . . . 14  |-  ( ( X : ( Base `  (ℤ/n `  N ) ) --> CC 
/\  x  e.  (
Base `  (ℤ/n `  N ) ) )  ->  ( ( *  o.  X ) `  x )  =  ( * `  ( X `
 x ) ) )
4322, 25, 42syl2anc 661 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( x  e.  (Unit `  (ℤ/n `  N ) )  /\  y  e.  (Unit `  (ℤ/n `  N
) ) ) )  ->  ( ( *  o.  X ) `  x )  =  ( * `  ( X `
 x ) ) )
44 fvco3 5787 . . . . . . . . . . . . . 14  |-  ( ( X : ( Base `  (ℤ/n `  N ) ) --> CC 
/\  y  e.  (
Base `  (ℤ/n `  N ) ) )  ->  ( ( *  o.  X ) `  y )  =  ( * `  ( X `
 y ) ) )
4522, 28, 44syl2anc 661 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( x  e.  (Unit `  (ℤ/n `  N ) )  /\  y  e.  (Unit `  (ℤ/n `  N
) ) ) )  ->  ( ( *  o.  X ) `  y )  =  ( * `  ( X `
 y ) ) )
4643, 45oveq12d 6128 . . . . . . . . . . . 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 2485 . . . . . . . . . . 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 2827 . . . . . . . . . 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 2443 . . . . . . . . . . . . . 14  |-  ( 1r
`  (ℤ/n `  N ) )  =  ( 1r `  (ℤ/n `  N
) )
507, 49rngidcl 16684 . . . . . . . . . . . . 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 5787 . . . . . . . . . . . 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 661 . . . . . . . . . . 11  |-  ( ph  ->  ( ( *  o.  X ) `  ( 1r `  (ℤ/n `  N ) ) )  =  ( * `  ( X `  ( 1r
`  (ℤ/n `  N ) ) ) ) )
5416simp2d 1001 . . . . . . . . . . . . 13  |-  ( ph  ->  ( X `  ( 1r `  (ℤ/n `  N ) ) )  =  1 )
5554fveq2d 5714 . . . . . . . . . . . 12  |-  ( ph  ->  ( * `  ( X `  ( 1r `  (ℤ/n `  N ) ) ) )  =  ( * `
 1 ) )
56 1re 9404 . . . . . . . . . . . . 13  |-  1  e.  RR
57 cjre 12647 . . . . . . . . . . . . 13  |-  ( 1  e.  RR  ->  (
* `  1 )  =  1 )
5856, 57ax-mp 5 . . . . . . . . . . . 12  |-  ( * `
 1 )  =  1
5955, 58syl6eq 2491 . . . . . . . . . . 11  |-  ( ph  ->  ( * `  ( X `  ( 1r `  (ℤ/n `  N ) ) ) )  =  1 )
6053, 59eqtrd 2475 . . . . . . . . . 10  |-  ( ph  ->  ( ( *  o.  X ) `  ( 1r `  (ℤ/n `  N ) ) )  =  1 )
6116simp3d 1002 . . . . . . . . . . 11  |-  ( ph  ->  A. x  e.  (
Base `  (ℤ/n `  N ) ) ( ( X `  x
)  =/=  0  ->  x  e.  (Unit `  (ℤ/n `  N
) ) ) )
628, 42sylan 471 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  ( Base `  (ℤ/n `  N ) ) )  ->  ( ( *  o.  X ) `  x )  =  ( * `  ( X `
 x ) ) )
63 cj0 12666 . . . . . . . . . . . . . . . . . 18  |-  ( * `
 0 )  =  0
6463eqcomi 2447 . . . . . . . . . . . . . . . . 17  |-  0  =  ( * ` 
0 )
6564a1i 11 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  ( Base `  (ℤ/n `  N ) ) )  ->  0  =  ( * `  0 ) )
6662, 65eqeq12d 2457 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  ( Base `  (ℤ/n `  N ) ) )  ->  ( ( ( *  o.  X ) `
 x )  =  0  <->  ( * `  ( X `  x ) )  =  ( * `
 0 ) ) )
678ffvelrnda 5862 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  ( Base `  (ℤ/n `  N ) ) )  ->  ( X `  x )  e.  CC )
68 0cn 9397 . . . . . . . . . . . . . . . 16  |-  0  e.  CC
69 cj11 12670 . . . . . . . . . . . . . . . 16  |-  ( ( ( X `  x
)  e.  CC  /\  0  e.  CC )  ->  ( ( * `  ( X `  x ) )  =  ( * `
 0 )  <->  ( X `  x )  =  0 ) )
7067, 68, 69sylancl 662 . . . . . . . . . . . . . . 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 2665 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ( Base `  (ℤ/n `  N ) ) )  ->  ( ( ( *  o.  X ) `
 x )  =/=  0  <->  ( X `  x )  =/=  0
) )
7372imbi1d 317 . . . . . . . . . . . 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 2750 . . . . . . . . . . 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 1168 . . . . . . . . 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 22596 . . . . . . . . 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 913 . . . . . . . 8  |-  ( ph  ->  ( *  o.  X
)  e.  D )
791, 2, 3, 4, 5, 78dchrmul 22606 . . . . . . 7  |-  ( ph  ->  ( X ( +g  `  G ) ( *  o.  X ) )  =  ( X  oF  x.  ( *  o.  X ) ) )
8079adantr 465 . . . . . 6  |-  ( (
ph  /\  x  e.  (Unit `  (ℤ/n `  N ) ) )  ->  ( X ( +g  `  G ) ( *  o.  X
) )  =  ( X  oF  x.  ( *  o.  X
) ) )
8180fveq1d 5712 . . . . 5  |-  ( (
ph  /\  x  e.  (Unit `  (ℤ/n `  N ) ) )  ->  ( ( X ( +g  `  G
) ( *  o.  X ) ) `  x )  =  ( ( X  oF  x.  ( *  o.  X ) ) `  x ) )
8223sseli 3371 . . . . . . . . 9  |-  ( x  e.  (Unit `  (ℤ/n `  N
) )  ->  x  e.  ( Base `  (ℤ/n `  N
) ) )
8382, 62sylan2 474 . . . . . . . 8  |-  ( (
ph  /\  x  e.  (Unit `  (ℤ/n `  N ) ) )  ->  ( ( *  o.  X ) `  x )  =  ( * `  ( X `
 x ) ) )
8483oveq2d 6126 . . . . . . 7  |-  ( (
ph  /\  x  e.  (Unit `  (ℤ/n `  N ) ) )  ->  ( ( X `
 x )  x.  ( ( *  o.  X ) `  x
) )  =  ( ( X `  x
)  x.  ( * `
 ( X `  x ) ) ) )
8582, 67sylan2 474 . . . . . . . 8  |-  ( (
ph  /\  x  e.  (Unit `  (ℤ/n `  N ) ) )  ->  ( X `  x )  e.  CC )
8685absvalsqd 12947 . . . . . . 7  |-  ( (
ph  /\  x  e.  (Unit `  (ℤ/n `  N ) ) )  ->  ( ( abs `  ( X `  x
) ) ^ 2 )  =  ( ( X `  x )  x.  ( * `  ( X `  x ) ) ) )
875adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  (Unit `  (ℤ/n `  N ) ) )  ->  X  e.  D
)
88 simpr 461 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  (Unit `  (ℤ/n `  N ) ) )  ->  x  e.  (Unit `  (ℤ/n `  N ) ) )
891, 3, 87, 2, 11, 88dchrabs 22618 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  (Unit `  (ℤ/n `  N ) ) )  ->  ( abs `  ( X `  x )
)  =  1 )
9089oveq1d 6125 . . . . . . . 8  |-  ( (
ph  /\  x  e.  (Unit `  (ℤ/n `  N ) ) )  ->  ( ( abs `  ( X `  x
) ) ^ 2 )  =  ( 1 ^ 2 ) )
91 sq1 11979 . . . . . . . 8  |-  ( 1 ^ 2 )  =  1
9290, 91syl6eq 2491 . . . . . . 7  |-  ( (
ph  /\  x  e.  (Unit `  (ℤ/n `  N ) ) )  ->  ( ( abs `  ( X `  x
) ) ^ 2 )  =  1 )
9384, 86, 923eqtr2d 2481 . . . . . 6  |-  ( (
ph  /\  x  e.  (Unit `  (ℤ/n `  N ) ) )  ->  ( ( X `
 x )  x.  ( ( *  o.  X ) `  x
) )  =  1 )
948adantr 465 . . . . . . . 8  |-  ( (
ph  /\  x  e.  (Unit `  (ℤ/n `  N ) ) )  ->  X : (
Base `  (ℤ/n `  N ) ) --> CC )
95 ffn 5578 . . . . . . . 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 5578 . . . . . . . . 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 465 . . . . . . 7  |-  ( (
ph  /\  x  e.  (Unit `  (ℤ/n `  N ) ) )  ->  ( *  o.  X )  Fn  ( Base `  (ℤ/n `  N ) ) )
100 fvex 5720 . . . . . . . 8  |-  ( Base `  (ℤ/n `  N ) )  e. 
_V
101100a1i 11 . . . . . . 7  |-  ( (
ph  /\  x  e.  (Unit `  (ℤ/n `  N ) ) )  ->  ( Base `  (ℤ/n `  N
) )  e.  _V )
10282adantl 466 . . . . . . 7  |-  ( (
ph  /\  x  e.  (Unit `  (ℤ/n `  N ) ) )  ->  x  e.  (
Base `  (ℤ/n `  N ) ) )
103 fnfvof 6352 . . . . . . 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 1219 . . . . . 6  |-  ( (
ph  /\  x  e.  (Unit `  (ℤ/n `  N ) ) )  ->  ( ( X  oF  x.  (
*  o.  X ) ) `  x )  =  ( ( X `
 x )  x.  ( ( *  o.  X ) `  x
) ) )
105 eqid 2443 . . . . . . 7  |-  ( 0g
`  G )  =  ( 0g `  G
)
10613adantr 465 . . . . . . 7  |-  ( (
ph  /\  x  e.  (Unit `  (ℤ/n `  N ) ) )  ->  N  e.  NN )
1071, 2, 105, 11, 106, 88dchr1 22615 . . . . . 6  |-  ( (
ph  /\  x  e.  (Unit `  (ℤ/n `  N ) ) )  ->  ( ( 0g
`  G ) `  x )  =  1 )
10893, 104, 1073eqtr4d 2485 . . . . 5  |-  ( (
ph  /\  x  e.  (Unit `  (ℤ/n `  N ) ) )  ->  ( ( X  oF  x.  (
*  o.  X ) ) `  x )  =  ( ( 0g
`  G ) `  x ) )
10981, 108eqtrd 2475 . . . 4  |-  ( (
ph  /\  x  e.  (Unit `  (ℤ/n `  N ) ) )  ->  ( ( X ( +g  `  G
) ( *  o.  X ) ) `  x )  =  ( ( 0g `  G
) `  x )
)
110109ralrimiva 2818 . . 3  |-  ( ph  ->  A. x  e.  (Unit `  (ℤ/n `  N ) ) ( ( X ( +g  `  G ) ( *  o.  X ) ) `
 x )  =  ( ( 0g `  G ) `  x
) )
1111, 2, 3, 4, 5, 78dchrmulcl 22607 . . . 4  |-  ( ph  ->  ( X ( +g  `  G ) ( *  o.  X ) )  e.  D )
1121dchrabl 22612 . . . . . 6  |-  ( N  e.  NN  ->  G  e.  Abel )
113 ablgrp 16301 . . . . . 6  |-  ( G  e.  Abel  ->  G  e. 
Grp )
11413, 112, 1133syl 20 . . . . 5  |-  ( ph  ->  G  e.  Grp )
1153, 105grpidcl 15585 . . . . 5  |-  ( G  e.  Grp  ->  ( 0g `  G )  e.  D )
116114, 115syl 16 . . . 4  |-  ( ph  ->  ( 0g `  G
)  e.  D )
1171, 2, 3, 11, 111, 116dchreq 22616 . . 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 15605 . . 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 1218 . 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 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2620   A.wral 2734   _Vcvv 2991    o. ccom 4863    Fn wfn 5432   -->wf 5433   ` cfv 5437  (class class class)co 6110    oFcof 6337   CCcc 9299   RRcr 9300   0cc0 9301   1c1 9302    x. cmul 9306   NNcn 10341   2c2 10390   NN0cn0 10598   ^cexp 11884   *ccj 12604   abscabs 12742   Basecbs 14193   +g cplusg 14257   .rcmulr 14258   0gc0g 14397   Grpcgrp 15429   invgcminusg 15430   Abelcabel 16297   1rcur 16622   Ringcrg 16664   CRingccrg 16665  Unitcui 16750  ℤ/nczn 17953  DChrcdchr 22590
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4422  ax-sep 4432  ax-nul 4440  ax-pow 4489  ax-pr 4550  ax-un 6391  ax-inf2 7866  ax-cnex 9357  ax-resscn 9358  ax-1cn 9359  ax-icn 9360  ax-addcl 9361  ax-addrcl 9362  ax-mulcl 9363  ax-mulrcl 9364  ax-mulcom 9365  ax-addass 9366  ax-mulass 9367  ax-distr 9368  ax-i2m1 9369  ax-1ne0 9370  ax-1rid 9371  ax-rnegex 9372  ax-rrecex 9373  ax-cnre 9374  ax-pre-lttri 9375  ax-pre-lttrn 9376  ax-pre-ltadd 9377  ax-pre-mulgt0 9378  ax-pre-sup 9379  ax-addf 9380  ax-mulf 9381
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2739  df-rex 2740  df-reu 2741  df-rmo 2742  df-rab 2743  df-v 2993  df-sbc 3206  df-csb 3308  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-pss 3363  df-nul 3657  df-if 3811  df-pw 3881  df-sn 3897  df-pr 3899  df-tp 3901  df-op 3903  df-uni 4111  df-int 4148  df-iun 4192  df-iin 4193  df-disj 4282  df-br 4312  df-opab 4370  df-mpt 4371  df-tr 4405  df-eprel 4651  df-id 4655  df-po 4660  df-so 4661  df-fr 4698  df-se 4699  df-we 4700  df-ord 4741  df-on 4742  df-lim 4743  df-suc 4744  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-rn 4870  df-res 4871  df-ima 4872  df-iota 5400  df-fun 5439  df-fn 5440  df-f 5441  df-f1 5442  df-fo 5443  df-f1o 5444  df-fv 5445  df-isom 5446  df-riota 6071  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-of 6339  df-om 6496  df-1st 6596  df-2nd 6597  df-supp 6710  df-tpos 6764  df-recs 6851  df-rdg 6885  df-1o 6939  df-2o 6940  df-oadd 6943  df-omul 6944  df-er 7120  df-ec 7122  df-qs 7126  df-map 7235  df-pm 7236  df-ixp 7283  df-en 7330  df-dom 7331  df-sdom 7332  df-fin 7333  df-fsupp 7640  df-fi 7680  df-sup 7710  df-oi 7743  df-card 8128  df-acn 8131  df-cda 8356  df-pnf 9439  df-mnf 9440  df-xr 9441  df-ltxr 9442  df-le 9443  df-sub 9616  df-neg 9617  df-div 10013  df-nn 10342  df-2 10399  df-3 10400  df-4 10401  df-5 10402  df-6 10403  df-7 10404  df-8 10405  df-9 10406  df-10 10407  df-n0 10599  df-z 10666  df-dec 10775  df-uz 10881  df-q 10973  df-rp 11011  df-xneg 11108  df-xadd 11109  df-xmul 11110  df-ioo 11323  df-ioc 11324  df-ico 11325  df-icc 11326  df-fz 11457  df-fzo 11568  df-fl 11661  df-mod 11728  df-seq 11826  df-exp 11885  df-fac 12071  df-bc 12098  df-hash 12123  df-shft 12575  df-cj 12607  df-re 12608  df-im 12609  df-sqr 12743  df-abs 12744  df-limsup 12968  df-clim 12985  df-rlim 12986  df-sum 13183  df-ef 13372  df-sin 13374  df-cos 13375  df-pi 13377  df-dvds 13555  df-struct 14195  df-ndx 14196  df-slot 14197  df-base 14198  df-sets 14199  df-ress 14200  df-plusg 14270  df-mulr 14271  df-starv 14272  df-sca 14273  df-vsca 14274  df-ip 14275  df-tset 14276  df-ple 14277  df-ds 14279  df-unif 14280  df-hom 14281  df-cco 14282  df-rest 14380  df-topn 14381  df-0g 14399  df-gsum 14400  df-topgen 14401  df-pt 14402  df-prds 14405  df-xrs 14459  df-qtop 14464  df-imas 14465  df-divs 14466  df-xps 14467  df-mre 14543  df-mrc 14544  df-acs 14546  df-mnd 15434  df-mhm 15483  df-submnd 15484  df-grp 15564  df-minusg 15565  df-sbg 15566  df-mulg 15567  df-subg 15697  df-nsg 15698  df-eqg 15699  df-ghm 15764  df-cntz 15854  df-od 16051  df-cmn 16298  df-abl 16299  df-mgp 16611  df-ur 16623  df-rng 16666  df-cring 16667  df-oppr 16734  df-dvdsr 16752  df-unit 16753  df-invr 16783  df-dvr 16794  df-rnghom 16825  df-drng 16853  df-subrg 16882  df-lmod 16969  df-lss 17033  df-lsp 17072  df-sra 17272  df-rgmod 17273  df-lidl 17274  df-rsp 17275  df-2idl 17333  df-psmet 17828  df-xmet 17829  df-met 17830  df-bl 17831  df-mopn 17832  df-fbas 17833  df-fg 17834  df-cnfld 17838  df-zring 17903  df-zrh 17954  df-zn 17957  df-top 18522  df-bases 18524  df-topon 18525  df-topsp 18526  df-cld 18642  df-ntr 18643  df-cls 18644  df-nei 18721  df-lp 18759  df-perf 18760  df-cn 18850  df-cnp 18851  df-haus 18938  df-tx 19154  df-hmeo 19347  df-fil 19438  df-fm 19530  df-flim 19531  df-flf 19532  df-xms 19914  df-ms 19915  df-tms 19916  df-cncf 20473  df-limc 21360  df-dv 21361  df-log 22027  df-cxp 22028  df-dchr 22591
This theorem is referenced by:  dchr2sum  22631  dchrisum0re  22781
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