MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dchrinv Structured version   Unicode version

Theorem dchrinv 21050
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 2438 . . . . . . . 8  |-  (ℤ/n `  N
)  =  (ℤ/n `  N
)
3 dchrabs.d . . . . . . . 8  |-  D  =  ( Base `  G
)
4 eqid 2438 . . . . . . . 8  |-  ( +g  `  G )  =  ( +g  `  G )
5 dchrabs.x . . . . . . . 8  |-  ( ph  ->  X  e.  D )
6 cjf 11914 . . . . . . . . . 10  |-  * : CC --> CC
7 eqid 2438 . . . . . . . . . . 11  |-  ( Base `  (ℤ/n `  N ) )  =  ( Base `  (ℤ/n `  N
) )
81, 2, 3, 7, 5dchrf 21031 . . . . . . . . . 10  |-  ( ph  ->  X : ( Base `  (ℤ/n `  N ) ) --> CC )
9 fco 5603 . . . . . . . . . 10  |-  ( ( * : CC --> CC  /\  X : ( Base `  (ℤ/n `  N
) ) --> CC )  ->  ( *  o.  X ) : (
Base `  (ℤ/n `  N ) ) --> CC )
106, 8, 9sylancr 646 . . . . . . . . 9  |-  ( ph  ->  ( *  o.  X
) : ( Base `  (ℤ/n `  N ) ) --> CC )
11 eqid 2438 . . . . . . . . . . . . . . . . . . . . 21  |-  (Unit `  (ℤ/n `  N ) )  =  (Unit `  (ℤ/n `  N ) )
121, 3dchrrcl 21029 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( X  e.  D  ->  N  e.  NN )
135, 12syl 16 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  N  e.  NN )
141, 2, 7, 11, 13, 3dchrelbas3 21027 . . . . . . . . . . . . . . . . . . . 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 203 . . . . . . . . . . . . . . . . . . 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 451 . . . . . . . . . . . . . . . . . 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 970 . . . . . . . . . . . . . . . . 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 2806 . . . . . . . . . . . . . . . 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 2806 . . . . . . . . . . . . . . 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 630 . . . . . . . . . . . . . 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 5735 . . . . . . . . . . . . 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 453 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( x  e.  (Unit `  (ℤ/n `  N ) )  /\  y  e.  (Unit `  (ℤ/n `  N
) ) ) )  ->  X : (
Base `  (ℤ/n `  N ) ) --> CC )
237, 11unitss 15770 . . . . . . . . . . . . . . . 16  |-  (Unit `  (ℤ/n `  N ) )  C_  ( Base `  (ℤ/n `  N ) )
24 simprl 734 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( x  e.  (Unit `  (ℤ/n `  N ) )  /\  y  e.  (Unit `  (ℤ/n `  N
) ) ) )  ->  x  e.  (Unit `  (ℤ/n `  N ) ) )
2523, 24sseldi 3348 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( x  e.  (Unit `  (ℤ/n `  N ) )  /\  y  e.  (Unit `  (ℤ/n `  N
) ) ) )  ->  x  e.  (
Base `  (ℤ/n `  N ) ) )
2622, 25ffvelrnd 5874 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( x  e.  (Unit `  (ℤ/n `  N ) )  /\  y  e.  (Unit `  (ℤ/n `  N
) ) ) )  ->  ( X `  x )  e.  CC )
27 simprr 735 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( x  e.  (Unit `  (ℤ/n `  N ) )  /\  y  e.  (Unit `  (ℤ/n `  N
) ) ) )  ->  y  e.  (Unit `  (ℤ/n `  N ) ) )
2823, 27sseldi 3348 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( x  e.  (Unit `  (ℤ/n `  N ) )  /\  y  e.  (Unit `  (ℤ/n `  N
) ) ) )  ->  y  e.  (
Base `  (ℤ/n `  N ) ) )
2922, 28ffvelrnd 5874 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( x  e.  (Unit `  (ℤ/n `  N ) )  /\  y  e.  (Unit `  (ℤ/n `  N
) ) ) )  ->  ( X `  y )  e.  CC )
3026, 29cjmuld 12031 . . . . . . . . . . . . 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 2470 . . . . . . . . . . . 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 10279 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  N  e.  NN0 )
332zncrng 16830 . . . . . . . . . . . . . . . 16  |-  ( N  e.  NN0  ->  (ℤ/n `  N
)  e.  CRing )
34 crngrng 15679 . . . . . . . . . . . . . . . 16  |-  ( (ℤ/n `  N )  e.  CRing  -> 
(ℤ/n `  N )  e.  Ring )
3532, 33, 343syl 19 . . . . . . . . . . . . . . 15  |-  ( ph  ->  (ℤ/n `  N )  e.  Ring )
3635adantr 453 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( x  e.  (Unit `  (ℤ/n `  N ) )  /\  y  e.  (Unit `  (ℤ/n `  N
) ) ) )  ->  (ℤ/n `  N )  e.  Ring )
37 eqid 2438 . . . . . . . . . . . . . . 15  |-  ( .r
`  (ℤ/n `  N ) )  =  ( .r `  (ℤ/n `  N
) )
387, 37rngcl 15682 . . . . . . . . . . . . . 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 1185 . . . . . . . . . . . . 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 5803 . . . . . . . . . . . . 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 644 . . . . . . . . . . . 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 5803 . . . . . . . . . . . . . 14  |-  ( ( X : ( Base `  (ℤ/n `  N ) ) --> CC 
/\  x  e.  (
Base `  (ℤ/n `  N ) ) )  ->  ( ( *  o.  X ) `  x )  =  ( * `  ( X `
 x ) ) )
4322, 25, 42syl2anc 644 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( x  e.  (Unit `  (ℤ/n `  N ) )  /\  y  e.  (Unit `  (ℤ/n `  N
) ) ) )  ->  ( ( *  o.  X ) `  x )  =  ( * `  ( X `
 x ) ) )
44 fvco3 5803 . . . . . . . . . . . . . 14  |-  ( ( X : ( Base `  (ℤ/n `  N ) ) --> CC 
/\  y  e.  (
Base `  (ℤ/n `  N ) ) )  ->  ( ( *  o.  X ) `  y )  =  ( * `  ( X `
 y ) ) )
4522, 28, 44syl2anc 644 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( x  e.  (Unit `  (ℤ/n `  N ) )  /\  y  e.  (Unit `  (ℤ/n `  N
) ) ) )  ->  ( ( *  o.  X ) `  y )  =  ( * `  ( X `
 y ) ) )
4643, 45oveq12d 6102 . . . . . . . . . . . 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 2480 . . . . . . . . . . 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 2800 . . . . . . . . . 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 2438 . . . . . . . . . . . . . 14  |-  ( 1r
`  (ℤ/n `  N ) )  =  ( 1r `  (ℤ/n `  N
) )
507, 49rngidcl 15689 . . . . . . . . . . . . 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 5803 . . . . . . . . . . . 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 644 . . . . . . . . . . 11  |-  ( ph  ->  ( ( *  o.  X ) `  ( 1r `  (ℤ/n `  N ) ) )  =  ( * `  ( X `  ( 1r
`  (ℤ/n `  N ) ) ) ) )
5416simp2d 971 . . . . . . . . . . . . 13  |-  ( ph  ->  ( X `  ( 1r `  (ℤ/n `  N ) ) )  =  1 )
5554fveq2d 5735 . . . . . . . . . . . 12  |-  ( ph  ->  ( * `  ( X `  ( 1r `  (ℤ/n `  N ) ) ) )  =  ( * `
 1 ) )
56 1re 9095 . . . . . . . . . . . . 13  |-  1  e.  RR
57 cjre 11949 . . . . . . . . . . . . 13  |-  ( 1  e.  RR  ->  (
* `  1 )  =  1 )
5856, 57ax-mp 5 . . . . . . . . . . . 12  |-  ( * `
 1 )  =  1
5955, 58syl6eq 2486 . . . . . . . . . . 11  |-  ( ph  ->  ( * `  ( X `  ( 1r `  (ℤ/n `  N ) ) ) )  =  1 )
6053, 59eqtrd 2470 . . . . . . . . . 10  |-  ( ph  ->  ( ( *  o.  X ) `  ( 1r `  (ℤ/n `  N ) ) )  =  1 )
6116simp3d 972 . . . . . . . . . . 11  |-  ( ph  ->  A. x  e.  (
Base `  (ℤ/n `  N ) ) ( ( X `  x
)  =/=  0  ->  x  e.  (Unit `  (ℤ/n `  N
) ) ) )
628, 42sylan 459 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  ( Base `  (ℤ/n `  N ) ) )  ->  ( ( *  o.  X ) `  x )  =  ( * `  ( X `
 x ) ) )
63 cj0 11968 . . . . . . . . . . . . . . . . . 18  |-  ( * `
 0 )  =  0
6463eqcomi 2442 . . . . . . . . . . . . . . . . 17  |-  0  =  ( * ` 
0 )
6564a1i 11 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  ( Base `  (ℤ/n `  N ) ) )  ->  0  =  ( * `  0 ) )
6662, 65eqeq12d 2452 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  ( Base `  (ℤ/n `  N ) ) )  ->  ( ( ( *  o.  X ) `
 x )  =  0  <->  ( * `  ( X `  x ) )  =  ( * `
 0 ) ) )
678ffvelrnda 5873 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  ( Base `  (ℤ/n `  N ) ) )  ->  ( X `  x )  e.  CC )
68 0cn 9089 . . . . . . . . . . . . . . . 16  |-  0  e.  CC
69 cj11 11972 . . . . . . . . . . . . . . . 16  |-  ( ( ( X `  x
)  e.  CC  /\  0  e.  CC )  ->  ( ( * `  ( X `  x ) )  =  ( * `
 0 )  <->  ( X `  x )  =  0 ) )
7067, 68, 69sylancl 645 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  ( Base `  (ℤ/n `  N ) ) )  ->  ( ( * `
 ( X `  x ) )  =  ( * `  0
)  <->  ( X `  x )  =  0 ) )
7166, 70bitrd 246 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  ( Base `  (ℤ/n `  N ) ) )  ->  ( ( ( *  o.  X ) `
 x )  =  0  <->  ( X `  x )  =  0 ) )
7271necon3bid 2638 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ( Base `  (ℤ/n `  N ) ) )  ->  ( ( ( *  o.  X ) `
 x )  =/=  0  <->  ( X `  x )  =/=  0
) )
7372imbi1d 310 . . . . . . . . . . . 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 2723 . . . . . . . . . . 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 225 . . . . . . . . . 10  |-  ( ph  ->  A. x  e.  (
Base `  (ℤ/n `  N ) ) ( ( ( *  o.  X ) `  x
)  =/=  0  ->  x  e.  (Unit `  (ℤ/n `  N
) ) ) )
7648, 60, 753jca 1135 . . . . . . . . 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 21027 . . . . . . . . 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 890 . . . . . . . 8  |-  ( ph  ->  ( *  o.  X
)  e.  D )
791, 2, 3, 4, 5, 78dchrmul 21037 . . . . . . 7  |-  ( ph  ->  ( X ( +g  `  G ) ( *  o.  X ) )  =  ( X  o F  x.  ( *  o.  X ) ) )
8079adantr 453 . . . . . 6  |-  ( (
ph  /\  x  e.  (Unit `  (ℤ/n `  N ) ) )  ->  ( X ( +g  `  G ) ( *  o.  X
) )  =  ( X  o F  x.  ( *  o.  X
) ) )
8180fveq1d 5733 . . . . 5  |-  ( (
ph  /\  x  e.  (Unit `  (ℤ/n `  N ) ) )  ->  ( ( X ( +g  `  G
) ( *  o.  X ) ) `  x )  =  ( ( X  o F  x.  ( *  o.  X ) ) `  x ) )
8223sseli 3346 . . . . . . . . 9  |-  ( x  e.  (Unit `  (ℤ/n `  N
) )  ->  x  e.  ( Base `  (ℤ/n `  N
) ) )
8382, 62sylan2 462 . . . . . . . 8  |-  ( (
ph  /\  x  e.  (Unit `  (ℤ/n `  N ) ) )  ->  ( ( *  o.  X ) `  x )  =  ( * `  ( X `
 x ) ) )
8483oveq2d 6100 . . . . . . 7  |-  ( (
ph  /\  x  e.  (Unit `  (ℤ/n `  N ) ) )  ->  ( ( X `
 x )  x.  ( ( *  o.  X ) `  x
) )  =  ( ( X `  x
)  x.  ( * `
 ( X `  x ) ) ) )
8582, 67sylan2 462 . . . . . . . 8  |-  ( (
ph  /\  x  e.  (Unit `  (ℤ/n `  N ) ) )  ->  ( X `  x )  e.  CC )
8685absvalsqd 12249 . . . . . . 7  |-  ( (
ph  /\  x  e.  (Unit `  (ℤ/n `  N ) ) )  ->  ( ( abs `  ( X `  x
) ) ^ 2 )  =  ( ( X `  x )  x.  ( * `  ( X `  x ) ) ) )
875adantr 453 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  (Unit `  (ℤ/n `  N ) ) )  ->  X  e.  D
)
88 simpr 449 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  (Unit `  (ℤ/n `  N ) ) )  ->  x  e.  (Unit `  (ℤ/n `  N ) ) )
891, 3, 87, 2, 11, 88dchrabs 21049 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  (Unit `  (ℤ/n `  N ) ) )  ->  ( abs `  ( X `  x )
)  =  1 )
9089oveq1d 6099 . . . . . . . 8  |-  ( (
ph  /\  x  e.  (Unit `  (ℤ/n `  N ) ) )  ->  ( ( abs `  ( X `  x
) ) ^ 2 )  =  ( 1 ^ 2 ) )
91 sq1 11481 . . . . . . . 8  |-  ( 1 ^ 2 )  =  1
9290, 91syl6eq 2486 . . . . . . 7  |-  ( (
ph  /\  x  e.  (Unit `  (ℤ/n `  N ) ) )  ->  ( ( abs `  ( X `  x
) ) ^ 2 )  =  1 )
9384, 86, 923eqtr2d 2476 . . . . . 6  |-  ( (
ph  /\  x  e.  (Unit `  (ℤ/n `  N ) ) )  ->  ( ( X `
 x )  x.  ( ( *  o.  X ) `  x
) )  =  1 )
948adantr 453 . . . . . . . 8  |-  ( (
ph  /\  x  e.  (Unit `  (ℤ/n `  N ) ) )  ->  X : (
Base `  (ℤ/n `  N ) ) --> CC )
95 ffn 5594 . . . . . . . 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 5594 . . . . . . . . 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 453 . . . . . . 7  |-  ( (
ph  /\  x  e.  (Unit `  (ℤ/n `  N ) ) )  ->  ( *  o.  X )  Fn  ( Base `  (ℤ/n `  N ) ) )
100 fvex 5745 . . . . . . . 8  |-  ( Base `  (ℤ/n `  N ) )  e. 
_V
101100a1i 11 . . . . . . 7  |-  ( (
ph  /\  x  e.  (Unit `  (ℤ/n `  N ) ) )  ->  ( Base `  (ℤ/n `  N
) )  e.  _V )
10282adantl 454 . . . . . . 7  |-  ( (
ph  /\  x  e.  (Unit `  (ℤ/n `  N ) ) )  ->  x  e.  (
Base `  (ℤ/n `  N ) ) )
103 fnfvof 6320 . . . . . . 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
) ) )
10496, 99, 101, 102, 103syl22anc 1186 . . . . . 6  |-  ( (
ph  /\  x  e.  (Unit `  (ℤ/n `  N ) ) )  ->  ( ( X  o F  x.  (
*  o.  X ) ) `  x )  =  ( ( X `
 x )  x.  ( ( *  o.  X ) `  x
) ) )
105 eqid 2438 . . . . . . 7  |-  ( 0g
`  G )  =  ( 0g `  G
)
10613adantr 453 . . . . . . 7  |-  ( (
ph  /\  x  e.  (Unit `  (ℤ/n `  N ) ) )  ->  N  e.  NN )
1071, 2, 105, 11, 106, 88dchr1 21046 . . . . . 6  |-  ( (
ph  /\  x  e.  (Unit `  (ℤ/n `  N ) ) )  ->  ( ( 0g
`  G ) `  x )  =  1 )
10893, 104, 1073eqtr4d 2480 . . . . 5  |-  ( (
ph  /\  x  e.  (Unit `  (ℤ/n `  N ) ) )  ->  ( ( X  o F  x.  (
*  o.  X ) ) `  x )  =  ( ( 0g
`  G ) `  x ) )
10981, 108eqtrd 2470 . . . 4  |-  ( (
ph  /\  x  e.  (Unit `  (ℤ/n `  N ) ) )  ->  ( ( X ( +g  `  G
) ( *  o.  X ) ) `  x )  =  ( ( 0g `  G
) `  x )
)
110109ralrimiva 2791 . . 3  |-  ( ph  ->  A. x  e.  (Unit `  (ℤ/n `  N ) ) ( ( X ( +g  `  G ) ( *  o.  X ) ) `
 x )  =  ( ( 0g `  G ) `  x
) )
1111, 2, 3, 4, 5, 78dchrmulcl 21038 . . . 4  |-  ( ph  ->  ( X ( +g  `  G ) ( *  o.  X ) )  e.  D )
1121dchrabl 21043 . . . . . 6  |-  ( N  e.  NN  ->  G  e.  Abel )
113 ablgrp 15422 . . . . . 6  |-  ( G  e.  Abel  ->  G  e. 
Grp )
11413, 112, 1133syl 19 . . . . 5  |-  ( ph  ->  G  e.  Grp )
1153, 105grpidcl 14838 . . . . 5  |-  ( G  e.  Grp  ->  ( 0g `  G )  e.  D )
116114, 115syl 16 . . . 4  |-  ( ph  ->  ( 0g `  G
)  e.  D )
1171, 2, 3, 11, 111, 116dchreq 21047 . . 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 225 . 2  |-  ( ph  ->  ( X ( +g  `  G ) ( *  o.  X ) )  =  ( 0g `  G ) )
119 dchrinv.i . . . 4  |-  I  =  ( inv g `  G )
1203, 4, 105, 119grpinvid1 14858 . . 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 1185 . 2  |-  ( ph  ->  ( ( I `  X )  =  ( *  o.  X )  <-> 
( X ( +g  `  G ) ( *  o.  X ) )  =  ( 0g `  G ) ) )
122118, 121mpbird 225 1  |-  ( ph  ->  ( I `  X
)  =  ( *  o.  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   A.wral 2707   _Vcvv 2958    o. ccom 4885    Fn wfn 5452   -->wf 5453   ` cfv 5457  (class class class)co 6084    o Fcof 6306   CCcc 8993   RRcr 8994   0cc0 8995   1c1 8996    x. cmul 9000   NNcn 10005   2c2 10054   NN0cn0 10226   ^cexp 11387   *ccj 11906   abscabs 12044   Basecbs 13474   +g cplusg 13534   .rcmulr 13535   0gc0g 13728   Grpcgrp 14690   inv gcminusg 14691   Abelcabel 15418   Ringcrg 15665   CRingccrg 15666   1rcur 15667  Unitcui 15749  ℤ/nczn 16786  DChrcdchr 21021
This theorem is referenced by:  dchr2sum  21062  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-iin 4098  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-ixp 7067  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-fi 7419  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-q 10580  df-rp 10618  df-xneg 10715  df-xadd 10716  df-xmul 10717  df-ioo 10925  df-ioc 10926  df-ico 10927  df-icc 10928  df-fz 11049  df-fzo 11141  df-fl 11207  df-mod 11256  df-seq 11329  df-exp 11388  df-fac 11572  df-bc 11599  df-hash 11624  df-shft 11887  df-cj 11909  df-re 11910  df-im 11911  df-sqr 12045  df-abs 12046  df-limsup 12270  df-clim 12287  df-rlim 12288  df-sum 12485  df-ef 12675  df-sin 12677  df-cos 12678  df-pi 12680  df-dvds 12858  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-hom 13558  df-cco 13559  df-rest 13655  df-topn 13656  df-topgen 13672  df-pt 13673  df-prds 13676  df-xrs 13731  df-0g 13732  df-gsum 13733  df-qtop 13738  df-imas 13739  df-divs 13740  df-xps 13741  df-mre 13816  df-mrc 13817  df-acs 13819  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-cntz 15121  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-dvr 15793  df-rnghom 15824  df-drng 15842  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-psmet 16699  df-xmet 16700  df-met 16701  df-bl 16702  df-mopn 16703  df-fbas 16704  df-fg 16705  df-cnfld 16709  df-zrh 16787  df-zn 16790  df-top 16968  df-bases 16970  df-topon 16971  df-topsp 16972  df-cld 17088  df-ntr 17089  df-cls 17090  df-nei 17167  df-lp 17205  df-perf 17206  df-cn 17296  df-cnp 17297  df-haus 17384  df-tx 17599  df-hmeo 17792  df-fil 17883  df-fm 17975  df-flim 17976  df-flf 17977  df-xms 18355  df-ms 18356  df-tms 18357  df-cncf 18913  df-limc 19758  df-dv 19759  df-log 20459  df-cxp 20460  df-dchr 21022
  Copyright terms: Public domain W3C validator