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Theorem dchrsum2 20434
Description: An orthogonality relation for Dirichlet characters: the sum of all the values of a Dirichlet character  X is  0 if  X is non-principal and  phi ( n ) otherwise. Part of Theorem 6.5.1 of [Shapiro] p. 230. (Contributed by Mario Carneiro, 28-Apr-2016.)
Hypotheses
Ref Expression
dchrsum.g  |-  G  =  (DChr `  N )
dchrsum.z  |-  Z  =  (ℤ/n `  N )
dchrsum.d  |-  D  =  ( Base `  G
)
dchrsum.1  |-  .1.  =  ( 0g `  G )
dchrsum.x  |-  ( ph  ->  X  e.  D )
dchrsum2.u  |-  U  =  (Unit `  Z )
Assertion
Ref Expression
dchrsum2  |-  ( ph  -> 
sum_ a  e.  U  ( X `  a )  =  if ( X  =  .1.  ,  ( phi `  N ) ,  0 ) )
Distinct variable groups:    .1. , a    ph, a    U, a    X, a    Z, a
Allowed substitution hints:    D( a)    G( a)    N( a)

Proof of Theorem dchrsum2
StepHypRef Expression
1 eqeq2 2265 . 2  |-  ( ( phi `  N )  =  if ( X  =  .1.  ,  ( phi `  N ) ,  0 )  -> 
( sum_ a  e.  U  ( X `  a )  =  ( phi `  N )  <->  sum_ a  e.  U  ( X `  a )  =  if ( X  =  .1. 
,  ( phi `  N ) ,  0 ) ) )
2 eqeq2 2265 . 2  |-  ( 0  =  if ( X  =  .1.  ,  ( phi `  N ) ,  0 )  -> 
( sum_ a  e.  U  ( X `  a )  =  0  <->  sum_ a  e.  U  ( X `  a )  =  if ( X  =  .1. 
,  ( phi `  N ) ,  0 ) ) )
3 fveq1 5422 . . . . . 6  |-  ( X  =  .1.  ->  ( X `  a )  =  (  .1.  `  a
) )
4 dchrsum.g . . . . . . 7  |-  G  =  (DChr `  N )
5 dchrsum.z . . . . . . 7  |-  Z  =  (ℤ/n `  N )
6 dchrsum.1 . . . . . . 7  |-  .1.  =  ( 0g `  G )
7 dchrsum2.u . . . . . . 7  |-  U  =  (Unit `  Z )
8 dchrsum.x . . . . . . . . 9  |-  ( ph  ->  X  e.  D )
9 dchrsum.d . . . . . . . . . 10  |-  D  =  ( Base `  G
)
104, 9dchrrcl 20406 . . . . . . . . 9  |-  ( X  e.  D  ->  N  e.  NN )
118, 10syl 17 . . . . . . . 8  |-  ( ph  ->  N  e.  NN )
1211adantr 453 . . . . . . 7  |-  ( (
ph  /\  a  e.  U )  ->  N  e.  NN )
13 simpr 449 . . . . . . 7  |-  ( (
ph  /\  a  e.  U )  ->  a  e.  U )
144, 5, 6, 7, 12, 13dchr1 20423 . . . . . 6  |-  ( (
ph  /\  a  e.  U )  ->  (  .1.  `  a )  =  1 )
153, 14sylan9eqr 2310 . . . . 5  |-  ( ( ( ph  /\  a  e.  U )  /\  X  =  .1.  )  ->  ( X `  a )  =  1 )
1615an32s 782 . . . 4  |-  ( ( ( ph  /\  X  =  .1.  )  /\  a  e.  U )  ->  ( X `  a )  =  1 )
1716sumeq2dv 12106 . . 3  |-  ( (
ph  /\  X  =  .1.  )  ->  sum_ a  e.  U  ( X `  a )  =  sum_ a  e.  U  1
)
185, 7znunithash 16445 . . . . . . . . 9  |-  ( N  e.  NN  ->  ( # `
 U )  =  ( phi `  N
) )
1911, 18syl 17 . . . . . . . 8  |-  ( ph  ->  ( # `  U
)  =  ( phi `  N ) )
2011phicld 12767 . . . . . . . . 9  |-  ( ph  ->  ( phi `  N
)  e.  NN )
2120nnnn0d 9950 . . . . . . . 8  |-  ( ph  ->  ( phi `  N
)  e.  NN0 )
2219, 21eqeltrd 2330 . . . . . . 7  |-  ( ph  ->  ( # `  U
)  e.  NN0 )
23 fvex 5437 . . . . . . . . 9  |-  (Unit `  Z )  e.  _V
247, 23eqeltri 2326 . . . . . . . 8  |-  U  e. 
_V
25 hashclb 11283 . . . . . . . 8  |-  ( U  e.  _V  ->  ( U  e.  Fin  <->  ( # `  U
)  e.  NN0 )
)
2624, 25ax-mp 10 . . . . . . 7  |-  ( U  e.  Fin  <->  ( # `  U
)  e.  NN0 )
2722, 26sylibr 205 . . . . . 6  |-  ( ph  ->  U  e.  Fin )
28 ax-1cn 8728 . . . . . 6  |-  1  e.  CC
29 fsumconst 12182 . . . . . 6  |-  ( ( U  e.  Fin  /\  1  e.  CC )  -> 
sum_ a  e.  U 
1  =  ( (
# `  U )  x.  1 ) )
3027, 28, 29sylancl 646 . . . . 5  |-  ( ph  -> 
sum_ a  e.  U 
1  =  ( (
# `  U )  x.  1 ) )
3119oveq1d 5772 . . . . 5  |-  ( ph  ->  ( ( # `  U
)  x.  1 )  =  ( ( phi `  N )  x.  1 ) )
3220nncnd 9695 . . . . . 6  |-  ( ph  ->  ( phi `  N
)  e.  CC )
3332mulid1d 8785 . . . . 5  |-  ( ph  ->  ( ( phi `  N )  x.  1 )  =  ( phi `  N ) )
3430, 31, 333eqtrd 2292 . . . 4  |-  ( ph  -> 
sum_ a  e.  U 
1  =  ( phi `  N ) )
3534adantr 453 . . 3  |-  ( (
ph  /\  X  =  .1.  )  ->  sum_ a  e.  U  1  =  ( phi `  N ) )
3617, 35eqtrd 2288 . 2  |-  ( (
ph  /\  X  =  .1.  )  ->  sum_ a  e.  U  ( X `  a )  =  ( phi `  N ) )
374dchrabl 20420 . . . . . . . . . 10  |-  ( N  e.  NN  ->  G  e.  Abel )
3811, 37syl 17 . . . . . . . . 9  |-  ( ph  ->  G  e.  Abel )
39 ablgrp 15021 . . . . . . . . 9  |-  ( G  e.  Abel  ->  G  e. 
Grp )
4038, 39syl 17 . . . . . . . 8  |-  ( ph  ->  G  e.  Grp )
419, 6grpidcl 14437 . . . . . . . 8  |-  ( G  e.  Grp  ->  .1.  e.  D )
4240, 41syl 17 . . . . . . 7  |-  ( ph  ->  .1.  e.  D )
434, 5, 9, 7, 8, 42dchreq 20424 . . . . . 6  |-  ( ph  ->  ( X  =  .1.  <->  A. k  e.  U  ( X `  k )  =  (  .1.  `  k ) ) )
4443notbid 287 . . . . 5  |-  ( ph  ->  ( -.  X  =  .1.  <->  -.  A. k  e.  U  ( X `  k )  =  (  .1.  `  k )
) )
45 rexnal 2525 . . . . 5  |-  ( E. k  e.  U  -.  ( X `  k )  =  (  .1.  `  k )  <->  -.  A. k  e.  U  ( X `  k )  =  (  .1.  `  k )
)
4644, 45syl6bbr 256 . . . 4  |-  ( ph  ->  ( -.  X  =  .1.  <->  E. k  e.  U  -.  ( X `  k
)  =  (  .1.  `  k ) ) )
47 df-ne 2421 . . . . . 6  |-  ( ( X `  k )  =/=  (  .1.  `  k )  <->  -.  ( X `  k )  =  (  .1.  `  k
) )
4811adantr 453 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  U )  ->  N  e.  NN )
49 simpr 449 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  U )  ->  k  e.  U )
504, 5, 6, 7, 48, 49dchr1 20423 . . . . . . . 8  |-  ( (
ph  /\  k  e.  U )  ->  (  .1.  `  k )  =  1 )
5150neeq2d 2433 . . . . . . 7  |-  ( (
ph  /\  k  e.  U )  ->  (
( X `  k
)  =/=  (  .1.  `  k )  <->  ( X `  k )  =/=  1
) )
52 oveq2 5765 . . . . . . . . . . . . . . . 16  |-  ( x  =  a  ->  (
k ( .r `  Z ) x )  =  ( k ( .r `  Z ) a ) )
5352fveq2d 5427 . . . . . . . . . . . . . . 15  |-  ( x  =  a  ->  ( X `  ( k
( .r `  Z
) x ) )  =  ( X `  ( k ( .r
`  Z ) a ) ) )
5453cbvsumv 12099 . . . . . . . . . . . . . 14  |-  sum_ x  e.  U  ( X `  ( k ( .r
`  Z ) x ) )  =  sum_ a  e.  U  ( X `  ( k
( .r `  Z
) a ) )
554, 5, 9dchrmhm 20407 . . . . . . . . . . . . . . . . . 18  |-  D  C_  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) )
5655, 8sseldi 3120 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
) )
5756ad2antrr 709 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
k  e.  U  /\  ( X `  k )  =/=  1 ) )  /\  a  e.  U
)  ->  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) ) )
58 eqid 2256 . . . . . . . . . . . . . . . . . . 19  |-  ( Base `  Z )  =  (
Base `  Z )
5958, 7unitss 15369 . . . . . . . . . . . . . . . . . 18  |-  U  C_  ( Base `  Z )
60 simprl 735 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  ( k  e.  U  /\  ( X `  k )  =/=  1 ) )  -> 
k  e.  U )
6159, 60sseldi 3120 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  ( k  e.  U  /\  ( X `  k )  =/=  1 ) )  -> 
k  e.  ( Base `  Z ) )
6261adantr 453 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
k  e.  U  /\  ( X `  k )  =/=  1 ) )  /\  a  e.  U
)  ->  k  e.  ( Base `  Z )
)
6359sseli 3118 . . . . . . . . . . . . . . . . 17  |-  ( a  e.  U  ->  a  e.  ( Base `  Z
) )
6463adantl 454 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
k  e.  U  /\  ( X `  k )  =/=  1 ) )  /\  a  e.  U
)  ->  a  e.  ( Base `  Z )
)
65 eqid 2256 . . . . . . . . . . . . . . . . . 18  |-  (mulGrp `  Z )  =  (mulGrp `  Z )
6665, 58mgpbas 15258 . . . . . . . . . . . . . . . . 17  |-  ( Base `  Z )  =  (
Base `  (mulGrp `  Z
) )
67 eqid 2256 . . . . . . . . . . . . . . . . . 18  |-  ( .r
`  Z )  =  ( .r `  Z
)
6865, 67mgpplusg 15256 . . . . . . . . . . . . . . . . 17  |-  ( .r
`  Z )  =  ( +g  `  (mulGrp `  Z ) )
69 eqid 2256 . . . . . . . . . . . . . . . . . 18  |-  (mulGrp ` fld )  =  (mulGrp ` fld )
70 cnfldmul 16312 . . . . . . . . . . . . . . . . . 18  |-  x.  =  ( .r ` fld )
7169, 70mgpplusg 15256 . . . . . . . . . . . . . . . . 17  |-  x.  =  ( +g  `  (mulGrp ` fld )
)
7266, 68, 71mhmlin 14349 . . . . . . . . . . . . . . . 16  |-  ( ( X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
)  /\  k  e.  ( Base `  Z )  /\  a  e.  ( Base `  Z ) )  ->  ( X `  ( k ( .r
`  Z ) a ) )  =  ( ( X `  k
)  x.  ( X `
 a ) ) )
7357, 62, 64, 72syl3anc 1187 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
k  e.  U  /\  ( X `  k )  =/=  1 ) )  /\  a  e.  U
)  ->  ( X `  ( k ( .r
`  Z ) a ) )  =  ( ( X `  k
)  x.  ( X `
 a ) ) )
7473sumeq2dv 12106 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( k  e.  U  /\  ( X `  k )  =/=  1 ) )  ->  sum_ a  e.  U  ( X `  ( k ( .r `  Z
) a ) )  =  sum_ a  e.  U  ( ( X `  k )  x.  ( X `  a )
) )
7554, 74syl5eq 2300 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( k  e.  U  /\  ( X `  k )  =/=  1 ) )  ->  sum_ x  e.  U  ( X `  ( k ( .r `  Z
) x ) )  =  sum_ a  e.  U  ( ( X `  k )  x.  ( X `  a )
) )
76 fveq2 5423 . . . . . . . . . . . . . 14  |-  ( a  =  ( k ( .r `  Z ) x )  ->  ( X `  a )  =  ( X `  ( k ( .r
`  Z ) x ) ) )
7727adantr 453 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( k  e.  U  /\  ( X `  k )  =/=  1 ) )  ->  U  e.  Fin )
7811nnnn0d 9950 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  N  e.  NN0 )
795zncrng 16425 . . . . . . . . . . . . . . . . . . 19  |-  ( N  e.  NN0  ->  Z  e. 
CRing )
8078, 79syl 17 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  Z  e.  CRing )
81 crngrng 15278 . . . . . . . . . . . . . . . . . 18  |-  ( Z  e.  CRing  ->  Z  e.  Ring )
8280, 81syl 17 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  Z  e.  Ring )
83 eqid 2256 . . . . . . . . . . . . . . . . . 18  |-  ( (mulGrp `  Z )s  U )  =  ( (mulGrp `  Z )s  U
)
847, 83unitgrp 15376 . . . . . . . . . . . . . . . . 17  |-  ( Z  e.  Ring  ->  ( (mulGrp `  Z )s  U )  e.  Grp )
8582, 84syl 17 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( (mulGrp `  Z
)s 
U )  e.  Grp )
8685adantr 453 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( k  e.  U  /\  ( X `  k )  =/=  1 ) )  -> 
( (mulGrp `  Z
)s 
U )  e.  Grp )
87 eqid 2256 . . . . . . . . . . . . . . . 16  |-  ( b  e.  U  |->  ( c  e.  U  |->  ( b ( .r `  Z
) c ) ) )  =  ( b  e.  U  |->  ( c  e.  U  |->  ( b ( .r `  Z
) c ) ) )
887, 83unitgrpbas 15375 . . . . . . . . . . . . . . . 16  |-  U  =  ( Base `  (
(mulGrp `  Z )s  U
) )
8983, 68ressplusg 13177 . . . . . . . . . . . . . . . . 17  |-  ( U  e.  _V  ->  ( .r `  Z )  =  ( +g  `  (
(mulGrp `  Z )s  U
) ) )
9024, 89ax-mp 10 . . . . . . . . . . . . . . . 16  |-  ( .r
`  Z )  =  ( +g  `  (
(mulGrp `  Z )s  U
) )
9187, 88, 90grplactf1o 14492 . . . . . . . . . . . . . . 15  |-  ( ( ( (mulGrp `  Z
)s 
U )  e.  Grp  /\  k  e.  U )  ->  ( ( b  e.  U  |->  ( c  e.  U  |->  ( b ( .r `  Z
) c ) ) ) `  k ) : U -1-1-onto-> U )
9286, 60, 91syl2anc 645 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( k  e.  U  /\  ( X `  k )  =/=  1 ) )  -> 
( ( b  e.  U  |->  ( c  e.  U  |->  ( b ( .r `  Z ) c ) ) ) `
 k ) : U -1-1-onto-> U )
9387, 88grplactval 14490 . . . . . . . . . . . . . . 15  |-  ( ( k  e.  U  /\  x  e.  U )  ->  ( ( ( b  e.  U  |->  ( c  e.  U  |->  ( b ( .r `  Z
) c ) ) ) `  k ) `
 x )  =  ( k ( .r
`  Z ) x ) )
9460, 93sylan 459 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
k  e.  U  /\  ( X `  k )  =/=  1 ) )  /\  x  e.  U
)  ->  ( (
( b  e.  U  |->  ( c  e.  U  |->  ( b ( .r
`  Z ) c ) ) ) `  k ) `  x
)  =  ( k ( .r `  Z
) x ) )
954, 5, 9, 58, 8dchrf 20408 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  X : ( Base `  Z ) --> CC )
96 ffvelrn 5562 . . . . . . . . . . . . . . . 16  |-  ( ( X : ( Base `  Z ) --> CC  /\  a  e.  ( Base `  Z ) )  -> 
( X `  a
)  e.  CC )
9795, 63, 96syl2an 465 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  a  e.  U )  ->  ( X `  a )  e.  CC )
9897adantlr 698 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
k  e.  U  /\  ( X `  k )  =/=  1 ) )  /\  a  e.  U
)  ->  ( X `  a )  e.  CC )
9976, 77, 92, 94, 98fsumf1o 12126 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( k  e.  U  /\  ( X `  k )  =/=  1 ) )  ->  sum_ a  e.  U  ( X `  a )  =  sum_ x  e.  U  ( X `  ( k ( .r `  Z
) x ) ) )
10095adantr 453 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( k  e.  U  /\  ( X `  k )  =/=  1 ) )  ->  X : ( Base `  Z
) --> CC )
101 ffvelrn 5562 . . . . . . . . . . . . . . 15  |-  ( ( X : ( Base `  Z ) --> CC  /\  k  e.  ( Base `  Z ) )  -> 
( X `  k
)  e.  CC )
102100, 61, 101syl2anc 645 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( k  e.  U  /\  ( X `  k )  =/=  1 ) )  -> 
( X `  k
)  e.  CC )
10377, 102, 98fsummulc2 12176 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( k  e.  U  /\  ( X `  k )  =/=  1 ) )  -> 
( ( X `  k )  x.  sum_ a  e.  U  ( X `  a )
)  =  sum_ a  e.  U  ( ( X `  k )  x.  ( X `  a
) ) )
10475, 99, 1033eqtr4rd 2299 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( k  e.  U  /\  ( X `  k )  =/=  1 ) )  -> 
( ( X `  k )  x.  sum_ a  e.  U  ( X `  a )
)  =  sum_ a  e.  U  ( X `  a ) )
10577, 98fsumcl 12136 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( k  e.  U  /\  ( X `  k )  =/=  1 ) )  ->  sum_ a  e.  U  ( X `  a )  e.  CC )
106105mulid2d 8786 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( k  e.  U  /\  ( X `  k )  =/=  1 ) )  -> 
( 1  x.  sum_ a  e.  U  ( X `  a )
)  =  sum_ a  e.  U  ( X `  a ) )
107104, 106oveq12d 5775 . . . . . . . . . . 11  |-  ( (
ph  /\  ( k  e.  U  /\  ( X `  k )  =/=  1 ) )  -> 
( ( ( X `
 k )  x. 
sum_ a  e.  U  ( X `  a ) )  -  ( 1  x.  sum_ a  e.  U  ( X `  a ) ) )  =  (
sum_ a  e.  U  ( X `  a )  -  sum_ a  e.  U  ( X `  a ) ) )
108105subidd 9078 . . . . . . . . . . 11  |-  ( (
ph  /\  ( k  e.  U  /\  ( X `  k )  =/=  1 ) )  -> 
( sum_ a  e.  U  ( X `  a )  -  sum_ a  e.  U  ( X `  a ) )  =  0 )
109107, 108eqtrd 2288 . . . . . . . . . 10  |-  ( (
ph  /\  ( k  e.  U  /\  ( X `  k )  =/=  1 ) )  -> 
( ( ( X `
 k )  x. 
sum_ a  e.  U  ( X `  a ) )  -  ( 1  x.  sum_ a  e.  U  ( X `  a ) ) )  =  0 )
11028a1i 12 . . . . . . . . . . 11  |-  ( (
ph  /\  ( k  e.  U  /\  ( X `  k )  =/=  1 ) )  -> 
1  e.  CC )
111102, 110, 105subdird 9169 . . . . . . . . . 10  |-  ( (
ph  /\  ( k  e.  U  /\  ( X `  k )  =/=  1 ) )  -> 
( ( ( X `
 k )  - 
1 )  x.  sum_ a  e.  U  ( X `  a )
)  =  ( ( ( X `  k
)  x.  sum_ a  e.  U  ( X `  a ) )  -  ( 1  x.  sum_ a  e.  U  ( X `  a )
) ) )
112 subcl 8984 . . . . . . . . . . . 12  |-  ( ( ( X `  k
)  e.  CC  /\  1  e.  CC )  ->  ( ( X `  k )  -  1 )  e.  CC )
113102, 28, 112sylancl 646 . . . . . . . . . . 11  |-  ( (
ph  /\  ( k  e.  U  /\  ( X `  k )  =/=  1 ) )  -> 
( ( X `  k )  -  1 )  e.  CC )
114113mul01d 8944 . . . . . . . . . 10  |-  ( (
ph  /\  ( k  e.  U  /\  ( X `  k )  =/=  1 ) )  -> 
( ( ( X `
 k )  - 
1 )  x.  0 )  =  0 )
115109, 111, 1143eqtr4d 2298 . . . . . . . . 9  |-  ( (
ph  /\  ( k  e.  U  /\  ( X `  k )  =/=  1 ) )  -> 
( ( ( X `
 k )  - 
1 )  x.  sum_ a  e.  U  ( X `  a )
)  =  ( ( ( X `  k
)  -  1 )  x.  0 ) )
116 0cn 8764 . . . . . . . . . . 11  |-  0  e.  CC
117116a1i 12 . . . . . . . . . 10  |-  ( (
ph  /\  ( k  e.  U  /\  ( X `  k )  =/=  1 ) )  -> 
0  e.  CC )
118 simprr 736 . . . . . . . . . . 11  |-  ( (
ph  /\  ( k  e.  U  /\  ( X `  k )  =/=  1 ) )  -> 
( X `  k
)  =/=  1 )
119 subeq0 9006 . . . . . . . . . . . . 13  |-  ( ( ( X `  k
)  e.  CC  /\  1  e.  CC )  ->  ( ( ( X `
 k )  - 
1 )  =  0  <-> 
( X `  k
)  =  1 ) )
120102, 28, 119sylancl 646 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( k  e.  U  /\  ( X `  k )  =/=  1 ) )  -> 
( ( ( X `
 k )  - 
1 )  =  0  <-> 
( X `  k
)  =  1 ) )
121120necon3bid 2454 . . . . . . . . . . 11  |-  ( (
ph  /\  ( k  e.  U  /\  ( X `  k )  =/=  1 ) )  -> 
( ( ( X `
 k )  - 
1 )  =/=  0  <->  ( X `  k )  =/=  1 ) )
122118, 121mpbird 225 . . . . . . . . . 10  |-  ( (
ph  /\  ( k  e.  U  /\  ( X `  k )  =/=  1 ) )  -> 
( ( X `  k )  -  1 )  =/=  0 )
123105, 117, 113, 122mulcand 9334 . . . . . . . . 9  |-  ( (
ph  /\  ( k  e.  U  /\  ( X `  k )  =/=  1 ) )  -> 
( ( ( ( X `  k )  -  1 )  x. 
sum_ a  e.  U  ( X `  a ) )  =  ( ( ( X `  k
)  -  1 )  x.  0 )  <->  sum_ a  e.  U  ( X `  a )  =  0 ) )
124115, 123mpbid 203 . . . . . . . 8  |-  ( (
ph  /\  ( k  e.  U  /\  ( X `  k )  =/=  1 ) )  ->  sum_ a  e.  U  ( X `  a )  =  0 )
125124expr 601 . . . . . . 7  |-  ( (
ph  /\  k  e.  U )  ->  (
( X `  k
)  =/=  1  ->  sum_ a  e.  U  ( X `  a )  =  0 ) )
12651, 125sylbid 208 . . . . . 6  |-  ( (
ph  /\  k  e.  U )  ->  (
( X `  k
)  =/=  (  .1.  `  k )  ->  sum_ a  e.  U  ( X `  a )  =  0 ) )
12747, 126syl5bir 211 . . . . 5  |-  ( (
ph  /\  k  e.  U )  ->  ( -.  ( X `  k
)  =  (  .1.  `  k )  ->  sum_ a  e.  U  ( X `  a )  =  0 ) )
128127rexlimdva 2638 . . . 4  |-  ( ph  ->  ( E. k  e.  U  -.  ( X `
 k )  =  (  .1.  `  k
)  ->  sum_ a  e.  U  ( X `  a )  =  0 ) )
12946, 128sylbid 208 . . 3  |-  ( ph  ->  ( -.  X  =  .1.  ->  sum_ a  e.  U  ( X `  a )  =  0 ) )
130129imp 420 . 2  |-  ( (
ph  /\  -.  X  =  .1.  )  ->  sum_ a  e.  U  ( X `  a )  =  0 )
1311, 2, 36, 130ifbothda 3536 1  |-  ( ph  -> 
sum_ a  e.  U  ( X `  a )  =  if ( X  =  .1.  ,  ( phi `  N ) ,  0 ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621    =/= wne 2419   A.wral 2516   E.wrex 2517   _Vcvv 2740   ifcif 3506    e. cmpt 4017   -->wf 4634   -1-1-onto->wf1o 4637   ` cfv 4638  (class class class)co 5757   Fincfn 6796   CCcc 8668   0cc0 8670   1c1 8671    x. cmul 8675    - cmin 8970   NNcn 9679   NN0cn0 9897   #chash 11268   sum_csu 12088   phicphi 12759   Basecbs 13075   ↾s cress 13076   +g cplusg 13135   .rcmulr 13136   0gc0g 13327   Grpcgrp 14289   MndHom cmhm 14340   Abelcabel 15017  mulGrpcmgp 15252   Ringcrg 15264   CRingccrg 15265  Unitcui 15348  ℂfldccnfld 16304  ℤ/nczn 16381  DChrcdchr 20398
This theorem is referenced by:  dchrsum  20435
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4071  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449  ax-inf2 7275  ax-cnex 8726  ax-resscn 8727  ax-1cn 8728  ax-icn 8729  ax-addcl 8730  ax-addrcl 8731  ax-mulcl 8732  ax-mulrcl 8733  ax-mulcom 8734  ax-addass 8735  ax-mulass 8736  ax-distr 8737  ax-i2m1 8738  ax-1ne0 8739  ax-1rid 8740  ax-rnegex 8741  ax-rrecex 8742  ax-cnre 8743  ax-pre-lttri 8744  ax-pre-lttrn 8745  ax-pre-ltadd 8746  ax-pre-mulgt0 8747  ax-pre-sup 8748  ax-addf 8749  ax-mulf 8750
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-uni 3769  df-int 3804  df-iun 3848  df-br 3964  df-opab 4018  df-mpt 4019  df-tr 4054  df-eprel 4242  df-id 4246  df-po 4251  df-so 4252  df-fr 4289  df-se 4290  df-we 4291  df-ord 4332  df-on 4333  df-lim 4334  df-suc 4335  df-om 4594  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-isom 4655  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-of 5977  df-1st 6021  df-2nd 6022  df-tpos 6133  df-iota 6190  df-riota 6237  df-recs 6321  df-rdg 6356  df-1o 6412  df-oadd 6416  df-er 6593  df-ec 6595  df-qs 6599  df-map 6707  df-en 6797  df-dom 6798  df-sdom 6799  df-fin 6800  df-sup 7127  df-oi 7158  df-card 7505  df-pnf 8802  df-mnf 8803  df-xr 8804  df-ltxr 8805  df-le 8806  df-sub 8972  df-neg 8973  df-div 9357  df-n 9680  df-2 9737  df-3 9738  df-4 9739  df-5 9740  df-6 9741  df-7 9742  df-8 9743  df-9 9744  df-10 9745  df-n0 9898  df-z 9957  df-dec 10057  df-uz 10163  df-rp 10287  df-fz 10714  df-fzo 10802  df-fl 10856  df-mod 10905  df-seq 10978  df-exp 11036  df-hash 11269  df-cj 11514  df-re 11515  df-im 11516  df-sqr 11650  df-abs 11651  df-clim 11892  df-sum 12089  df-divides 12459  df-gcd 12613  df-phi 12761  df-struct 13077  df-ndx 13078  df-slot 13079  df-base 13080  df-sets 13081  df-ress 13082  df-plusg 13148  df-mulr 13149  df-starv 13150  df-sca 13151  df-vsca 13152  df-tset 13154  df-ple 13155  df-ds 13157  df-0g 13331  df-imas 13338  df-divs 13339  df-mnd 14294  df-mhm 14342  df-grp 14416  df-minusg 14417  df-sbg 14418  df-mulg 14419  df-subg 14545  df-nsg 14546  df-eqg 14547  df-ghm 14608  df-cmn 15018  df-abl 15019  df-mgp 15253  df-ring 15267  df-cring 15268  df-ur 15269  df-oppr 15332  df-dvdsr 15350  df-unit 15351  df-invr 15381  df-rnghom 15423  df-subrg 15470  df-lmod 15556  df-lss 15617  df-lsp 15656  df-sra 15852  df-rgmod 15853  df-lidl 15854  df-rsp 15855  df-2idl 15911  df-cnfld 16305  df-zrh 16382  df-zn 16385  df-dchr 20399
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