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Theorem dchrsum2 20469
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 2267 . 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 2267 . 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 5457 . . . . . 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 20441 . . . . . . . . 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 20458 . . . . . 6  |-  ( (
ph  /\  a  e.  U )  ->  (  .1.  `  a )  =  1 )
153, 14sylan9eqr 2312 . . . . 5  |-  ( ( ( ph  /\  a  e.  U )  /\  X  =  .1.  )  ->  ( X `  a )  =  1 )
1615an32s 782 . . . 4  |-  ( ( ( ph  /\  X  =  .1.  )  /\  a  e.  U )  ->  ( X `  a )  =  1 )
1716sumeq2dv 12141 . . 3  |-  ( (
ph  /\  X  =  .1.  )  ->  sum_ a  e.  U  ( X `  a )  =  sum_ a  e.  U  1
)
185, 7znunithash 16480 . . . . . . . . 9  |-  ( N  e.  NN  ->  ( # `
 U )  =  ( phi `  N
) )
1911, 18syl 17 . . . . . . . 8  |-  ( ph  ->  ( # `  U
)  =  ( phi `  N ) )
2011phicld 12802 . . . . . . . . 9  |-  ( ph  ->  ( phi `  N
)  e.  NN )
2120nnnn0d 9985 . . . . . . . 8  |-  ( ph  ->  ( phi `  N
)  e.  NN0 )
2219, 21eqeltrd 2332 . . . . . . 7  |-  ( ph  ->  ( # `  U
)  e.  NN0 )
23 fvex 5472 . . . . . . . . 9  |-  (Unit `  Z )  e.  _V
247, 23eqeltri 2328 . . . . . . . 8  |-  U  e. 
_V
25 hashclb 11318 . . . . . . . 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 8763 . . . . . 6  |-  1  e.  CC
29 fsumconst 12217 . . . . . 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 5807 . . . . 5  |-  ( ph  ->  ( ( # `  U
)  x.  1 )  =  ( ( phi `  N )  x.  1 ) )
3220nncnd 9730 . . . . . 6  |-  ( ph  ->  ( phi `  N
)  e.  CC )
3332mulid1d 8820 . . . . 5  |-  ( ph  ->  ( ( phi `  N )  x.  1 )  =  ( phi `  N ) )
3430, 31, 333eqtrd 2294 . . . 4  |-  ( ph  -> 
sum_ a  e.  U 
1  =  ( phi `  N ) )
3534adantr 453 . . 3  |-  ( (
ph  /\  X  =  .1.  )  ->  sum_ a  e.  U  1  =  ( phi `  N ) )
3617, 35eqtrd 2290 . 2  |-  ( (
ph  /\  X  =  .1.  )  ->  sum_ a  e.  U  ( X `  a )  =  ( phi `  N ) )
374dchrabl 20455 . . . . . . . . . 10  |-  ( N  e.  NN  ->  G  e.  Abel )
3811, 37syl 17 . . . . . . . . 9  |-  ( ph  ->  G  e.  Abel )
39 ablgrp 15056 . . . . . . . . 9  |-  ( G  e.  Abel  ->  G  e. 
Grp )
4038, 39syl 17 . . . . . . . 8  |-  ( ph  ->  G  e.  Grp )
419, 6grpidcl 14472 . . . . . . . 8  |-  ( G  e.  Grp  ->  .1.  e.  D )
4240, 41syl 17 . . . . . . 7  |-  ( ph  ->  .1.  e.  D )
434, 5, 9, 7, 8, 42dchreq 20459 . . . . . 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 2529 . . . . 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 2423 . . . . . 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 20458 . . . . . . . 8  |-  ( (
ph  /\  k  e.  U )  ->  (  .1.  `  k )  =  1 )
5150neeq2d 2435 . . . . . . 7  |-  ( (
ph  /\  k  e.  U )  ->  (
( X `  k
)  =/=  (  .1.  `  k )  <->  ( X `  k )  =/=  1
) )
52 oveq2 5800 . . . . . . . . . . . . . . . 16  |-  ( x  =  a  ->  (
k ( .r `  Z ) x )  =  ( k ( .r `  Z ) a ) )
5352fveq2d 5462 . . . . . . . . . . . . . . 15  |-  ( x  =  a  ->  ( X `  ( k
( .r `  Z
) x ) )  =  ( X `  ( k ( .r
`  Z ) a ) ) )
5453cbvsumv 12134 . . . . . . . . . . . . . 14  |-  sum_ x  e.  U  ( X `  ( k ( .r
`  Z ) x ) )  =  sum_ a  e.  U  ( X `  ( k
( .r `  Z
) a ) )
554, 5, 9dchrmhm 20442 . . . . . . . . . . . . . . . . . 18  |-  D  C_  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) )
5655, 8sseldi 3153 . . . . . . . . . . . . . . . . 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 2258 . . . . . . . . . . . . . . . . . . 19  |-  ( Base `  Z )  =  (
Base `  Z )
5958, 7unitss 15404 . . . . . . . . . . . . . . . . . 18  |-  U  C_  ( Base `  Z )
60 simprl 735 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  ( k  e.  U  /\  ( X `  k )  =/=  1 ) )  -> 
k  e.  U )
6159, 60sseldi 3153 . . . . . . . . . . . . . . . . 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 3151 . . . . . . . . . . . . . . . . 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 2258 . . . . . . . . . . . . . . . . . 18  |-  (mulGrp `  Z )  =  (mulGrp `  Z )
6665, 58mgpbas 15293 . . . . . . . . . . . . . . . . 17  |-  ( Base `  Z )  =  (
Base `  (mulGrp `  Z
) )
67 eqid 2258 . . . . . . . . . . . . . . . . . 18  |-  ( .r
`  Z )  =  ( .r `  Z
)
6865, 67mgpplusg 15291 . . . . . . . . . . . . . . . . 17  |-  ( .r
`  Z )  =  ( +g  `  (mulGrp `  Z ) )
69 eqid 2258 . . . . . . . . . . . . . . . . . 18  |-  (mulGrp ` fld )  =  (mulGrp ` fld )
70 cnfldmul 16347 . . . . . . . . . . . . . . . . . 18  |-  x.  =  ( .r ` fld )
7169, 70mgpplusg 15291 . . . . . . . . . . . . . . . . 17  |-  x.  =  ( +g  `  (mulGrp ` fld )
)
7266, 68, 71mhmlin 14384 . . . . . . . . . . . . . . . 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 12141 . . . . . . . . . . . . . 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 2302 . . . . . . . . . . . . 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 5458 . . . . . . . . . . . . . 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 9985 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  N  e.  NN0 )
795zncrng 16460 . . . . . . . . . . . . . . . . . . 19  |-  ( N  e.  NN0  ->  Z  e. 
CRing )
8078, 79syl 17 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  Z  e.  CRing )
81 crngrng 15313 . . . . . . . . . . . . . . . . . 18  |-  ( Z  e.  CRing  ->  Z  e.  Ring )
8280, 81syl 17 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  Z  e.  Ring )
83 eqid 2258 . . . . . . . . . . . . . . . . . 18  |-  ( (mulGrp `  Z )s  U )  =  ( (mulGrp `  Z )s  U
)
847, 83unitgrp 15411 . . . . . . . . . . . . . . . . 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 2258 . . . . . . . . . . . . . . . 16  |-  ( b  e.  U  |->  ( c  e.  U  |->  ( b ( .r `  Z
) c ) ) )  =  ( b  e.  U  |->  ( c  e.  U  |->  ( b ( .r `  Z
) c ) ) )
887, 83unitgrpbas 15410 . . . . . . . . . . . . . . . 16  |-  U  =  ( Base `  (
(mulGrp `  Z )s  U
) )
8983, 68ressplusg 13212 . . . . . . . . . . . . . . . . 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 14527 . . . . . . . . . . . . . . 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 14525 . . . . . . . . . . . . . . 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 20443 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  X : ( Base `  Z ) --> CC )
96 ffvelrn 5597 . . . . . . . . . . . . . . . 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 12161 . . . . . . . . . . . . 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 5597 . . . . . . . . . . . . . . 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 12211 . . . . . . . . . . . . 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 2301 . . . . . . . . . . . 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 12171 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( k  e.  U  /\  ( X `  k )  =/=  1 ) )  ->  sum_ a  e.  U  ( X `  a )  e.  CC )
106105mulid2d 8821 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( k  e.  U  /\  ( X `  k )  =/=  1 ) )  -> 
( 1  x.  sum_ a  e.  U  ( X `  a )
)  =  sum_ a  e.  U  ( X `  a ) )
107104, 106oveq12d 5810 . . . . . . . . . . 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 9113 . . . . . . . . . . 11  |-  ( (
ph  /\  ( k  e.  U  /\  ( X `  k )  =/=  1 ) )  -> 
( sum_ a  e.  U  ( X `  a )  -  sum_ a  e.  U  ( X `  a ) )  =  0 )
109107, 108eqtrd 2290 . . . . . . . . . 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 9204 . . . . . . . . . 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 9019 . . . . . . . . . . . 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 8979 . . . . . . . . . 10  |-  ( (
ph  /\  ( k  e.  U  /\  ( X `  k )  =/=  1 ) )  -> 
( ( ( X `
 k )  - 
1 )  x.  0 )  =  0 )
115109, 111, 1143eqtr4d 2300 . . . . . . . . 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 8799 . . . . . . . . . . 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 9041 . . . . . . . . . . . . 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 2456 . . . . . . . . . . 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 9369 . . . . . . . . 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 2642 . . . 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 3569 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 2421   A.wral 2518   E.wrex 2519   _Vcvv 2763   ifcif 3539    e. cmpt 4051   -->wf 4669   -1-1-onto->wf1o 4672   ` cfv 4673  (class class class)co 5792   Fincfn 6831   CCcc 8703   0cc0 8705   1c1 8706    x. cmul 8710    - cmin 9005   NNcn 9714   NN0cn0 9932   #chash 11303   sum_csu 12123   phicphi 12794   Basecbs 13110   ↾s cress 13111   +g cplusg 13170   .rcmulr 13171   0gc0g 13362   Grpcgrp 14324   MndHom cmhm 14375   Abelcabel 15052  mulGrpcmgp 15287   Ringcrg 15299   CRingccrg 15300  Unitcui 15383  ℂfldccnfld 16339  ℤ/nczn 16416  DChrcdchr 20433
This theorem is referenced by:  dchrsum  20470
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 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-inf2 7310  ax-cnex 8761  ax-resscn 8762  ax-1cn 8763  ax-icn 8764  ax-addcl 8765  ax-addrcl 8766  ax-mulcl 8767  ax-mulrcl 8768  ax-mulcom 8769  ax-addass 8770  ax-mulass 8771  ax-distr 8772  ax-i2m1 8773  ax-1ne0 8774  ax-1rid 8775  ax-rnegex 8776  ax-rrecex 8777  ax-cnre 8778  ax-pre-lttri 8779  ax-pre-lttrn 8780  ax-pre-ltadd 8781  ax-pre-mulgt0 8782  ax-pre-sup 8783  ax-addf 8784  ax-mulf 8785
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 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-int 3837  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-se 4325  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-isom 4690  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-of 6012  df-1st 6056  df-2nd 6057  df-tpos 6168  df-iota 6225  df-riota 6272  df-recs 6356  df-rdg 6391  df-1o 6447  df-oadd 6451  df-er 6628  df-ec 6630  df-qs 6634  df-map 6742  df-en 6832  df-dom 6833  df-sdom 6834  df-fin 6835  df-sup 7162  df-oi 7193  df-card 7540  df-pnf 8837  df-mnf 8838  df-xr 8839  df-ltxr 8840  df-le 8841  df-sub 9007  df-neg 9008  df-div 9392  df-n 9715  df-2 9772  df-3 9773  df-4 9774  df-5 9775  df-6 9776  df-7 9777  df-8 9778  df-9 9779  df-10 9780  df-n0 9933  df-z 9992  df-dec 10092  df-uz 10198  df-rp 10322  df-fz 10749  df-fzo 10837  df-fl 10891  df-mod 10940  df-seq 11013  df-exp 11071  df-hash 11304  df-cj 11549  df-re 11550  df-im 11551  df-sqr 11685  df-abs 11686  df-clim 11927  df-sum 12124  df-divides 12494  df-gcd 12648  df-phi 12796  df-struct 13112  df-ndx 13113  df-slot 13114  df-base 13115  df-sets 13116  df-ress 13117  df-plusg 13183  df-mulr 13184  df-starv 13185  df-sca 13186  df-vsca 13187  df-tset 13189  df-ple 13190  df-ds 13192  df-0g 13366  df-imas 13373  df-divs 13374  df-mnd 14329  df-mhm 14377  df-grp 14451  df-minusg 14452  df-sbg 14453  df-mulg 14454  df-subg 14580  df-nsg 14581  df-eqg 14582  df-ghm 14643  df-cmn 15053  df-abl 15054  df-mgp 15288  df-ring 15302  df-cring 15303  df-ur 15304  df-oppr 15367  df-dvdsr 15385  df-unit 15386  df-invr 15416  df-rnghom 15458  df-subrg 15505  df-lmod 15591  df-lss 15652  df-lsp 15691  df-sra 15887  df-rgmod 15888  df-lidl 15889  df-rsp 15890  df-2idl 15946  df-cnfld 16340  df-zrh 16417  df-zn 16420  df-dchr 20434
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