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Theorem dchrsum2 20339
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 2262 . 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 2262 . 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 5376 . . . . . 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 20311 . . . . . . . . 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 20328 . . . . . 6  |-  ( (
ph  /\  a  e.  U )  ->  (  .1.  `  a )  =  1 )
153, 14sylan9eqr 2307 . . . . 5  |-  ( ( ( ph  /\  a  e.  U )  /\  X  =  .1.  )  ->  ( X `  a )  =  1 )
1615an32s 782 . . . 4  |-  ( ( ( ph  /\  X  =  .1.  )  /\  a  e.  U )  ->  ( X `  a )  =  1 )
1716sumeq2dv 12053 . . 3  |-  ( (
ph  /\  X  =  .1.  )  ->  sum_ a  e.  U  ( X `  a )  =  sum_ a  e.  U  1
)
185, 7znunithash 16350 . . . . . . . . 9  |-  ( N  e.  NN  ->  ( # `
 U )  =  ( phi `  N
) )
1911, 18syl 17 . . . . . . . 8  |-  ( ph  ->  ( # `  U
)  =  ( phi `  N ) )
2011phicld 12714 . . . . . . . . 9  |-  ( ph  ->  ( phi `  N
)  e.  NN )
2120nnnn0d 9897 . . . . . . . 8  |-  ( ph  ->  ( phi `  N
)  e.  NN0 )
2219, 21eqeltrd 2327 . . . . . . 7  |-  ( ph  ->  ( # `  U
)  e.  NN0 )
23 fvex 5391 . . . . . . . . 9  |-  (Unit `  Z )  e.  _V
247, 23eqeltri 2323 . . . . . . . 8  |-  U  e. 
_V
25 hashclb 11230 . . . . . . . 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 8675 . . . . . 6  |-  1  e.  CC
29 fsumconst 12129 . . . . . 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 5725 . . . . 5  |-  ( ph  ->  ( ( # `  U
)  x.  1 )  =  ( ( phi `  N )  x.  1 ) )
3220nncnd 9642 . . . . . 6  |-  ( ph  ->  ( phi `  N
)  e.  CC )
3332mulid1d 8732 . . . . 5  |-  ( ph  ->  ( ( phi `  N )  x.  1 )  =  ( phi `  N ) )
3430, 31, 333eqtrd 2289 . . . 4  |-  ( ph  -> 
sum_ a  e.  U 
1  =  ( phi `  N ) )
3534adantr 453 . . 3  |-  ( (
ph  /\  X  =  .1.  )  ->  sum_ a  e.  U  1  =  ( phi `  N ) )
3617, 35eqtrd 2285 . 2  |-  ( (
ph  /\  X  =  .1.  )  ->  sum_ a  e.  U  ( X `  a )  =  ( phi `  N ) )
374dchrabl 20325 . . . . . . . . . 10  |-  ( N  e.  NN  ->  G  e.  Abel )
3811, 37syl 17 . . . . . . . . 9  |-  ( ph  ->  G  e.  Abel )
39 ablgrp 14929 . . . . . . . . 9  |-  ( G  e.  Abel  ->  G  e. 
Grp )
4038, 39syl 17 . . . . . . . 8  |-  ( ph  ->  G  e.  Grp )
419, 6grpidcl 14345 . . . . . . . 8  |-  ( G  e.  Grp  ->  .1.  e.  D )
4240, 41syl 17 . . . . . . 7  |-  ( ph  ->  .1.  e.  D )
434, 5, 9, 7, 8, 42dchreq 20329 . . . . . 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 2518 . . . . 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 2414 . . . . . 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 20328 . . . . . . . 8  |-  ( (
ph  /\  k  e.  U )  ->  (  .1.  `  k )  =  1 )
5150neeq2d 2426 . . . . . . 7  |-  ( (
ph  /\  k  e.  U )  ->  (
( X `  k
)  =/=  (  .1.  `  k )  <->  ( X `  k )  =/=  1
) )
52 oveq2 5718 . . . . . . . . . . . . . . . 16  |-  ( x  =  a  ->  (
k ( .r `  Z ) x )  =  ( k ( .r `  Z ) a ) )
5352fveq2d 5381 . . . . . . . . . . . . . . 15  |-  ( x  =  a  ->  ( X `  ( k
( .r `  Z
) x ) )  =  ( X `  ( k ( .r
`  Z ) a ) ) )
5453cbvsumv 12046 . . . . . . . . . . . . . 14  |-  sum_ x  e.  U  ( X `  ( k ( .r
`  Z ) x ) )  =  sum_ a  e.  U  ( X `  ( k
( .r `  Z
) a ) )
554, 5, 9dchrmhm 20312 . . . . . . . . . . . . . . . . . 18  |-  D  C_  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) )
5655, 8sseldi 3101 . . . . . . . . . . . . . . . . 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 2253 . . . . . . . . . . . . . . . . . . 19  |-  ( Base `  Z )  =  (
Base `  Z )
5958, 7unitss 15277 . . . . . . . . . . . . . . . . . 18  |-  U  C_  ( Base `  Z )
60 simprl 735 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  ( k  e.  U  /\  ( X `  k )  =/=  1 ) )  -> 
k  e.  U )
6159, 60sseldi 3101 . . . . . . . . . . . . . . . . 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 3099 . . . . . . . . . . . . . . . . 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 2253 . . . . . . . . . . . . . . . . . 18  |-  (mulGrp `  Z )  =  (mulGrp `  Z )
6665, 58mgpbas 15166 . . . . . . . . . . . . . . . . 17  |-  ( Base `  Z )  =  (
Base `  (mulGrp `  Z
) )
67 eqid 2253 . . . . . . . . . . . . . . . . . 18  |-  ( .r
`  Z )  =  ( .r `  Z
)
6865, 67mgpplusg 15164 . . . . . . . . . . . . . . . . 17  |-  ( .r
`  Z )  =  ( +g  `  (mulGrp `  Z ) )
69 eqid 2253 . . . . . . . . . . . . . . . . . 18  |-  (mulGrp ` fld )  =  (mulGrp ` fld )
70 cnfldmul 16217 . . . . . . . . . . . . . . . . . 18  |-  x.  =  ( .r ` fld )
7169, 70mgpplusg 15164 . . . . . . . . . . . . . . . . 17  |-  x.  =  ( +g  `  (mulGrp ` fld )
)
7266, 68, 71mhmlin 14257 . . . . . . . . . . . . . . . 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 12053 . . . . . . . . . . . . . 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 2297 . . . . . . . . . . . . 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 5377 . . . . . . . . . . . . . 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 9897 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  N  e.  NN0 )
795zncrng 16330 . . . . . . . . . . . . . . . . . . 19  |-  ( N  e.  NN0  ->  Z  e. 
CRing )
8078, 79syl 17 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  Z  e.  CRing )
81 crngrng 15186 . . . . . . . . . . . . . . . . . 18  |-  ( Z  e.  CRing  ->  Z  e.  Ring )
8280, 81syl 17 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  Z  e.  Ring )
83 eqid 2253 . . . . . . . . . . . . . . . . . 18  |-  ( (mulGrp `  Z )s  U )  =  ( (mulGrp `  Z )s  U
)
847, 83unitgrp 15284 . . . . . . . . . . . . . . . . 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 2253 . . . . . . . . . . . . . . . 16  |-  ( b  e.  U  |->  ( c  e.  U  |->  ( b ( .r `  Z
) c ) ) )  =  ( b  e.  U  |->  ( c  e.  U  |->  ( b ( .r `  Z
) c ) ) )
887, 83unitgrpbas 15283 . . . . . . . . . . . . . . . 16  |-  U  =  ( Base `  (
(mulGrp `  Z )s  U
) )
8983, 68ressplusg 13124 . . . . . . . . . . . . . . . . 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 14400 . . . . . . . . . . . . . . 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 14398 . . . . . . . . . . . . . . 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 20313 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  X : ( Base `  Z ) --> CC )
96 ffvelrn 5515 . . . . . . . . . . . . . . . 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 12073 . . . . . . . . . . . . 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 5515 . . . . . . . . . . . . . . 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 12123 . . . . . . . . . . . . 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 2296 . . . . . . . . . . . 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 12083 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( k  e.  U  /\  ( X `  k )  =/=  1 ) )  ->  sum_ a  e.  U  ( X `  a )  e.  CC )
106105mulid2d 8733 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( k  e.  U  /\  ( X `  k )  =/=  1 ) )  -> 
( 1  x.  sum_ a  e.  U  ( X `  a )
)  =  sum_ a  e.  U  ( X `  a ) )
107104, 106oveq12d 5728 . . . . . . . . . . 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 9025 . . . . . . . . . . 11  |-  ( (
ph  /\  ( k  e.  U  /\  ( X `  k )  =/=  1 ) )  -> 
( sum_ a  e.  U  ( X `  a )  -  sum_ a  e.  U  ( X `  a ) )  =  0 )
109107, 108eqtrd 2285 . . . . . . . . . 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 9116 . . . . . . . . . 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 8931 . . . . . . . . . . . 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 8891 . . . . . . . . . 10  |-  ( (
ph  /\  ( k  e.  U  /\  ( X `  k )  =/=  1 ) )  -> 
( ( ( X `
 k )  - 
1 )  x.  0 )  =  0 )
115109, 111, 1143eqtr4d 2295 . . . . . . . . 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 8711 . . . . . . . . . . 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 8953 . . . . . . . . . . . . 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 2447 . . . . . . . . . . 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 9281 . . . . . . . . 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 2629 . . . 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 3500 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 2412   A.wral 2509   E.wrex 2510   _Vcvv 2727   ifcif 3470    e. cmpt 3974   -->wf 4588   -1-1-onto->wf1o 4591   ` cfv 4592  (class class class)co 5710   Fincfn 6749   CCcc 8615   0cc0 8617   1c1 8618    x. cmul 8622    - cmin 8917   NNcn 9626   NN0cn0 9844   #chash 11215   sum_csu 12035   phicphi 12706   Basecbs 13022   ↾s cress 13023   +g cplusg 13082   .rcmulr 13083   0gc0g 13274   Grpcgrp 14197   MndHom cmhm 14248   Abelcabel 14925  mulGrpcmgp 15160   Ringcrg 15172   CRingccrg 15173  Unitcui 15256  ℂfldccnfld 16209  ℤ/nczn 16286  DChrcdchr 20303
This theorem is referenced by:  dchrsum  20340
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 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-inf2 7226  ax-cnex 8673  ax-resscn 8674  ax-1cn 8675  ax-icn 8676  ax-addcl 8677  ax-addrcl 8678  ax-mulcl 8679  ax-mulrcl 8680  ax-mulcom 8681  ax-addass 8682  ax-mulass 8683  ax-distr 8684  ax-i2m1 8685  ax-1ne0 8686  ax-1rid 8687  ax-rnegex 8688  ax-rrecex 8689  ax-cnre 8690  ax-pre-lttri 8691  ax-pre-lttrn 8692  ax-pre-ltadd 8693  ax-pre-mulgt0 8694  ax-pre-sup 8695  ax-addf 8696  ax-mulf 8697
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-int 3761  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-se 4246  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-isom 4609  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-of 5930  df-1st 5974  df-2nd 5975  df-tpos 6086  df-iota 6143  df-riota 6190  df-recs 6274  df-rdg 6309  df-1o 6365  df-oadd 6369  df-er 6546  df-ec 6548  df-qs 6552  df-map 6660  df-en 6750  df-dom 6751  df-sdom 6752  df-fin 6753  df-sup 7078  df-oi 7109  df-card 7456  df-pnf 8749  df-mnf 8750  df-xr 8751  df-ltxr 8752  df-le 8753  df-sub 8919  df-neg 8920  df-div 9304  df-n 9627  df-2 9684  df-3 9685  df-4 9686  df-5 9687  df-6 9688  df-7 9689  df-8 9690  df-9 9691  df-10 9692  df-n0 9845  df-z 9904  df-dec 10004  df-uz 10110  df-rp 10234  df-fz 10661  df-fzo 10749  df-fl 10803  df-mod 10852  df-seq 10925  df-exp 10983  df-hash 11216  df-cj 11461  df-re 11462  df-im 11463  df-sqr 11597  df-abs 11598  df-clim 11839  df-sum 12036  df-divides 12406  df-gcd 12560  df-phi 12708  df-struct 13024  df-ndx 13025  df-slot 13026  df-base 13027  df-sets 13028  df-ress 13029  df-plusg 13095  df-mulr 13096  df-starv 13097  df-sca 13098  df-vsca 13099  df-tset 13101  df-ple 13102  df-ds 13104  df-0g 13278  df-imas 13285  df-divs 13286  df-mnd 14202  df-mhm 14250  df-grp 14324  df-minusg 14325  df-sbg 14326  df-mulg 14327  df-subg 14453  df-nsg 14454  df-eqg 14455  df-ghm 14516  df-cmn 14926  df-abl 14927  df-mgp 15161  df-ring 15175  df-cring 15176  df-ur 15177  df-oppr 15240  df-dvdsr 15258  df-unit 15259  df-invr 15289  df-rnghom 15331  df-subrg 15378  df-lmod 15464  df-lss 15525  df-lsp 15564  df-sra 15757  df-rgmod 15758  df-lidl 15759  df-rsp 15760  df-2idl 15816  df-cnfld 16210  df-zrh 16287  df-zn 16290  df-dchr 20304
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