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Theorem dchrsum2 21005
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
Dummy variables  k  x  b  c are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqeq2 2413 . 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 2413 . 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 5686 . . . . . 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 20977 . . . . . . . . 9  |-  ( X  e.  D  ->  N  e.  NN )
118, 10syl 16 . . . . . . . 8  |-  ( ph  ->  N  e.  NN )
1211adantr 452 . . . . . . 7  |-  ( (
ph  /\  a  e.  U )  ->  N  e.  NN )
13 simpr 448 . . . . . . 7  |-  ( (
ph  /\  a  e.  U )  ->  a  e.  U )
144, 5, 6, 7, 12, 13dchr1 20994 . . . . . 6  |-  ( (
ph  /\  a  e.  U )  ->  (  .1.  `  a )  =  1 )
153, 14sylan9eqr 2458 . . . . 5  |-  ( ( ( ph  /\  a  e.  U )  /\  X  =  .1.  )  ->  ( X `  a )  =  1 )
1615an32s 780 . . . 4  |-  ( ( ( ph  /\  X  =  .1.  )  /\  a  e.  U )  ->  ( X `  a )  =  1 )
1716sumeq2dv 12452 . . 3  |-  ( (
ph  /\  X  =  .1.  )  ->  sum_ a  e.  U  ( X `  a )  =  sum_ a  e.  U  1
)
185, 7znunithash 16800 . . . . . . . . 9  |-  ( N  e.  NN  ->  ( # `
 U )  =  ( phi `  N
) )
1911, 18syl 16 . . . . . . . 8  |-  ( ph  ->  ( # `  U
)  =  ( phi `  N ) )
2011phicld 13116 . . . . . . . . 9  |-  ( ph  ->  ( phi `  N
)  e.  NN )
2120nnnn0d 10230 . . . . . . . 8  |-  ( ph  ->  ( phi `  N
)  e.  NN0 )
2219, 21eqeltrd 2478 . . . . . . 7  |-  ( ph  ->  ( # `  U
)  e.  NN0 )
23 fvex 5701 . . . . . . . . 9  |-  (Unit `  Z )  e.  _V
247, 23eqeltri 2474 . . . . . . . 8  |-  U  e. 
_V
25 hashclb 11596 . . . . . . . 8  |-  ( U  e.  _V  ->  ( U  e.  Fin  <->  ( # `  U
)  e.  NN0 )
)
2624, 25ax-mp 8 . . . . . . 7  |-  ( U  e.  Fin  <->  ( # `  U
)  e.  NN0 )
2722, 26sylibr 204 . . . . . 6  |-  ( ph  ->  U  e.  Fin )
28 ax-1cn 9004 . . . . . 6  |-  1  e.  CC
29 fsumconst 12528 . . . . . 6  |-  ( ( U  e.  Fin  /\  1  e.  CC )  -> 
sum_ a  e.  U 
1  =  ( (
# `  U )  x.  1 ) )
3027, 28, 29sylancl 644 . . . . 5  |-  ( ph  -> 
sum_ a  e.  U 
1  =  ( (
# `  U )  x.  1 ) )
3119oveq1d 6055 . . . . 5  |-  ( ph  ->  ( ( # `  U
)  x.  1 )  =  ( ( phi `  N )  x.  1 ) )
3220nncnd 9972 . . . . . 6  |-  ( ph  ->  ( phi `  N
)  e.  CC )
3332mulid1d 9061 . . . . 5  |-  ( ph  ->  ( ( phi `  N )  x.  1 )  =  ( phi `  N ) )
3430, 31, 333eqtrd 2440 . . . 4  |-  ( ph  -> 
sum_ a  e.  U 
1  =  ( phi `  N ) )
3534adantr 452 . . 3  |-  ( (
ph  /\  X  =  .1.  )  ->  sum_ a  e.  U  1  =  ( phi `  N ) )
3617, 35eqtrd 2436 . 2  |-  ( (
ph  /\  X  =  .1.  )  ->  sum_ a  e.  U  ( X `  a )  =  ( phi `  N ) )
374dchrabl 20991 . . . . . . . . . 10  |-  ( N  e.  NN  ->  G  e.  Abel )
3811, 37syl 16 . . . . . . . . 9  |-  ( ph  ->  G  e.  Abel )
39 ablgrp 15372 . . . . . . . . 9  |-  ( G  e.  Abel  ->  G  e. 
Grp )
4038, 39syl 16 . . . . . . . 8  |-  ( ph  ->  G  e.  Grp )
419, 6grpidcl 14788 . . . . . . . 8  |-  ( G  e.  Grp  ->  .1.  e.  D )
4240, 41syl 16 . . . . . . 7  |-  ( ph  ->  .1.  e.  D )
434, 5, 9, 7, 8, 42dchreq 20995 . . . . . 6  |-  ( ph  ->  ( X  =  .1.  <->  A. k  e.  U  ( X `  k )  =  (  .1.  `  k ) ) )
4443notbid 286 . . . . 5  |-  ( ph  ->  ( -.  X  =  .1.  <->  -.  A. k  e.  U  ( X `  k )  =  (  .1.  `  k )
) )
45 rexnal 2677 . . . . 5  |-  ( E. k  e.  U  -.  ( X `  k )  =  (  .1.  `  k )  <->  -.  A. k  e.  U  ( X `  k )  =  (  .1.  `  k )
)
4644, 45syl6bbr 255 . . . 4  |-  ( ph  ->  ( -.  X  =  .1.  <->  E. k  e.  U  -.  ( X `  k
)  =  (  .1.  `  k ) ) )
47 df-ne 2569 . . . . . 6  |-  ( ( X `  k )  =/=  (  .1.  `  k )  <->  -.  ( X `  k )  =  (  .1.  `  k
) )
4811adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  U )  ->  N  e.  NN )
49 simpr 448 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  U )  ->  k  e.  U )
504, 5, 6, 7, 48, 49dchr1 20994 . . . . . . . 8  |-  ( (
ph  /\  k  e.  U )  ->  (  .1.  `  k )  =  1 )
5150neeq2d 2581 . . . . . . 7  |-  ( (
ph  /\  k  e.  U )  ->  (
( X `  k
)  =/=  (  .1.  `  k )  <->  ( X `  k )  =/=  1
) )
5227adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  ( k  e.  U  /\  ( X `  k )  =/=  1 ) )  ->  U  e.  Fin )
53 eqid 2404 . . . . . . . . . . . . 13  |-  ( Base `  Z )  =  (
Base `  Z )
544, 5, 9, 53, 8dchrf 20979 . . . . . . . . . . . 12  |-  ( ph  ->  X : ( Base `  Z ) --> CC )
5553, 7unitss 15720 . . . . . . . . . . . . 13  |-  U  C_  ( Base `  Z )
5655sseli 3304 . . . . . . . . . . . 12  |-  ( a  e.  U  ->  a  e.  ( Base `  Z
) )
57 ffvelrn 5827 . . . . . . . . . . . 12  |-  ( ( X : ( Base `  Z ) --> CC  /\  a  e.  ( Base `  Z ) )  -> 
( X `  a
)  e.  CC )
5854, 56, 57syl2an 464 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  U )  ->  ( X `  a )  e.  CC )
5958adantlr 696 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
k  e.  U  /\  ( X `  k )  =/=  1 ) )  /\  a  e.  U
)  ->  ( X `  a )  e.  CC )
6052, 59fsumcl 12482 . . . . . . . . 9  |-  ( (
ph  /\  ( k  e.  U  /\  ( X `  k )  =/=  1 ) )  ->  sum_ a  e.  U  ( X `  a )  e.  CC )
61 0cn 9040 . . . . . . . . . 10  |-  0  e.  CC
6261a1i 11 . . . . . . . . 9  |-  ( (
ph  /\  ( k  e.  U  /\  ( X `  k )  =/=  1 ) )  -> 
0  e.  CC )
6354adantr 452 . . . . . . . . . . 11  |-  ( (
ph  /\  ( k  e.  U  /\  ( X `  k )  =/=  1 ) )  ->  X : ( Base `  Z
) --> CC )
64 simprl 733 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( k  e.  U  /\  ( X `  k )  =/=  1 ) )  -> 
k  e.  U )
6555, 64sseldi 3306 . . . . . . . . . . 11  |-  ( (
ph  /\  ( k  e.  U  /\  ( X `  k )  =/=  1 ) )  -> 
k  e.  ( Base `  Z ) )
6663, 65ffvelrnd 5830 . . . . . . . . . 10  |-  ( (
ph  /\  ( k  e.  U  /\  ( X `  k )  =/=  1 ) )  -> 
( X `  k
)  e.  CC )
67 subcl 9261 . . . . . . . . . 10  |-  ( ( ( X `  k
)  e.  CC  /\  1  e.  CC )  ->  ( ( X `  k )  -  1 )  e.  CC )
6866, 28, 67sylancl 644 . . . . . . . . 9  |-  ( (
ph  /\  ( k  e.  U  /\  ( X `  k )  =/=  1 ) )  -> 
( ( X `  k )  -  1 )  e.  CC )
69 simprr 734 . . . . . . . . . 10  |-  ( (
ph  /\  ( k  e.  U  /\  ( X `  k )  =/=  1 ) )  -> 
( X `  k
)  =/=  1 )
70 subeq0 9283 . . . . . . . . . . . 12  |-  ( ( ( X `  k
)  e.  CC  /\  1  e.  CC )  ->  ( ( ( X `
 k )  - 
1 )  =  0  <-> 
( X `  k
)  =  1 ) )
7166, 28, 70sylancl 644 . . . . . . . . . . 11  |-  ( (
ph  /\  ( k  e.  U  /\  ( X `  k )  =/=  1 ) )  -> 
( ( ( X `
 k )  - 
1 )  =  0  <-> 
( X `  k
)  =  1 ) )
7271necon3bid 2602 . . . . . . . . . 10  |-  ( (
ph  /\  ( k  e.  U  /\  ( X `  k )  =/=  1 ) )  -> 
( ( ( X `
 k )  - 
1 )  =/=  0  <->  ( X `  k )  =/=  1 ) )
7369, 72mpbird 224 . . . . . . . . 9  |-  ( (
ph  /\  ( k  e.  U  /\  ( X `  k )  =/=  1 ) )  -> 
( ( X `  k )  -  1 )  =/=  0 )
74 oveq2 6048 . . . . . . . . . . . . . . . 16  |-  ( x  =  a  ->  (
k ( .r `  Z ) x )  =  ( k ( .r `  Z ) a ) )
7574fveq2d 5691 . . . . . . . . . . . . . . 15  |-  ( x  =  a  ->  ( X `  ( k
( .r `  Z
) x ) )  =  ( X `  ( k ( .r
`  Z ) a ) ) )
7675cbvsumv 12445 . . . . . . . . . . . . . 14  |-  sum_ x  e.  U  ( X `  ( k ( .r
`  Z ) x ) )  =  sum_ a  e.  U  ( X `  ( k
( .r `  Z
) a ) )
774, 5, 9dchrmhm 20978 . . . . . . . . . . . . . . . . . 18  |-  D  C_  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) )
7877, 8sseldi 3306 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
) )
7978ad2antrr 707 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
k  e.  U  /\  ( X `  k )  =/=  1 ) )  /\  a  e.  U
)  ->  X  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) ) )
8065adantr 452 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
k  e.  U  /\  ( X `  k )  =/=  1 ) )  /\  a  e.  U
)  ->  k  e.  ( Base `  Z )
)
8156adantl 453 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
k  e.  U  /\  ( X `  k )  =/=  1 ) )  /\  a  e.  U
)  ->  a  e.  ( Base `  Z )
)
82 eqid 2404 . . . . . . . . . . . . . . . . . 18  |-  (mulGrp `  Z )  =  (mulGrp `  Z )
8382, 53mgpbas 15609 . . . . . . . . . . . . . . . . 17  |-  ( Base `  Z )  =  (
Base `  (mulGrp `  Z
) )
84 eqid 2404 . . . . . . . . . . . . . . . . . 18  |-  ( .r
`  Z )  =  ( .r `  Z
)
8582, 84mgpplusg 15607 . . . . . . . . . . . . . . . . 17  |-  ( .r
`  Z )  =  ( +g  `  (mulGrp `  Z ) )
86 eqid 2404 . . . . . . . . . . . . . . . . . 18  |-  (mulGrp ` fld )  =  (mulGrp ` fld )
87 cnfldmul 16664 . . . . . . . . . . . . . . . . . 18  |-  x.  =  ( .r ` fld )
8886, 87mgpplusg 15607 . . . . . . . . . . . . . . . . 17  |-  x.  =  ( +g  `  (mulGrp ` fld )
)
8983, 85, 88mhmlin 14700 . . . . . . . . . . . . . . . 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 ) ) )
9079, 80, 81, 89syl3anc 1184 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
k  e.  U  /\  ( X `  k )  =/=  1 ) )  /\  a  e.  U
)  ->  ( X `  ( k ( .r
`  Z ) a ) )  =  ( ( X `  k
)  x.  ( X `
 a ) ) )
9190sumeq2dv 12452 . . . . . . . . . . . . . 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 )
) )
9276, 91syl5eq 2448 . . . . . . . . . . . . 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 )
) )
93 fveq2 5687 . . . . . . . . . . . . . 14  |-  ( a  =  ( k ( .r `  Z ) x )  ->  ( X `  a )  =  ( X `  ( k ( .r
`  Z ) x ) ) )
9411nnnn0d 10230 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  N  e.  NN0 )
955zncrng 16780 . . . . . . . . . . . . . . . . . . 19  |-  ( N  e.  NN0  ->  Z  e. 
CRing )
9694, 95syl 16 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  Z  e.  CRing )
97 crngrng 15629 . . . . . . . . . . . . . . . . . 18  |-  ( Z  e.  CRing  ->  Z  e.  Ring )
9896, 97syl 16 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  Z  e.  Ring )
99 eqid 2404 . . . . . . . . . . . . . . . . . 18  |-  ( (mulGrp `  Z )s  U )  =  ( (mulGrp `  Z )s  U
)
1007, 99unitgrp 15727 . . . . . . . . . . . . . . . . 17  |-  ( Z  e.  Ring  ->  ( (mulGrp `  Z )s  U )  e.  Grp )
10198, 100syl 16 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( (mulGrp `  Z
)s 
U )  e.  Grp )
102101adantr 452 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( k  e.  U  /\  ( X `  k )  =/=  1 ) )  -> 
( (mulGrp `  Z
)s 
U )  e.  Grp )
103 eqid 2404 . . . . . . . . . . . . . . . 16  |-  ( b  e.  U  |->  ( c  e.  U  |->  ( b ( .r `  Z
) c ) ) )  =  ( b  e.  U  |->  ( c  e.  U  |->  ( b ( .r `  Z
) c ) ) )
1047, 99unitgrpbas 15726 . . . . . . . . . . . . . . . 16  |-  U  =  ( Base `  (
(mulGrp `  Z )s  U
) )
10599, 85ressplusg 13526 . . . . . . . . . . . . . . . . 17  |-  ( U  e.  _V  ->  ( .r `  Z )  =  ( +g  `  (
(mulGrp `  Z )s  U
) ) )
10624, 105ax-mp 8 . . . . . . . . . . . . . . . 16  |-  ( .r
`  Z )  =  ( +g  `  (
(mulGrp `  Z )s  U
) )
107103, 104, 106grplactf1o 14843 . . . . . . . . . . . . . . 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 )
108102, 64, 107syl2anc 643 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( k  e.  U  /\  ( X `  k )  =/=  1 ) )  -> 
( ( b  e.  U  |->  ( c  e.  U  |->  ( b ( .r `  Z ) c ) ) ) `
 k ) : U -1-1-onto-> U )
109103, 104grplactval 14841 . . . . . . . . . . . . . . 15  |-  ( ( k  e.  U  /\  x  e.  U )  ->  ( ( ( b  e.  U  |->  ( c  e.  U  |->  ( b ( .r `  Z
) c ) ) ) `  k ) `
 x )  =  ( k ( .r
`  Z ) x ) )
11064, 109sylan 458 . . . . . . . . . . . . . 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 ) )
11193, 52, 108, 110, 59fsumf1o 12472 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( k  e.  U  /\  ( X `  k )  =/=  1 ) )  ->  sum_ a  e.  U  ( X `  a )  =  sum_ x  e.  U  ( X `  ( k ( .r `  Z
) x ) ) )
11252, 66, 59fsummulc2 12522 . . . . . . . . . . . . 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
) ) )
11392, 111, 1123eqtr4rd 2447 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( k  e.  U  /\  ( X `  k )  =/=  1 ) )  -> 
( ( X `  k )  x.  sum_ a  e.  U  ( X `  a )
)  =  sum_ a  e.  U  ( X `  a ) )
11460mulid2d 9062 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( k  e.  U  /\  ( X `  k )  =/=  1 ) )  -> 
( 1  x.  sum_ a  e.  U  ( X `  a )
)  =  sum_ a  e.  U  ( X `  a ) )
115113, 114oveq12d 6058 . . . . . . . . . . 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 ) ) )
11660subidd 9355 . . . . . . . . . . 11  |-  ( (
ph  /\  ( k  e.  U  /\  ( X `  k )  =/=  1 ) )  -> 
( sum_ a  e.  U  ( X `  a )  -  sum_ a  e.  U  ( X `  a ) )  =  0 )
117115, 116eqtrd 2436 . . . . . . . . . 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 )
11828a1i 11 . . . . . . . . . . 11  |-  ( (
ph  /\  ( k  e.  U  /\  ( X `  k )  =/=  1 ) )  -> 
1  e.  CC )
11966, 118, 60subdird 9446 . . . . . . . . . 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 )
) ) )
12068mul01d 9221 . . . . . . . . . 10  |-  ( (
ph  /\  ( k  e.  U  /\  ( X `  k )  =/=  1 ) )  -> 
( ( ( X `
 k )  - 
1 )  x.  0 )  =  0 )
121117, 119, 1203eqtr4d 2446 . . . . . . . . 9  |-  ( (
ph  /\  ( k  e.  U  /\  ( X `  k )  =/=  1 ) )  -> 
( ( ( X `
 k )  - 
1 )  x.  sum_ a  e.  U  ( X `  a )
)  =  ( ( ( X `  k
)  -  1 )  x.  0 ) )
12260, 62, 68, 73, 121mulcanad 9613 . . . . . . . 8  |-  ( (
ph  /\  ( k  e.  U  /\  ( X `  k )  =/=  1 ) )  ->  sum_ a  e.  U  ( X `  a )  =  0 )
123122expr 599 . . . . . . 7  |-  ( (
ph  /\  k  e.  U )  ->  (
( X `  k
)  =/=  1  ->  sum_ a  e.  U  ( X `  a )  =  0 ) )
12451, 123sylbid 207 . . . . . 6  |-  ( (
ph  /\  k  e.  U )  ->  (
( X `  k
)  =/=  (  .1.  `  k )  ->  sum_ a  e.  U  ( X `  a )  =  0 ) )
12547, 124syl5bir 210 . . . . 5  |-  ( (
ph  /\  k  e.  U )  ->  ( -.  ( X `  k
)  =  (  .1.  `  k )  ->  sum_ a  e.  U  ( X `  a )  =  0 ) )
126125rexlimdva 2790 . . . 4  |-  ( ph  ->  ( E. k  e.  U  -.  ( X `
 k )  =  (  .1.  `  k
)  ->  sum_ a  e.  U  ( X `  a )  =  0 ) )
12746, 126sylbid 207 . . 3  |-  ( ph  ->  ( -.  X  =  .1.  ->  sum_ a  e.  U  ( X `  a )  =  0 ) )
128127imp 419 . 2  |-  ( (
ph  /\  -.  X  =  .1.  )  ->  sum_ a  e.  U  ( X `  a )  =  0 )
1291, 2, 36, 128ifbothda 3729 1  |-  ( ph  -> 
sum_ a  e.  U  ( X `  a )  =  if ( X  =  .1.  ,  ( phi `  N ) ,  0 ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2567   A.wral 2666   E.wrex 2667   _Vcvv 2916   ifcif 3699    e. cmpt 4226   -->wf 5409   -1-1-onto->wf1o 5412   ` cfv 5413  (class class class)co 6040   Fincfn 7068   CCcc 8944   0cc0 8946   1c1 8947    x. cmul 8951    - cmin 9247   NNcn 9956   NN0cn0 10177   #chash 11573   sum_csu 12434   phicphi 13108   Basecbs 13424   ↾s cress 13425   +g cplusg 13484   .rcmulr 13485   0gc0g 13678   Grpcgrp 14640   MndHom cmhm 14691   Abelcabel 15368  mulGrpcmgp 15603   Ringcrg 15615   CRingccrg 15616  Unitcui 15699  ℂfldccnfld 16658  ℤ/nczn 16736  DChrcdchr 20969
This theorem is referenced by:  dchrsum  21006
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-tpos 6438  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-ec 6866  df-qs 6870  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-oi 7435  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-rp 10569  df-fz 11000  df-fzo 11091  df-fl 11157  df-mod 11206  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-clim 12237  df-sum 12435  df-dvds 12808  df-gcd 12962  df-phi 13110  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-0g 13682  df-imas 13689  df-divs 13690  df-mnd 14645  df-mhm 14693  df-grp 14767  df-minusg 14768  df-sbg 14769  df-mulg 14770  df-subg 14896  df-nsg 14897  df-eqg 14898  df-ghm 14959  df-cmn 15369  df-abl 15370  df-mgp 15604  df-rng 15618  df-cring 15619  df-ur 15620  df-oppr 15683  df-dvdsr 15701  df-unit 15702  df-invr 15732  df-rnghom 15774  df-subrg 15821  df-lmod 15907  df-lss 15964  df-lsp 16003  df-sra 16199  df-rgmod 16200  df-lidl 16201  df-rsp 16202  df-2idl 16258  df-cnfld 16659  df-zrh 16737  df-zn 16740  df-dchr 20970
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