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Theorem dchrsum2 21035
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 2439 . 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 2439 . 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 5713 . . . . . 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 21007 . . . . . . . . 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 21024 . . . . . 6  |-  ( (
ph  /\  a  e.  U )  ->  (  .1.  `  a )  =  1 )
153, 14sylan9eqr 2484 . . . . 5  |-  ( ( ( ph  /\  a  e.  U )  /\  X  =  .1.  )  ->  ( X `  a )  =  1 )
1615an32s 780 . . . 4  |-  ( ( ( ph  /\  X  =  .1.  )  /\  a  e.  U )  ->  ( X `  a )  =  1 )
1716sumeq2dv 12480 . . 3  |-  ( (
ph  /\  X  =  .1.  )  ->  sum_ a  e.  U  ( X `  a )  =  sum_ a  e.  U  1
)
185, 7znunithash 16828 . . . . . . . . 9  |-  ( N  e.  NN  ->  ( # `
 U )  =  ( phi `  N
) )
1911, 18syl 16 . . . . . . . 8  |-  ( ph  ->  ( # `  U
)  =  ( phi `  N ) )
2011phicld 13144 . . . . . . . . 9  |-  ( ph  ->  ( phi `  N
)  e.  NN )
2120nnnn0d 10258 . . . . . . . 8  |-  ( ph  ->  ( phi `  N
)  e.  NN0 )
2219, 21eqeltrd 2504 . . . . . . 7  |-  ( ph  ->  ( # `  U
)  e.  NN0 )
23 fvex 5728 . . . . . . . . 9  |-  (Unit `  Z )  e.  _V
247, 23eqeltri 2500 . . . . . . . 8  |-  U  e. 
_V
25 hashclb 11624 . . . . . . . 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 9032 . . . . . 6  |-  1  e.  CC
29 fsumconst 12556 . . . . . 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 6082 . . . . 5  |-  ( ph  ->  ( ( # `  U
)  x.  1 )  =  ( ( phi `  N )  x.  1 ) )
3220nncnd 10000 . . . . . 6  |-  ( ph  ->  ( phi `  N
)  e.  CC )
3332mulid1d 9089 . . . . 5  |-  ( ph  ->  ( ( phi `  N )  x.  1 )  =  ( phi `  N ) )
3430, 31, 333eqtrd 2466 . . . 4  |-  ( ph  -> 
sum_ a  e.  U 
1  =  ( phi `  N ) )
3534adantr 452 . . 3  |-  ( (
ph  /\  X  =  .1.  )  ->  sum_ a  e.  U  1  =  ( phi `  N ) )
3617, 35eqtrd 2462 . 2  |-  ( (
ph  /\  X  =  .1.  )  ->  sum_ a  e.  U  ( X `  a )  =  ( phi `  N ) )
374dchrabl 21021 . . . . . . . . . 10  |-  ( N  e.  NN  ->  G  e.  Abel )
3811, 37syl 16 . . . . . . . . 9  |-  ( ph  ->  G  e.  Abel )
39 ablgrp 15400 . . . . . . . . 9  |-  ( G  e.  Abel  ->  G  e. 
Grp )
4038, 39syl 16 . . . . . . . 8  |-  ( ph  ->  G  e.  Grp )
419, 6grpidcl 14816 . . . . . . . 8  |-  ( G  e.  Grp  ->  .1.  e.  D )
4240, 41syl 16 . . . . . . 7  |-  ( ph  ->  .1.  e.  D )
434, 5, 9, 7, 8, 42dchreq 21025 . . . . . 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 2703 . . . . 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 2595 . . . . . 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 21024 . . . . . . . 8  |-  ( (
ph  /\  k  e.  U )  ->  (  .1.  `  k )  =  1 )
5150neeq2d 2607 . . . . . . 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 2430 . . . . . . . . . . . . 13  |-  ( Base `  Z )  =  (
Base `  Z )
544, 5, 9, 53, 8dchrf 21009 . . . . . . . . . . . 12  |-  ( ph  ->  X : ( Base `  Z ) --> CC )
5553, 7unitss 15748 . . . . . . . . . . . . 13  |-  U  C_  ( Base `  Z )
5655sseli 3331 . . . . . . . . . . . 12  |-  ( a  e.  U  ->  a  e.  ( Base `  Z
) )
57 ffvelrn 5854 . . . . . . . . . . . 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 12510 . . . . . . . . 9  |-  ( (
ph  /\  ( k  e.  U  /\  ( X `  k )  =/=  1 ) )  ->  sum_ a  e.  U  ( X `  a )  e.  CC )
61 0cn 9068 . . . . . . . . . 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 3333 . . . . . . . . . . 11  |-  ( (
ph  /\  ( k  e.  U  /\  ( X `  k )  =/=  1 ) )  -> 
k  e.  ( Base `  Z ) )
6663, 65ffvelrnd 5857 . . . . . . . . . 10  |-  ( (
ph  /\  ( k  e.  U  /\  ( X `  k )  =/=  1 ) )  -> 
( X `  k
)  e.  CC )
67 subcl 9289 . . . . . . . . . 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 9311 . . . . . . . . . . . 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 2628 . . . . . . . . . 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 6075 . . . . . . . . . . . . . . . 16  |-  ( x  =  a  ->  (
k ( .r `  Z ) x )  =  ( k ( .r `  Z ) a ) )
7574fveq2d 5718 . . . . . . . . . . . . . . 15  |-  ( x  =  a  ->  ( X `  ( k
( .r `  Z
) x ) )  =  ( X `  ( k ( .r
`  Z ) a ) ) )
7675cbvsumv 12473 . . . . . . . . . . . . . 14  |-  sum_ x  e.  U  ( X `  ( k ( .r
`  Z ) x ) )  =  sum_ a  e.  U  ( X `  ( k
( .r `  Z
) a ) )
774, 5, 9dchrmhm 21008 . . . . . . . . . . . . . . . . . 18  |-  D  C_  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) )
7877, 8sseldi 3333 . . . . . . . . . . . . . . . . 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 2430 . . . . . . . . . . . . . . . . . 18  |-  (mulGrp `  Z )  =  (mulGrp `  Z )
8382, 53mgpbas 15637 . . . . . . . . . . . . . . . . 17  |-  ( Base `  Z )  =  (
Base `  (mulGrp `  Z
) )
84 eqid 2430 . . . . . . . . . . . . . . . . . 18  |-  ( .r
`  Z )  =  ( .r `  Z
)
8582, 84mgpplusg 15635 . . . . . . . . . . . . . . . . 17  |-  ( .r
`  Z )  =  ( +g  `  (mulGrp `  Z ) )
86 eqid 2430 . . . . . . . . . . . . . . . . . 18  |-  (mulGrp ` fld )  =  (mulGrp ` fld )
87 cnfldmul 16692 . . . . . . . . . . . . . . . . . 18  |-  x.  =  ( .r ` fld )
8886, 87mgpplusg 15635 . . . . . . . . . . . . . . . . 17  |-  x.  =  ( +g  `  (mulGrp ` fld )
)
8983, 85, 88mhmlin 14728 . . . . . . . . . . . . . . . 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 12480 . . . . . . . . . . . . . 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 2474 . . . . . . . . . . . . 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 5714 . . . . . . . . . . . . . 14  |-  ( a  =  ( k ( .r `  Z ) x )  ->  ( X `  a )  =  ( X `  ( k ( .r
`  Z ) x ) ) )
9411nnnn0d 10258 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  N  e.  NN0 )
955zncrng 16808 . . . . . . . . . . . . . . . . . . 19  |-  ( N  e.  NN0  ->  Z  e. 
CRing )
9694, 95syl 16 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  Z  e.  CRing )
97 crngrng 15657 . . . . . . . . . . . . . . . . . 18  |-  ( Z  e.  CRing  ->  Z  e.  Ring )
9896, 97syl 16 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  Z  e.  Ring )
99 eqid 2430 . . . . . . . . . . . . . . . . . 18  |-  ( (mulGrp `  Z )s  U )  =  ( (mulGrp `  Z )s  U
)
1007, 99unitgrp 15755 . . . . . . . . . . . . . . . . 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 2430 . . . . . . . . . . . . . . . 16  |-  ( b  e.  U  |->  ( c  e.  U  |->  ( b ( .r `  Z
) c ) ) )  =  ( b  e.  U  |->  ( c  e.  U  |->  ( b ( .r `  Z
) c ) ) )
1047, 99unitgrpbas 15754 . . . . . . . . . . . . . . . 16  |-  U  =  ( Base `  (
(mulGrp `  Z )s  U
) )
10599, 85ressplusg 13554 . . . . . . . . . . . . . . . . 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 14871 . . . . . . . . . . . . . . 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 14869 . . . . . . . . . . . . . . 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 12500 . . . . . . . . . . . . 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 12550 . . . . . . . . . . . . 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 2473 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( k  e.  U  /\  ( X `  k )  =/=  1 ) )  -> 
( ( X `  k )  x.  sum_ a  e.  U  ( X `  a )
)  =  sum_ a  e.  U  ( X `  a ) )
11460mulid2d 9090 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( k  e.  U  /\  ( X `  k )  =/=  1 ) )  -> 
( 1  x.  sum_ a  e.  U  ( X `  a )
)  =  sum_ a  e.  U  ( X `  a ) )
115113, 114oveq12d 6085 . . . . . . . . . . 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 9383 . . . . . . . . . . 11  |-  ( (
ph  /\  ( k  e.  U  /\  ( X `  k )  =/=  1 ) )  -> 
( sum_ a  e.  U  ( X `  a )  -  sum_ a  e.  U  ( X `  a ) )  =  0 )
117115, 116eqtrd 2462 . . . . . . . . . 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 9474 . . . . . . . . . 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 9249 . . . . . . . . . 10  |-  ( (
ph  /\  ( k  e.  U  /\  ( X `  k )  =/=  1 ) )  -> 
( ( ( X `
 k )  - 
1 )  x.  0 )  =  0 )
121117, 119, 1203eqtr4d 2472 . . . . . . . . 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 9641 . . . . . . . 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 2817 . . . 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 3756 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 1652    e. wcel 1725    =/= wne 2593   A.wral 2692   E.wrex 2693   _Vcvv 2943   ifcif 3726    e. cmpt 4253   -->wf 5436   -1-1-onto->wf1o 5439   ` cfv 5440  (class class class)co 6067   Fincfn 7095   CCcc 8972   0cc0 8974   1c1 8975    x. cmul 8979    - cmin 9275   NNcn 9984   NN0cn0 10205   #chash 11601   sum_csu 12462   phicphi 13136   Basecbs 13452   ↾s cress 13453   +g cplusg 13512   .rcmulr 13513   0gc0g 13706   Grpcgrp 14668   MndHom cmhm 14719   Abelcabel 15396  mulGrpcmgp 15631   Ringcrg 15643   CRingccrg 15644  Unitcui 15727  ℂfldccnfld 16686  ℤ/nczn 16764  DChrcdchr 20999
This theorem is referenced by:  dchrsum  21036
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2411  ax-rep 4307  ax-sep 4317  ax-nul 4325  ax-pow 4364  ax-pr 4390  ax-un 4687  ax-inf2 7580  ax-cnex 9030  ax-resscn 9031  ax-1cn 9032  ax-icn 9033  ax-addcl 9034  ax-addrcl 9035  ax-mulcl 9036  ax-mulrcl 9037  ax-mulcom 9038  ax-addass 9039  ax-mulass 9040  ax-distr 9041  ax-i2m1 9042  ax-1ne0 9043  ax-1rid 9044  ax-rnegex 9045  ax-rrecex 9046  ax-cnre 9047  ax-pre-lttri 9048  ax-pre-lttrn 9049  ax-pre-ltadd 9050  ax-pre-mulgt0 9051  ax-pre-sup 9052  ax-addf 9053  ax-mulf 9054
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-ne 2595  df-nel 2596  df-ral 2697  df-rex 2698  df-reu 2699  df-rmo 2700  df-rab 2701  df-v 2945  df-sbc 3149  df-csb 3239  df-dif 3310  df-un 3312  df-in 3314  df-ss 3321  df-pss 3323  df-nul 3616  df-if 3727  df-pw 3788  df-sn 3807  df-pr 3808  df-tp 3809  df-op 3810  df-uni 4003  df-int 4038  df-iun 4082  df-br 4200  df-opab 4254  df-mpt 4255  df-tr 4290  df-eprel 4481  df-id 4485  df-po 4490  df-so 4491  df-fr 4528  df-se 4529  df-we 4530  df-ord 4571  df-on 4572  df-lim 4573  df-suc 4574  df-om 4832  df-xp 4870  df-rel 4871  df-cnv 4872  df-co 4873  df-dm 4874  df-rn 4875  df-res 4876  df-ima 4877  df-iota 5404  df-fun 5442  df-fn 5443  df-f 5444  df-f1 5445  df-fo 5446  df-f1o 5447  df-fv 5448  df-isom 5449  df-ov 6070  df-oprab 6071  df-mpt2 6072  df-of 6291  df-1st 6335  df-2nd 6336  df-tpos 6465  df-riota 6535  df-recs 6619  df-rdg 6654  df-1o 6710  df-oadd 6714  df-er 6891  df-ec 6893  df-qs 6897  df-map 7006  df-en 7096  df-dom 7097  df-sdom 7098  df-fin 7099  df-sup 7432  df-oi 7463  df-card 7810  df-pnf 9106  df-mnf 9107  df-xr 9108  df-ltxr 9109  df-le 9110  df-sub 9277  df-neg 9278  df-div 9662  df-nn 9985  df-2 10042  df-3 10043  df-4 10044  df-5 10045  df-6 10046  df-7 10047  df-8 10048  df-9 10049  df-10 10050  df-n0 10206  df-z 10267  df-dec 10367  df-uz 10473  df-rp 10597  df-fz 11028  df-fzo 11119  df-fl 11185  df-mod 11234  df-seq 11307  df-exp 11366  df-hash 11602  df-cj 11887  df-re 11888  df-im 11889  df-sqr 12023  df-abs 12024  df-clim 12265  df-sum 12463  df-dvds 12836  df-gcd 12990  df-phi 13138  df-struct 13454  df-ndx 13455  df-slot 13456  df-base 13457  df-sets 13458  df-ress 13459  df-plusg 13525  df-mulr 13526  df-starv 13527  df-sca 13528  df-vsca 13529  df-tset 13531  df-ple 13532  df-ds 13534  df-unif 13535  df-0g 13710  df-imas 13717  df-divs 13718  df-mnd 14673  df-mhm 14721  df-grp 14795  df-minusg 14796  df-sbg 14797  df-mulg 14798  df-subg 14924  df-nsg 14925  df-eqg 14926  df-ghm 14987  df-cmn 15397  df-abl 15398  df-mgp 15632  df-rng 15646  df-cring 15647  df-ur 15648  df-oppr 15711  df-dvdsr 15729  df-unit 15730  df-invr 15760  df-rnghom 15802  df-subrg 15849  df-lmod 15935  df-lss 15992  df-lsp 16031  df-sra 16227  df-rgmod 16228  df-lidl 16229  df-rsp 16230  df-2idl 16286  df-cnfld 16687  df-zrh 16765  df-zn 16768  df-dchr 21000
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