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Theorem sumdchr2 20562
Description: Lemma for sumdchr 20564. (Contributed by Mario Carneiro, 28-Apr-2016.)
Hypotheses
Ref Expression
sumdchr.g  |-  G  =  (DChr `  N )
sumdchr.d  |-  D  =  ( Base `  G
)
sumdchr2.z  |-  Z  =  (ℤ/n `  N )
sumdchr2.1  |-  .1.  =  ( 1r `  Z )
sumdchr2.b  |-  B  =  ( Base `  Z
)
sumdchr2.n  |-  ( ph  ->  N  e.  NN )
sumdchr2.x  |-  ( ph  ->  A  e.  B )
Assertion
Ref Expression
sumdchr2  |-  ( ph  -> 
sum_ x  e.  D  ( x `  A
)  =  if ( A  =  .1.  , 
( # `  D ) ,  0 ) )
Distinct variable groups:    x,  .1.    x, A    x, D    x, N    x, G    ph, x
Allowed substitution hints:    B( x)    Z( x)

Proof of Theorem sumdchr2
Dummy variables  y 
z  a  b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqeq2 2325 . 2  |-  ( (
# `  D )  =  if ( A  =  .1.  ,  ( # `  D ) ,  0 )  ->  ( sum_ x  e.  D  ( x `
 A )  =  ( # `  D
)  <->  sum_ x  e.  D  ( x `  A
)  =  if ( A  =  .1.  , 
( # `  D ) ,  0 ) ) )
2 eqeq2 2325 . 2  |-  ( 0  =  if ( A  =  .1.  ,  (
# `  D ) ,  0 )  -> 
( sum_ x  e.  D  ( x `  A
)  =  0  <->  sum_ x  e.  D  ( x `
 A )  =  if ( A  =  .1.  ,  ( # `  D ) ,  0 ) ) )
3 fveq2 5563 . . . . . 6  |-  ( A  =  .1.  ->  (
x `  A )  =  ( x `  .1.  ) )
4 sumdchr.g . . . . . . . . 9  |-  G  =  (DChr `  N )
5 sumdchr2.z . . . . . . . . 9  |-  Z  =  (ℤ/n `  N )
6 sumdchr.d . . . . . . . . 9  |-  D  =  ( Base `  G
)
74, 5, 6dchrmhm 20533 . . . . . . . 8  |-  D  C_  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) )
8 simpr 447 . . . . . . . 8  |-  ( (
ph  /\  x  e.  D )  ->  x  e.  D )
97, 8sseldi 3212 . . . . . . 7  |-  ( (
ph  /\  x  e.  D )  ->  x  e.  ( (mulGrp `  Z
) MndHom  (mulGrp ` fld ) ) )
10 eqid 2316 . . . . . . . . 9  |-  (mulGrp `  Z )  =  (mulGrp `  Z )
11 sumdchr2.1 . . . . . . . . 9  |-  .1.  =  ( 1r `  Z )
1210, 11rngidval 15392 . . . . . . . 8  |-  .1.  =  ( 0g `  (mulGrp `  Z ) )
13 eqid 2316 . . . . . . . . 9  |-  (mulGrp ` fld )  =  (mulGrp ` fld )
14 cnfld1 16455 . . . . . . . . 9  |-  1  =  ( 1r ` fld )
1513, 14rngidval 15392 . . . . . . . 8  |-  1  =  ( 0g `  (mulGrp ` fld ) )
1612, 15mhm0 14472 . . . . . . 7  |-  ( x  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) )  ->  (
x `  .1.  )  =  1 )
179, 16syl 15 . . . . . 6  |-  ( (
ph  /\  x  e.  D )  ->  (
x `  .1.  )  =  1 )
183, 17sylan9eqr 2370 . . . . 5  |-  ( ( ( ph  /\  x  e.  D )  /\  A  =  .1.  )  ->  (
x `  A )  =  1 )
1918an32s 779 . . . 4  |-  ( ( ( ph  /\  A  =  .1.  )  /\  x  e.  D )  ->  (
x `  A )  =  1 )
2019sumeq2dv 12223 . . 3  |-  ( (
ph  /\  A  =  .1.  )  ->  sum_ x  e.  D  ( x `  A )  =  sum_ x  e.  D  1 )
21 sumdchr2.n . . . . . . 7  |-  ( ph  ->  N  e.  NN )
224, 6dchrfi 20547 . . . . . . 7  |-  ( N  e.  NN  ->  D  e.  Fin )
2321, 22syl 15 . . . . . 6  |-  ( ph  ->  D  e.  Fin )
24 ax-1cn 8840 . . . . . 6  |-  1  e.  CC
25 fsumconst 12299 . . . . . 6  |-  ( ( D  e.  Fin  /\  1  e.  CC )  -> 
sum_ x  e.  D 
1  =  ( (
# `  D )  x.  1 ) )
2623, 24, 25sylancl 643 . . . . 5  |-  ( ph  -> 
sum_ x  e.  D 
1  =  ( (
# `  D )  x.  1 ) )
27 hashcl 11397 . . . . . . . 8  |-  ( D  e.  Fin  ->  ( # `
 D )  e. 
NN0 )
2821, 22, 273syl 18 . . . . . . 7  |-  ( ph  ->  ( # `  D
)  e.  NN0 )
2928nn0cnd 10067 . . . . . 6  |-  ( ph  ->  ( # `  D
)  e.  CC )
3029mulid1d 8897 . . . . 5  |-  ( ph  ->  ( ( # `  D
)  x.  1 )  =  ( # `  D
) )
3126, 30eqtrd 2348 . . . 4  |-  ( ph  -> 
sum_ x  e.  D 
1  =  ( # `  D ) )
3231adantr 451 . . 3  |-  ( (
ph  /\  A  =  .1.  )  ->  sum_ x  e.  D  1  =  ( # `  D ) )
3320, 32eqtrd 2348 . 2  |-  ( (
ph  /\  A  =  .1.  )  ->  sum_ x  e.  D  ( x `  A )  =  (
# `  D )
)
34 df-ne 2481 . . 3  |-  ( A  =/=  .1.  <->  -.  A  =  .1.  )
35 sumdchr2.b . . . . 5  |-  B  =  ( Base `  Z
)
3621adantr 451 . . . . 5  |-  ( (
ph  /\  A  =/=  .1.  )  ->  N  e.  NN )
37 simpr 447 . . . . 5  |-  ( (
ph  /\  A  =/=  .1.  )  ->  A  =/= 
.1.  )
38 sumdchr2.x . . . . . 6  |-  ( ph  ->  A  e.  B )
3938adantr 451 . . . . 5  |-  ( (
ph  /\  A  =/=  .1.  )  ->  A  e.  B )
404, 5, 6, 35, 11, 36, 37, 39dchrpt 20559 . . . 4  |-  ( (
ph  /\  A  =/=  .1.  )  ->  E. y  e.  D  ( y `  A )  =/=  1
)
41 oveq2 5908 . . . . . . . . . . . . . 14  |-  ( z  =  x  ->  (
y ( +g  `  G
) z )  =  ( y ( +g  `  G ) x ) )
4241fveq1d 5565 . . . . . . . . . . . . 13  |-  ( z  =  x  ->  (
( y ( +g  `  G ) z ) `
 A )  =  ( ( y ( +g  `  G ) x ) `  A
) )
4342cbvsumv 12216 . . . . . . . . . . . 12  |-  sum_ z  e.  D  ( (
y ( +g  `  G
) z ) `  A )  =  sum_ x  e.  D  ( ( y ( +g  `  G
) x ) `  A )
44 eqid 2316 . . . . . . . . . . . . . . . 16  |-  ( +g  `  G )  =  ( +g  `  G )
45 simprl 732 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  A  =/=  .1.  )  /\  (
y  e.  D  /\  ( y `  A
)  =/=  1 ) )  ->  y  e.  D )
4645adantr 451 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  A  =/=  .1.  )  /\  ( y  e.  D  /\  ( y `  A
)  =/=  1 ) )  /\  x  e.  D )  ->  y  e.  D )
47 simpr 447 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  A  =/=  .1.  )  /\  ( y  e.  D  /\  ( y `  A
)  =/=  1 ) )  /\  x  e.  D )  ->  x  e.  D )
484, 5, 6, 44, 46, 47dchrmul 20540 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  A  =/=  .1.  )  /\  ( y  e.  D  /\  ( y `  A
)  =/=  1 ) )  /\  x  e.  D )  ->  (
y ( +g  `  G
) x )  =  ( y  o F  x.  x ) )
4948fveq1d 5565 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  A  =/=  .1.  )  /\  ( y  e.  D  /\  ( y `  A
)  =/=  1 ) )  /\  x  e.  D )  ->  (
( y ( +g  `  G ) x ) `
 A )  =  ( ( y  o F  x.  x ) `
 A ) )
504, 5, 6, 35, 45dchrf 20534 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  A  =/=  .1.  )  /\  (
y  e.  D  /\  ( y `  A
)  =/=  1 ) )  ->  y : B
--> CC )
5150adantr 451 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  A  =/=  .1.  )  /\  ( y  e.  D  /\  ( y `  A
)  =/=  1 ) )  /\  x  e.  D )  ->  y : B --> CC )
52 ffn 5427 . . . . . . . . . . . . . . . 16  |-  ( y : B --> CC  ->  y  Fn  B )
5351, 52syl 15 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  A  =/=  .1.  )  /\  ( y  e.  D  /\  ( y `  A
)  =/=  1 ) )  /\  x  e.  D )  ->  y  Fn  B )
544, 5, 6, 35, 47dchrf 20534 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  A  =/=  .1.  )  /\  ( y  e.  D  /\  ( y `  A
)  =/=  1 ) )  /\  x  e.  D )  ->  x : B --> CC )
55 ffn 5427 . . . . . . . . . . . . . . . 16  |-  ( x : B --> CC  ->  x  Fn  B )
5654, 55syl 15 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  A  =/=  .1.  )  /\  ( y  e.  D  /\  ( y `  A
)  =/=  1 ) )  /\  x  e.  D )  ->  x  Fn  B )
57 fvex 5577 . . . . . . . . . . . . . . . . 17  |-  ( Base `  Z )  e.  _V
5835, 57eqeltri 2386 . . . . . . . . . . . . . . . 16  |-  B  e. 
_V
5958a1i 10 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  A  =/=  .1.  )  /\  ( y  e.  D  /\  ( y `  A
)  =/=  1 ) )  /\  x  e.  D )  ->  B  e.  _V )
6039adantr 451 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  A  =/=  .1.  )  /\  (
y  e.  D  /\  ( y `  A
)  =/=  1 ) )  ->  A  e.  B )
6160adantr 451 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  A  =/=  .1.  )  /\  ( y  e.  D  /\  ( y `  A
)  =/=  1 ) )  /\  x  e.  D )  ->  A  e.  B )
62 fnfvof 6132 . . . . . . . . . . . . . . 15  |-  ( ( ( y  Fn  B  /\  x  Fn  B
)  /\  ( B  e.  _V  /\  A  e.  B ) )  -> 
( ( y  o F  x.  x ) `
 A )  =  ( ( y `  A )  x.  (
x `  A )
) )
6353, 56, 59, 61, 62syl22anc 1183 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  A  =/=  .1.  )  /\  ( y  e.  D  /\  ( y `  A
)  =/=  1 ) )  /\  x  e.  D )  ->  (
( y  o F  x.  x ) `  A )  =  ( ( y `  A
)  x.  ( x `
 A ) ) )
6449, 63eqtrd 2348 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  A  =/=  .1.  )  /\  ( y  e.  D  /\  ( y `  A
)  =/=  1 ) )  /\  x  e.  D )  ->  (
( y ( +g  `  G ) x ) `
 A )  =  ( ( y `  A )  x.  (
x `  A )
) )
6564sumeq2dv 12223 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  A  =/=  .1.  )  /\  (
y  e.  D  /\  ( y `  A
)  =/=  1 ) )  ->  sum_ x  e.  D  ( ( y ( +g  `  G
) x ) `  A )  =  sum_ x  e.  D  ( ( y `  A )  x.  ( x `  A ) ) )
6643, 65syl5eq 2360 . . . . . . . . . . 11  |-  ( ( ( ph  /\  A  =/=  .1.  )  /\  (
y  e.  D  /\  ( y `  A
)  =/=  1 ) )  ->  sum_ z  e.  D  ( ( y ( +g  `  G
) z ) `  A )  =  sum_ x  e.  D  ( ( y `  A )  x.  ( x `  A ) ) )
67 fveq1 5562 . . . . . . . . . . . 12  |-  ( x  =  ( y ( +g  `  G ) z )  ->  (
x `  A )  =  ( ( y ( +g  `  G
) z ) `  A ) )
6836adantr 451 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  A  =/=  .1.  )  /\  (
y  e.  D  /\  ( y `  A
)  =/=  1 ) )  ->  N  e.  NN )
6968, 22syl 15 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  A  =/=  .1.  )  /\  (
y  e.  D  /\  ( y `  A
)  =/=  1 ) )  ->  D  e.  Fin )
704dchrabl 20546 . . . . . . . . . . . . . 14  |-  ( N  e.  NN  ->  G  e.  Abel )
71 ablgrp 15143 . . . . . . . . . . . . . 14  |-  ( G  e.  Abel  ->  G  e. 
Grp )
7268, 70, 713syl 18 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  A  =/=  .1.  )  /\  (
y  e.  D  /\  ( y `  A
)  =/=  1 ) )  ->  G  e.  Grp )
73 eqid 2316 . . . . . . . . . . . . . 14  |-  ( a  e.  D  |->  ( b  e.  D  |->  ( a ( +g  `  G
) b ) ) )  =  ( a  e.  D  |->  ( b  e.  D  |->  ( a ( +g  `  G
) b ) ) )
7473, 6, 44grplactf1o 14614 . . . . . . . . . . . . 13  |-  ( ( G  e.  Grp  /\  y  e.  D )  ->  ( ( a  e.  D  |->  ( b  e.  D  |->  ( a ( +g  `  G ) b ) ) ) `
 y ) : D -1-1-onto-> D )
7572, 45, 74syl2anc 642 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  A  =/=  .1.  )  /\  (
y  e.  D  /\  ( y `  A
)  =/=  1 ) )  ->  ( (
a  e.  D  |->  ( b  e.  D  |->  ( a ( +g  `  G
) b ) ) ) `  y ) : D -1-1-onto-> D )
7673, 6grplactval 14612 . . . . . . . . . . . . 13  |-  ( ( y  e.  D  /\  z  e.  D )  ->  ( ( ( a  e.  D  |->  ( b  e.  D  |->  ( a ( +g  `  G
) b ) ) ) `  y ) `
 z )  =  ( y ( +g  `  G ) z ) )
7745, 76sylan 457 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  A  =/=  .1.  )  /\  ( y  e.  D  /\  ( y `  A
)  =/=  1 ) )  /\  z  e.  D )  ->  (
( ( a  e.  D  |->  ( b  e.  D  |->  ( a ( +g  `  G ) b ) ) ) `
 y ) `  z )  =  ( y ( +g  `  G
) z ) )
78 ffvelrn 5701 . . . . . . . . . . . . 13  |-  ( ( x : B --> CC  /\  A  e.  B )  ->  ( x `  A
)  e.  CC )
7954, 61, 78syl2anc 642 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  A  =/=  .1.  )  /\  ( y  e.  D  /\  ( y `  A
)  =/=  1 ) )  /\  x  e.  D )  ->  (
x `  A )  e.  CC )
8067, 69, 75, 77, 79fsumf1o 12243 . . . . . . . . . . 11  |-  ( ( ( ph  /\  A  =/=  .1.  )  /\  (
y  e.  D  /\  ( y `  A
)  =/=  1 ) )  ->  sum_ x  e.  D  ( x `  A )  =  sum_ z  e.  D  (
( y ( +g  `  G ) z ) `
 A ) )
81 ffvelrn 5701 . . . . . . . . . . . . 13  |-  ( ( y : B --> CC  /\  A  e.  B )  ->  ( y `  A
)  e.  CC )
8250, 60, 81syl2anc 642 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  A  =/=  .1.  )  /\  (
y  e.  D  /\  ( y `  A
)  =/=  1 ) )  ->  ( y `  A )  e.  CC )
8369, 82, 79fsummulc2 12293 . . . . . . . . . . 11  |-  ( ( ( ph  /\  A  =/=  .1.  )  /\  (
y  e.  D  /\  ( y `  A
)  =/=  1 ) )  ->  ( (
y `  A )  x.  sum_ x  e.  D  ( x `  A
) )  =  sum_ x  e.  D  ( ( y `  A )  x.  ( x `  A ) ) )
8466, 80, 833eqtr4rd 2359 . . . . . . . . . 10  |-  ( ( ( ph  /\  A  =/=  .1.  )  /\  (
y  e.  D  /\  ( y `  A
)  =/=  1 ) )  ->  ( (
y `  A )  x.  sum_ x  e.  D  ( x `  A
) )  =  sum_ x  e.  D  ( x `
 A ) )
8569, 79fsumcl 12253 . . . . . . . . . . 11  |-  ( ( ( ph  /\  A  =/=  .1.  )  /\  (
y  e.  D  /\  ( y `  A
)  =/=  1 ) )  ->  sum_ x  e.  D  ( x `  A )  e.  CC )
8685mulid2d 8898 . . . . . . . . . 10  |-  ( ( ( ph  /\  A  =/=  .1.  )  /\  (
y  e.  D  /\  ( y `  A
)  =/=  1 ) )  ->  ( 1  x.  sum_ x  e.  D  ( x `  A
) )  =  sum_ x  e.  D  ( x `
 A ) )
8784, 86oveq12d 5918 . . . . . . . . 9  |-  ( ( ( ph  /\  A  =/=  .1.  )  /\  (
y  e.  D  /\  ( y `  A
)  =/=  1 ) )  ->  ( (
( y `  A
)  x.  sum_ x  e.  D  ( x `  A ) )  -  ( 1  x.  sum_ x  e.  D  ( x `
 A ) ) )  =  ( sum_ x  e.  D  ( x `
 A )  -  sum_ x  e.  D  ( x `  A ) ) )
8885subidd 9190 . . . . . . . . 9  |-  ( ( ( ph  /\  A  =/=  .1.  )  /\  (
y  e.  D  /\  ( y `  A
)  =/=  1 ) )  ->  ( sum_ x  e.  D  ( x `
 A )  -  sum_ x  e.  D  ( x `  A ) )  =  0 )
8987, 88eqtrd 2348 . . . . . . . 8  |-  ( ( ( ph  /\  A  =/=  .1.  )  /\  (
y  e.  D  /\  ( y `  A
)  =/=  1 ) )  ->  ( (
( y `  A
)  x.  sum_ x  e.  D  ( x `  A ) )  -  ( 1  x.  sum_ x  e.  D  ( x `
 A ) ) )  =  0 )
9024a1i 10 . . . . . . . . 9  |-  ( ( ( ph  /\  A  =/=  .1.  )  /\  (
y  e.  D  /\  ( y `  A
)  =/=  1 ) )  ->  1  e.  CC )
9182, 90, 85subdird 9281 . . . . . . . 8  |-  ( ( ( ph  /\  A  =/=  .1.  )  /\  (
y  e.  D  /\  ( y `  A
)  =/=  1 ) )  ->  ( (
( y `  A
)  -  1 )  x.  sum_ x  e.  D  ( x `  A
) )  =  ( ( ( y `  A )  x.  sum_ x  e.  D  ( x `
 A ) )  -  ( 1  x. 
sum_ x  e.  D  ( x `  A
) ) ) )
92 subcl 9096 . . . . . . . . . 10  |-  ( ( ( y `  A
)  e.  CC  /\  1  e.  CC )  ->  ( ( y `  A )  -  1 )  e.  CC )
9382, 24, 92sylancl 643 . . . . . . . . 9  |-  ( ( ( ph  /\  A  =/=  .1.  )  /\  (
y  e.  D  /\  ( y `  A
)  =/=  1 ) )  ->  ( (
y `  A )  -  1 )  e.  CC )
9493mul01d 9056 . . . . . . . 8  |-  ( ( ( ph  /\  A  =/=  .1.  )  /\  (
y  e.  D  /\  ( y `  A
)  =/=  1 ) )  ->  ( (
( y `  A
)  -  1 )  x.  0 )  =  0 )
9589, 91, 943eqtr4d 2358 . . . . . . 7  |-  ( ( ( ph  /\  A  =/=  .1.  )  /\  (
y  e.  D  /\  ( y `  A
)  =/=  1 ) )  ->  ( (
( y `  A
)  -  1 )  x.  sum_ x  e.  D  ( x `  A
) )  =  ( ( ( y `  A )  -  1 )  x.  0 ) )
96 0cn 8876 . . . . . . . . 9  |-  0  e.  CC
9796a1i 10 . . . . . . . 8  |-  ( ( ( ph  /\  A  =/=  .1.  )  /\  (
y  e.  D  /\  ( y `  A
)  =/=  1 ) )  ->  0  e.  CC )
98 simprr 733 . . . . . . . . 9  |-  ( ( ( ph  /\  A  =/=  .1.  )  /\  (
y  e.  D  /\  ( y `  A
)  =/=  1 ) )  ->  ( y `  A )  =/=  1
)
99 subeq0 9118 . . . . . . . . . . 11  |-  ( ( ( y `  A
)  e.  CC  /\  1  e.  CC )  ->  ( ( ( y `
 A )  - 
1 )  =  0  <-> 
( y `  A
)  =  1 ) )
10082, 24, 99sylancl 643 . . . . . . . . . 10  |-  ( ( ( ph  /\  A  =/=  .1.  )  /\  (
y  e.  D  /\  ( y `  A
)  =/=  1 ) )  ->  ( (
( y `  A
)  -  1 )  =  0  <->  ( y `  A )  =  1 ) )
101100necon3bid 2514 . . . . . . . . 9  |-  ( ( ( ph  /\  A  =/=  .1.  )  /\  (
y  e.  D  /\  ( y `  A
)  =/=  1 ) )  ->  ( (
( y `  A
)  -  1 )  =/=  0  <->  ( y `  A )  =/=  1
) )
10298, 101mpbird 223 . . . . . . . 8  |-  ( ( ( ph  /\  A  =/=  .1.  )  /\  (
y  e.  D  /\  ( y `  A
)  =/=  1 ) )  ->  ( (
y `  A )  -  1 )  =/=  0 )
10385, 97, 93, 102mulcand 9446 . . . . . . 7  |-  ( ( ( ph  /\  A  =/=  .1.  )  /\  (
y  e.  D  /\  ( y `  A
)  =/=  1 ) )  ->  ( (
( ( y `  A )  -  1 )  x.  sum_ x  e.  D  ( x `  A ) )  =  ( ( ( y `
 A )  - 
1 )  x.  0 )  <->  sum_ x  e.  D  ( x `  A
)  =  0 ) )
10495, 103mpbid 201 . . . . . 6  |-  ( ( ( ph  /\  A  =/=  .1.  )  /\  (
y  e.  D  /\  ( y `  A
)  =/=  1 ) )  ->  sum_ x  e.  D  ( x `  A )  =  0 )
105104expr 598 . . . . 5  |-  ( ( ( ph  /\  A  =/=  .1.  )  /\  y  e.  D )  ->  (
( y `  A
)  =/=  1  ->  sum_ x  e.  D  ( x `  A )  =  0 ) )
106105rexlimdva 2701 . . . 4  |-  ( (
ph  /\  A  =/=  .1.  )  ->  ( E. y  e.  D  ( y `  A )  =/=  1  ->  sum_ x  e.  D  ( x `  A )  =  0 ) )
10740, 106mpd 14 . . 3  |-  ( (
ph  /\  A  =/=  .1.  )  ->  sum_ x  e.  D  ( x `  A )  =  0 )
10834, 107sylan2br 462 . 2  |-  ( (
ph  /\  -.  A  =  .1.  )  ->  sum_ x  e.  D  ( x `  A )  =  0 )
1091, 2, 33, 108ifbothda 3629 1  |-  ( ph  -> 
sum_ x  e.  D  ( x `  A
)  =  if ( A  =  .1.  , 
( # `  D ) ,  0 ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1633    e. wcel 1701    =/= wne 2479   E.wrex 2578   _Vcvv 2822   ifcif 3599    e. cmpt 4114    Fn wfn 5287   -->wf 5288   -1-1-onto->wf1o 5291   ` cfv 5292  (class class class)co 5900    o Fcof 6118   Fincfn 6906   CCcc 8780   0cc0 8782   1c1 8783    x. cmul 8787    - cmin 9082   NNcn 9791   NN0cn0 10012   #chash 11384   sum_csu 12205   Basecbs 13195   +g cplusg 13255   Grpcgrp 14411   MndHom cmhm 14462   Abelcabel 15139  mulGrpcmgp 15374   1rcur 15388  ℂfldccnfld 16432  ℤ/nczn 16510  DChrcdchr 20524
This theorem is referenced by:  dchrhash  20563  sumdchr  20564
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-inf2 7387  ax-cnex 8838  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859  ax-pre-sup 8860  ax-addf 8861  ax-mulf 8862
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-int 3900  df-iun 3944  df-iin 3945  df-disj 4031  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-se 4390  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-isom 5301  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-of 6120  df-1st 6164  df-2nd 6165  df-tpos 6276  df-rpss 6319  df-riota 6346  df-recs 6430  df-rdg 6465  df-1o 6521  df-2o 6522  df-oadd 6525  df-omul 6526  df-er 6702  df-ec 6704  df-qs 6708  df-map 6817  df-pm 6818  df-ixp 6861  df-en 6907  df-dom 6908  df-sdom 6909  df-fin 6910  df-fi 7210  df-sup 7239  df-oi 7270  df-card 7617  df-acn 7620  df-cda 7839  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085  df-div 9469  df-nn 9792  df-2 9849  df-3 9850  df-4 9851  df-5 9852  df-6 9853  df-7 9854  df-8 9855  df-9 9856  df-10 9857  df-n0 10013  df-z 10072  df-dec 10172  df-uz 10278  df-q 10364  df-rp 10402  df-xneg 10499  df-xadd 10500  df-xmul 10501  df-ioo 10707  df-ioc 10708  df-ico 10709  df-icc 10710  df-fz 10830  df-fzo 10918  df-fl 10972  df-mod 11021  df-seq 11094  df-exp 11152  df-fac 11336  df-bc 11363  df-hash 11385  df-word 11456  df-concat 11457  df-s1 11458  df-shft 11609  df-cj 11631  df-re 11632  df-im 11633  df-sqr 11767  df-abs 11768  df-limsup 11992  df-clim 12009  df-rlim 12010  df-sum 12206  df-ef 12396  df-sin 12398  df-cos 12399  df-pi 12401  df-dvds 12579  df-gcd 12733  df-prm 12806  df-phi 12881  df-pc 12937  df-struct 13197  df-ndx 13198  df-slot 13199  df-base 13200  df-sets 13201  df-ress 13202  df-plusg 13268  df-mulr 13269  df-starv 13270  df-sca 13271  df-vsca 13272  df-tset 13274  df-ple 13275  df-ds 13277  df-unif 13278  df-hom 13279  df-cco 13280  df-rest 13376  df-topn 13377  df-topgen 13393  df-pt 13394  df-prds 13397  df-xrs 13452  df-0g 13453  df-gsum 13454  df-qtop 13459  df-imas 13460  df-divs 13461  df-xps 13462  df-mre 13537  df-mrc 13538  df-acs 13540  df-mnd 14416  df-mhm 14464  df-submnd 14465  df-grp 14538  df-minusg 14539  df-sbg 14540  df-mulg 14541  df-subg 14667  df-nsg 14668  df-eqg 14669  df-ghm 14730  df-gim 14772  df-ga 14793  df-cntz 14842  df-oppg 14868  df-od 14893  df-gex 14894  df-pgp 14895  df-lsm 14996  df-pj1 14997  df-cmn 15140  df-abl 15141  df-cyg 15214  df-dprd 15282  df-dpj 15283  df-mgp 15375  df-rng 15389  df-cring 15390  df-ur 15391  df-oppr 15454  df-dvdsr 15472  df-unit 15473  df-invr 15503  df-rnghom 15545  df-subrg 15592  df-lmod 15678  df-lss 15739  df-lsp 15778  df-sra 15974  df-rgmod 15975  df-lidl 15976  df-rsp 15977  df-2idl 16033  df-xmet 16425  df-met 16426  df-bl 16427  df-mopn 16428  df-fbas 16429  df-fg 16430  df-cnfld 16433  df-zrh 16511  df-zn 16514  df-top 16692  df-bases 16694  df-topon 16695  df-topsp 16696  df-cld 16812  df-ntr 16813  df-cls 16814  df-nei 16891  df-lp 16924  df-perf 16925  df-cn 17013  df-cnp 17014  df-haus 17099  df-tx 17313  df-hmeo 17502  df-fil 17593  df-fm 17685  df-flim 17686  df-flf 17687  df-xms 17937  df-ms 17938  df-tms 17939  df-cncf 18434  df-0p 19078  df-limc 19269  df-dv 19270  df-ply 19623  df-idp 19624  df-coe 19625  df-dgr 19626  df-quot 19724  df-log 19967  df-cxp 19968  df-dchr 20525
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