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Theorem sumdchr2 20509
Description: Lemma for sumdchr 20511. (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 2292 . 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 2292 . 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 5525 . . . . . 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 20480 . . . . . . . 8  |-  D  C_  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) )
8 simpr 447 . . . . . . . 8  |-  ( (
ph  /\  x  e.  D )  ->  x  e.  D )
97, 8sseldi 3178 . . . . . . 7  |-  ( (
ph  /\  x  e.  D )  ->  x  e.  ( (mulGrp `  Z
) MndHom  (mulGrp ` fld ) ) )
10 eqid 2283 . . . . . . . . 9  |-  (mulGrp `  Z )  =  (mulGrp `  Z )
11 sumdchr2.1 . . . . . . . . 9  |-  .1.  =  ( 1r `  Z )
1210, 11rngidval 15343 . . . . . . . 8  |-  .1.  =  ( 0g `  (mulGrp `  Z ) )
13 eqid 2283 . . . . . . . . 9  |-  (mulGrp ` fld )  =  (mulGrp ` fld )
14 cnfld1 16399 . . . . . . . . 9  |-  1  =  ( 1r ` fld )
1513, 14rngidval 15343 . . . . . . . 8  |-  1  =  ( 0g `  (mulGrp ` fld ) )
1612, 15mhm0 14423 . . . . . . 7  |-  ( x  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) )  ->  (
x `  .1.  )  =  1 )
179, 16syl 15 . . . . . 6  |-  ( (
ph  /\  x  e.  D )  ->  (
x `  .1.  )  =  1 )
183, 17sylan9eqr 2337 . . . . 5  |-  ( ( ( ph  /\  x  e.  D )  /\  A  =  .1.  )  ->  (
x `  A )  =  1 )
1918an32s 779 . . . 4  |-  ( ( ( ph  /\  A  =  .1.  )  /\  x  e.  D )  ->  (
x `  A )  =  1 )
2019sumeq2dv 12176 . . 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 20494 . . . . . . 7  |-  ( N  e.  NN  ->  D  e.  Fin )
2321, 22syl 15 . . . . . 6  |-  ( ph  ->  D  e.  Fin )
24 ax-1cn 8795 . . . . . 6  |-  1  e.  CC
25 fsumconst 12252 . . . . . 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 11350 . . . . . . . 8  |-  ( D  e.  Fin  ->  ( # `
 D )  e. 
NN0 )
2821, 22, 273syl 18 . . . . . . 7  |-  ( ph  ->  ( # `  D
)  e.  NN0 )
2928nn0cnd 10020 . . . . . 6  |-  ( ph  ->  ( # `  D
)  e.  CC )
3029mulid1d 8852 . . . . 5  |-  ( ph  ->  ( ( # `  D
)  x.  1 )  =  ( # `  D
) )
3126, 30eqtrd 2315 . . . 4  |-  ( ph  -> 
sum_ x  e.  D 
1  =  ( # `  D ) )
3231adantr 451 . . 3  |-  ( (
ph  /\  A  =  .1.  )  ->  sum_ x  e.  D  1  =  ( # `  D ) )
3320, 32eqtrd 2315 . 2  |-  ( (
ph  /\  A  =  .1.  )  ->  sum_ x  e.  D  ( x `  A )  =  (
# `  D )
)
34 df-ne 2448 . . 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 20506 . . . 4  |-  ( (
ph  /\  A  =/=  .1.  )  ->  E. y  e.  D  ( y `  A )  =/=  1
)
41 oveq2 5866 . . . . . . . . . . . . . 14  |-  ( z  =  x  ->  (
y ( +g  `  G
) z )  =  ( y ( +g  `  G ) x ) )
4241fveq1d 5527 . . . . . . . . . . . . 13  |-  ( z  =  x  ->  (
( y ( +g  `  G ) z ) `
 A )  =  ( ( y ( +g  `  G ) x ) `  A
) )
4342cbvsumv 12169 . . . . . . . . . . . 12  |-  sum_ z  e.  D  ( (
y ( +g  `  G
) z ) `  A )  =  sum_ x  e.  D  ( ( y ( +g  `  G
) x ) `  A )
44 eqid 2283 . . . . . . . . . . . . . . . 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 20487 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  A  =/=  .1.  )  /\  ( y  e.  D  /\  ( y `  A
)  =/=  1 ) )  /\  x  e.  D )  ->  (
y ( +g  `  G
) x )  =  ( y  o F  x.  x ) )
4948fveq1d 5527 . . . . . . . . . . . . . 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 20481 . . . . . . . . . . . . . . . . 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 5389 . . . . . . . . . . . . . . . 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 20481 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  A  =/=  .1.  )  /\  ( y  e.  D  /\  ( y `  A
)  =/=  1 ) )  /\  x  e.  D )  ->  x : B --> CC )
55 ffn 5389 . . . . . . . . . . . . . . . 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 5539 . . . . . . . . . . . . . . . . 17  |-  ( Base `  Z )  e.  _V
5835, 57eqeltri 2353 . . . . . . . . . . . . . . . 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 6090 . . . . . . . . . . . . . . 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 2315 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  A  =/=  .1.  )  /\  ( y  e.  D  /\  ( y `  A
)  =/=  1 ) )  /\  x  e.  D )  ->  (
( y ( +g  `  G ) x ) `
 A )  =  ( ( y `  A )  x.  (
x `  A )
) )
6564sumeq2dv 12176 . . . . . . . . . . . 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 2327 . . . . . . . . . . 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 5524 . . . . . . . . . . . 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 20493 . . . . . . . . . . . . . 14  |-  ( N  e.  NN  ->  G  e.  Abel )
71 ablgrp 15094 . . . . . . . . . . . . . 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 2283 . . . . . . . . . . . . . 14  |-  ( a  e.  D  |->  ( b  e.  D  |->  ( a ( +g  `  G
) b ) ) )  =  ( a  e.  D  |->  ( b  e.  D  |->  ( a ( +g  `  G
) b ) ) )
7473, 6, 44grplactf1o 14565 . . . . . . . . . . . . 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 14563 . . . . . . . . . . . . 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 5663 . . . . . . . . . . . . 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 12196 . . . . . . . . . . 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 5663 . . . . . . . . . . . . 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 12246 . . . . . . . . . . 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 2326 . . . . . . . . . 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 12206 . . . . . . . . . . 11  |-  ( ( ( ph  /\  A  =/=  .1.  )  /\  (
y  e.  D  /\  ( y `  A
)  =/=  1 ) )  ->  sum_ x  e.  D  ( x `  A )  e.  CC )
8685mulid2d 8853 . . . . . . . . . 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 5876 . . . . . . . . 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 9145 . . . . . . . . 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 2315 . . . . . . . 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 9236 . . . . . . . 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 9051 . . . . . . . . . 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 9011 . . . . . . . 8  |-  ( ( ( ph  /\  A  =/=  .1.  )  /\  (
y  e.  D  /\  ( y `  A
)  =/=  1 ) )  ->  ( (
( y `  A
)  -  1 )  x.  0 )  =  0 )
9589, 91, 943eqtr4d 2325 . . . . . . 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 8831 . . . . . . . . 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 9073 . . . . . . . . . . 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 2481 . . . . . . . . 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 9401 . . . . . . 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 2667 . . . 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 3595 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 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544   _Vcvv 2788   ifcif 3565    e. cmpt 4077    Fn wfn 5250   -->wf 5251   -1-1-onto->wf1o 5254   ` cfv 5255  (class class class)co 5858    o Fcof 6076   Fincfn 6863   CCcc 8735   0cc0 8737   1c1 8738    x. cmul 8742    - cmin 9037   NNcn 9746   NN0cn0 9965   #chash 11337   sum_csu 12158   Basecbs 13148   +g cplusg 13208   Grpcgrp 14362   MndHom cmhm 14413   Abelcabel 15090  mulGrpcmgp 15325   1rcur 15339  ℂfldccnfld 16377  ℤ/nczn 16454  DChrcdchr 20471
This theorem is referenced by:  dchrhash  20510  sumdchr  20511
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-disj 3994  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-tpos 6234  df-rpss 6277  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-omul 6484  df-er 6660  df-ec 6662  df-qs 6666  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-acn 7575  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-ioc 10661  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-fac 11289  df-bc 11316  df-hash 11338  df-word 11409  df-concat 11410  df-s1 11411  df-shft 11562  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-limsup 11945  df-clim 11962  df-rlim 11963  df-sum 12159  df-ef 12349  df-sin 12351  df-cos 12352  df-pi 12354  df-dvds 12532  df-gcd 12686  df-prm 12759  df-phi 12834  df-pc 12890  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-rest 13327  df-topn 13328  df-topgen 13344  df-pt 13345  df-prds 13348  df-xrs 13403  df-0g 13404  df-gsum 13405  df-qtop 13410  df-imas 13411  df-divs 13412  df-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-mhm 14415  df-submnd 14416  df-grp 14489  df-minusg 14490  df-sbg 14491  df-mulg 14492  df-subg 14618  df-nsg 14619  df-eqg 14620  df-ghm 14681  df-gim 14723  df-ga 14744  df-cntz 14793  df-oppg 14819  df-od 14844  df-gex 14845  df-pgp 14846  df-lsm 14947  df-pj1 14948  df-cmn 15091  df-abl 15092  df-cyg 15165  df-dprd 15233  df-dpj 15234  df-mgp 15326  df-rng 15340  df-cring 15341  df-ur 15342  df-oppr 15405  df-dvdsr 15423  df-unit 15424  df-invr 15454  df-rnghom 15496  df-subrg 15543  df-lmod 15629  df-lss 15690  df-lsp 15729  df-sra 15925  df-rgmod 15926  df-lidl 15927  df-rsp 15928  df-2idl 15984  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-zrh 16455  df-zn 16458  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cld 16756  df-ntr 16757  df-cls 16758  df-nei 16835  df-lp 16868  df-perf 16869  df-cn 16957  df-cnp 16958  df-haus 17043  df-tx 17257  df-hmeo 17446  df-fbas 17520  df-fg 17521  df-fil 17541  df-fm 17633  df-flim 17634  df-flf 17635  df-xms 17885  df-ms 17886  df-tms 17887  df-cncf 18382  df-0p 19025  df-limc 19216  df-dv 19217  df-ply 19570  df-idp 19571  df-coe 19572  df-dgr 19573  df-quot 19671  df-log 19914  df-cxp 19915  df-dchr 20472
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