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Theorem sumdchr2 24277
Description: Lemma for sumdchr 24279. (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 2482 . 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 2482 . 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 5879 . . . . . 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 24248 . . . . . . . 8  |-  D  C_  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) )
8 simpr 468 . . . . . . . 8  |-  ( (
ph  /\  x  e.  D )  ->  x  e.  D )
97, 8sseldi 3416 . . . . . . 7  |-  ( (
ph  /\  x  e.  D )  ->  x  e.  ( (mulGrp `  Z
) MndHom  (mulGrp ` fld ) ) )
10 eqid 2471 . . . . . . . . 9  |-  (mulGrp `  Z )  =  (mulGrp `  Z )
11 sumdchr2.1 . . . . . . . . 9  |-  .1.  =  ( 1r `  Z )
1210, 11ringidval 17815 . . . . . . . 8  |-  .1.  =  ( 0g `  (mulGrp `  Z ) )
13 eqid 2471 . . . . . . . . 9  |-  (mulGrp ` fld )  =  (mulGrp ` fld )
14 cnfld1 19070 . . . . . . . . 9  |-  1  =  ( 1r ` fld )
1513, 14ringidval 17815 . . . . . . . 8  |-  1  =  ( 0g `  (mulGrp ` fld ) )
1612, 15mhm0 16668 . . . . . . 7  |-  ( x  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) )  ->  (
x `  .1.  )  =  1 )
179, 16syl 17 . . . . . 6  |-  ( (
ph  /\  x  e.  D )  ->  (
x `  .1.  )  =  1 )
183, 17sylan9eqr 2527 . . . . 5  |-  ( ( ( ph  /\  x  e.  D )  /\  A  =  .1.  )  ->  (
x `  A )  =  1 )
1918an32s 821 . . . 4  |-  ( ( ( ph  /\  A  =  .1.  )  /\  x  e.  D )  ->  (
x `  A )  =  1 )
2019sumeq2dv 13846 . . 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 24262 . . . . . . 7  |-  ( N  e.  NN  ->  D  e.  Fin )
2321, 22syl 17 . . . . . 6  |-  ( ph  ->  D  e.  Fin )
24 ax-1cn 9615 . . . . . 6  |-  1  e.  CC
25 fsumconst 13928 . . . . . 6  |-  ( ( D  e.  Fin  /\  1  e.  CC )  -> 
sum_ x  e.  D 
1  =  ( (
# `  D )  x.  1 ) )
2623, 24, 25sylancl 675 . . . . 5  |-  ( ph  -> 
sum_ x  e.  D 
1  =  ( (
# `  D )  x.  1 ) )
27 hashcl 12576 . . . . . . . 8  |-  ( D  e.  Fin  ->  ( # `
 D )  e. 
NN0 )
2821, 22, 273syl 18 . . . . . . 7  |-  ( ph  ->  ( # `  D
)  e.  NN0 )
2928nn0cnd 10951 . . . . . 6  |-  ( ph  ->  ( # `  D
)  e.  CC )
3029mulid1d 9678 . . . . 5  |-  ( ph  ->  ( ( # `  D
)  x.  1 )  =  ( # `  D
) )
3126, 30eqtrd 2505 . . . 4  |-  ( ph  -> 
sum_ x  e.  D 
1  =  ( # `  D ) )
3231adantr 472 . . 3  |-  ( (
ph  /\  A  =  .1.  )  ->  sum_ x  e.  D  1  =  ( # `  D ) )
3320, 32eqtrd 2505 . 2  |-  ( (
ph  /\  A  =  .1.  )  ->  sum_ x  e.  D  ( x `  A )  =  (
# `  D )
)
34 df-ne 2643 . . 3  |-  ( A  =/=  .1.  <->  -.  A  =  .1.  )
35 sumdchr2.b . . . . 5  |-  B  =  ( Base `  Z
)
3621adantr 472 . . . . 5  |-  ( (
ph  /\  A  =/=  .1.  )  ->  N  e.  NN )
37 simpr 468 . . . . 5  |-  ( (
ph  /\  A  =/=  .1.  )  ->  A  =/= 
.1.  )
38 sumdchr2.x . . . . . 6  |-  ( ph  ->  A  e.  B )
3938adantr 472 . . . . 5  |-  ( (
ph  /\  A  =/=  .1.  )  ->  A  e.  B )
404, 5, 6, 35, 11, 36, 37, 39dchrpt 24274 . . . 4  |-  ( (
ph  /\  A  =/=  .1.  )  ->  E. y  e.  D  ( y `  A )  =/=  1
)
4136adantr 472 . . . . . . 7  |-  ( ( ( ph  /\  A  =/=  .1.  )  /\  (
y  e.  D  /\  ( y `  A
)  =/=  1 ) )  ->  N  e.  NN )
4241, 22syl 17 . . . . . 6  |-  ( ( ( ph  /\  A  =/=  .1.  )  /\  (
y  e.  D  /\  ( y `  A
)  =/=  1 ) )  ->  D  e.  Fin )
43 simpr 468 . . . . . . . 8  |-  ( ( ( ( ph  /\  A  =/=  .1.  )  /\  ( y  e.  D  /\  ( y `  A
)  =/=  1 ) )  /\  x  e.  D )  ->  x  e.  D )
444, 5, 6, 35, 43dchrf 24249 . . . . . . 7  |-  ( ( ( ( ph  /\  A  =/=  .1.  )  /\  ( y  e.  D  /\  ( y `  A
)  =/=  1 ) )  /\  x  e.  D )  ->  x : B --> CC )
4539adantr 472 . . . . . . . 8  |-  ( ( ( ph  /\  A  =/=  .1.  )  /\  (
y  e.  D  /\  ( y `  A
)  =/=  1 ) )  ->  A  e.  B )
4645adantr 472 . . . . . . 7  |-  ( ( ( ( ph  /\  A  =/=  .1.  )  /\  ( y  e.  D  /\  ( y `  A
)  =/=  1 ) )  /\  x  e.  D )  ->  A  e.  B )
4744, 46ffvelrnd 6038 . . . . . 6  |-  ( ( ( ( ph  /\  A  =/=  .1.  )  /\  ( y  e.  D  /\  ( y `  A
)  =/=  1 ) )  /\  x  e.  D )  ->  (
x `  A )  e.  CC )
4842, 47fsumcl 13876 . . . . 5  |-  ( ( ( ph  /\  A  =/=  .1.  )  /\  (
y  e.  D  /\  ( y `  A
)  =/=  1 ) )  ->  sum_ x  e.  D  ( x `  A )  e.  CC )
49 0cnd 9654 . . . . 5  |-  ( ( ( ph  /\  A  =/=  .1.  )  /\  (
y  e.  D  /\  ( y `  A
)  =/=  1 ) )  ->  0  e.  CC )
50 simprl 772 . . . . . . . 8  |-  ( ( ( ph  /\  A  =/=  .1.  )  /\  (
y  e.  D  /\  ( y `  A
)  =/=  1 ) )  ->  y  e.  D )
514, 5, 6, 35, 50dchrf 24249 . . . . . . 7  |-  ( ( ( ph  /\  A  =/=  .1.  )  /\  (
y  e.  D  /\  ( y `  A
)  =/=  1 ) )  ->  y : B
--> CC )
5251, 45ffvelrnd 6038 . . . . . 6  |-  ( ( ( ph  /\  A  =/=  .1.  )  /\  (
y  e.  D  /\  ( y `  A
)  =/=  1 ) )  ->  ( y `  A )  e.  CC )
53 subcl 9894 . . . . . 6  |-  ( ( ( y `  A
)  e.  CC  /\  1  e.  CC )  ->  ( ( y `  A )  -  1 )  e.  CC )
5452, 24, 53sylancl 675 . . . . 5  |-  ( ( ( ph  /\  A  =/=  .1.  )  /\  (
y  e.  D  /\  ( y `  A
)  =/=  1 ) )  ->  ( (
y `  A )  -  1 )  e.  CC )
55 simprr 774 . . . . . 6  |-  ( ( ( ph  /\  A  =/=  .1.  )  /\  (
y  e.  D  /\  ( y `  A
)  =/=  1 ) )  ->  ( y `  A )  =/=  1
)
56 subeq0 9920 . . . . . . . 8  |-  ( ( ( y `  A
)  e.  CC  /\  1  e.  CC )  ->  ( ( ( y `
 A )  - 
1 )  =  0  <-> 
( y `  A
)  =  1 ) )
5752, 24, 56sylancl 675 . . . . . . 7  |-  ( ( ( ph  /\  A  =/=  .1.  )  /\  (
y  e.  D  /\  ( y `  A
)  =/=  1 ) )  ->  ( (
( y `  A
)  -  1 )  =  0  <->  ( y `  A )  =  1 ) )
5857necon3bid 2687 . . . . . 6  |-  ( ( ( ph  /\  A  =/=  .1.  )  /\  (
y  e.  D  /\  ( y `  A
)  =/=  1 ) )  ->  ( (
( y `  A
)  -  1 )  =/=  0  <->  ( y `  A )  =/=  1
) )
5955, 58mpbird 240 . . . . 5  |-  ( ( ( ph  /\  A  =/=  .1.  )  /\  (
y  e.  D  /\  ( y `  A
)  =/=  1 ) )  ->  ( (
y `  A )  -  1 )  =/=  0 )
60 oveq2 6316 . . . . . . . . . . . 12  |-  ( z  =  x  ->  (
y ( +g  `  G
) z )  =  ( y ( +g  `  G ) x ) )
6160fveq1d 5881 . . . . . . . . . . 11  |-  ( z  =  x  ->  (
( y ( +g  `  G ) z ) `
 A )  =  ( ( y ( +g  `  G ) x ) `  A
) )
6261cbvsumv 13839 . . . . . . . . . 10  |-  sum_ z  e.  D  ( (
y ( +g  `  G
) z ) `  A )  =  sum_ x  e.  D  ( ( y ( +g  `  G
) x ) `  A )
63 eqid 2471 . . . . . . . . . . . . . 14  |-  ( +g  `  G )  =  ( +g  `  G )
6450adantr 472 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  A  =/=  .1.  )  /\  ( y  e.  D  /\  ( y `  A
)  =/=  1 ) )  /\  x  e.  D )  ->  y  e.  D )
654, 5, 6, 63, 64, 43dchrmul 24255 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  A  =/=  .1.  )  /\  ( y  e.  D  /\  ( y `  A
)  =/=  1 ) )  /\  x  e.  D )  ->  (
y ( +g  `  G
) x )  =  ( y  oF  x.  x ) )
6665fveq1d 5881 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  A  =/=  .1.  )  /\  ( y  e.  D  /\  ( y `  A
)  =/=  1 ) )  /\  x  e.  D )  ->  (
( y ( +g  `  G ) x ) `
 A )  =  ( ( y  oF  x.  x ) `
 A ) )
6751adantr 472 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  A  =/=  .1.  )  /\  ( y  e.  D  /\  ( y `  A
)  =/=  1 ) )  /\  x  e.  D )  ->  y : B --> CC )
68 ffn 5739 . . . . . . . . . . . . . 14  |-  ( y : B --> CC  ->  y  Fn  B )
6967, 68syl 17 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  A  =/=  .1.  )  /\  ( y  e.  D  /\  ( y `  A
)  =/=  1 ) )  /\  x  e.  D )  ->  y  Fn  B )
70 ffn 5739 . . . . . . . . . . . . . 14  |-  ( x : B --> CC  ->  x  Fn  B )
7144, 70syl 17 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  A  =/=  .1.  )  /\  ( y  e.  D  /\  ( y `  A
)  =/=  1 ) )  /\  x  e.  D )  ->  x  Fn  B )
72 fvex 5889 . . . . . . . . . . . . . . 15  |-  ( Base `  Z )  e.  _V
7335, 72eqeltri 2545 . . . . . . . . . . . . . 14  |-  B  e. 
_V
7473a1i 11 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  A  =/=  .1.  )  /\  ( y  e.  D  /\  ( y `  A
)  =/=  1 ) )  /\  x  e.  D )  ->  B  e.  _V )
75 fnfvof 6564 . . . . . . . . . . . . 13  |-  ( ( ( y  Fn  B  /\  x  Fn  B
)  /\  ( B  e.  _V  /\  A  e.  B ) )  -> 
( ( y  oF  x.  x ) `
 A )  =  ( ( y `  A )  x.  (
x `  A )
) )
7669, 71, 74, 46, 75syl22anc 1293 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  A  =/=  .1.  )  /\  ( y  e.  D  /\  ( y `  A
)  =/=  1 ) )  /\  x  e.  D )  ->  (
( y  oF  x.  x ) `  A )  =  ( ( y `  A
)  x.  ( x `
 A ) ) )
7766, 76eqtrd 2505 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  A  =/=  .1.  )  /\  ( y  e.  D  /\  ( y `  A
)  =/=  1 ) )  /\  x  e.  D )  ->  (
( y ( +g  `  G ) x ) `
 A )  =  ( ( y `  A )  x.  (
x `  A )
) )
7877sumeq2dv 13846 . . . . . . . . . 10  |-  ( ( ( 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 ) ) )
7962, 78syl5eq 2517 . . . . . . . . 9  |-  ( ( ( 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 ) ) )
80 fveq1 5878 . . . . . . . . . 10  |-  ( x  =  ( y ( +g  `  G ) z )  ->  (
x `  A )  =  ( ( y ( +g  `  G
) z ) `  A ) )
814dchrabl 24261 . . . . . . . . . . . 12  |-  ( N  e.  NN  ->  G  e.  Abel )
82 ablgrp 17513 . . . . . . . . . . . 12  |-  ( G  e.  Abel  ->  G  e. 
Grp )
8341, 81, 823syl 18 . . . . . . . . . . 11  |-  ( ( ( ph  /\  A  =/=  .1.  )  /\  (
y  e.  D  /\  ( y `  A
)  =/=  1 ) )  ->  G  e.  Grp )
84 eqid 2471 . . . . . . . . . . . 12  |-  ( a  e.  D  |->  ( b  e.  D  |->  ( a ( +g  `  G
) b ) ) )  =  ( a  e.  D  |->  ( b  e.  D  |->  ( a ( +g  `  G
) b ) ) )
8584, 6, 63grplactf1o 16833 . . . . . . . . . . 11  |-  ( ( G  e.  Grp  /\  y  e.  D )  ->  ( ( a  e.  D  |->  ( b  e.  D  |->  ( a ( +g  `  G ) b ) ) ) `
 y ) : D -1-1-onto-> D )
8683, 50, 85syl2anc 673 . . . . . . . . . 10  |-  ( ( ( 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 )
8784, 6grplactval 16831 . . . . . . . . . . 11  |-  ( ( y  e.  D  /\  z  e.  D )  ->  ( ( ( a  e.  D  |->  ( b  e.  D  |->  ( a ( +g  `  G
) b ) ) ) `  y ) `
 z )  =  ( y ( +g  `  G ) z ) )
8850, 87sylan 479 . . . . . . . . . 10  |-  ( ( ( ( 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 ) )
8980, 42, 86, 88, 47fsumf1o 13866 . . . . . . . . 9  |-  ( ( ( ph  /\  A  =/=  .1.  )  /\  (
y  e.  D  /\  ( y `  A
)  =/=  1 ) )  ->  sum_ x  e.  D  ( x `  A )  =  sum_ z  e.  D  (
( y ( +g  `  G ) z ) `
 A ) )
9042, 52, 47fsummulc2 13922 . . . . . . . . 9  |-  ( ( ( 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 ) ) )
9179, 89, 903eqtr4rd 2516 . . . . . . . 8  |-  ( ( ( ph  /\  A  =/=  .1.  )  /\  (
y  e.  D  /\  ( y `  A
)  =/=  1 ) )  ->  ( (
y `  A )  x.  sum_ x  e.  D  ( x `  A
) )  =  sum_ x  e.  D  ( x `
 A ) )
9248mulid2d 9679 . . . . . . . 8  |-  ( ( ( ph  /\  A  =/=  .1.  )  /\  (
y  e.  D  /\  ( y `  A
)  =/=  1 ) )  ->  ( 1  x.  sum_ x  e.  D  ( x `  A
) )  =  sum_ x  e.  D  ( x `
 A ) )
9391, 92oveq12d 6326 . . . . . . 7  |-  ( ( ( 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 ) ) )
9448subidd 9993 . . . . . . 7  |-  ( ( ( ph  /\  A  =/=  .1.  )  /\  (
y  e.  D  /\  ( y `  A
)  =/=  1 ) )  ->  ( sum_ x  e.  D  ( x `
 A )  -  sum_ x  e.  D  ( x `  A ) )  =  0 )
9593, 94eqtrd 2505 . . . . . 6  |-  ( ( ( 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 )
9624a1i 11 . . . . . . 7  |-  ( ( ( ph  /\  A  =/=  .1.  )  /\  (
y  e.  D  /\  ( y `  A
)  =/=  1 ) )  ->  1  e.  CC )
9752, 96, 48subdird 10096 . . . . . 6  |-  ( ( ( 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
) ) ) )
9854mul01d 9850 . . . . . 6  |-  ( ( ( ph  /\  A  =/=  .1.  )  /\  (
y  e.  D  /\  ( y `  A
)  =/=  1 ) )  ->  ( (
( y `  A
)  -  1 )  x.  0 )  =  0 )
9995, 97, 983eqtr4d 2515 . . . . 5  |-  ( ( ( ph  /\  A  =/=  .1.  )  /\  (
y  e.  D  /\  ( y `  A
)  =/=  1 ) )  ->  ( (
( y `  A
)  -  1 )  x.  sum_ x  e.  D  ( x `  A
) )  =  ( ( ( y `  A )  -  1 )  x.  0 ) )
10048, 49, 54, 59, 99mulcanad 10269 . . . 4  |-  ( ( ( ph  /\  A  =/=  .1.  )  /\  (
y  e.  D  /\  ( y `  A
)  =/=  1 ) )  ->  sum_ x  e.  D  ( x `  A )  =  0 )
10140, 100rexlimddv 2875 . . 3  |-  ( (
ph  /\  A  =/=  .1.  )  ->  sum_ x  e.  D  ( x `  A )  =  0 )
10234, 101sylan2br 484 . 2  |-  ( (
ph  /\  -.  A  =  .1.  )  ->  sum_ x  e.  D  ( x `  A )  =  0 )
1031, 2, 33, 102ifbothda 3907 1  |-  ( ph  -> 
sum_ x  e.  D  ( x `  A
)  =  if ( A  =  .1.  , 
( # `  D ) ,  0 ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    /\ wa 376    = wceq 1452    e. wcel 1904    =/= wne 2641   _Vcvv 3031   ifcif 3872    |-> cmpt 4454    Fn wfn 5584   -->wf 5585   -1-1-onto->wf1o 5588   ` cfv 5589  (class class class)co 6308    oFcof 6548   Fincfn 7587   CCcc 9555   0cc0 9557   1c1 9558    x. cmul 9562    - cmin 9880   NNcn 10631   NN0cn0 10893   #chash 12553   sum_csu 13829   Basecbs 15199   +g cplusg 15268   MndHom cmhm 16658   Grpcgrp 16747   Abelcabl 17509  mulGrpcmgp 17801   1rcur 17813  ℂfldccnfld 19047  ℤ/nczn 19151  DChrcdchr 24239
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635  ax-addf 9636  ax-mulf 9637
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-disj 4367  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-rpss 6590  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-tpos 6991  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-omul 7205  df-er 7381  df-ec 7383  df-qs 7387  df-map 7492  df-pm 7493  df-ixp 7541  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-fi 7943  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-acn 8394  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-q 11288  df-rp 11326  df-xneg 11432  df-xadd 11433  df-xmul 11434  df-ioo 11664  df-ioc 11665  df-ico 11666  df-icc 11667  df-fz 11811  df-fzo 11943  df-fl 12061  df-mod 12130  df-seq 12252  df-exp 12311  df-fac 12498  df-bc 12526  df-hash 12554  df-word 12711  df-concat 12713  df-s1 12714  df-shft 13207  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-limsup 13603  df-clim 13629  df-rlim 13630  df-sum 13830  df-ef 14198  df-sin 14200  df-cos 14201  df-pi 14203  df-dvds 14383  df-gcd 14548  df-prm 14702  df-phi 14793  df-pc 14866  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-starv 15283  df-sca 15284  df-vsca 15285  df-ip 15286  df-tset 15287  df-ple 15288  df-ds 15290  df-unif 15291  df-hom 15292  df-cco 15293  df-rest 15399  df-topn 15400  df-0g 15418  df-gsum 15419  df-topgen 15420  df-pt 15421  df-prds 15424  df-xrs 15478  df-qtop 15484  df-imas 15485  df-qus 15487  df-xps 15488  df-mre 15570  df-mrc 15571  df-acs 15573  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-mhm 16660  df-submnd 16661  df-grp 16751  df-minusg 16752  df-sbg 16753  df-mulg 16754  df-subg 16892  df-nsg 16893  df-eqg 16894  df-ghm 16959  df-gim 17001  df-ga 17022  df-cntz 17049  df-oppg 17075  df-od 17250  df-gex 17252  df-pgp 17254  df-lsm 17366  df-pj1 17367  df-cmn 17510  df-abl 17511  df-cyg 17591  df-dprd 17705  df-dpj 17706  df-mgp 17802  df-ur 17814  df-ring 17860  df-cring 17861  df-oppr 17929  df-dvdsr 17947  df-unit 17948  df-invr 17978  df-rnghom 18021  df-subrg 18084  df-lmod 18171  df-lss 18234  df-lsp 18273  df-sra 18473  df-rgmod 18474  df-lidl 18475  df-rsp 18476  df-2idl 18533  df-psmet 19039  df-xmet 19040  df-met 19041  df-bl 19042  df-mopn 19043  df-fbas 19044  df-fg 19045  df-cnfld 19048  df-zring 19117  df-zrh 19152  df-zn 19155  df-top 19998  df-bases 19999  df-topon 20000  df-topsp 20001  df-cld 20111  df-ntr 20112  df-cls 20113  df-nei 20191  df-lp 20229  df-perf 20230  df-cn 20320  df-cnp 20321  df-haus 20408  df-tx 20654  df-hmeo 20847  df-fil 20939  df-fm 21031  df-flim 21032  df-flf 21033  df-xms 21413  df-ms 21414  df-tms 21415  df-cncf 21988  df-0p 22707  df-limc 22900  df-dv 22901  df-ply 23221  df-idp 23222  df-coe 23223  df-dgr 23224  df-quot 23323  df-log 23585  df-cxp 23586  df-dchr 24240
This theorem is referenced by:  dchrhash  24278  sumdchr  24279
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