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Theorem sumdchr2 22743
Description: Lemma for sumdchr 22745. (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 2469 . 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 2469 . 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 5800 . . . . . 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 22714 . . . . . . . 8  |-  D  C_  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) )
8 simpr 461 . . . . . . . 8  |-  ( (
ph  /\  x  e.  D )  ->  x  e.  D )
97, 8sseldi 3463 . . . . . . 7  |-  ( (
ph  /\  x  e.  D )  ->  x  e.  ( (mulGrp `  Z
) MndHom  (mulGrp ` fld ) ) )
10 eqid 2454 . . . . . . . . 9  |-  (mulGrp `  Z )  =  (mulGrp `  Z )
11 sumdchr2.1 . . . . . . . . 9  |-  .1.  =  ( 1r `  Z )
1210, 11rngidval 16728 . . . . . . . 8  |-  .1.  =  ( 0g `  (mulGrp `  Z ) )
13 eqid 2454 . . . . . . . . 9  |-  (mulGrp ` fld )  =  (mulGrp ` fld )
14 cnfld1 17967 . . . . . . . . 9  |-  1  =  ( 1r ` fld )
1513, 14rngidval 16728 . . . . . . . 8  |-  1  =  ( 0g `  (mulGrp ` fld ) )
1612, 15mhm0 15592 . . . . . . 7  |-  ( x  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) )  ->  (
x `  .1.  )  =  1 )
179, 16syl 16 . . . . . 6  |-  ( (
ph  /\  x  e.  D )  ->  (
x `  .1.  )  =  1 )
183, 17sylan9eqr 2517 . . . . 5  |-  ( ( ( ph  /\  x  e.  D )  /\  A  =  .1.  )  ->  (
x `  A )  =  1 )
1918an32s 802 . . . 4  |-  ( ( ( ph  /\  A  =  .1.  )  /\  x  e.  D )  ->  (
x `  A )  =  1 )
2019sumeq2dv 13299 . . 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 22728 . . . . . . 7  |-  ( N  e.  NN  ->  D  e.  Fin )
2321, 22syl 16 . . . . . 6  |-  ( ph  ->  D  e.  Fin )
24 ax-1cn 9452 . . . . . 6  |-  1  e.  CC
25 fsumconst 13376 . . . . . 6  |-  ( ( D  e.  Fin  /\  1  e.  CC )  -> 
sum_ x  e.  D 
1  =  ( (
# `  D )  x.  1 ) )
2623, 24, 25sylancl 662 . . . . 5  |-  ( ph  -> 
sum_ x  e.  D 
1  =  ( (
# `  D )  x.  1 ) )
27 hashcl 12244 . . . . . . . 8  |-  ( D  e.  Fin  ->  ( # `
 D )  e. 
NN0 )
2821, 22, 273syl 20 . . . . . . 7  |-  ( ph  ->  ( # `  D
)  e.  NN0 )
2928nn0cnd 10750 . . . . . 6  |-  ( ph  ->  ( # `  D
)  e.  CC )
3029mulid1d 9515 . . . . 5  |-  ( ph  ->  ( ( # `  D
)  x.  1 )  =  ( # `  D
) )
3126, 30eqtrd 2495 . . . 4  |-  ( ph  -> 
sum_ x  e.  D 
1  =  ( # `  D ) )
3231adantr 465 . . 3  |-  ( (
ph  /\  A  =  .1.  )  ->  sum_ x  e.  D  1  =  ( # `  D ) )
3320, 32eqtrd 2495 . 2  |-  ( (
ph  /\  A  =  .1.  )  ->  sum_ x  e.  D  ( x `  A )  =  (
# `  D )
)
34 df-ne 2650 . . 3  |-  ( A  =/=  .1.  <->  -.  A  =  .1.  )
35 sumdchr2.b . . . . 5  |-  B  =  ( Base `  Z
)
3621adantr 465 . . . . 5  |-  ( (
ph  /\  A  =/=  .1.  )  ->  N  e.  NN )
37 simpr 461 . . . . 5  |-  ( (
ph  /\  A  =/=  .1.  )  ->  A  =/= 
.1.  )
38 sumdchr2.x . . . . . 6  |-  ( ph  ->  A  e.  B )
3938adantr 465 . . . . 5  |-  ( (
ph  /\  A  =/=  .1.  )  ->  A  e.  B )
404, 5, 6, 35, 11, 36, 37, 39dchrpt 22740 . . . 4  |-  ( (
ph  /\  A  =/=  .1.  )  ->  E. y  e.  D  ( y `  A )  =/=  1
)
4136adantr 465 . . . . . . 7  |-  ( ( ( ph  /\  A  =/=  .1.  )  /\  (
y  e.  D  /\  ( y `  A
)  =/=  1 ) )  ->  N  e.  NN )
4241, 22syl 16 . . . . . 6  |-  ( ( ( ph  /\  A  =/=  .1.  )  /\  (
y  e.  D  /\  ( y `  A
)  =/=  1 ) )  ->  D  e.  Fin )
43 simpr 461 . . . . . . . 8  |-  ( ( ( ( ph  /\  A  =/=  .1.  )  /\  ( y  e.  D  /\  ( y `  A
)  =/=  1 ) )  /\  x  e.  D )  ->  x  e.  D )
444, 5, 6, 35, 43dchrf 22715 . . . . . . 7  |-  ( ( ( ( ph  /\  A  =/=  .1.  )  /\  ( y  e.  D  /\  ( y `  A
)  =/=  1 ) )  /\  x  e.  D )  ->  x : B --> CC )
4539adantr 465 . . . . . . . 8  |-  ( ( ( ph  /\  A  =/=  .1.  )  /\  (
y  e.  D  /\  ( y `  A
)  =/=  1 ) )  ->  A  e.  B )
4645adantr 465 . . . . . . 7  |-  ( ( ( ( ph  /\  A  =/=  .1.  )  /\  ( y  e.  D  /\  ( y `  A
)  =/=  1 ) )  /\  x  e.  D )  ->  A  e.  B )
4744, 46ffvelrnd 5954 . . . . . 6  |-  ( ( ( ( ph  /\  A  =/=  .1.  )  /\  ( y  e.  D  /\  ( y `  A
)  =/=  1 ) )  /\  x  e.  D )  ->  (
x `  A )  e.  CC )
4842, 47fsumcl 13329 . . . . 5  |-  ( ( ( ph  /\  A  =/=  .1.  )  /\  (
y  e.  D  /\  ( y `  A
)  =/=  1 ) )  ->  sum_ x  e.  D  ( x `  A )  e.  CC )
49 0cnd 9491 . . . . 5  |-  ( ( ( ph  /\  A  =/=  .1.  )  /\  (
y  e.  D  /\  ( y `  A
)  =/=  1 ) )  ->  0  e.  CC )
50 simprl 755 . . . . . . . 8  |-  ( ( ( ph  /\  A  =/=  .1.  )  /\  (
y  e.  D  /\  ( y `  A
)  =/=  1 ) )  ->  y  e.  D )
514, 5, 6, 35, 50dchrf 22715 . . . . . . 7  |-  ( ( ( ph  /\  A  =/=  .1.  )  /\  (
y  e.  D  /\  ( y `  A
)  =/=  1 ) )  ->  y : B
--> CC )
5251, 45ffvelrnd 5954 . . . . . 6  |-  ( ( ( ph  /\  A  =/=  .1.  )  /\  (
y  e.  D  /\  ( y `  A
)  =/=  1 ) )  ->  ( y `  A )  e.  CC )
53 subcl 9721 . . . . . 6  |-  ( ( ( y `  A
)  e.  CC  /\  1  e.  CC )  ->  ( ( y `  A )  -  1 )  e.  CC )
5452, 24, 53sylancl 662 . . . . 5  |-  ( ( ( ph  /\  A  =/=  .1.  )  /\  (
y  e.  D  /\  ( y `  A
)  =/=  1 ) )  ->  ( (
y `  A )  -  1 )  e.  CC )
55 simprr 756 . . . . . 6  |-  ( ( ( ph  /\  A  =/=  .1.  )  /\  (
y  e.  D  /\  ( y `  A
)  =/=  1 ) )  ->  ( y `  A )  =/=  1
)
56 subeq0 9747 . . . . . . . 8  |-  ( ( ( y `  A
)  e.  CC  /\  1  e.  CC )  ->  ( ( ( y `
 A )  - 
1 )  =  0  <-> 
( y `  A
)  =  1 ) )
5752, 24, 56sylancl 662 . . . . . . 7  |-  ( ( ( ph  /\  A  =/=  .1.  )  /\  (
y  e.  D  /\  ( y `  A
)  =/=  1 ) )  ->  ( (
( y `  A
)  -  1 )  =  0  <->  ( y `  A )  =  1 ) )
5857necon3bid 2710 . . . . . 6  |-  ( ( ( ph  /\  A  =/=  .1.  )  /\  (
y  e.  D  /\  ( y `  A
)  =/=  1 ) )  ->  ( (
( y `  A
)  -  1 )  =/=  0  <->  ( y `  A )  =/=  1
) )
5955, 58mpbird 232 . . . . 5  |-  ( ( ( ph  /\  A  =/=  .1.  )  /\  (
y  e.  D  /\  ( y `  A
)  =/=  1 ) )  ->  ( (
y `  A )  -  1 )  =/=  0 )
60 oveq2 6209 . . . . . . . . . . . 12  |-  ( z  =  x  ->  (
y ( +g  `  G
) z )  =  ( y ( +g  `  G ) x ) )
6160fveq1d 5802 . . . . . . . . . . 11  |-  ( z  =  x  ->  (
( y ( +g  `  G ) z ) `
 A )  =  ( ( y ( +g  `  G ) x ) `  A
) )
6261cbvsumv 13292 . . . . . . . . . 10  |-  sum_ z  e.  D  ( (
y ( +g  `  G
) z ) `  A )  =  sum_ x  e.  D  ( ( y ( +g  `  G
) x ) `  A )
63 eqid 2454 . . . . . . . . . . . . . 14  |-  ( +g  `  G )  =  ( +g  `  G )
6450adantr 465 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  A  =/=  .1.  )  /\  ( y  e.  D  /\  ( y `  A
)  =/=  1 ) )  /\  x  e.  D )  ->  y  e.  D )
654, 5, 6, 63, 64, 43dchrmul 22721 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  A  =/=  .1.  )  /\  ( y  e.  D  /\  ( y `  A
)  =/=  1 ) )  /\  x  e.  D )  ->  (
y ( +g  `  G
) x )  =  ( y  oF  x.  x ) )
6665fveq1d 5802 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  A  =/=  .1.  )  /\  ( y  e.  D  /\  ( y `  A
)  =/=  1 ) )  /\  x  e.  D )  ->  (
( y ( +g  `  G ) x ) `
 A )  =  ( ( y  oF  x.  x ) `
 A ) )
6751adantr 465 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  A  =/=  .1.  )  /\  ( y  e.  D  /\  ( y `  A
)  =/=  1 ) )  /\  x  e.  D )  ->  y : B --> CC )
68 ffn 5668 . . . . . . . . . . . . . 14  |-  ( y : B --> CC  ->  y  Fn  B )
6967, 68syl 16 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  A  =/=  .1.  )  /\  ( y  e.  D  /\  ( y `  A
)  =/=  1 ) )  /\  x  e.  D )  ->  y  Fn  B )
70 ffn 5668 . . . . . . . . . . . . . 14  |-  ( x : B --> CC  ->  x  Fn  B )
7144, 70syl 16 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  A  =/=  .1.  )  /\  ( y  e.  D  /\  ( y `  A
)  =/=  1 ) )  /\  x  e.  D )  ->  x  Fn  B )
72 fvex 5810 . . . . . . . . . . . . . . 15  |-  ( Base `  Z )  e.  _V
7335, 72eqeltri 2538 . . . . . . . . . . . . . 14  |-  B  e. 
_V
7473a1i 11 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  A  =/=  .1.  )  /\  ( y  e.  D  /\  ( y `  A
)  =/=  1 ) )  /\  x  e.  D )  ->  B  e.  _V )
75 fnfvof 6444 . . . . . . . . . . . . 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 1220 . . . . . . . . . . . 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 2495 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  A  =/=  .1.  )  /\  ( y  e.  D  /\  ( y `  A
)  =/=  1 ) )  /\  x  e.  D )  ->  (
( y ( +g  `  G ) x ) `
 A )  =  ( ( y `  A )  x.  (
x `  A )
) )
7877sumeq2dv 13299 . . . . . . . . . 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 2507 . . . . . . . . 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 5799 . . . . . . . . . 10  |-  ( x  =  ( y ( +g  `  G ) z )  ->  (
x `  A )  =  ( ( y ( +g  `  G
) z ) `  A ) )
814dchrabl 22727 . . . . . . . . . . . 12  |-  ( N  e.  NN  ->  G  e.  Abel )
82 ablgrp 16404 . . . . . . . . . . . 12  |-  ( G  e.  Abel  ->  G  e. 
Grp )
8341, 81, 823syl 20 . . . . . . . . . . 11  |-  ( ( ( ph  /\  A  =/=  .1.  )  /\  (
y  e.  D  /\  ( y `  A
)  =/=  1 ) )  ->  G  e.  Grp )
84 eqid 2454 . . . . . . . . . . . 12  |-  ( a  e.  D  |->  ( b  e.  D  |->  ( a ( +g  `  G
) b ) ) )  =  ( a  e.  D  |->  ( b  e.  D  |->  ( a ( +g  `  G
) b ) ) )
8584, 6, 63grplactf1o 15745 . . . . . . . . . . 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 661 . . . . . . . . . 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 15743 . . . . . . . . . . 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 471 . . . . . . . . . 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 13319 . . . . . . . . 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 13370 . . . . . . . . 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 2506 . . . . . . . 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 9516 . . . . . . . 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 6219 . . . . . . 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 9819 . . . . . . 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 2495 . . . . . 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 9913 . . . . . 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 9680 . . . . . 6  |-  ( ( ( ph  /\  A  =/=  .1.  )  /\  (
y  e.  D  /\  ( y `  A
)  =/=  1 ) )  ->  ( (
( y `  A
)  -  1 )  x.  0 )  =  0 )
9995, 97, 983eqtr4d 2505 . . . . 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 10083 . . . 4  |-  ( ( ( ph  /\  A  =/=  .1.  )  /\  (
y  e.  D  /\  ( y `  A
)  =/=  1 ) )  ->  sum_ x  e.  D  ( x `  A )  =  0 )
10140, 100rexlimddv 2951 . . 3  |-  ( (
ph  /\  A  =/=  .1.  )  ->  sum_ x  e.  D  ( x `  A )  =  0 )
10234, 101sylan2br 476 . 2  |-  ( (
ph  /\  -.  A  =  .1.  )  ->  sum_ x  e.  D  ( x `  A )  =  0 )
1031, 2, 33, 102ifbothda 3933 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 184    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2648   _Vcvv 3078   ifcif 3900    |-> cmpt 4459    Fn wfn 5522   -->wf 5523   -1-1-onto->wf1o 5526   ` cfv 5527  (class class class)co 6201    oFcof 6429   Fincfn 7421   CCcc 9392   0cc0 9394   1c1 9395    x. cmul 9399    - cmin 9707   NNcn 10434   NN0cn0 10691   #chash 12221   sum_csu 13282   Basecbs 14293   +g cplusg 14358   Grpcgrp 15530   MndHom cmhm 15582   Abelcabel 16400  mulGrpcmgp 16714   1rcur 16726  ℂfldccnfld 17944  ℤ/nczn 18060  DChrcdchr 22705
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-inf2 7959  ax-cnex 9450  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470  ax-pre-mulgt0 9471  ax-pre-sup 9472  ax-addf 9473  ax-mulf 9474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-int 4238  df-iun 4282  df-iin 4283  df-disj 4372  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-se 4789  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-isom 5536  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-of 6431  df-rpss 6471  df-om 6588  df-1st 6688  df-2nd 6689  df-supp 6802  df-tpos 6856  df-recs 6943  df-rdg 6977  df-1o 7031  df-2o 7032  df-oadd 7035  df-omul 7036  df-er 7212  df-ec 7214  df-qs 7218  df-map 7327  df-pm 7328  df-ixp 7375  df-en 7422  df-dom 7423  df-sdom 7424  df-fin 7425  df-fsupp 7733  df-fi 7773  df-sup 7803  df-oi 7836  df-card 8221  df-acn 8224  df-cda 8449  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710  df-div 10106  df-nn 10435  df-2 10492  df-3 10493  df-4 10494  df-5 10495  df-6 10496  df-7 10497  df-8 10498  df-9 10499  df-10 10500  df-n0 10692  df-z 10759  df-dec 10868  df-uz 10974  df-q 11066  df-rp 11104  df-xneg 11201  df-xadd 11202  df-xmul 11203  df-ioo 11416  df-ioc 11417  df-ico 11418  df-icc 11419  df-fz 11556  df-fzo 11667  df-fl 11760  df-mod 11827  df-seq 11925  df-exp 11984  df-fac 12170  df-bc 12197  df-hash 12222  df-word 12348  df-concat 12350  df-s1 12351  df-shft 12675  df-cj 12707  df-re 12708  df-im 12709  df-sqr 12843  df-abs 12844  df-limsup 13068  df-clim 13085  df-rlim 13086  df-sum 13283  df-ef 13472  df-sin 13474  df-cos 13475  df-pi 13477  df-dvds 13655  df-gcd 13810  df-prm 13883  df-phi 13960  df-pc 14023  df-struct 14295  df-ndx 14296  df-slot 14297  df-base 14298  df-sets 14299  df-ress 14300  df-plusg 14371  df-mulr 14372  df-starv 14373  df-sca 14374  df-vsca 14375  df-ip 14376  df-tset 14377  df-ple 14378  df-ds 14380  df-unif 14381  df-hom 14382  df-cco 14383  df-rest 14481  df-topn 14482  df-0g 14500  df-gsum 14501  df-topgen 14502  df-pt 14503  df-prds 14506  df-xrs 14560  df-qtop 14565  df-imas 14566  df-divs 14567  df-xps 14568  df-mre 14644  df-mrc 14645  df-acs 14647  df-mnd 15535  df-mhm 15584  df-submnd 15585  df-grp 15665  df-minusg 15666  df-sbg 15667  df-mulg 15668  df-subg 15798  df-nsg 15799  df-eqg 15800  df-ghm 15865  df-gim 15907  df-ga 15928  df-cntz 15955  df-oppg 15981  df-od 16154  df-gex 16155  df-pgp 16156  df-lsm 16257  df-pj1 16258  df-cmn 16401  df-abl 16402  df-cyg 16477  df-dprd 16600  df-dpj 16601  df-mgp 16715  df-ur 16727  df-rng 16771  df-cring 16772  df-oppr 16839  df-dvdsr 16857  df-unit 16858  df-invr 16888  df-rnghom 16930  df-subrg 16987  df-lmod 17074  df-lss 17138  df-lsp 17177  df-sra 17377  df-rgmod 17378  df-lidl 17379  df-rsp 17380  df-2idl 17438  df-psmet 17935  df-xmet 17936  df-met 17937  df-bl 17938  df-mopn 17939  df-fbas 17940  df-fg 17941  df-cnfld 17945  df-zring 18010  df-zrh 18061  df-zn 18064  df-top 18636  df-bases 18638  df-topon 18639  df-topsp 18640  df-cld 18756  df-ntr 18757  df-cls 18758  df-nei 18835  df-lp 18873  df-perf 18874  df-cn 18964  df-cnp 18965  df-haus 19052  df-tx 19268  df-hmeo 19461  df-fil 19552  df-fm 19644  df-flim 19645  df-flf 19646  df-xms 20028  df-ms 20029  df-tms 20030  df-cncf 20587  df-0p 21282  df-limc 21475  df-dv 21476  df-ply 21790  df-idp 21791  df-coe 21792  df-dgr 21793  df-quot 21891  df-log 22142  df-cxp 22143  df-dchr 22706
This theorem is referenced by:  dchrhash  22744  sumdchr  22745
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