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Theorem sumdchr2 23266
Description: Lemma for sumdchr 23268. (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 2475 . 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 2475 . 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 5857 . . . . . 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 23237 . . . . . . . 8  |-  D  C_  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) )
8 simpr 461 . . . . . . . 8  |-  ( (
ph  /\  x  e.  D )  ->  x  e.  D )
97, 8sseldi 3495 . . . . . . 7  |-  ( (
ph  /\  x  e.  D )  ->  x  e.  ( (mulGrp `  Z
) MndHom  (mulGrp ` fld ) ) )
10 eqid 2460 . . . . . . . . 9  |-  (mulGrp `  Z )  =  (mulGrp `  Z )
11 sumdchr2.1 . . . . . . . . 9  |-  .1.  =  ( 1r `  Z )
1210, 11rngidval 16938 . . . . . . . 8  |-  .1.  =  ( 0g `  (mulGrp `  Z ) )
13 eqid 2460 . . . . . . . . 9  |-  (mulGrp ` fld )  =  (mulGrp ` fld )
14 cnfld1 18207 . . . . . . . . 9  |-  1  =  ( 1r ` fld )
1513, 14rngidval 16938 . . . . . . . 8  |-  1  =  ( 0g `  (mulGrp ` fld ) )
1612, 15mhm0 15778 . . . . . . 7  |-  ( x  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) )  ->  (
x `  .1.  )  =  1 )
179, 16syl 16 . . . . . 6  |-  ( (
ph  /\  x  e.  D )  ->  (
x `  .1.  )  =  1 )
183, 17sylan9eqr 2523 . . . . 5  |-  ( ( ( ph  /\  x  e.  D )  /\  A  =  .1.  )  ->  (
x `  A )  =  1 )
1918an32s 802 . . . 4  |-  ( ( ( ph  /\  A  =  .1.  )  /\  x  e.  D )  ->  (
x `  A )  =  1 )
2019sumeq2dv 13474 . . 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 23251 . . . . . . 7  |-  ( N  e.  NN  ->  D  e.  Fin )
2321, 22syl 16 . . . . . 6  |-  ( ph  ->  D  e.  Fin )
24 ax-1cn 9539 . . . . . 6  |-  1  e.  CC
25 fsumconst 13554 . . . . . 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 12383 . . . . . . . 8  |-  ( D  e.  Fin  ->  ( # `
 D )  e. 
NN0 )
2821, 22, 273syl 20 . . . . . . 7  |-  ( ph  ->  ( # `  D
)  e.  NN0 )
2928nn0cnd 10843 . . . . . 6  |-  ( ph  ->  ( # `  D
)  e.  CC )
3029mulid1d 9602 . . . . 5  |-  ( ph  ->  ( ( # `  D
)  x.  1 )  =  ( # `  D
) )
3126, 30eqtrd 2501 . . . 4  |-  ( ph  -> 
sum_ x  e.  D 
1  =  ( # `  D ) )
3231adantr 465 . . 3  |-  ( (
ph  /\  A  =  .1.  )  ->  sum_ x  e.  D  1  =  ( # `  D ) )
3320, 32eqtrd 2501 . 2  |-  ( (
ph  /\  A  =  .1.  )  ->  sum_ x  e.  D  ( x `  A )  =  (
# `  D )
)
34 df-ne 2657 . . 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 23263 . . . 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 23238 . . . . . . 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 6013 . . . . . 6  |-  ( ( ( ( ph  /\  A  =/=  .1.  )  /\  ( y  e.  D  /\  ( y `  A
)  =/=  1 ) )  /\  x  e.  D )  ->  (
x `  A )  e.  CC )
4842, 47fsumcl 13504 . . . . 5  |-  ( ( ( ph  /\  A  =/=  .1.  )  /\  (
y  e.  D  /\  ( y `  A
)  =/=  1 ) )  ->  sum_ x  e.  D  ( x `  A )  e.  CC )
49 0cnd 9578 . . . . 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 23238 . . . . . . 7  |-  ( ( ( ph  /\  A  =/=  .1.  )  /\  (
y  e.  D  /\  ( y `  A
)  =/=  1 ) )  ->  y : B
--> CC )
5251, 45ffvelrnd 6013 . . . . . 6  |-  ( ( ( ph  /\  A  =/=  .1.  )  /\  (
y  e.  D  /\  ( y `  A
)  =/=  1 ) )  ->  ( y `  A )  e.  CC )
53 subcl 9808 . . . . . 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 9834 . . . . . . . 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 2718 . . . . . 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 6283 . . . . . . . . . . . 12  |-  ( z  =  x  ->  (
y ( +g  `  G
) z )  =  ( y ( +g  `  G ) x ) )
6160fveq1d 5859 . . . . . . . . . . 11  |-  ( z  =  x  ->  (
( y ( +g  `  G ) z ) `
 A )  =  ( ( y ( +g  `  G ) x ) `  A
) )
6261cbvsumv 13467 . . . . . . . . . 10  |-  sum_ z  e.  D  ( (
y ( +g  `  G
) z ) `  A )  =  sum_ x  e.  D  ( ( y ( +g  `  G
) x ) `  A )
63 eqid 2460 . . . . . . . . . . . . . 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 23244 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  A  =/=  .1.  )  /\  ( y  e.  D  /\  ( y `  A
)  =/=  1 ) )  /\  x  e.  D )  ->  (
y ( +g  `  G
) x )  =  ( y  oF  x.  x ) )
6665fveq1d 5859 . . . . . . . . . . . 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 5722 . . . . . . . . . . . . . 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 5722 . . . . . . . . . . . . . 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 5867 . . . . . . . . . . . . . . 15  |-  ( Base `  Z )  e.  _V
7335, 72eqeltri 2544 . . . . . . . . . . . . . 14  |-  B  e. 
_V
7473a1i 11 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  A  =/=  .1.  )  /\  ( y  e.  D  /\  ( y `  A
)  =/=  1 ) )  /\  x  e.  D )  ->  B  e.  _V )
75 fnfvof 6528 . . . . . . . . . . . . 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 1224 . . . . . . . . . . . 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 2501 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  A  =/=  .1.  )  /\  ( y  e.  D  /\  ( y `  A
)  =/=  1 ) )  /\  x  e.  D )  ->  (
( y ( +g  `  G ) x ) `
 A )  =  ( ( y `  A )  x.  (
x `  A )
) )
7877sumeq2dv 13474 . . . . . . . . . 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 2513 . . . . . . . . 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 5856 . . . . . . . . . 10  |-  ( x  =  ( y ( +g  `  G ) z )  ->  (
x `  A )  =  ( ( y ( +g  `  G
) z ) `  A ) )
814dchrabl 23250 . . . . . . . . . . . 12  |-  ( N  e.  NN  ->  G  e.  Abel )
82 ablgrp 16592 . . . . . . . . . . . 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 2460 . . . . . . . . . . . 12  |-  ( a  e.  D  |->  ( b  e.  D  |->  ( a ( +g  `  G
) b ) ) )  =  ( a  e.  D  |->  ( b  e.  D  |->  ( a ( +g  `  G
) b ) ) )
8584, 6, 63grplactf1o 15933 . . . . . . . . . . 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 15931 . . . . . . . . . . 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 13494 . . . . . . . . 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 13548 . . . . . . . . 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 2512 . . . . . . . 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 9603 . . . . . . . 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 6293 . . . . . . 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 9907 . . . . . . 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 2501 . . . . . 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 10002 . . . . . 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 9767 . . . . . 6  |-  ( ( ( ph  /\  A  =/=  .1.  )  /\  (
y  e.  D  /\  ( y `  A
)  =/=  1 ) )  ->  ( (
( y `  A
)  -  1 )  x.  0 )  =  0 )
9995, 97, 983eqtr4d 2511 . . . . 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 10173 . . . 4  |-  ( ( ( ph  /\  A  =/=  .1.  )  /\  (
y  e.  D  /\  ( y `  A
)  =/=  1 ) )  ->  sum_ x  e.  D  ( x `  A )  =  0 )
10140, 100rexlimddv 2952 . . 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 3967 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 1374    e. wcel 1762    =/= wne 2655   _Vcvv 3106   ifcif 3932    |-> cmpt 4498    Fn wfn 5574   -->wf 5575   -1-1-onto->wf1o 5578   ` cfv 5579  (class class class)co 6275    oFcof 6513   Fincfn 7506   CCcc 9479   0cc0 9481   1c1 9482    x. cmul 9486    - cmin 9794   NNcn 10525   NN0cn0 10784   #chash 12360   sum_csu 13457   Basecbs 14479   +g cplusg 14544   Grpcgrp 15716   MndHom cmhm 15768   Abelcabel 16588  mulGrpcmgp 16924   1rcur 16936  ℂfldccnfld 18184  ℤ/nczn 18300  DChrcdchr 23228
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-inf2 8047  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559  ax-addf 9560  ax-mulf 9561
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-iin 4321  df-disj 4411  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-se 4832  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-isom 5588  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-of 6515  df-rpss 6555  df-om 6672  df-1st 6774  df-2nd 6775  df-supp 6892  df-tpos 6945  df-recs 7032  df-rdg 7066  df-1o 7120  df-2o 7121  df-oadd 7124  df-omul 7125  df-er 7301  df-ec 7303  df-qs 7307  df-map 7412  df-pm 7413  df-ixp 7460  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-fsupp 7819  df-fi 7860  df-sup 7890  df-oi 7924  df-card 8309  df-acn 8312  df-cda 8537  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-2 10583  df-3 10584  df-4 10585  df-5 10586  df-6 10587  df-7 10588  df-8 10589  df-9 10590  df-10 10591  df-n0 10785  df-z 10854  df-dec 10966  df-uz 11072  df-q 11172  df-rp 11210  df-xneg 11307  df-xadd 11308  df-xmul 11309  df-ioo 11522  df-ioc 11523  df-ico 11524  df-icc 11525  df-fz 11662  df-fzo 11782  df-fl 11886  df-mod 11953  df-seq 12064  df-exp 12123  df-fac 12309  df-bc 12336  df-hash 12361  df-word 12495  df-concat 12497  df-s1 12498  df-shft 12850  df-cj 12882  df-re 12883  df-im 12884  df-sqr 13018  df-abs 13019  df-limsup 13243  df-clim 13260  df-rlim 13261  df-sum 13458  df-ef 13654  df-sin 13656  df-cos 13657  df-pi 13659  df-dvds 13837  df-gcd 13993  df-prm 14066  df-phi 14144  df-pc 14209  df-struct 14481  df-ndx 14482  df-slot 14483  df-base 14484  df-sets 14485  df-ress 14486  df-plusg 14557  df-mulr 14558  df-starv 14559  df-sca 14560  df-vsca 14561  df-ip 14562  df-tset 14563  df-ple 14564  df-ds 14566  df-unif 14567  df-hom 14568  df-cco 14569  df-rest 14667  df-topn 14668  df-0g 14686  df-gsum 14687  df-topgen 14688  df-pt 14689  df-prds 14692  df-xrs 14746  df-qtop 14751  df-imas 14752  df-divs 14753  df-xps 14754  df-mre 14830  df-mrc 14831  df-acs 14833  df-mnd 15721  df-mhm 15770  df-submnd 15771  df-grp 15851  df-minusg 15852  df-sbg 15853  df-mulg 15854  df-subg 15986  df-nsg 15987  df-eqg 15988  df-ghm 16053  df-gim 16095  df-ga 16116  df-cntz 16143  df-oppg 16169  df-od 16342  df-gex 16343  df-pgp 16344  df-lsm 16445  df-pj1 16446  df-cmn 16589  df-abl 16590  df-cyg 16665  df-dprd 16810  df-dpj 16811  df-mgp 16925  df-ur 16937  df-rng 16981  df-cring 16982  df-oppr 17049  df-dvdsr 17067  df-unit 17068  df-invr 17098  df-rnghom 17141  df-subrg 17203  df-lmod 17290  df-lss 17355  df-lsp 17394  df-sra 17594  df-rgmod 17595  df-lidl 17596  df-rsp 17597  df-2idl 17655  df-psmet 18175  df-xmet 18176  df-met 18177  df-bl 18178  df-mopn 18179  df-fbas 18180  df-fg 18181  df-cnfld 18185  df-zring 18250  df-zrh 18301  df-zn 18304  df-top 19159  df-bases 19161  df-topon 19162  df-topsp 19163  df-cld 19279  df-ntr 19280  df-cls 19281  df-nei 19358  df-lp 19396  df-perf 19397  df-cn 19487  df-cnp 19488  df-haus 19575  df-tx 19791  df-hmeo 19984  df-fil 20075  df-fm 20167  df-flim 20168  df-flf 20169  df-xms 20551  df-ms 20552  df-tms 20553  df-cncf 21110  df-0p 21805  df-limc 21998  df-dv 21999  df-ply 22313  df-idp 22314  df-coe 22315  df-dgr 22316  df-quot 22414  df-log 22665  df-cxp 22666  df-dchr 23229
This theorem is referenced by:  dchrhash  23267  sumdchr  23268
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