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Theorem dchrpt 23414
Description: For any element other than 1, there is a Dirichlet character that is not one at the given element. (Contributed by Mario Carneiro, 28-Apr-2016.)
Hypotheses
Ref Expression
dchrpt.g  |-  G  =  (DChr `  N )
dchrpt.z  |-  Z  =  (ℤ/n `  N )
dchrpt.d  |-  D  =  ( Base `  G
)
dchrpt.b  |-  B  =  ( Base `  Z
)
dchrpt.1  |-  .1.  =  ( 1r `  Z )
dchrpt.n  |-  ( ph  ->  N  e.  NN )
dchrpt.n1  |-  ( ph  ->  A  =/=  .1.  )
dchrpt.a  |-  ( ph  ->  A  e.  B )
Assertion
Ref Expression
dchrpt  |-  ( ph  ->  E. x  e.  D  ( x `  A
)  =/=  1 )
Distinct variable groups:    x,  .1.    x, A    x, B    x, G    x, N    x, Z    x, D    ph, x

Proof of Theorem dchrpt
Dummy variables  a 
b  k  n  u  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dchrpt.g . . . . 5  |-  G  =  (DChr `  N )
2 dchrpt.z . . . . 5  |-  Z  =  (ℤ/n `  N )
3 dchrpt.d . . . . 5  |-  D  =  ( Base `  G
)
4 dchrpt.b . . . . 5  |-  B  =  ( Base `  Z
)
5 dchrpt.1 . . . . 5  |-  .1.  =  ( 1r `  Z )
6 dchrpt.n . . . . . 6  |-  ( ph  ->  N  e.  NN )
76ad3antrrr 729 . . . . 5  |-  ( ( ( ( ph  /\  A  e.  (Unit `  Z
) )  /\  w  e. Word  (Unit `  Z )
)  /\  ( (
(mulGrp `  Z )s  (Unit `  Z ) ) dom DProd  ( k  e.  dom  w  |->  ran  ( n  e.  ZZ  |->  ( n (.g `  ( (mulGrp `  Z
)s  (Unit `  Z )
) ) ( w `
 k ) ) ) )  /\  (
( (mulGrp `  Z
)s  (Unit `  Z )
) DProd  ( k  e. 
dom  w  |->  ran  (
n  e.  ZZ  |->  ( n (.g `  ( (mulGrp `  Z )s  (Unit `  Z )
) ) ( w `
 k ) ) ) ) )  =  (Unit `  Z )
) )  ->  N  e.  NN )
8 dchrpt.n1 . . . . . 6  |-  ( ph  ->  A  =/=  .1.  )
98ad3antrrr 729 . . . . 5  |-  ( ( ( ( ph  /\  A  e.  (Unit `  Z
) )  /\  w  e. Word  (Unit `  Z )
)  /\  ( (
(mulGrp `  Z )s  (Unit `  Z ) ) dom DProd  ( k  e.  dom  w  |->  ran  ( n  e.  ZZ  |->  ( n (.g `  ( (mulGrp `  Z
)s  (Unit `  Z )
) ) ( w `
 k ) ) ) )  /\  (
( (mulGrp `  Z
)s  (Unit `  Z )
) DProd  ( k  e. 
dom  w  |->  ran  (
n  e.  ZZ  |->  ( n (.g `  ( (mulGrp `  Z )s  (Unit `  Z )
) ) ( w `
 k ) ) ) ) )  =  (Unit `  Z )
) )  ->  A  =/=  .1.  )
10 eqid 2443 . . . . 5  |-  (Unit `  Z )  =  (Unit `  Z )
11 eqid 2443 . . . . 5  |-  ( (mulGrp `  Z )s  (Unit `  Z )
)  =  ( (mulGrp `  Z )s  (Unit `  Z )
)
12 eqid 2443 . . . . 5  |-  (.g `  (
(mulGrp `  Z )s  (Unit `  Z ) ) )  =  (.g `  ( (mulGrp `  Z )s  (Unit `  Z )
) )
13 oveq1 6288 . . . . . . . . 9  |-  ( n  =  b  ->  (
n (.g `  ( (mulGrp `  Z )s  (Unit `  Z )
) ) ( w `
 k ) )  =  ( b (.g `  ( (mulGrp `  Z
)s  (Unit `  Z )
) ) ( w `
 k ) ) )
1413cbvmptv 4528 . . . . . . . 8  |-  ( n  e.  ZZ  |->  ( n (.g `  ( (mulGrp `  Z )s  (Unit `  Z )
) ) ( w `
 k ) ) )  =  ( b  e.  ZZ  |->  ( b (.g `  ( (mulGrp `  Z )s  (Unit `  Z )
) ) ( w `
 k ) ) )
15 fveq2 5856 . . . . . . . . . 10  |-  ( k  =  a  ->  (
w `  k )  =  ( w `  a ) )
1615oveq2d 6297 . . . . . . . . 9  |-  ( k  =  a  ->  (
b (.g `  ( (mulGrp `  Z )s  (Unit `  Z )
) ) ( w `
 k ) )  =  ( b (.g `  ( (mulGrp `  Z
)s  (Unit `  Z )
) ) ( w `
 a ) ) )
1716mpteq2dv 4524 . . . . . . . 8  |-  ( k  =  a  ->  (
b  e.  ZZ  |->  ( b (.g `  ( (mulGrp `  Z )s  (Unit `  Z )
) ) ( w `
 k ) ) )  =  ( b  e.  ZZ  |->  ( b (.g `  ( (mulGrp `  Z )s  (Unit `  Z )
) ) ( w `
 a ) ) ) )
1814, 17syl5eq 2496 . . . . . . 7  |-  ( k  =  a  ->  (
n  e.  ZZ  |->  ( n (.g `  ( (mulGrp `  Z )s  (Unit `  Z )
) ) ( w `
 k ) ) )  =  ( b  e.  ZZ  |->  ( b (.g `  ( (mulGrp `  Z )s  (Unit `  Z )
) ) ( w `
 a ) ) ) )
1918rneqd 5220 . . . . . 6  |-  ( k  =  a  ->  ran  ( n  e.  ZZ  |->  ( n (.g `  (
(mulGrp `  Z )s  (Unit `  Z ) ) ) ( w `  k
) ) )  =  ran  ( b  e.  ZZ  |->  ( b (.g `  ( (mulGrp `  Z
)s  (Unit `  Z )
) ) ( w `
 a ) ) ) )
2019cbvmptv 4528 . . . . 5  |-  ( k  e.  dom  w  |->  ran  ( n  e.  ZZ  |->  ( n (.g `  (
(mulGrp `  Z )s  (Unit `  Z ) ) ) ( w `  k
) ) ) )  =  ( a  e. 
dom  w  |->  ran  (
b  e.  ZZ  |->  ( b (.g `  ( (mulGrp `  Z )s  (Unit `  Z )
) ) ( w `
 a ) ) ) )
21 simpllr 760 . . . . 5  |-  ( ( ( ( ph  /\  A  e.  (Unit `  Z
) )  /\  w  e. Word  (Unit `  Z )
)  /\  ( (
(mulGrp `  Z )s  (Unit `  Z ) ) dom DProd  ( k  e.  dom  w  |->  ran  ( n  e.  ZZ  |->  ( n (.g `  ( (mulGrp `  Z
)s  (Unit `  Z )
) ) ( w `
 k ) ) ) )  /\  (
( (mulGrp `  Z
)s  (Unit `  Z )
) DProd  ( k  e. 
dom  w  |->  ran  (
n  e.  ZZ  |->  ( n (.g `  ( (mulGrp `  Z )s  (Unit `  Z )
) ) ( w `
 k ) ) ) ) )  =  (Unit `  Z )
) )  ->  A  e.  (Unit `  Z )
)
22 simplr 755 . . . . 5  |-  ( ( ( ( ph  /\  A  e.  (Unit `  Z
) )  /\  w  e. Word  (Unit `  Z )
)  /\  ( (
(mulGrp `  Z )s  (Unit `  Z ) ) dom DProd  ( k  e.  dom  w  |->  ran  ( n  e.  ZZ  |->  ( n (.g `  ( (mulGrp `  Z
)s  (Unit `  Z )
) ) ( w `
 k ) ) ) )  /\  (
( (mulGrp `  Z
)s  (Unit `  Z )
) DProd  ( k  e. 
dom  w  |->  ran  (
n  e.  ZZ  |->  ( n (.g `  ( (mulGrp `  Z )s  (Unit `  Z )
) ) ( w `
 k ) ) ) ) )  =  (Unit `  Z )
) )  ->  w  e. Word  (Unit `  Z )
)
23 simprl 756 . . . . 5  |-  ( ( ( ( ph  /\  A  e.  (Unit `  Z
) )  /\  w  e. Word  (Unit `  Z )
)  /\  ( (
(mulGrp `  Z )s  (Unit `  Z ) ) dom DProd  ( k  e.  dom  w  |->  ran  ( n  e.  ZZ  |->  ( n (.g `  ( (mulGrp `  Z
)s  (Unit `  Z )
) ) ( w `
 k ) ) ) )  /\  (
( (mulGrp `  Z
)s  (Unit `  Z )
) DProd  ( k  e. 
dom  w  |->  ran  (
n  e.  ZZ  |->  ( n (.g `  ( (mulGrp `  Z )s  (Unit `  Z )
) ) ( w `
 k ) ) ) ) )  =  (Unit `  Z )
) )  ->  (
(mulGrp `  Z )s  (Unit `  Z ) ) dom DProd  ( k  e.  dom  w  |->  ran  ( n  e.  ZZ  |->  ( n (.g `  ( (mulGrp `  Z
)s  (Unit `  Z )
) ) ( w `
 k ) ) ) ) )
24 simprr 757 . . . . 5  |-  ( ( ( ( ph  /\  A  e.  (Unit `  Z
) )  /\  w  e. Word  (Unit `  Z )
)  /\  ( (
(mulGrp `  Z )s  (Unit `  Z ) ) dom DProd  ( k  e.  dom  w  |->  ran  ( n  e.  ZZ  |->  ( n (.g `  ( (mulGrp `  Z
)s  (Unit `  Z )
) ) ( w `
 k ) ) ) )  /\  (
( (mulGrp `  Z
)s  (Unit `  Z )
) DProd  ( k  e. 
dom  w  |->  ran  (
n  e.  ZZ  |->  ( n (.g `  ( (mulGrp `  Z )s  (Unit `  Z )
) ) ( w `
 k ) ) ) ) )  =  (Unit `  Z )
) )  ->  (
( (mulGrp `  Z
)s  (Unit `  Z )
) DProd  ( k  e. 
dom  w  |->  ran  (
n  e.  ZZ  |->  ( n (.g `  ( (mulGrp `  Z )s  (Unit `  Z )
) ) ( w `
 k ) ) ) ) )  =  (Unit `  Z )
)
251, 2, 3, 4, 5, 7, 9, 10, 11, 12, 20, 21, 22, 23, 24dchrptlem3 23413 . . . 4  |-  ( ( ( ( ph  /\  A  e.  (Unit `  Z
) )  /\  w  e. Word  (Unit `  Z )
)  /\  ( (
(mulGrp `  Z )s  (Unit `  Z ) ) dom DProd  ( k  e.  dom  w  |->  ran  ( n  e.  ZZ  |->  ( n (.g `  ( (mulGrp `  Z
)s  (Unit `  Z )
) ) ( w `
 k ) ) ) )  /\  (
( (mulGrp `  Z
)s  (Unit `  Z )
) DProd  ( k  e. 
dom  w  |->  ran  (
n  e.  ZZ  |->  ( n (.g `  ( (mulGrp `  Z )s  (Unit `  Z )
) ) ( w `
 k ) ) ) ) )  =  (Unit `  Z )
) )  ->  E. x  e.  D  ( x `  A )  =/=  1
)
26253adantr1 1156 . . 3  |-  ( ( ( ( ph  /\  A  e.  (Unit `  Z
) )  /\  w  e. Word  (Unit `  Z )
)  /\  ( (
k  e.  dom  w  |->  ran  ( n  e.  ZZ  |->  ( n (.g `  ( (mulGrp `  Z
)s  (Unit `  Z )
) ) ( w `
 k ) ) ) ) : dom  w
--> { u  e.  (SubGrp `  ( (mulGrp `  Z
)s  (Unit `  Z )
) )  |  ( ( (mulGrp `  Z
)s  (Unit `  Z )
)s  u )  e.  (CycGrp 
i^i  ran pGrp  ) }  /\  ( (mulGrp `  Z )s  (Unit `  Z ) ) dom DProd  ( k  e.  dom  w  |->  ran  ( n  e.  ZZ  |->  ( n (.g `  ( (mulGrp `  Z
)s  (Unit `  Z )
) ) ( w `
 k ) ) ) )  /\  (
( (mulGrp `  Z
)s  (Unit `  Z )
) DProd  ( k  e. 
dom  w  |->  ran  (
n  e.  ZZ  |->  ( n (.g `  ( (mulGrp `  Z )s  (Unit `  Z )
) ) ( w `
 k ) ) ) ) )  =  (Unit `  Z )
) )  ->  E. x  e.  D  ( x `  A )  =/=  1
)
2710, 11unitgrpbas 17189 . . . 4  |-  (Unit `  Z )  =  (
Base `  ( (mulGrp `  Z )s  (Unit `  Z )
) )
28 eqid 2443 . . . 4  |-  { u  e.  (SubGrp `  ( (mulGrp `  Z )s  (Unit `  Z )
) )  |  ( ( (mulGrp `  Z
)s  (Unit `  Z )
)s  u )  e.  (CycGrp 
i^i  ran pGrp  ) }  =  { u  e.  (SubGrp `  ( (mulGrp `  Z
)s  (Unit `  Z )
) )  |  ( ( (mulGrp `  Z
)s  (Unit `  Z )
)s  u )  e.  (CycGrp 
i^i  ran pGrp  ) }
296nnnn0d 10858 . . . . . 6  |-  ( ph  ->  N  e.  NN0 )
302zncrng 18456 . . . . . 6  |-  ( N  e.  NN0  ->  Z  e. 
CRing )
3110, 11unitabl 17191 . . . . . 6  |-  ( Z  e.  CRing  ->  ( (mulGrp `  Z )s  (Unit `  Z )
)  e.  Abel )
3229, 30, 313syl 20 . . . . 5  |-  ( ph  ->  ( (mulGrp `  Z
)s  (Unit `  Z )
)  e.  Abel )
3332adantr 465 . . . 4  |-  ( (
ph  /\  A  e.  (Unit `  Z ) )  ->  ( (mulGrp `  Z )s  (Unit `  Z )
)  e.  Abel )
342, 4znfi 18471 . . . . . . 7  |-  ( N  e.  NN  ->  B  e.  Fin )
356, 34syl 16 . . . . . 6  |-  ( ph  ->  B  e.  Fin )
364, 10unitss 17183 . . . . . 6  |-  (Unit `  Z )  C_  B
37 ssfi 7742 . . . . . 6  |-  ( ( B  e.  Fin  /\  (Unit `  Z )  C_  B )  ->  (Unit `  Z )  e.  Fin )
3835, 36, 37sylancl 662 . . . . 5  |-  ( ph  ->  (Unit `  Z )  e.  Fin )
3938adantr 465 . . . 4  |-  ( (
ph  /\  A  e.  (Unit `  Z ) )  ->  (Unit `  Z
)  e.  Fin )
40 eqid 2443 . . . 4  |-  ( k  e.  dom  w  |->  ran  ( n  e.  ZZ  |->  ( n (.g `  (
(mulGrp `  Z )s  (Unit `  Z ) ) ) ( w `  k
) ) ) )  =  ( k  e. 
dom  w  |->  ran  (
n  e.  ZZ  |->  ( n (.g `  ( (mulGrp `  Z )s  (Unit `  Z )
) ) ( w `
 k ) ) ) )
4127, 28, 33, 39, 12, 40ablfac2 17014 . . 3  |-  ( (
ph  /\  A  e.  (Unit `  Z ) )  ->  E. w  e. Word  (Unit `  Z ) ( ( k  e.  dom  w  |->  ran  ( n  e.  ZZ  |->  ( n (.g `  ( (mulGrp `  Z
)s  (Unit `  Z )
) ) ( w `
 k ) ) ) ) : dom  w
--> { u  e.  (SubGrp `  ( (mulGrp `  Z
)s  (Unit `  Z )
) )  |  ( ( (mulGrp `  Z
)s  (Unit `  Z )
)s  u )  e.  (CycGrp 
i^i  ran pGrp  ) }  /\  ( (mulGrp `  Z )s  (Unit `  Z ) ) dom DProd  ( k  e.  dom  w  |->  ran  ( n  e.  ZZ  |->  ( n (.g `  ( (mulGrp `  Z
)s  (Unit `  Z )
) ) ( w `
 k ) ) ) )  /\  (
( (mulGrp `  Z
)s  (Unit `  Z )
) DProd  ( k  e. 
dom  w  |->  ran  (
n  e.  ZZ  |->  ( n (.g `  ( (mulGrp `  Z )s  (Unit `  Z )
) ) ( w `
 k ) ) ) ) )  =  (Unit `  Z )
) )
4226, 41r19.29a 2985 . 2  |-  ( (
ph  /\  A  e.  (Unit `  Z ) )  ->  E. x  e.  D  ( x `  A
)  =/=  1 )
431dchrabl 23401 . . . . 5  |-  ( N  e.  NN  ->  G  e.  Abel )
44 ablgrp 16677 . . . . 5  |-  ( G  e.  Abel  ->  G  e. 
Grp )
45 eqid 2443 . . . . . 6  |-  ( 0g
`  G )  =  ( 0g `  G
)
463, 45grpidcl 15952 . . . . 5  |-  ( G  e.  Grp  ->  ( 0g `  G )  e.  D )
476, 43, 44, 464syl 21 . . . 4  |-  ( ph  ->  ( 0g `  G
)  e.  D )
4847adantr 465 . . 3  |-  ( (
ph  /\  -.  A  e.  (Unit `  Z )
)  ->  ( 0g `  G )  e.  D
)
49 0ne1 10609 . . . 4  |-  0  =/=  1
50 dchrpt.a . . . . . . . 8  |-  ( ph  ->  A  e.  B )
511, 2, 3, 4, 10, 47, 50dchrn0 23397 . . . . . . 7  |-  ( ph  ->  ( ( ( 0g
`  G ) `  A )  =/=  0  <->  A  e.  (Unit `  Z
) ) )
5251necon1bbid 2693 . . . . . 6  |-  ( ph  ->  ( -.  A  e.  (Unit `  Z )  <->  ( ( 0g `  G
) `  A )  =  0 ) )
5352biimpa 484 . . . . 5  |-  ( (
ph  /\  -.  A  e.  (Unit `  Z )
)  ->  ( ( 0g `  G ) `  A )  =  0 )
5453neeq1d 2720 . . . 4  |-  ( (
ph  /\  -.  A  e.  (Unit `  Z )
)  ->  ( (
( 0g `  G
) `  A )  =/=  1  <->  0  =/=  1
) )
5549, 54mpbiri 233 . . 3  |-  ( (
ph  /\  -.  A  e.  (Unit `  Z )
)  ->  ( ( 0g `  G ) `  A )  =/=  1
)
56 fveq1 5855 . . . . 5  |-  ( x  =  ( 0g `  G )  ->  (
x `  A )  =  ( ( 0g
`  G ) `  A ) )
5756neeq1d 2720 . . . 4  |-  ( x  =  ( 0g `  G )  ->  (
( x `  A
)  =/=  1  <->  (
( 0g `  G
) `  A )  =/=  1 ) )
5857rspcev 3196 . . 3  |-  ( ( ( 0g `  G
)  e.  D  /\  ( ( 0g `  G ) `  A
)  =/=  1 )  ->  E. x  e.  D  ( x `  A
)  =/=  1 )
5948, 55, 58syl2anc 661 . 2  |-  ( (
ph  /\  -.  A  e.  (Unit `  Z )
)  ->  E. x  e.  D  ( x `  A )  =/=  1
)
6042, 59pm2.61dan 791 1  |-  ( ph  ->  E. x  e.  D  ( x `  A
)  =/=  1 )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804    =/= wne 2638   E.wrex 2794   {crab 2797    i^i cin 3460    C_ wss 3461   class class class wbr 4437    |-> cmpt 4495   dom cdm 4989   ran crn 4990   -->wf 5574   ` cfv 5578  (class class class)co 6281   Fincfn 7518   0cc0 9495   1c1 9496   NNcn 10542   NN0cn0 10801   ZZcz 10870  Word cword 12513   Basecbs 14509   ↾s cress 14510   0gc0g 14714   Grpcgrp 15927  .gcmg 15930  SubGrpcsubg 16069   pGrp cpgp 16425   Abelcabl 16673  CycGrpccyg 16754   DProd cdprd 16898  mulGrpcmgp 17015   1rcur 17027   CRingccrg 17073  Unitcui 17162  ℤ/nczn 18413  DChrcdchr 23379
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-pre-sup 9573  ax-addf 9574  ax-mulf 9575
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-fal 1389  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-iin 4318  df-disj 4408  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-of 6525  df-rpss 6565  df-om 6686  df-1st 6785  df-2nd 6786  df-supp 6904  df-tpos 6957  df-recs 7044  df-rdg 7078  df-1o 7132  df-2o 7133  df-oadd 7136  df-omul 7137  df-er 7313  df-ec 7315  df-qs 7319  df-map 7424  df-pm 7425  df-ixp 7472  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-fsupp 7832  df-fi 7873  df-sup 7903  df-oi 7938  df-card 8323  df-acn 8326  df-cda 8551  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10213  df-nn 10543  df-2 10600  df-3 10601  df-4 10602  df-5 10603  df-6 10604  df-7 10605  df-8 10606  df-9 10607  df-10 10608  df-n0 10802  df-z 10871  df-dec 10985  df-uz 11091  df-q 11192  df-rp 11230  df-xneg 11327  df-xadd 11328  df-xmul 11329  df-ioo 11542  df-ioc 11543  df-ico 11544  df-icc 11545  df-fz 11682  df-fzo 11804  df-fl 11908  df-mod 11976  df-seq 12087  df-exp 12146  df-fac 12333  df-bc 12360  df-hash 12385  df-word 12521  df-concat 12523  df-s1 12524  df-shft 12879  df-cj 12911  df-re 12912  df-im 12913  df-sqrt 13047  df-abs 13048  df-limsup 13273  df-clim 13290  df-rlim 13291  df-sum 13488  df-ef 13681  df-sin 13683  df-cos 13684  df-pi 13686  df-dvds 13864  df-gcd 14022  df-prm 14095  df-pc 14238  df-struct 14511  df-ndx 14512  df-slot 14513  df-base 14514  df-sets 14515  df-ress 14516  df-plusg 14587  df-mulr 14588  df-starv 14589  df-sca 14590  df-vsca 14591  df-ip 14592  df-tset 14593  df-ple 14594  df-ds 14596  df-unif 14597  df-hom 14598  df-cco 14599  df-rest 14697  df-topn 14698  df-0g 14716  df-gsum 14717  df-topgen 14718  df-pt 14719  df-prds 14722  df-xrs 14776  df-qtop 14781  df-imas 14782  df-qus 14783  df-xps 14784  df-mre 14860  df-mrc 14861  df-acs 14863  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-mhm 15840  df-submnd 15841  df-grp 15931  df-minusg 15932  df-sbg 15933  df-mulg 15934  df-subg 16072  df-nsg 16073  df-eqg 16074  df-ghm 16139  df-gim 16181  df-ga 16202  df-cntz 16229  df-oppg 16255  df-od 16427  df-gex 16428  df-pgp 16429  df-lsm 16530  df-pj1 16531  df-cmn 16674  df-abl 16675  df-cyg 16755  df-dprd 16900  df-dpj 16901  df-mgp 17016  df-ur 17028  df-ring 17074  df-cring 17075  df-oppr 17146  df-dvdsr 17164  df-unit 17165  df-invr 17195  df-rnghom 17238  df-subrg 17301  df-lmod 17388  df-lss 17453  df-lsp 17492  df-sra 17692  df-rgmod 17693  df-lidl 17694  df-rsp 17695  df-2idl 17754  df-psmet 18285  df-xmet 18286  df-met 18287  df-bl 18288  df-mopn 18289  df-fbas 18290  df-fg 18291  df-cnfld 18295  df-zring 18363  df-zrh 18414  df-zn 18417  df-top 19272  df-bases 19274  df-topon 19275  df-topsp 19276  df-cld 19393  df-ntr 19394  df-cls 19395  df-nei 19472  df-lp 19510  df-perf 19511  df-cn 19601  df-cnp 19602  df-haus 19689  df-tx 19936  df-hmeo 20129  df-fil 20220  df-fm 20312  df-flim 20313  df-flf 20314  df-xms 20696  df-ms 20697  df-tms 20698  df-cncf 21255  df-limc 22143  df-dv 22144  df-log 22816  df-cxp 22817  df-dchr 23380
This theorem is referenced by:  sumdchr2  23417
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