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Theorem dchrpt 24194
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 734 . . . . 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 734 . . . . 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 2422 . . . . 5  |-  (Unit `  Z )  =  (Unit `  Z )
11 eqid 2422 . . . . 5  |-  ( (mulGrp `  Z )s  (Unit `  Z )
)  =  ( (mulGrp `  Z )s  (Unit `  Z )
)
12 eqid 2422 . . . . 5  |-  (.g `  (
(mulGrp `  Z )s  (Unit `  Z ) ) )  =  (.g `  ( (mulGrp `  Z )s  (Unit `  Z )
) )
13 oveq1 6313 . . . . . . . . 9  |-  ( n  =  b  ->  (
n (.g `  ( (mulGrp `  Z )s  (Unit `  Z )
) ) ( w `
 k ) )  =  ( b (.g `  ( (mulGrp `  Z
)s  (Unit `  Z )
) ) ( w `
 k ) ) )
1413cbvmptv 4516 . . . . . . . 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 5882 . . . . . . . . . 10  |-  ( k  =  a  ->  (
w `  k )  =  ( w `  a ) )
1615oveq2d 6322 . . . . . . . . 9  |-  ( k  =  a  ->  (
b (.g `  ( (mulGrp `  Z )s  (Unit `  Z )
) ) ( w `
 k ) )  =  ( b (.g `  ( (mulGrp `  Z
)s  (Unit `  Z )
) ) ( w `
 a ) ) )
1716mpteq2dv 4511 . . . . . . . 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 2475 . . . . . . 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 5081 . . . . . 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 4516 . . . . 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 767 . . . . 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 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 )
) )  ->  w  e. Word  (Unit `  Z )
)
23 simprl 762 . . . . 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 764 . . . . 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 24193 . . . 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 1164 . . 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 17894 . . . 4  |-  (Unit `  Z )  =  (
Base `  ( (mulGrp `  Z )s  (Unit `  Z )
) )
28 eqid 2422 . . . 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 10933 . . . . . 6  |-  ( ph  ->  N  e.  NN0 )
302zncrng 19114 . . . . . 6  |-  ( N  e.  NN0  ->  Z  e. 
CRing )
3110, 11unitabl 17896 . . . . . 6  |-  ( Z  e.  CRing  ->  ( (mulGrp `  Z )s  (Unit `  Z )
)  e.  Abel )
3229, 30, 313syl 18 . . . . 5  |-  ( ph  ->  ( (mulGrp `  Z
)s  (Unit `  Z )
)  e.  Abel )
3332adantr 466 . . . 4  |-  ( (
ph  /\  A  e.  (Unit `  Z ) )  ->  ( (mulGrp `  Z )s  (Unit `  Z )
)  e.  Abel )
342, 4znfi 19129 . . . . . . 7  |-  ( N  e.  NN  ->  B  e.  Fin )
356, 34syl 17 . . . . . 6  |-  ( ph  ->  B  e.  Fin )
364, 10unitss 17888 . . . . . 6  |-  (Unit `  Z )  C_  B
37 ssfi 7802 . . . . . 6  |-  ( ( B  e.  Fin  /\  (Unit `  Z )  C_  B )  ->  (Unit `  Z )  e.  Fin )
3835, 36, 37sylancl 666 . . . . 5  |-  ( ph  ->  (Unit `  Z )  e.  Fin )
3938adantr 466 . . . 4  |-  ( (
ph  /\  A  e.  (Unit `  Z ) )  ->  (Unit `  Z
)  e.  Fin )
40 eqid 2422 . . . 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 17722 . . 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 2967 . 2  |-  ( (
ph  /\  A  e.  (Unit `  Z ) )  ->  E. x  e.  D  ( x `  A
)  =/=  1 )
431dchrabl 24181 . . . . 5  |-  ( N  e.  NN  ->  G  e.  Abel )
44 ablgrp 17435 . . . . 5  |-  ( G  e.  Abel  ->  G  e. 
Grp )
45 eqid 2422 . . . . . 6  |-  ( 0g
`  G )  =  ( 0g `  G
)
463, 45grpidcl 16694 . . . . 5  |-  ( G  e.  Grp  ->  ( 0g `  G )  e.  D )
476, 43, 44, 464syl 19 . . . 4  |-  ( ph  ->  ( 0g `  G
)  e.  D )
4847adantr 466 . . 3  |-  ( (
ph  /\  -.  A  e.  (Unit `  Z )
)  ->  ( 0g `  G )  e.  D
)
49 0ne1 10685 . . . 4  |-  0  =/=  1
50 dchrpt.a . . . . . . . 8  |-  ( ph  ->  A  e.  B )
511, 2, 3, 4, 10, 47, 50dchrn0 24177 . . . . . . 7  |-  ( ph  ->  ( ( ( 0g
`  G ) `  A )  =/=  0  <->  A  e.  (Unit `  Z
) ) )
5251necon1bbid 2670 . . . . . 6  |-  ( ph  ->  ( -.  A  e.  (Unit `  Z )  <->  ( ( 0g `  G
) `  A )  =  0 ) )
5352biimpa 486 . . . . 5  |-  ( (
ph  /\  -.  A  e.  (Unit `  Z )
)  ->  ( ( 0g `  G ) `  A )  =  0 )
5453neeq1d 2697 . . . 4  |-  ( (
ph  /\  -.  A  e.  (Unit `  Z )
)  ->  ( (
( 0g `  G
) `  A )  =/=  1  <->  0  =/=  1
) )
5549, 54mpbiri 236 . . 3  |-  ( (
ph  /\  -.  A  e.  (Unit `  Z )
)  ->  ( ( 0g `  G ) `  A )  =/=  1
)
56 fveq1 5881 . . . . 5  |-  ( x  =  ( 0g `  G )  ->  (
x `  A )  =  ( ( 0g
`  G ) `  A ) )
5756neeq1d 2697 . . . 4  |-  ( x  =  ( 0g `  G )  ->  (
( x `  A
)  =/=  1  <->  (
( 0g `  G
) `  A )  =/=  1 ) )
5857rspcev 3182 . . 3  |-  ( ( ( 0g `  G
)  e.  D  /\  ( ( 0g `  G ) `  A
)  =/=  1 )  ->  E. x  e.  D  ( x `  A
)  =/=  1 )
5948, 55, 58syl2anc 665 . 2  |-  ( (
ph  /\  -.  A  e.  (Unit `  Z )
)  ->  E. x  e.  D  ( x `  A )  =/=  1
)
6042, 59pm2.61dan 798 1  |-  ( ph  ->  E. x  e.  D  ( x `  A
)  =/=  1 )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872    =/= wne 2614   E.wrex 2772   {crab 2775    i^i cin 3435    C_ wss 3436   class class class wbr 4423    |-> cmpt 4482   dom cdm 4853   ran crn 4854   -->wf 5597   ` cfv 5601  (class class class)co 6306   Fincfn 7581   0cc0 9547   1c1 9548   NNcn 10617   NN0cn0 10877   ZZcz 10945  Word cword 12661   Basecbs 15121   ↾s cress 15122   0gc0g 15338   Grpcgrp 16669  .gcmg 16672  SubGrpcsubg 16811   pGrp cpgp 17169   Abelcabl 17431  CycGrpccyg 17512   DProd cdprd 17625  mulGrpcmgp 17723   1rcur 17735   CRingccrg 17781  Unitcui 17867  ℤ/nczn 19073  DChrcdchr 24159
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6598  ax-inf2 8156  ax-cnex 9603  ax-resscn 9604  ax-1cn 9605  ax-icn 9606  ax-addcl 9607  ax-addrcl 9608  ax-mulcl 9609  ax-mulrcl 9610  ax-mulcom 9611  ax-addass 9612  ax-mulass 9613  ax-distr 9614  ax-i2m1 9615  ax-1ne0 9616  ax-1rid 9617  ax-rnegex 9618  ax-rrecex 9619  ax-cnre 9620  ax-pre-lttri 9621  ax-pre-lttrn 9622  ax-pre-ltadd 9623  ax-pre-mulgt0 9624  ax-pre-sup 9625  ax-addf 9626  ax-mulf 9627
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rmo 2779  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-int 4256  df-iun 4301  df-iin 4302  df-disj 4395  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-se 4813  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6268  df-ov 6309  df-oprab 6310  df-mpt2 6311  df-of 6546  df-rpss 6586  df-om 6708  df-1st 6808  df-2nd 6809  df-supp 6927  df-tpos 6985  df-wrecs 7040  df-recs 7102  df-rdg 7140  df-1o 7194  df-2o 7195  df-oadd 7198  df-omul 7199  df-er 7375  df-ec 7377  df-qs 7381  df-map 7486  df-pm 7487  df-ixp 7535  df-en 7582  df-dom 7583  df-sdom 7584  df-fin 7585  df-fsupp 7894  df-fi 7935  df-sup 7966  df-inf 7967  df-oi 8035  df-card 8382  df-acn 8385  df-cda 8606  df-pnf 9685  df-mnf 9686  df-xr 9687  df-ltxr 9688  df-le 9689  df-sub 9870  df-neg 9871  df-div 10278  df-nn 10618  df-2 10676  df-3 10677  df-4 10678  df-5 10679  df-6 10680  df-7 10681  df-8 10682  df-9 10683  df-10 10684  df-n0 10878  df-z 10946  df-dec 11060  df-uz 11168  df-q 11273  df-rp 11311  df-xneg 11417  df-xadd 11418  df-xmul 11419  df-ioo 11647  df-ioc 11648  df-ico 11649  df-icc 11650  df-fz 11793  df-fzo 11924  df-fl 12035  df-mod 12104  df-seq 12221  df-exp 12280  df-fac 12467  df-bc 12495  df-hash 12523  df-word 12669  df-concat 12671  df-s1 12672  df-shft 13131  df-cj 13163  df-re 13164  df-im 13165  df-sqrt 13299  df-abs 13300  df-limsup 13526  df-clim 13552  df-rlim 13553  df-sum 13753  df-ef 14121  df-sin 14123  df-cos 14124  df-pi 14126  df-dvds 14306  df-gcd 14469  df-prm 14623  df-pc 14787  df-struct 15123  df-ndx 15124  df-slot 15125  df-base 15126  df-sets 15127  df-ress 15128  df-plusg 15203  df-mulr 15204  df-starv 15205  df-sca 15206  df-vsca 15207  df-ip 15208  df-tset 15209  df-ple 15210  df-ds 15212  df-unif 15213  df-hom 15214  df-cco 15215  df-rest 15321  df-topn 15322  df-0g 15340  df-gsum 15341  df-topgen 15342  df-pt 15343  df-prds 15346  df-xrs 15400  df-qtop 15406  df-imas 15407  df-qus 15409  df-xps 15410  df-mre 15492  df-mrc 15493  df-acs 15495  df-mgm 16488  df-sgrp 16527  df-mnd 16537  df-mhm 16582  df-submnd 16583  df-grp 16673  df-minusg 16674  df-sbg 16675  df-mulg 16676  df-subg 16814  df-nsg 16815  df-eqg 16816  df-ghm 16881  df-gim 16923  df-ga 16944  df-cntz 16971  df-oppg 16997  df-od 17172  df-gex 17174  df-pgp 17176  df-lsm 17288  df-pj1 17289  df-cmn 17432  df-abl 17433  df-cyg 17513  df-dprd 17627  df-dpj 17628  df-mgp 17724  df-ur 17736  df-ring 17782  df-cring 17783  df-oppr 17851  df-dvdsr 17869  df-unit 17870  df-invr 17900  df-rnghom 17943  df-subrg 18006  df-lmod 18093  df-lss 18156  df-lsp 18195  df-sra 18395  df-rgmod 18396  df-lidl 18397  df-rsp 18398  df-2idl 18456  df-psmet 18962  df-xmet 18963  df-met 18964  df-bl 18965  df-mopn 18966  df-fbas 18967  df-fg 18968  df-cnfld 18971  df-zring 19039  df-zrh 19074  df-zn 19077  df-top 19920  df-bases 19921  df-topon 19922  df-topsp 19923  df-cld 20033  df-ntr 20034  df-cls 20035  df-nei 20113  df-lp 20151  df-perf 20152  df-cn 20242  df-cnp 20243  df-haus 20330  df-tx 20576  df-hmeo 20769  df-fil 20860  df-fm 20952  df-flim 20953  df-flf 20954  df-xms 21334  df-ms 21335  df-tms 21336  df-cncf 21909  df-limc 22820  df-dv 22821  df-log 23505  df-cxp 23506  df-dchr 24160
This theorem is referenced by:  sumdchr2  24197
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