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Theorem dchrpt 23263
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 eqid 2460 . . . . 5  |-  (Unit `  Z )  =  (Unit `  Z )
2 eqid 2460 . . . . 5  |-  ( (mulGrp `  Z )s  (Unit `  Z )
)  =  ( (mulGrp `  Z )s  (Unit `  Z )
)
31, 2unitgrpbas 17092 . . . 4  |-  (Unit `  Z )  =  (
Base `  ( (mulGrp `  Z )s  (Unit `  Z )
) )
4 eqid 2460 . . . 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  ) }
5 dchrpt.n . . . . . . 7  |-  ( ph  ->  N  e.  NN )
65nnnn0d 10841 . . . . . 6  |-  ( ph  ->  N  e.  NN0 )
7 dchrpt.z . . . . . . 7  |-  Z  =  (ℤ/n `  N )
87zncrng 18343 . . . . . 6  |-  ( N  e.  NN0  ->  Z  e. 
CRing )
91, 2unitabl 17094 . . . . . 6  |-  ( Z  e.  CRing  ->  ( (mulGrp `  Z )s  (Unit `  Z )
)  e.  Abel )
106, 8, 93syl 20 . . . . 5  |-  ( ph  ->  ( (mulGrp `  Z
)s  (Unit `  Z )
)  e.  Abel )
1110adantr 465 . . . 4  |-  ( (
ph  /\  A  e.  (Unit `  Z ) )  ->  ( (mulGrp `  Z )s  (Unit `  Z )
)  e.  Abel )
12 dchrpt.b . . . . . . . 8  |-  B  =  ( Base `  Z
)
137, 12znfi 18358 . . . . . . 7  |-  ( N  e.  NN  ->  B  e.  Fin )
145, 13syl 16 . . . . . 6  |-  ( ph  ->  B  e.  Fin )
1512, 1unitss 17086 . . . . . 6  |-  (Unit `  Z )  C_  B
16 ssfi 7730 . . . . . 6  |-  ( ( B  e.  Fin  /\  (Unit `  Z )  C_  B )  ->  (Unit `  Z )  e.  Fin )
1714, 15, 16sylancl 662 . . . . 5  |-  ( ph  ->  (Unit `  Z )  e.  Fin )
1817adantr 465 . . . 4  |-  ( (
ph  /\  A  e.  (Unit `  Z ) )  ->  (Unit `  Z
)  e.  Fin )
19 eqid 2460 . . . 4  |-  (.g `  (
(mulGrp `  Z )s  (Unit `  Z ) ) )  =  (.g `  ( (mulGrp `  Z )s  (Unit `  Z )
) )
20 eqid 2460 . . . 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 ) ) ) )
213, 4, 11, 18, 19, 20ablfac2 16923 . . 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 )
) )
22 dchrpt.g . . . . . . 7  |-  G  =  (DChr `  N )
23 dchrpt.d . . . . . . 7  |-  D  =  ( Base `  G
)
24 dchrpt.1 . . . . . . 7  |-  .1.  =  ( 1r `  Z )
255ad3antrrr 729 . . . . . . 7  |-  ( ( ( ( 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 )
26 dchrpt.n1 . . . . . . . 8  |-  ( ph  ->  A  =/=  .1.  )
2726ad3antrrr 729 . . . . . . 7  |-  ( ( ( ( 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.  )
28 oveq1 6282 . . . . . . . . . . 11  |-  ( n  =  b  ->  (
n (.g `  ( (mulGrp `  Z )s  (Unit `  Z )
) ) ( w `
 k ) )  =  ( b (.g `  ( (mulGrp `  Z
)s  (Unit `  Z )
) ) ( w `
 k ) ) )
2928cbvmptv 4531 . . . . . . . . . 10  |-  ( n  e.  ZZ  |->  ( n (.g `  ( (mulGrp `  Z )s  (Unit `  Z )
) ) ( w `
 k ) ) )  =  ( b  e.  ZZ  |->  ( b (.g `  ( (mulGrp `  Z )s  (Unit `  Z )
) ) ( w `
 k ) ) )
30 fveq2 5857 . . . . . . . . . . . 12  |-  ( k  =  a  ->  (
w `  k )  =  ( w `  a ) )
3130oveq2d 6291 . . . . . . . . . . 11  |-  ( k  =  a  ->  (
b (.g `  ( (mulGrp `  Z )s  (Unit `  Z )
) ) ( w `
 k ) )  =  ( b (.g `  ( (mulGrp `  Z
)s  (Unit `  Z )
) ) ( w `
 a ) ) )
3231mpteq2dv 4527 . . . . . . . . . 10  |-  ( 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 ) ) ) )
3329, 32syl5eq 2513 . . . . . . . . 9  |-  ( 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 ) ) ) )
3433rneqd 5221 . . . . . . . 8  |-  ( 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 ) ) ) )
3534cbvmptv 4531 . . . . . . 7  |-  ( 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 ) ) ) )
36 simpllr 758 . . . . . . 7  |-  ( ( ( ( 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 )
)
37 simplr 754 . . . . . . 7  |-  ( ( ( ( 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 )
)
38 simprl 755 . . . . . . 7  |-  ( ( ( ( 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 ) ) ) ) )
39 simprr 756 . . . . . . 7  |-  ( ( ( ( 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 )
)
4022, 7, 23, 12, 24, 25, 27, 1, 2, 19, 35, 36, 37, 38, 39dchrptlem3 23262 . . . . . 6  |-  ( ( ( ( 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
)
41403adantr1 1150 . . . . 5  |-  ( ( ( ( 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
)
4241ex 434 . . . 4  |-  ( ( ( 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
) )
4342rexlimdva 2948 . . 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 )
)  ->  E. x  e.  D  ( x `  A )  =/=  1
) )
4421, 43mpd 15 . 2  |-  ( (
ph  /\  A  e.  (Unit `  Z ) )  ->  E. x  e.  D  ( x `  A
)  =/=  1 )
4522dchrabl 23250 . . . . 5  |-  ( N  e.  NN  ->  G  e.  Abel )
46 ablgrp 16592 . . . . 5  |-  ( G  e.  Abel  ->  G  e. 
Grp )
47 eqid 2460 . . . . . 6  |-  ( 0g
`  G )  =  ( 0g `  G
)
4823, 47grpidcl 15872 . . . . 5  |-  ( G  e.  Grp  ->  ( 0g `  G )  e.  D )
495, 45, 46, 484syl 21 . . . 4  |-  ( ph  ->  ( 0g `  G
)  e.  D )
5049adantr 465 . . 3  |-  ( (
ph  /\  -.  A  e.  (Unit `  Z )
)  ->  ( 0g `  G )  e.  D
)
51 0ne1 10592 . . . 4  |-  0  =/=  1
52 dchrpt.a . . . . . . . 8  |-  ( ph  ->  A  e.  B )
5322, 7, 23, 12, 1, 49, 52dchrn0 23246 . . . . . . 7  |-  ( ph  ->  ( ( ( 0g
`  G ) `  A )  =/=  0  <->  A  e.  (Unit `  Z
) ) )
5453necon1bbid 2710 . . . . . 6  |-  ( ph  ->  ( -.  A  e.  (Unit `  Z )  <->  ( ( 0g `  G
) `  A )  =  0 ) )
5554biimpa 484 . . . . 5  |-  ( (
ph  /\  -.  A  e.  (Unit `  Z )
)  ->  ( ( 0g `  G ) `  A )  =  0 )
5655neeq1d 2737 . . . 4  |-  ( (
ph  /\  -.  A  e.  (Unit `  Z )
)  ->  ( (
( 0g `  G
) `  A )  =/=  1  <->  0  =/=  1
) )
5751, 56mpbiri 233 . . 3  |-  ( (
ph  /\  -.  A  e.  (Unit `  Z )
)  ->  ( ( 0g `  G ) `  A )  =/=  1
)
58 fveq1 5856 . . . . 5  |-  ( x  =  ( 0g `  G )  ->  (
x `  A )  =  ( ( 0g
`  G ) `  A ) )
5958neeq1d 2737 . . . 4  |-  ( x  =  ( 0g `  G )  ->  (
( x `  A
)  =/=  1  <->  (
( 0g `  G
) `  A )  =/=  1 ) )
6059rspcev 3207 . . 3  |-  ( ( ( 0g `  G
)  e.  D  /\  ( ( 0g `  G ) `  A
)  =/=  1 )  ->  E. x  e.  D  ( x `  A
)  =/=  1 )
6150, 57, 60syl2anc 661 . 2  |-  ( (
ph  /\  -.  A  e.  (Unit `  Z )
)  ->  E. x  e.  D  ( x `  A )  =/=  1
)
6244, 61pm2.61dan 789 1  |-  ( ph  ->  E. x  e.  D  ( x `  A
)  =/=  1 )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762    =/= wne 2655   E.wrex 2808   {crab 2811    i^i cin 3468    C_ wss 3469   class class class wbr 4440    |-> cmpt 4498   dom cdm 4992   ran crn 4993   -->wf 5575   ` cfv 5579  (class class class)co 6275   Fincfn 7506   0cc0 9481   1c1 9482   NNcn 10525   NN0cn0 10784   ZZcz 10853  Word cword 12487   Basecbs 14479   ↾s cress 14480   0gc0g 14684   Grpcgrp 15716  .gcmg 15720  SubGrpcsubg 15983   pGrp cpgp 16340   Abelcabel 16588  CycGrpccyg 16664   DProd cdprd 16808  mulGrpcmgp 16924   1rcur 16936   CRingccrg 16980  Unitcui 17065  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-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-limc 21998  df-dv 21999  df-log 22665  df-cxp 22666  df-dchr 23229
This theorem is referenced by:  sumdchr2  23266
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