MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dchrpt Structured version   Unicode version

Theorem dchrpt 22586
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 2438 . . . . 5  |-  (Unit `  Z )  =  (Unit `  Z )
2 eqid 2438 . . . . 5  |-  ( (mulGrp `  Z )s  (Unit `  Z )
)  =  ( (mulGrp `  Z )s  (Unit `  Z )
)
31, 2unitgrpbas 16748 . . . 4  |-  (Unit `  Z )  =  (
Base `  ( (mulGrp `  Z )s  (Unit `  Z )
) )
4 eqid 2438 . . . 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 10628 . . . . . 6  |-  ( ph  ->  N  e.  NN0 )
7 dchrpt.z . . . . . . 7  |-  Z  =  (ℤ/n `  N )
87zncrng 17957 . . . . . 6  |-  ( N  e.  NN0  ->  Z  e. 
CRing )
91, 2unitabl 16750 . . . . . 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 17972 . . . . . . 7  |-  ( N  e.  NN  ->  B  e.  Fin )
145, 13syl 16 . . . . . 6  |-  ( ph  ->  B  e.  Fin )
1512, 1unitss 16742 . . . . . 6  |-  (Unit `  Z )  C_  B
16 ssfi 7525 . . . . . 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 2438 . . . 4  |-  (.g `  (
(mulGrp `  Z )s  (Unit `  Z ) ) )  =  (.g `  ( (mulGrp `  Z )s  (Unit `  Z )
) )
20 eqid 2438 . . . 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 16580 . . 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 6093 . . . . . . . . . . 11  |-  ( n  =  b  ->  (
n (.g `  ( (mulGrp `  Z )s  (Unit `  Z )
) ) ( w `
 k ) )  =  ( b (.g `  ( (mulGrp `  Z
)s  (Unit `  Z )
) ) ( w `
 k ) ) )
2928cbvmptv 4378 . . . . . . . . . 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 5686 . . . . . . . . . . . 12  |-  ( k  =  a  ->  (
w `  k )  =  ( w `  a ) )
3130oveq2d 6102 . . . . . . . . . . 11  |-  ( k  =  a  ->  (
b (.g `  ( (mulGrp `  Z )s  (Unit `  Z )
) ) ( w `
 k ) )  =  ( b (.g `  ( (mulGrp `  Z
)s  (Unit `  Z )
) ) ( w `
 a ) ) )
3231mpteq2dv 4374 . . . . . . . . . 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 2482 . . . . . . . . 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 5062 . . . . . . . 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 4378 . . . . . . 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 22585 . . . . . 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 1147 . . . . 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 2836 . . 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 22573 . . . . 5  |-  ( N  e.  NN  ->  G  e.  Abel )
46 ablgrp 16273 . . . . 5  |-  ( G  e.  Abel  ->  G  e. 
Grp )
47 eqid 2438 . . . . . 6  |-  ( 0g
`  G )  =  ( 0g `  G
)
4823, 47grpidcl 15557 . . . . 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 10381 . . . 4  |-  0  =/=  1
52 dchrpt.a . . . . . . . 8  |-  ( ph  ->  A  e.  B )
5322, 7, 23, 12, 1, 49, 52dchrn0 22569 . . . . . . 7  |-  ( ph  ->  ( ( ( 0g
`  G ) `  A )  =/=  0  <->  A  e.  (Unit `  Z
) ) )
5453necon1bbid 2660 . . . . . 6  |-  ( ph  ->  ( -.  A  e.  (Unit `  Z )  <->  ( ( 0g `  G
) `  A )  =  0 ) )
5554biimpa 484 . . . . 5  |-  ( (
ph  /\  -.  A  e.  (Unit `  Z )
)  ->  ( ( 0g `  G ) `  A )  =  0 )
5655neeq1d 2616 . . . 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 5685 . . . . 5  |-  ( x  =  ( 0g `  G )  ->  (
x `  A )  =  ( ( 0g
`  G ) `  A ) )
5958neeq1d 2616 . . . 4  |-  ( x  =  ( 0g `  G )  ->  (
( x `  A
)  =/=  1  <->  (
( 0g `  G
) `  A )  =/=  1 ) )
6059rspcev 3068 . . 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 965    = wceq 1369    e. wcel 1756    =/= wne 2601   E.wrex 2711   {crab 2714    i^i cin 3322    C_ wss 3323   class class class wbr 4287    e. cmpt 4345   dom cdm 4835   ran crn 4836   -->wf 5409   ` cfv 5413  (class class class)co 6086   Fincfn 7302   0cc0 9274   1c1 9275   NNcn 10314   NN0cn0 10571   ZZcz 10638  Word cword 12213   Basecbs 14166   ↾s cress 14167   0gc0g 14370   Grpcgrp 15402  .gcmg 15406  SubGrpcsubg 15666   pGrp cpgp 16021   Abelcabel 16269  CycGrpccyg 16345   DProd cdprd 16465  mulGrpcmgp 16581   1rcur 16593   CRingccrg 16636  Unitcui 16721  ℤ/nczn 17914  DChrcdchr 22551
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352  ax-addf 9353  ax-mulf 9354
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-iin 4169  df-disj 4258  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-of 6315  df-rpss 6355  df-om 6472  df-1st 6572  df-2nd 6573  df-supp 6686  df-tpos 6740  df-recs 6824  df-rdg 6858  df-1o 6912  df-2o 6913  df-oadd 6916  df-omul 6917  df-er 7093  df-ec 7095  df-qs 7099  df-map 7208  df-pm 7209  df-ixp 7256  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-fsupp 7613  df-fi 7653  df-sup 7683  df-oi 7716  df-card 8101  df-acn 8104  df-cda 8329  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-5 10375  df-6 10376  df-7 10377  df-8 10378  df-9 10379  df-10 10380  df-n0 10572  df-z 10639  df-dec 10748  df-uz 10854  df-q 10946  df-rp 10984  df-xneg 11081  df-xadd 11082  df-xmul 11083  df-ioo 11296  df-ioc 11297  df-ico 11298  df-icc 11299  df-fz 11430  df-fzo 11541  df-fl 11634  df-mod 11701  df-seq 11799  df-exp 11858  df-fac 12044  df-bc 12071  df-hash 12096  df-word 12221  df-concat 12223  df-s1 12224  df-shft 12548  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-limsup 12941  df-clim 12958  df-rlim 12959  df-sum 13156  df-ef 13345  df-sin 13347  df-cos 13348  df-pi 13350  df-dvds 13528  df-gcd 13683  df-prm 13756  df-pc 13896  df-struct 14168  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-ress 14173  df-plusg 14243  df-mulr 14244  df-starv 14245  df-sca 14246  df-vsca 14247  df-ip 14248  df-tset 14249  df-ple 14250  df-ds 14252  df-unif 14253  df-hom 14254  df-cco 14255  df-rest 14353  df-topn 14354  df-0g 14372  df-gsum 14373  df-topgen 14374  df-pt 14375  df-prds 14378  df-xrs 14432  df-qtop 14437  df-imas 14438  df-divs 14439  df-xps 14440  df-mre 14516  df-mrc 14517  df-acs 14519  df-mnd 15407  df-mhm 15456  df-submnd 15457  df-grp 15536  df-minusg 15537  df-sbg 15538  df-mulg 15539  df-subg 15669  df-nsg 15670  df-eqg 15671  df-ghm 15736  df-gim 15778  df-ga 15799  df-cntz 15826  df-oppg 15852  df-od 16023  df-gex 16024  df-pgp 16025  df-lsm 16126  df-pj1 16127  df-cmn 16270  df-abl 16271  df-cyg 16346  df-dprd 16467  df-dpj 16468  df-mgp 16582  df-ur 16594  df-rng 16637  df-cring 16638  df-oppr 16705  df-dvdsr 16723  df-unit 16724  df-invr 16754  df-rnghom 16796  df-subrg 16843  df-lmod 16930  df-lss 16994  df-lsp 17033  df-sra 17233  df-rgmod 17234  df-lidl 17235  df-rsp 17236  df-2idl 17294  df-psmet 17789  df-xmet 17790  df-met 17791  df-bl 17792  df-mopn 17793  df-fbas 17794  df-fg 17795  df-cnfld 17799  df-zring 17864  df-zrh 17915  df-zn 17918  df-top 18483  df-bases 18485  df-topon 18486  df-topsp 18487  df-cld 18603  df-ntr 18604  df-cls 18605  df-nei 18682  df-lp 18720  df-perf 18721  df-cn 18811  df-cnp 18812  df-haus 18899  df-tx 19115  df-hmeo 19308  df-fil 19399  df-fm 19491  df-flim 19492  df-flf 19493  df-xms 19875  df-ms 19876  df-tms 19877  df-cncf 20434  df-limc 21321  df-dv 21322  df-log 21988  df-cxp 21989  df-dchr 22552
This theorem is referenced by:  sumdchr2  22589
  Copyright terms: Public domain W3C validator