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Theorem dchrpt 21004
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 2404 . . . . 5  |-  (Unit `  Z )  =  (Unit `  Z )
2 eqid 2404 . . . . 5  |-  ( (mulGrp `  Z )s  (Unit `  Z )
)  =  ( (mulGrp `  Z )s  (Unit `  Z )
)
31, 2unitgrpbas 15726 . . . 4  |-  (Unit `  Z )  =  (
Base `  ( (mulGrp `  Z )s  (Unit `  Z )
) )
4 eqid 2404 . . . 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 10230 . . . . . 6  |-  ( ph  ->  N  e.  NN0 )
7 dchrpt.z . . . . . . 7  |-  Z  =  (ℤ/n `  N )
87zncrng 16780 . . . . . 6  |-  ( N  e.  NN0  ->  Z  e. 
CRing )
91, 2unitabl 15728 . . . . . 6  |-  ( Z  e.  CRing  ->  ( (mulGrp `  Z )s  (Unit `  Z )
)  e.  Abel )
106, 8, 93syl 19 . . . . 5  |-  ( ph  ->  ( (mulGrp `  Z
)s  (Unit `  Z )
)  e.  Abel )
1110adantr 452 . . . 4  |-  ( (
ph  /\  A  e.  (Unit `  Z ) )  ->  ( (mulGrp `  Z )s  (Unit `  Z )
)  e.  Abel )
12 dchrpt.b . . . . . . . 8  |-  B  =  ( Base `  Z
)
137, 12znfi 16795 . . . . . . 7  |-  ( N  e.  NN  ->  B  e.  Fin )
145, 13syl 16 . . . . . 6  |-  ( ph  ->  B  e.  Fin )
1512, 1unitss 15720 . . . . . 6  |-  (Unit `  Z )  C_  B
16 ssfi 7288 . . . . . 6  |-  ( ( B  e.  Fin  /\  (Unit `  Z )  C_  B )  ->  (Unit `  Z )  e.  Fin )
1714, 15, 16sylancl 644 . . . . 5  |-  ( ph  ->  (Unit `  Z )  e.  Fin )
1817adantr 452 . . . 4  |-  ( (
ph  /\  A  e.  (Unit `  Z ) )  ->  (Unit `  Z
)  e.  Fin )
19 eqid 2404 . . . 4  |-  (.g `  (
(mulGrp `  Z )s  (Unit `  Z ) ) )  =  (.g `  ( (mulGrp `  Z )s  (Unit `  Z )
) )
20 eqid 2404 . . . 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 15602 . . 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 711 . . . . . . 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 711 . . . . . . 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 6047 . . . . . . . . . . 11  |-  ( n  =  b  ->  (
n (.g `  ( (mulGrp `  Z )s  (Unit `  Z )
) ) ( w `
 k ) )  =  ( b (.g `  ( (mulGrp `  Z
)s  (Unit `  Z )
) ) ( w `
 k ) ) )
2928cbvmptv 4260 . . . . . . . . . 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 5687 . . . . . . . . . . . 12  |-  ( k  =  a  ->  (
w `  k )  =  ( w `  a ) )
3130oveq2d 6056 . . . . . . . . . . 11  |-  ( k  =  a  ->  (
b (.g `  ( (mulGrp `  Z )s  (Unit `  Z )
) ) ( w `
 k ) )  =  ( b (.g `  ( (mulGrp `  Z
)s  (Unit `  Z )
) ) ( w `
 a ) ) )
3231mpteq2dv 4256 . . . . . . . . . 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 2448 . . . . . . . . 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 5056 . . . . . . . 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 4260 . . . . . . 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 736 . . . . . . 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 732 . . . . . . 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 733 . . . . . . 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 734 . . . . . . 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 21003 . . . . . 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 1116 . . . . 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 424 . . . 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 2790 . . 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 20991 . . . . . 6  |-  ( N  e.  NN  ->  G  e.  Abel )
46 ablgrp 15372 . . . . . 6  |-  ( G  e.  Abel  ->  G  e. 
Grp )
475, 45, 463syl 19 . . . . 5  |-  ( ph  ->  G  e.  Grp )
48 eqid 2404 . . . . . 6  |-  ( 0g
`  G )  =  ( 0g `  G
)
4923, 48grpidcl 14788 . . . . 5  |-  ( G  e.  Grp  ->  ( 0g `  G )  e.  D )
5047, 49syl 16 . . . 4  |-  ( ph  ->  ( 0g `  G
)  e.  D )
5150adantr 452 . . 3  |-  ( (
ph  /\  -.  A  e.  (Unit `  Z )
)  ->  ( 0g `  G )  e.  D
)
52 ax-1ne0 9015 . . . . 5  |-  1  =/=  0
5352necomi 2649 . . . 4  |-  0  =/=  1
54 dchrpt.a . . . . . . . 8  |-  ( ph  ->  A  e.  B )
5522, 7, 23, 12, 1, 50, 54dchrn0 20987 . . . . . . 7  |-  ( ph  ->  ( ( ( 0g
`  G ) `  A )  =/=  0  <->  A  e.  (Unit `  Z
) ) )
5655necon1bbid 2621 . . . . . 6  |-  ( ph  ->  ( -.  A  e.  (Unit `  Z )  <->  ( ( 0g `  G
) `  A )  =  0 ) )
5756biimpa 471 . . . . 5  |-  ( (
ph  /\  -.  A  e.  (Unit `  Z )
)  ->  ( ( 0g `  G ) `  A )  =  0 )
5857neeq1d 2580 . . . 4  |-  ( (
ph  /\  -.  A  e.  (Unit `  Z )
)  ->  ( (
( 0g `  G
) `  A )  =/=  1  <->  0  =/=  1
) )
5953, 58mpbiri 225 . . 3  |-  ( (
ph  /\  -.  A  e.  (Unit `  Z )
)  ->  ( ( 0g `  G ) `  A )  =/=  1
)
60 fveq1 5686 . . . . 5  |-  ( x  =  ( 0g `  G )  ->  (
x `  A )  =  ( ( 0g
`  G ) `  A ) )
6160neeq1d 2580 . . . 4  |-  ( x  =  ( 0g `  G )  ->  (
( x `  A
)  =/=  1  <->  (
( 0g `  G
) `  A )  =/=  1 ) )
6261rspcev 3012 . . 3  |-  ( ( ( 0g `  G
)  e.  D  /\  ( ( 0g `  G ) `  A
)  =/=  1 )  ->  E. x  e.  D  ( x `  A
)  =/=  1 )
6351, 59, 62syl2anc 643 . 2  |-  ( (
ph  /\  -.  A  e.  (Unit `  Z )
)  ->  E. x  e.  D  ( x `  A )  =/=  1
)
6444, 63pm2.61dan 767 1  |-  ( ph  ->  E. x  e.  D  ( x `  A
)  =/=  1 )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567   E.wrex 2667   {crab 2670    i^i cin 3279    C_ wss 3280   class class class wbr 4172    e. cmpt 4226   dom cdm 4837   ran crn 4838   -->wf 5409   ` cfv 5413  (class class class)co 6040   Fincfn 7068   0cc0 8946   1c1 8947   NNcn 9956   NN0cn0 10177   ZZcz 10238  Word cword 11672   Basecbs 13424   ↾s cress 13425   0gc0g 13678   Grpcgrp 14640  .gcmg 14644  SubGrpcsubg 14893   pGrp cpgp 15120   Abelcabel 15368  CycGrpccyg 15442   DProd cdprd 15509  mulGrpcmgp 15603   CRingccrg 15616   1rcur 15617  Unitcui 15699  ℤ/nczn 16736  DChrcdchr 20969
This theorem is referenced by:  sumdchr2  21007
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-disj 4143  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  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-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-tpos 6438  df-rpss 6481  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-omul 6688  df-er 6864  df-ec 6866  df-qs 6870  df-map 6979  df-pm 6980  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-oi 7435  df-card 7782  df-acn 7785  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ioo 10876  df-ioc 10877  df-ico 10878  df-icc 10879  df-fz 11000  df-fzo 11091  df-fl 11157  df-mod 11206  df-seq 11279  df-exp 11338  df-fac 11522  df-bc 11549  df-hash 11574  df-word 11678  df-concat 11679  df-s1 11680  df-shft 11837  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-limsup 12220  df-clim 12237  df-rlim 12238  df-sum 12435  df-ef 12625  df-sin 12627  df-cos 12628  df-pi 12630  df-dvds 12808  df-gcd 12962  df-prm 13035  df-pc 13166  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-hom 13508  df-cco 13509  df-rest 13605  df-topn 13606  df-topgen 13622  df-pt 13623  df-prds 13626  df-xrs 13681  df-0g 13682  df-gsum 13683  df-qtop 13688  df-imas 13689  df-divs 13690  df-xps 13691  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-mhm 14693  df-submnd 14694  df-grp 14767  df-minusg 14768  df-sbg 14769  df-mulg 14770  df-subg 14896  df-nsg 14897  df-eqg 14898  df-ghm 14959  df-gim 15001  df-ga 15022  df-cntz 15071  df-oppg 15097  df-od 15122  df-gex 15123  df-pgp 15124  df-lsm 15225  df-pj1 15226  df-cmn 15369  df-abl 15370  df-cyg 15443  df-dprd 15511  df-dpj 15512  df-mgp 15604  df-rng 15618  df-cring 15619  df-ur 15620  df-oppr 15683  df-dvdsr 15701  df-unit 15702  df-invr 15732  df-rnghom 15774  df-subrg 15821  df-lmod 15907  df-lss 15964  df-lsp 16003  df-sra 16199  df-rgmod 16200  df-lidl 16201  df-rsp 16202  df-2idl 16258  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-fbas 16654  df-fg 16655  df-cnfld 16659  df-zrh 16737  df-zn 16740  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-cld 17038  df-ntr 17039  df-cls 17040  df-nei 17117  df-lp 17155  df-perf 17156  df-cn 17245  df-cnp 17246  df-haus 17333  df-tx 17547  df-hmeo 17740  df-fil 17831  df-fm 17923  df-flim 17924  df-flf 17925  df-xms 18303  df-ms 18304  df-tms 18305  df-cncf 18861  df-limc 19706  df-dv 19707  df-log 20407  df-cxp 20408  df-dchr 20970
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