MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  odf1o1 Structured version   Unicode version

Theorem odf1o1 15198
Description: An element with zero order has infinitely many multiples. (Contributed by Stefan O'Rear, 6-Sep-2015.)
Hypotheses
Ref Expression
odf1o1.x  |-  X  =  ( Base `  G
)
odf1o1.t  |-  .x.  =  (.g
`  G )
odf1o1.o  |-  O  =  ( od `  G
)
odf1o1.k  |-  K  =  (mrCls `  (SubGrp `  G
) )
Assertion
Ref Expression
odf1o1  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  =  0 )  -> 
( x  e.  ZZ  |->  ( x  .x.  A ) ) : ZZ -1-1-onto-> ( K `  { A } ) )
Distinct variable groups:    x, A    x, G    x, K    x, O    x,  .x.    x, X

Proof of Theorem odf1o1
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 simpl1 960 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  =  0 )  /\  x  e.  ZZ )  ->  G  e.  Grp )
2 odf1o1.x . . . . . . . 8  |-  X  =  ( Base `  G
)
32subgacs 14967 . . . . . . 7  |-  ( G  e.  Grp  ->  (SubGrp `  G )  e.  (ACS
`  X ) )
4 acsmre 13869 . . . . . . 7  |-  ( (SubGrp `  G )  e.  (ACS
`  X )  -> 
(SubGrp `  G )  e.  (Moore `  X )
)
51, 3, 43syl 19 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  =  0 )  /\  x  e.  ZZ )  ->  (SubGrp `  G )  e.  (Moore `  X )
)
6 simpl2 961 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  =  0 )  /\  x  e.  ZZ )  ->  A  e.  X )
76snssd 3935 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  =  0 )  /\  x  e.  ZZ )  ->  { A }  C_  X )
8 odf1o1.k . . . . . . 7  |-  K  =  (mrCls `  (SubGrp `  G
) )
98mrccl 13828 . . . . . 6  |-  ( ( (SubGrp `  G )  e.  (Moore `  X )  /\  { A }  C_  X )  ->  ( K `  { A } )  e.  (SubGrp `  G ) )
105, 7, 9syl2anc 643 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  =  0 )  /\  x  e.  ZZ )  ->  ( K `  { A } )  e.  (SubGrp `  G ) )
11 simpr 448 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  =  0 )  /\  x  e.  ZZ )  ->  x  e.  ZZ )
125, 8, 7mrcssidd 13842 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  =  0 )  /\  x  e.  ZZ )  ->  { A }  C_  ( K `  { A } ) )
13 snidg 3831 . . . . . . 7  |-  ( A  e.  X  ->  A  e.  { A } )
146, 13syl 16 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  =  0 )  /\  x  e.  ZZ )  ->  A  e.  { A } )
1512, 14sseldd 3341 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  =  0 )  /\  x  e.  ZZ )  ->  A  e.  ( K `
 { A }
) )
16 odf1o1.t . . . . . 6  |-  .x.  =  (.g
`  G )
1716subgmulgcl 14949 . . . . 5  |-  ( ( ( K `  { A } )  e.  (SubGrp `  G )  /\  x  e.  ZZ  /\  A  e.  ( K `  { A } ) )  -> 
( x  .x.  A
)  e.  ( K `
 { A }
) )
1810, 11, 15, 17syl3anc 1184 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  =  0 )  /\  x  e.  ZZ )  ->  ( x  .x.  A
)  e.  ( K `
 { A }
) )
1918ex 424 . . 3  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  =  0 )  -> 
( x  e.  ZZ  ->  ( x  .x.  A
)  e.  ( K `
 { A }
) ) )
20 simpl3 962 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  =  0 )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  ->  ( O `  A )  =  0 )
2120breq1d 4214 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  =  0 )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  ->  ( ( O `  A )  ||  ( x  -  y
)  <->  0  ||  (
x  -  y ) ) )
22 zsubcl 10311 . . . . . . . 8  |-  ( ( x  e.  ZZ  /\  y  e.  ZZ )  ->  ( x  -  y
)  e.  ZZ )
2322adantl 453 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  =  0 )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  ->  ( x  -  y )  e.  ZZ )
24 0dvds 12862 . . . . . . 7  |-  ( ( x  -  y )  e.  ZZ  ->  (
0  ||  ( x  -  y )  <->  ( x  -  y )  =  0 ) )
2523, 24syl 16 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  =  0 )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  ->  ( 0 
||  ( x  -  y )  <->  ( x  -  y )  =  0 ) )
2621, 25bitrd 245 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  =  0 )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  ->  ( ( O `  A )  ||  ( x  -  y
)  <->  ( x  -  y )  =  0 ) )
27 simpl1 960 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  =  0 )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  ->  G  e.  Grp )
28 simpl2 961 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  =  0 )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  ->  A  e.  X )
29 simprl 733 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  =  0 )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  ->  x  e.  ZZ )
30 simprr 734 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  =  0 )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  ->  y  e.  ZZ )
31 odf1o1.o . . . . . . 7  |-  O  =  ( od `  G
)
32 eqid 2435 . . . . . . 7  |-  ( 0g
`  G )  =  ( 0g `  G
)
332, 31, 16, 32odcong 15179 . . . . . 6  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  ->  ( ( O `  A )  ||  ( x  -  y
)  <->  ( x  .x.  A )  =  ( y  .x.  A ) ) )
3427, 28, 29, 30, 33syl112anc 1188 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  =  0 )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  ->  ( ( O `  A )  ||  ( x  -  y
)  <->  ( x  .x.  A )  =  ( y  .x.  A ) ) )
35 zcn 10279 . . . . . . 7  |-  ( x  e.  ZZ  ->  x  e.  CC )
36 zcn 10279 . . . . . . 7  |-  ( y  e.  ZZ  ->  y  e.  CC )
37 subeq0 9319 . . . . . . 7  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( ( x  -  y )  =  0  <-> 
x  =  y ) )
3835, 36, 37syl2an 464 . . . . . 6  |-  ( ( x  e.  ZZ  /\  y  e.  ZZ )  ->  ( ( x  -  y )  =  0  <-> 
x  =  y ) )
3938adantl 453 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  =  0 )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  ->  ( (
x  -  y )  =  0  <->  x  =  y ) )
4026, 34, 393bitr3d 275 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  =  0 )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  ->  ( (
x  .x.  A )  =  ( y  .x.  A )  <->  x  =  y ) )
4140ex 424 . . 3  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  =  0 )  -> 
( ( x  e.  ZZ  /\  y  e.  ZZ )  ->  (
( x  .x.  A
)  =  ( y 
.x.  A )  <->  x  =  y ) ) )
4219, 41dom2lem 7139 . 2  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  =  0 )  -> 
( x  e.  ZZ  |->  ( x  .x.  A ) ) : ZZ -1-1-> ( K `  { A } ) )
43 f1f 5631 . . . 4  |-  ( ( x  e.  ZZ  |->  ( x  .x.  A ) ) : ZZ -1-1-> ( K `  { A } )  ->  (
x  e.  ZZ  |->  ( x  .x.  A ) ) : ZZ --> ( K `
 { A }
) )
4442, 43syl 16 . . 3  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  =  0 )  -> 
( x  e.  ZZ  |->  ( x  .x.  A ) ) : ZZ --> ( K `
 { A }
) )
45 eqid 2435 . . . . . 6  |-  ( x  e.  ZZ  |->  ( x 
.x.  A ) )  =  ( x  e.  ZZ  |->  ( x  .x.  A ) )
462, 16, 45, 8cycsubg2 14969 . . . . 5  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( K `  { A } )  =  ran  ( x  e.  ZZ  |->  ( x  .x.  A ) ) )
47463adant3 977 . . . 4  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  =  0 )  -> 
( K `  { A } )  =  ran  ( x  e.  ZZ  |->  ( x  .x.  A ) ) )
4847eqcomd 2440 . . 3  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  =  0 )  ->  ran  ( x  e.  ZZ  |->  ( x  .x.  A ) )  =  ( K `
 { A }
) )
49 dffo2 5649 . . 3  |-  ( ( x  e.  ZZ  |->  ( x  .x.  A ) ) : ZZ -onto-> ( K `  { A } )  <->  ( (
x  e.  ZZ  |->  ( x  .x.  A ) ) : ZZ --> ( K `
 { A }
)  /\  ran  ( x  e.  ZZ  |->  ( x 
.x.  A ) )  =  ( K `  { A } ) ) )
5044, 48, 49sylanbrc 646 . 2  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  =  0 )  -> 
( x  e.  ZZ  |->  ( x  .x.  A ) ) : ZZ -onto-> ( K `  { A } ) )
51 df-f1o 5453 . 2  |-  ( ( x  e.  ZZ  |->  ( x  .x.  A ) ) : ZZ -1-1-onto-> ( K `  { A } )  <->  ( (
x  e.  ZZ  |->  ( x  .x.  A ) ) : ZZ -1-1-> ( K `  { A } )  /\  (
x  e.  ZZ  |->  ( x  .x.  A ) ) : ZZ -onto-> ( K `  { A } ) ) )
5242, 50, 51sylanbrc 646 1  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  =  0 )  -> 
( x  e.  ZZ  |->  ( x  .x.  A ) ) : ZZ -1-1-onto-> ( K `  { A } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    C_ wss 3312   {csn 3806   class class class wbr 4204    e. cmpt 4258   ran crn 4871   -->wf 5442   -1-1->wf1 5443   -onto->wfo 5444   -1-1-onto->wf1o 5445   ` cfv 5446  (class class class)co 6073   CCcc 8980   0cc0 8982    - cmin 9283   ZZcz 10274    || cdivides 12844   Basecbs 13461   0gc0g 13715  Moorecmre 13799  mrClscmrc 13800  ACScacs 13802   Grpcgrp 14677  .gcmg 14681  SubGrpcsubg 14930   odcod 15155
This theorem is referenced by:  odhash  15200
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-rp 10605  df-fz 11036  df-fl 11194  df-mod 11243  df-seq 11316  df-exp 11375  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-dvds 12845  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-0g 13719  df-mre 13803  df-mrc 13804  df-acs 13806  df-mnd 14682  df-submnd 14731  df-grp 14804  df-minusg 14805  df-sbg 14806  df-mulg 14807  df-subg 14933  df-od 15159
  Copyright terms: Public domain W3C validator