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Theorem odf1o1 17159
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 1008 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  =  0 )  /\  x  e.  ZZ )  ->  G  e.  Grp )
2 odf1o1.x . . . . . . . 8  |-  X  =  ( Base `  G
)
32subgacs 16803 . . . . . . 7  |-  ( G  e.  Grp  ->  (SubGrp `  G )  e.  (ACS
`  X ) )
4 acsmre 15509 . . . . . . 7  |-  ( (SubGrp `  G )  e.  (ACS
`  X )  -> 
(SubGrp `  G )  e.  (Moore `  X )
)
51, 3, 43syl 18 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  =  0 )  /\  x  e.  ZZ )  ->  (SubGrp `  G )  e.  (Moore `  X )
)
6 simpl2 1009 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  =  0 )  /\  x  e.  ZZ )  ->  A  e.  X )
76snssd 4148 . . . . . 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 15468 . . . . . 6  |-  ( ( (SubGrp `  G )  e.  (Moore `  X )  /\  { A }  C_  X )  ->  ( K `  { A } )  e.  (SubGrp `  G ) )
105, 7, 9syl2anc 665 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  =  0 )  /\  x  e.  ZZ )  ->  ( K `  { A } )  e.  (SubGrp `  G ) )
11 simpr 462 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  =  0 )  /\  x  e.  ZZ )  ->  x  e.  ZZ )
125, 8, 7mrcssidd 15482 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  =  0 )  /\  x  e.  ZZ )  ->  { A }  C_  ( K `  { A } ) )
13 snidg 4028 . . . . . . 7  |-  ( A  e.  X  ->  A  e.  { A } )
146, 13syl 17 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  =  0 )  /\  x  e.  ZZ )  ->  A  e.  { A } )
1512, 14sseldd 3471 . . . . 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 16781 . . . . 5  |-  ( ( ( K `  { A } )  e.  (SubGrp `  G )  /\  x  e.  ZZ  /\  A  e.  ( K `  { A } ) )  -> 
( x  .x.  A
)  e.  ( K `
 { A }
) )
1810, 11, 15, 17syl3anc 1264 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  =  0 )  /\  x  e.  ZZ )  ->  ( x  .x.  A
)  e.  ( K `
 { A }
) )
1918ex 435 . . 3  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  =  0 )  -> 
( x  e.  ZZ  ->  ( x  .x.  A
)  e.  ( K `
 { A }
) ) )
20 simpl3 1010 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  =  0 )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  ->  ( O `  A )  =  0 )
2120breq1d 4436 . . . . . 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 10979 . . . . . . . 8  |-  ( ( x  e.  ZZ  /\  y  e.  ZZ )  ->  ( x  -  y
)  e.  ZZ )
2322adantl 467 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  =  0 )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  ->  ( x  -  y )  e.  ZZ )
24 0dvds 14301 . . . . . . 7  |-  ( ( x  -  y )  e.  ZZ  ->  (
0  ||  ( x  -  y )  <->  ( x  -  y )  =  0 ) )
2523, 24syl 17 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  =  0 )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  ->  ( 0 
||  ( x  -  y )  <->  ( x  -  y )  =  0 ) )
2621, 25bitrd 256 . . . . 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 1008 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  =  0 )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  ->  G  e.  Grp )
28 simpl2 1009 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  =  0 )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  ->  A  e.  X )
29 simprl 762 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  =  0 )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  ->  x  e.  ZZ )
30 simprr 764 . . . . . 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 2429 . . . . . . 7  |-  ( 0g
`  G )  =  ( 0g `  G
)
332, 31, 16, 32odcong 17140 . . . . . 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 1268 . . . . 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 10942 . . . . . . 7  |-  ( x  e.  ZZ  ->  x  e.  CC )
36 zcn 10942 . . . . . . 7  |-  ( y  e.  ZZ  ->  y  e.  CC )
37 subeq0 9899 . . . . . . 7  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( ( x  -  y )  =  0  <-> 
x  =  y ) )
3835, 36, 37syl2an 479 . . . . . 6  |-  ( ( x  e.  ZZ  /\  y  e.  ZZ )  ->  ( ( x  -  y )  =  0  <-> 
x  =  y ) )
3938adantl 467 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  =  0 )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  ->  ( (
x  -  y )  =  0  <->  x  =  y ) )
4026, 34, 393bitr3d 286 . . . 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 435 . . 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 7616 . 2  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  =  0 )  -> 
( x  e.  ZZ  |->  ( x  .x.  A ) ) : ZZ -1-1-> ( K `  { A } ) )
43 f1f 5796 . . . 4  |-  ( ( x  e.  ZZ  |->  ( x  .x.  A ) ) : ZZ -1-1-> ( K `  { A } )  ->  (
x  e.  ZZ  |->  ( x  .x.  A ) ) : ZZ --> ( K `
 { A }
) )
4442, 43syl 17 . . 3  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  =  0 )  -> 
( x  e.  ZZ  |->  ( x  .x.  A ) ) : ZZ --> ( K `
 { A }
) )
45 eqid 2429 . . . . . 6  |-  ( x  e.  ZZ  |->  ( x 
.x.  A ) )  =  ( x  e.  ZZ  |->  ( x  .x.  A ) )
462, 16, 45, 8cycsubg2 16805 . . . . 5  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( K `  { A } )  =  ran  ( x  e.  ZZ  |->  ( x  .x.  A ) ) )
47463adant3 1025 . . . 4  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  =  0 )  -> 
( K `  { A } )  =  ran  ( x  e.  ZZ  |->  ( x  .x.  A ) ) )
4847eqcomd 2437 . . 3  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  =  0 )  ->  ran  ( x  e.  ZZ  |->  ( x  .x.  A ) )  =  ( K `
 { A }
) )
49 dffo2 5814 . . 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 668 . 2  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  =  0 )  -> 
( x  e.  ZZ  |->  ( x  .x.  A ) ) : ZZ -onto-> ( K `  { A } ) )
51 df-f1o 5608 . 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 668 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 setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870    C_ wss 3442   {csn 4002   class class class wbr 4426    |-> cmpt 4484   ran crn 4855   -->wf 5597   -1-1->wf1 5598   -onto->wfo 5599   -1-1-onto->wf1o 5600   ` cfv 5601  (class class class)co 6305   CCcc 9536   0cc0 9538    - cmin 9859   ZZcz 10937    || cdvds 14283   Basecbs 15084   0gc0g 15297  Moorecmre 15439  mrClscmrc 15440  ACScacs 15442   Grpcgrp 16620  .gcmg 16623  SubGrpcsubg 16762   odcod 17116
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-inf2 8146  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-pre-sup 9616
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-iin 4305  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  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-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-oadd 7194  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-sup 7962  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-rp 11303  df-fz 11783  df-fl 12025  df-mod 12094  df-seq 12211  df-exp 12270  df-cj 13141  df-re 13142  df-im 13143  df-sqrt 13277  df-abs 13278  df-dvds 14284  df-ndx 15087  df-slot 15088  df-base 15089  df-sets 15090  df-ress 15091  df-plusg 15165  df-0g 15299  df-mre 15443  df-mrc 15444  df-acs 15446  df-mgm 16439  df-sgrp 16478  df-mnd 16488  df-submnd 16534  df-grp 16624  df-minusg 16625  df-sbg 16626  df-mulg 16627  df-subg 16765  df-od 17120
This theorem is referenced by:  odhash  17161
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