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Theorem odf1o1 15133
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 14902 . . . . . . 7  |-  ( G  e.  Grp  ->  (SubGrp `  G )  e.  (ACS
`  X ) )
4 acsmre 13804 . . . . . . 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 3886 . . . . . 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 13763 . . . . . 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 13777 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  =  0 )  /\  x  e.  ZZ )  ->  { A }  C_  ( K `  { A } ) )
13 snidg 3782 . . . . . . 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 3292 . . . . 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 14884 . . . . 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 4163 . . . . . 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 10251 . . . . . . . 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 12797 . . . . . . 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 2387 . . . . . . 7  |-  ( 0g
`  G )  =  ( 0g `  G
)
332, 31, 16, 32odcong 15114 . . . . . 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 10219 . . . . . . 7  |-  ( x  e.  ZZ  ->  x  e.  CC )
36 zcn 10219 . . . . . . 7  |-  ( y  e.  ZZ  ->  y  e.  CC )
37 subeq0 9259 . . . . . . 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 7083 . 2  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  =  0 )  -> 
( x  e.  ZZ  |->  ( x  .x.  A ) ) : ZZ -1-1-> ( K `  { A } ) )
43 f1f 5579 . . . 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 2387 . . . . . 6  |-  ( x  e.  ZZ  |->  ( x 
.x.  A ) )  =  ( x  e.  ZZ  |->  ( x  .x.  A ) )
462, 16, 45, 8cycsubg2 14904 . . . . 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 2392 . . 3  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  =  0 )  ->  ran  ( x  e.  ZZ  |->  ( x  .x.  A ) )  =  ( K `
 { A }
) )
49 dffo2 5597 . . 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 5401 . 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 1649    e. wcel 1717    C_ wss 3263   {csn 3757   class class class wbr 4153    e. cmpt 4207   ran crn 4819   -->wf 5390   -1-1->wf1 5391   -onto->wfo 5392   -1-1-onto->wf1o 5393   ` cfv 5394  (class class class)co 6020   CCcc 8921   0cc0 8923    - cmin 9223   ZZcz 10214    || cdivides 12779   Basecbs 13396   0gc0g 13650  Moorecmre 13734  mrClscmrc 13735  ACScacs 13737   Grpcgrp 14612  .gcmg 14616  SubGrpcsubg 14865   odcod 15090
This theorem is referenced by:  odhash  15135
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-inf2 7529  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000  ax-pre-sup 9001
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-iin 4038  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-1o 6660  df-oadd 6664  df-er 6841  df-en 7046  df-dom 7047  df-sdom 7048  df-fin 7049  df-sup 7381  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-div 9610  df-nn 9933  df-2 9990  df-3 9991  df-n0 10154  df-z 10215  df-uz 10421  df-rp 10545  df-fz 10976  df-fl 11129  df-mod 11178  df-seq 11251  df-exp 11310  df-cj 11831  df-re 11832  df-im 11833  df-sqr 11967  df-abs 11968  df-dvds 12780  df-ndx 13399  df-slot 13400  df-base 13401  df-sets 13402  df-ress 13403  df-plusg 13469  df-0g 13654  df-mre 13738  df-mrc 13739  df-acs 13741  df-mnd 14617  df-submnd 14666  df-grp 14739  df-minusg 14740  df-sbg 14741  df-mulg 14742  df-subg 14868  df-od 15094
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