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Theorem odf1o1 16076
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 991 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  =  0 )  /\  x  e.  ZZ )  ->  G  e.  Grp )
2 odf1o1.x . . . . . . . 8  |-  X  =  ( Base `  G
)
32subgacs 15721 . . . . . . 7  |-  ( G  e.  Grp  ->  (SubGrp `  G )  e.  (ACS
`  X ) )
4 acsmre 14595 . . . . . . 7  |-  ( (SubGrp `  G )  e.  (ACS
`  X )  -> 
(SubGrp `  G )  e.  (Moore `  X )
)
51, 3, 43syl 20 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  =  0 )  /\  x  e.  ZZ )  ->  (SubGrp `  G )  e.  (Moore `  X )
)
6 simpl2 992 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  =  0 )  /\  x  e.  ZZ )  ->  A  e.  X )
76snssd 4023 . . . . . 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 14554 . . . . . 6  |-  ( ( (SubGrp `  G )  e.  (Moore `  X )  /\  { A }  C_  X )  ->  ( K `  { A } )  e.  (SubGrp `  G ) )
105, 7, 9syl2anc 661 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  =  0 )  /\  x  e.  ZZ )  ->  ( K `  { A } )  e.  (SubGrp `  G ) )
11 simpr 461 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  =  0 )  /\  x  e.  ZZ )  ->  x  e.  ZZ )
125, 8, 7mrcssidd 14568 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  =  0 )  /\  x  e.  ZZ )  ->  { A }  C_  ( K `  { A } ) )
13 snidg 3908 . . . . . . 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 3362 . . . . 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 15699 . . . . 5  |-  ( ( ( K `  { A } )  e.  (SubGrp `  G )  /\  x  e.  ZZ  /\  A  e.  ( K `  { A } ) )  -> 
( x  .x.  A
)  e.  ( K `
 { A }
) )
1810, 11, 15, 17syl3anc 1218 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  =  0 )  /\  x  e.  ZZ )  ->  ( x  .x.  A
)  e.  ( K `
 { A }
) )
1918ex 434 . . 3  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  =  0 )  -> 
( x  e.  ZZ  ->  ( x  .x.  A
)  e.  ( K `
 { A }
) ) )
20 simpl3 993 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  =  0 )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  ->  ( O `  A )  =  0 )
2120breq1d 4307 . . . . . 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 10692 . . . . . . . 8  |-  ( ( x  e.  ZZ  /\  y  e.  ZZ )  ->  ( x  -  y
)  e.  ZZ )
2322adantl 466 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  =  0 )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  ->  ( x  -  y )  e.  ZZ )
24 0dvds 13558 . . . . . . 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 253 . . . . 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 991 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  =  0 )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  ->  G  e.  Grp )
28 simpl2 992 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  =  0 )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  ->  A  e.  X )
29 simprl 755 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  =  0 )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  ->  x  e.  ZZ )
30 simprr 756 . . . . . 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 2443 . . . . . . 7  |-  ( 0g
`  G )  =  ( 0g `  G
)
332, 31, 16, 32odcong 16057 . . . . . 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 1222 . . . . 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 10656 . . . . . . 7  |-  ( x  e.  ZZ  ->  x  e.  CC )
36 zcn 10656 . . . . . . 7  |-  ( y  e.  ZZ  ->  y  e.  CC )
37 subeq0 9640 . . . . . . 7  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( ( x  -  y )  =  0  <-> 
x  =  y ) )
3835, 36, 37syl2an 477 . . . . . 6  |-  ( ( x  e.  ZZ  /\  y  e.  ZZ )  ->  ( ( x  -  y )  =  0  <-> 
x  =  y ) )
3938adantl 466 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  =  0 )  /\  ( x  e.  ZZ  /\  y  e.  ZZ ) )  ->  ( (
x  -  y )  =  0  <->  x  =  y ) )
4026, 34, 393bitr3d 283 . . . 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 434 . . 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 7354 . 2  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  =  0 )  -> 
( x  e.  ZZ  |->  ( x  .x.  A ) ) : ZZ -1-1-> ( K `  { A } ) )
43 f1f 5611 . . . 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 2443 . . . . . 6  |-  ( x  e.  ZZ  |->  ( x 
.x.  A ) )  =  ( x  e.  ZZ  |->  ( x  .x.  A ) )
462, 16, 45, 8cycsubg2 15723 . . . . 5  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( K `  { A } )  =  ran  ( x  e.  ZZ  |->  ( x  .x.  A ) ) )
47463adant3 1008 . . . 4  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  =  0 )  -> 
( K `  { A } )  =  ran  ( x  e.  ZZ  |->  ( x  .x.  A ) ) )
4847eqcomd 2448 . . 3  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  =  0 )  ->  ran  ( x  e.  ZZ  |->  ( x  .x.  A ) )  =  ( K `
 { A }
) )
49 dffo2 5629 . . 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 664 . 2  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  =  0 )  -> 
( x  e.  ZZ  |->  ( x  .x.  A ) ) : ZZ -onto-> ( K `  { A } ) )
51 df-f1o 5430 . 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 664 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 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    C_ wss 3333   {csn 3882   class class class wbr 4297    e. cmpt 4355   ran crn 4846   -->wf 5419   -1-1->wf1 5420   -onto->wfo 5421   -1-1-onto->wf1o 5422   ` cfv 5423  (class class class)co 6096   CCcc 9285   0cc0 9287    - cmin 9600   ZZcz 10651    || cdivides 13540   Basecbs 14179   0gc0g 14383  Moorecmre 14525  mrClscmrc 14526  ACScacs 14528   Grpcgrp 15415  .gcmg 15419  SubGrpcsubg 15680   odcod 16033
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 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-inf2 7852  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364  ax-pre-sup 9365
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-iin 4179  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-oadd 6929  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-sup 7696  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-nn 10328  df-2 10385  df-3 10386  df-n0 10585  df-z 10652  df-uz 10867  df-rp 10997  df-fz 11443  df-fl 11647  df-mod 11714  df-seq 11812  df-exp 11871  df-cj 12593  df-re 12594  df-im 12595  df-sqr 12729  df-abs 12730  df-dvds 13541  df-ndx 14182  df-slot 14183  df-base 14184  df-sets 14185  df-ress 14186  df-plusg 14256  df-0g 14385  df-mre 14529  df-mrc 14530  df-acs 14532  df-mnd 15420  df-submnd 15470  df-grp 15550  df-minusg 15551  df-sbg 15552  df-mulg 15553  df-subg 15683  df-od 16037
This theorem is referenced by:  odhash  16078
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