MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  odf1 Unicode version

Theorem odf1 15153
Description: The multiples of an element with infinite order form an infinite cyclic subgroup of  G. (Contributed by Mario Carneiro, 14-Jan-2015.) (Revised by Mario Carneiro, 23-Sep-2015.)
Hypotheses
Ref Expression
odf1.1  |-  X  =  ( Base `  G
)
odf1.2  |-  O  =  ( od `  G
)
odf1.3  |-  .x.  =  (.g
`  G )
odf1.4  |-  F  =  ( x  e.  ZZ  |->  ( x  .x.  A ) )
Assertion
Ref Expression
odf1  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( ( O `  A )  =  0  <-> 
F : ZZ -1-1-> X
) )
Distinct variable groups:    x, A    x, G    x, O    x,  .x.    x, X
Allowed substitution hint:    F( x)

Proof of Theorem odf1
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 odf1.1 . . . . . . . 8  |-  X  =  ( Base `  G
)
2 odf1.3 . . . . . . . 8  |-  .x.  =  (.g
`  G )
31, 2mulgcl 14862 . . . . . . 7  |-  ( ( G  e.  Grp  /\  x  e.  ZZ  /\  A  e.  X )  ->  (
x  .x.  A )  e.  X )
433expa 1153 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  x  e.  ZZ )  /\  A  e.  X
)  ->  ( x  .x.  A )  e.  X
)
54an32s 780 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  x  e.  ZZ )  ->  ( x  .x.  A )  e.  X
)
6 odf1.4 . . . . 5  |-  F  =  ( x  e.  ZZ  |->  ( x  .x.  A ) )
75, 6fmptd 5852 . . . 4  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  F : ZZ --> X )
87adantr 452 . . 3  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  ( O `  A )  =  0 )  ->  F : ZZ
--> X )
9 oveq1 6047 . . . . . . . . 9  |-  ( x  =  y  ->  (
x  .x.  A )  =  ( y  .x.  A ) )
10 ovex 6065 . . . . . . . . 9  |-  ( x 
.x.  A )  e. 
_V
119, 6, 10fvmpt3i 5768 . . . . . . . 8  |-  ( y  e.  ZZ  ->  ( F `  y )  =  ( y  .x.  A ) )
12 oveq1 6047 . . . . . . . . 9  |-  ( x  =  z  ->  (
x  .x.  A )  =  ( z  .x.  A ) )
1312, 6, 10fvmpt3i 5768 . . . . . . . 8  |-  ( z  e.  ZZ  ->  ( F `  z )  =  ( z  .x.  A ) )
1411, 13eqeqan12d 2419 . . . . . . 7  |-  ( ( y  e.  ZZ  /\  z  e.  ZZ )  ->  ( ( F `  y )  =  ( F `  z )  <-> 
( y  .x.  A
)  =  ( z 
.x.  A ) ) )
1514adantl 453 . . . . . 6  |-  ( ( ( ( G  e. 
Grp  /\  A  e.  X )  /\  ( O `  A )  =  0 )  /\  ( y  e.  ZZ  /\  z  e.  ZZ ) )  ->  ( ( F `  y )  =  ( F `  z )  <->  ( y  .x.  A )  =  ( z  .x.  A ) ) )
16 simplr 732 . . . . . . . 8  |-  ( ( ( ( G  e. 
Grp  /\  A  e.  X )  /\  ( O `  A )  =  0 )  /\  ( y  e.  ZZ  /\  z  e.  ZZ ) )  ->  ( O `  A )  =  0 )
1716breq1d 4182 . . . . . . 7  |-  ( ( ( ( G  e. 
Grp  /\  A  e.  X )  /\  ( O `  A )  =  0 )  /\  ( y  e.  ZZ  /\  z  e.  ZZ ) )  ->  ( ( O `  A )  ||  ( y  -  z
)  <->  0  ||  (
y  -  z ) ) )
18 odf1.2 . . . . . . . . . 10  |-  O  =  ( od `  G
)
19 eqid 2404 . . . . . . . . . 10  |-  ( 0g
`  G )  =  ( 0g `  G
)
201, 18, 2, 19odcong 15142 . . . . . . . . 9  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  ( y  e.  ZZ  /\  z  e.  ZZ ) )  ->  ( ( O `  A )  ||  ( y  -  z
)  <->  ( y  .x.  A )  =  ( z  .x.  A ) ) )
21203expa 1153 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  ( y  e.  ZZ  /\  z  e.  ZZ ) )  -> 
( ( O `  A )  ||  (
y  -  z )  <-> 
( y  .x.  A
)  =  ( z 
.x.  A ) ) )
2221adantlr 696 . . . . . . 7  |-  ( ( ( ( G  e. 
Grp  /\  A  e.  X )  /\  ( O `  A )  =  0 )  /\  ( y  e.  ZZ  /\  z  e.  ZZ ) )  ->  ( ( O `  A )  ||  ( y  -  z
)  <->  ( y  .x.  A )  =  ( z  .x.  A ) ) )
23 zsubcl 10275 . . . . . . . . 9  |-  ( ( y  e.  ZZ  /\  z  e.  ZZ )  ->  ( y  -  z
)  e.  ZZ )
2423adantl 453 . . . . . . . 8  |-  ( ( ( ( G  e. 
Grp  /\  A  e.  X )  /\  ( O `  A )  =  0 )  /\  ( y  e.  ZZ  /\  z  e.  ZZ ) )  ->  ( y  -  z )  e.  ZZ )
25 0dvds 12825 . . . . . . . 8  |-  ( ( y  -  z )  e.  ZZ  ->  (
0  ||  ( y  -  z )  <->  ( y  -  z )  =  0 ) )
2624, 25syl 16 . . . . . . 7  |-  ( ( ( ( G  e. 
Grp  /\  A  e.  X )  /\  ( O `  A )  =  0 )  /\  ( y  e.  ZZ  /\  z  e.  ZZ ) )  ->  ( 0 
||  ( y  -  z )  <->  ( y  -  z )  =  0 ) )
2717, 22, 263bitr3d 275 . . . . . 6  |-  ( ( ( ( G  e. 
Grp  /\  A  e.  X )  /\  ( O `  A )  =  0 )  /\  ( y  e.  ZZ  /\  z  e.  ZZ ) )  ->  ( (
y  .x.  A )  =  ( z  .x.  A )  <->  ( y  -  z )  =  0 ) )
28 zcn 10243 . . . . . . . 8  |-  ( y  e.  ZZ  ->  y  e.  CC )
29 zcn 10243 . . . . . . . 8  |-  ( z  e.  ZZ  ->  z  e.  CC )
30 subeq0 9283 . . . . . . . 8  |-  ( ( y  e.  CC  /\  z  e.  CC )  ->  ( ( y  -  z )  =  0  <-> 
y  =  z ) )
3128, 29, 30syl2an 464 . . . . . . 7  |-  ( ( y  e.  ZZ  /\  z  e.  ZZ )  ->  ( ( y  -  z )  =  0  <-> 
y  =  z ) )
3231adantl 453 . . . . . 6  |-  ( ( ( ( G  e. 
Grp  /\  A  e.  X )  /\  ( O `  A )  =  0 )  /\  ( y  e.  ZZ  /\  z  e.  ZZ ) )  ->  ( (
y  -  z )  =  0  <->  y  =  z ) )
3315, 27, 323bitrd 271 . . . . 5  |-  ( ( ( ( G  e. 
Grp  /\  A  e.  X )  /\  ( O `  A )  =  0 )  /\  ( y  e.  ZZ  /\  z  e.  ZZ ) )  ->  ( ( F `  y )  =  ( F `  z )  <->  y  =  z ) )
3433biimpd 199 . . . 4  |-  ( ( ( ( G  e. 
Grp  /\  A  e.  X )  /\  ( O `  A )  =  0 )  /\  ( y  e.  ZZ  /\  z  e.  ZZ ) )  ->  ( ( F `  y )  =  ( F `  z )  ->  y  =  z ) )
3534ralrimivva 2758 . . 3  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  ( O `  A )  =  0 )  ->  A. y  e.  ZZ  A. z  e.  ZZ  ( ( F `
 y )  =  ( F `  z
)  ->  y  =  z ) )
36 dff13 5963 . . 3  |-  ( F : ZZ -1-1-> X  <->  ( F : ZZ --> X  /\  A. y  e.  ZZ  A. z  e.  ZZ  ( ( F `
 y )  =  ( F `  z
)  ->  y  =  z ) ) )
378, 35, 36sylanbrc 646 . 2  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  ( O `  A )  =  0 )  ->  F : ZZ
-1-1-> X )
381, 18, 2, 19odid 15131 . . . . . 6  |-  ( A  e.  X  ->  (
( O `  A
)  .x.  A )  =  ( 0g `  G ) )
391, 19, 2mulg0 14850 . . . . . 6  |-  ( A  e.  X  ->  (
0  .x.  A )  =  ( 0g `  G ) )
4038, 39eqtr4d 2439 . . . . 5  |-  ( A  e.  X  ->  (
( O `  A
)  .x.  A )  =  ( 0  .x. 
A ) )
4140ad2antlr 708 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  F : ZZ -1-1-> X )  ->  ( ( O `  A )  .x.  A )  =  ( 0  .x.  A ) )
421, 18odcl 15129 . . . . . . 7  |-  ( A  e.  X  ->  ( O `  A )  e.  NN0 )
4342ad2antlr 708 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  F : ZZ -1-1-> X )  ->  ( O `  A )  e.  NN0 )
4443nn0zd 10329 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  F : ZZ -1-1-> X )  ->  ( O `  A )  e.  ZZ )
45 oveq1 6047 . . . . . 6  |-  ( x  =  ( O `  A )  ->  (
x  .x.  A )  =  ( ( O `
 A )  .x.  A ) )
4645, 6, 10fvmpt3i 5768 . . . . 5  |-  ( ( O `  A )  e.  ZZ  ->  ( F `  ( O `  A ) )  =  ( ( O `  A )  .x.  A
) )
4744, 46syl 16 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  F : ZZ -1-1-> X )  ->  ( F `  ( O `  A
) )  =  ( ( O `  A
)  .x.  A )
)
48 0z 10249 . . . . . 6  |-  0  e.  ZZ
4948a1i 11 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  F : ZZ -1-1-> X )  ->  0  e.  ZZ )
50 oveq1 6047 . . . . . 6  |-  ( x  =  0  ->  (
x  .x.  A )  =  ( 0  .x. 
A ) )
5150, 6, 10fvmpt3i 5768 . . . . 5  |-  ( 0  e.  ZZ  ->  ( F `  0 )  =  ( 0  .x. 
A ) )
5249, 51syl 16 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  F : ZZ -1-1-> X )  ->  ( F `  0 )  =  ( 0  .x.  A
) )
5341, 47, 523eqtr4d 2446 . . 3  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  F : ZZ -1-1-> X )  ->  ( F `  ( O `  A
) )  =  ( F `  0 ) )
54 simpr 448 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  F : ZZ -1-1-> X )  ->  F : ZZ
-1-1-> X )
55 f1fveq 5967 . . . 4  |-  ( ( F : ZZ -1-1-> X  /\  ( ( O `  A )  e.  ZZ  /\  0  e.  ZZ ) )  ->  ( ( F `  ( O `  A ) )  =  ( F `  0
)  <->  ( O `  A )  =  0 ) )
5654, 44, 49, 55syl12anc 1182 . . 3  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  F : ZZ -1-1-> X )  ->  ( ( F `  ( O `  A ) )  =  ( F `  0
)  <->  ( O `  A )  =  0 ) )
5753, 56mpbid 202 . 2  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  F : ZZ -1-1-> X )  ->  ( O `  A )  =  0 )
5837, 57impbida 806 1  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( ( O `  A )  =  0  <-> 
F : ZZ -1-1-> X
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2666   class class class wbr 4172    e. cmpt 4226   -->wf 5409   -1-1->wf1 5410   ` cfv 5413  (class class class)co 6040   CCcc 8944   0cc0 8946    - cmin 9247   NN0cn0 10177   ZZcz 10238    || cdivides 12807   Basecbs 13424   0gc0g 13678   Grpcgrp 14640  .gcmg 14644   odcod 15118
This theorem is referenced by:  odinf  15154  odcl2  15156  zrhchr  24313
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
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-iun 4055  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-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-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-sup 7404  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-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-fz 11000  df-fl 11157  df-mod 11206  df-seq 11279  df-exp 11338  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-dvds 12808  df-0g 13682  df-mnd 14645  df-grp 14767  df-minusg 14768  df-sbg 14769  df-mulg 14770  df-od 15122
  Copyright terms: Public domain W3C validator