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

Theorem odf1 14891
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 14600 . . . . . . 7  |-  ( ( G  e.  Grp  /\  x  e.  ZZ  /\  A  e.  X )  ->  (
x  .x.  A )  e.  X )
433expa 1151 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  x  e.  ZZ )  /\  A  e.  X
)  ->  ( x  .x.  A )  e.  X
)
54an32s 779 . . . . 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 5700 . . . 4  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  F : ZZ --> X )
87adantr 451 . . 3  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  ( O `  A )  =  0 )  ->  F : ZZ
--> X )
9 oveq1 5881 . . . . . . . . 9  |-  ( x  =  y  ->  (
x  .x.  A )  =  ( y  .x.  A ) )
10 ovex 5899 . . . . . . . . 9  |-  ( x 
.x.  A )  e. 
_V
119, 6, 10fvmpt3i 5621 . . . . . . . 8  |-  ( y  e.  ZZ  ->  ( F `  y )  =  ( y  .x.  A ) )
12 oveq1 5881 . . . . . . . . 9  |-  ( x  =  z  ->  (
x  .x.  A )  =  ( z  .x.  A ) )
1312, 6, 10fvmpt3i 5621 . . . . . . . 8  |-  ( z  e.  ZZ  ->  ( F `  z )  =  ( z  .x.  A ) )
1411, 13eqeqan12d 2311 . . . . . . 7  |-  ( ( y  e.  ZZ  /\  z  e.  ZZ )  ->  ( ( F `  y )  =  ( F `  z )  <-> 
( y  .x.  A
)  =  ( z 
.x.  A ) ) )
1514adantl 452 . . . . . 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 731 . . . . . . . 8  |-  ( ( ( ( G  e. 
Grp  /\  A  e.  X )  /\  ( O `  A )  =  0 )  /\  ( y  e.  ZZ  /\  z  e.  ZZ ) )  ->  ( O `  A )  =  0 )
1716breq1d 4049 . . . . . . 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 2296 . . . . . . . . . 10  |-  ( 0g
`  G )  =  ( 0g `  G
)
201, 18, 2, 19odcong 14880 . . . . . . . . 9  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  ( y  e.  ZZ  /\  z  e.  ZZ ) )  ->  ( ( O `  A )  ||  ( y  -  z
)  <->  ( y  .x.  A )  =  ( z  .x.  A ) ) )
21203expa 1151 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  ( y  e.  ZZ  /\  z  e.  ZZ ) )  -> 
( ( O `  A )  ||  (
y  -  z )  <-> 
( y  .x.  A
)  =  ( z 
.x.  A ) ) )
2221adantlr 695 . . . . . . 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 10077 . . . . . . . . 9  |-  ( ( y  e.  ZZ  /\  z  e.  ZZ )  ->  ( y  -  z
)  e.  ZZ )
2423adantl 452 . . . . . . . 8  |-  ( ( ( ( G  e. 
Grp  /\  A  e.  X )  /\  ( O `  A )  =  0 )  /\  ( y  e.  ZZ  /\  z  e.  ZZ ) )  ->  ( y  -  z )  e.  ZZ )
25 0dvds 12565 . . . . . . . 8  |-  ( ( y  -  z )  e.  ZZ  ->  (
0  ||  ( y  -  z )  <->  ( y  -  z )  =  0 ) )
2624, 25syl 15 . . . . . . 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 274 . . . . . 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 10045 . . . . . . . 8  |-  ( y  e.  ZZ  ->  y  e.  CC )
29 zcn 10045 . . . . . . . 8  |-  ( z  e.  ZZ  ->  z  e.  CC )
30 subeq0 9089 . . . . . . . 8  |-  ( ( y  e.  CC  /\  z  e.  CC )  ->  ( ( y  -  z )  =  0  <-> 
y  =  z ) )
3128, 29, 30syl2an 463 . . . . . . 7  |-  ( ( y  e.  ZZ  /\  z  e.  ZZ )  ->  ( ( y  -  z )  =  0  <-> 
y  =  z ) )
3231adantl 452 . . . . . 6  |-  ( ( ( ( G  e. 
Grp  /\  A  e.  X )  /\  ( O `  A )  =  0 )  /\  ( y  e.  ZZ  /\  z  e.  ZZ ) )  ->  ( (
y  -  z )  =  0  <->  y  =  z ) )
3315, 27, 323bitrd 270 . . . . 5  |-  ( ( ( ( G  e. 
Grp  /\  A  e.  X )  /\  ( O `  A )  =  0 )  /\  ( y  e.  ZZ  /\  z  e.  ZZ ) )  ->  ( ( F `  y )  =  ( F `  z )  <->  y  =  z ) )
3433biimpd 198 . . . 4  |-  ( ( ( ( G  e. 
Grp  /\  A  e.  X )  /\  ( O `  A )  =  0 )  /\  ( y  e.  ZZ  /\  z  e.  ZZ ) )  ->  ( ( F `  y )  =  ( F `  z )  ->  y  =  z ) )
3534ralrimivva 2648 . . 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 5799 . . 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 645 . 2  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  ( O `  A )  =  0 )  ->  F : ZZ
-1-1-> X )
381, 18, 2, 19odid 14869 . . . . . 6  |-  ( A  e.  X  ->  (
( O `  A
)  .x.  A )  =  ( 0g `  G ) )
391, 19, 2mulg0 14588 . . . . . 6  |-  ( A  e.  X  ->  (
0  .x.  A )  =  ( 0g `  G ) )
4038, 39eqtr4d 2331 . . . . 5  |-  ( A  e.  X  ->  (
( O `  A
)  .x.  A )  =  ( 0  .x. 
A ) )
4140ad2antlr 707 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  F : ZZ -1-1-> X )  ->  ( ( O `  A )  .x.  A )  =  ( 0  .x.  A ) )
421, 18odcl 14867 . . . . . . 7  |-  ( A  e.  X  ->  ( O `  A )  e.  NN0 )
4342ad2antlr 707 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  F : ZZ -1-1-> X )  ->  ( O `  A )  e.  NN0 )
4443nn0zd 10131 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  F : ZZ -1-1-> X )  ->  ( O `  A )  e.  ZZ )
45 oveq1 5881 . . . . . 6  |-  ( x  =  ( O `  A )  ->  (
x  .x.  A )  =  ( ( O `
 A )  .x.  A ) )
4645, 6, 10fvmpt3i 5621 . . . . 5  |-  ( ( O `  A )  e.  ZZ  ->  ( F `  ( O `  A ) )  =  ( ( O `  A )  .x.  A
) )
4744, 46syl 15 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  F : ZZ -1-1-> X )  ->  ( F `  ( O `  A
) )  =  ( ( O `  A
)  .x.  A )
)
48 0z 10051 . . . . . 6  |-  0  e.  ZZ
4948a1i 10 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  F : ZZ -1-1-> X )  ->  0  e.  ZZ )
50 oveq1 5881 . . . . . 6  |-  ( x  =  0  ->  (
x  .x.  A )  =  ( 0  .x. 
A ) )
5150, 6, 10fvmpt3i 5621 . . . . 5  |-  ( 0  e.  ZZ  ->  ( F `  0 )  =  ( 0  .x. 
A ) )
5249, 51syl 15 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  F : ZZ -1-1-> X )  ->  ( F `  0 )  =  ( 0  .x.  A
) )
5341, 47, 523eqtr4d 2338 . . 3  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  F : ZZ -1-1-> X )  ->  ( F `  ( O `  A
) )  =  ( F `  0 ) )
54 simpr 447 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  F : ZZ -1-1-> X )  ->  F : ZZ
-1-1-> X )
55 f1fveq 5802 . . . 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 1180 . . 3  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  F : ZZ -1-1-> X )  ->  ( ( F `  ( O `  A ) )  =  ( F `  0
)  <->  ( O `  A )  =  0 ) )
5753, 56mpbid 201 . 2  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  F : ZZ -1-1-> X )  ->  ( O `  A )  =  0 )
5837, 57impbida 805 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 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   class class class wbr 4039    e. cmpt 4093   -->wf 5267   -1-1->wf1 5268   ` cfv 5271  (class class class)co 5874   CCcc 8751   0cc0 8753    - cmin 9053   NN0cn0 9981   ZZcz 10040    || cdivides 12547   Basecbs 13164   0gc0g 13416   Grpcgrp 14378  .gcmg 14382   odcod 14856
This theorem is referenced by:  odinf  14892  odcl2  14894
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-fz 10799  df-fl 10941  df-mod 10990  df-seq 11063  df-exp 11121  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-dvds 12548  df-0g 13420  df-mnd 14383  df-grp 14505  df-minusg 14506  df-sbg 14507  df-mulg 14508  df-od 14860
  Copyright terms: Public domain W3C validator