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Theorem odf1 16701
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 16276 . . . . . . 7  |-  ( ( G  e.  Grp  /\  x  e.  ZZ  /\  A  e.  X )  ->  (
x  .x.  A )  e.  X )
433expa 1194 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  x  e.  ZZ )  /\  A  e.  X
)  ->  ( x  .x.  A )  e.  X
)
54an32s 802 . . . . 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 5957 . . . 4  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  F : ZZ --> X )
87adantr 463 . . 3  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  ( O `  A )  =  0 )  ->  F : ZZ
--> X )
9 oveq1 6203 . . . . . . . . 9  |-  ( x  =  y  ->  (
x  .x.  A )  =  ( y  .x.  A ) )
10 ovex 6224 . . . . . . . . 9  |-  ( x 
.x.  A )  e. 
_V
119, 6, 10fvmpt3i 5861 . . . . . . . 8  |-  ( y  e.  ZZ  ->  ( F `  y )  =  ( y  .x.  A ) )
12 oveq1 6203 . . . . . . . . 9  |-  ( x  =  z  ->  (
x  .x.  A )  =  ( z  .x.  A ) )
1312, 6, 10fvmpt3i 5861 . . . . . . . 8  |-  ( z  e.  ZZ  ->  ( F `  z )  =  ( z  .x.  A ) )
1411, 13eqeqan12d 2405 . . . . . . 7  |-  ( ( y  e.  ZZ  /\  z  e.  ZZ )  ->  ( ( F `  y )  =  ( F `  z )  <-> 
( y  .x.  A
)  =  ( z 
.x.  A ) ) )
1514adantl 464 . . . . . 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 753 . . . . . . . 8  |-  ( ( ( ( G  e. 
Grp  /\  A  e.  X )  /\  ( O `  A )  =  0 )  /\  ( y  e.  ZZ  /\  z  e.  ZZ ) )  ->  ( O `  A )  =  0 )
1716breq1d 4377 . . . . . . 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 2382 . . . . . . . . . 10  |-  ( 0g
`  G )  =  ( 0g `  G
)
201, 18, 2, 19odcong 16690 . . . . . . . . 9  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  ( y  e.  ZZ  /\  z  e.  ZZ ) )  ->  ( ( O `  A )  ||  ( y  -  z
)  <->  ( y  .x.  A )  =  ( z  .x.  A ) ) )
21203expa 1194 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  ( y  e.  ZZ  /\  z  e.  ZZ ) )  -> 
( ( O `  A )  ||  (
y  -  z )  <-> 
( y  .x.  A
)  =  ( z 
.x.  A ) ) )
2221adantlr 712 . . . . . . 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 10823 . . . . . . . . 9  |-  ( ( y  e.  ZZ  /\  z  e.  ZZ )  ->  ( y  -  z
)  e.  ZZ )
2423adantl 464 . . . . . . . 8  |-  ( ( ( ( G  e. 
Grp  /\  A  e.  X )  /\  ( O `  A )  =  0 )  /\  ( y  e.  ZZ  /\  z  e.  ZZ ) )  ->  ( y  -  z )  e.  ZZ )
25 0dvds 14006 . . . . . . . 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 283 . . . . . 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 10786 . . . . . . . 8  |-  ( y  e.  ZZ  ->  y  e.  CC )
29 zcn 10786 . . . . . . . 8  |-  ( z  e.  ZZ  ->  z  e.  CC )
30 subeq0 9758 . . . . . . . 8  |-  ( ( y  e.  CC  /\  z  e.  CC )  ->  ( ( y  -  z )  =  0  <-> 
y  =  z ) )
3128, 29, 30syl2an 475 . . . . . . 7  |-  ( ( y  e.  ZZ  /\  z  e.  ZZ )  ->  ( ( y  -  z )  =  0  <-> 
y  =  z ) )
3231adantl 464 . . . . . 6  |-  ( ( ( ( G  e. 
Grp  /\  A  e.  X )  /\  ( O `  A )  =  0 )  /\  ( y  e.  ZZ  /\  z  e.  ZZ ) )  ->  ( (
y  -  z )  =  0  <->  y  =  z ) )
3315, 27, 323bitrd 279 . . . . 5  |-  ( ( ( ( G  e. 
Grp  /\  A  e.  X )  /\  ( O `  A )  =  0 )  /\  ( y  e.  ZZ  /\  z  e.  ZZ ) )  ->  ( ( F `  y )  =  ( F `  z )  <->  y  =  z ) )
3433biimpd 207 . . . 4  |-  ( ( ( ( G  e. 
Grp  /\  A  e.  X )  /\  ( O `  A )  =  0 )  /\  ( y  e.  ZZ  /\  z  e.  ZZ ) )  ->  ( ( F `  y )  =  ( F `  z )  ->  y  =  z ) )
3534ralrimivva 2803 . . 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 6067 . . 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 662 . 2  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  ( O `  A )  =  0 )  ->  F : ZZ
-1-1-> X )
381, 18, 2, 19odid 16679 . . . . . 6  |-  ( A  e.  X  ->  (
( O `  A
)  .x.  A )  =  ( 0g `  G ) )
391, 19, 2mulg0 16264 . . . . . 6  |-  ( A  e.  X  ->  (
0  .x.  A )  =  ( 0g `  G ) )
4038, 39eqtr4d 2426 . . . . 5  |-  ( A  e.  X  ->  (
( O `  A
)  .x.  A )  =  ( 0  .x. 
A ) )
4140ad2antlr 724 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  F : ZZ -1-1-> X )  ->  ( ( O `  A )  .x.  A )  =  ( 0  .x.  A ) )
421, 18odcl 16677 . . . . . . 7  |-  ( A  e.  X  ->  ( O `  A )  e.  NN0 )
4342ad2antlr 724 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  F : ZZ -1-1-> X )  ->  ( O `  A )  e.  NN0 )
4443nn0zd 10882 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  F : ZZ -1-1-> X )  ->  ( O `  A )  e.  ZZ )
45 oveq1 6203 . . . . . 6  |-  ( x  =  ( O `  A )  ->  (
x  .x.  A )  =  ( ( O `
 A )  .x.  A ) )
4645, 6, 10fvmpt3i 5861 . . . . 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 0zd 10793 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  F : ZZ -1-1-> X )  ->  0  e.  ZZ )
49 oveq1 6203 . . . . . 6  |-  ( x  =  0  ->  (
x  .x.  A )  =  ( 0  .x. 
A ) )
5049, 6, 10fvmpt3i 5861 . . . . 5  |-  ( 0  e.  ZZ  ->  ( F `  0 )  =  ( 0  .x. 
A ) )
5148, 50syl 16 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  F : ZZ -1-1-> X )  ->  ( F `  0 )  =  ( 0  .x.  A
) )
5241, 47, 513eqtr4d 2433 . . 3  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  F : ZZ -1-1-> X )  ->  ( F `  ( O `  A
) )  =  ( F `  0 ) )
53 simpr 459 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  F : ZZ -1-1-> X )  ->  F : ZZ
-1-1-> X )
54 f1fveq 6071 . . . 4  |-  ( ( F : ZZ -1-1-> X  /\  ( ( O `  A )  e.  ZZ  /\  0  e.  ZZ ) )  ->  ( ( F `  ( O `  A ) )  =  ( F `  0
)  <->  ( O `  A )  =  0 ) )
5553, 44, 48, 54syl12anc 1224 . . 3  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  F : ZZ -1-1-> X )  ->  ( ( F `  ( O `  A ) )  =  ( F `  0
)  <->  ( O `  A )  =  0 ) )
5652, 55mpbid 210 . 2  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  F : ZZ -1-1-> X )  ->  ( O `  A )  =  0 )
5737, 56impbida 830 1  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( ( O `  A )  =  0  <-> 
F : ZZ -1-1-> X
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1399    e. wcel 1826   A.wral 2732   class class class wbr 4367    |-> cmpt 4425   -->wf 5492   -1-1->wf1 5493   ` cfv 5496  (class class class)co 6196   CCcc 9401   0cc0 9403    - cmin 9718   NN0cn0 10712   ZZcz 10781    || cdvds 13988   Basecbs 14634   0gc0g 14847   Grpcgrp 16170  .gcmg 16173   odcod 16666
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-inf2 7972  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480  ax-pre-sup 9481
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-1st 6699  df-2nd 6700  df-recs 6960  df-rdg 6994  df-er 7229  df-en 7436  df-dom 7437  df-sdom 7438  df-sup 7816  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-div 10124  df-nn 10453  df-2 10511  df-3 10512  df-n0 10713  df-z 10782  df-uz 11002  df-rp 11140  df-fz 11594  df-fl 11828  df-mod 11897  df-seq 12011  df-exp 12070  df-cj 12934  df-re 12935  df-im 12936  df-sqrt 13070  df-abs 13071  df-dvds 13989  df-0g 14849  df-mgm 15989  df-sgrp 16028  df-mnd 16038  df-grp 16174  df-minusg 16175  df-sbg 16176  df-mulg 16177  df-od 16670
This theorem is referenced by:  odinf  16702  odcl2  16704  zrhchr  28110
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