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Theorem odf1 16373
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 15952 . . . . . . 7  |-  ( ( G  e.  Grp  /\  x  e.  ZZ  /\  A  e.  X )  ->  (
x  .x.  A )  e.  X )
433expa 1191 . . . . . 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 6036 . . . 4  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  F : ZZ --> X )
87adantr 465 . . 3  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  ( O `  A )  =  0 )  ->  F : ZZ
--> X )
9 oveq1 6282 . . . . . . . . 9  |-  ( x  =  y  ->  (
x  .x.  A )  =  ( y  .x.  A ) )
10 ovex 6300 . . . . . . . . 9  |-  ( x 
.x.  A )  e. 
_V
119, 6, 10fvmpt3i 5945 . . . . . . . 8  |-  ( y  e.  ZZ  ->  ( F `  y )  =  ( y  .x.  A ) )
12 oveq1 6282 . . . . . . . . 9  |-  ( x  =  z  ->  (
x  .x.  A )  =  ( z  .x.  A ) )
1312, 6, 10fvmpt3i 5945 . . . . . . . 8  |-  ( z  e.  ZZ  ->  ( F `  z )  =  ( z  .x.  A ) )
1411, 13eqeqan12d 2483 . . . . . . 7  |-  ( ( y  e.  ZZ  /\  z  e.  ZZ )  ->  ( ( F `  y )  =  ( F `  z )  <-> 
( y  .x.  A
)  =  ( z 
.x.  A ) ) )
1514adantl 466 . . . . . 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 754 . . . . . . . 8  |-  ( ( ( ( G  e. 
Grp  /\  A  e.  X )  /\  ( O `  A )  =  0 )  /\  ( y  e.  ZZ  /\  z  e.  ZZ ) )  ->  ( O `  A )  =  0 )
1716breq1d 4450 . . . . . . 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 2460 . . . . . . . . . 10  |-  ( 0g
`  G )  =  ( 0g `  G
)
201, 18, 2, 19odcong 16362 . . . . . . . . 9  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  ( y  e.  ZZ  /\  z  e.  ZZ ) )  ->  ( ( O `  A )  ||  ( y  -  z
)  <->  ( y  .x.  A )  =  ( z  .x.  A ) ) )
21203expa 1191 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  ( y  e.  ZZ  /\  z  e.  ZZ ) )  -> 
( ( O `  A )  ||  (
y  -  z )  <-> 
( y  .x.  A
)  =  ( z 
.x.  A ) ) )
2221adantlr 714 . . . . . . 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 10894 . . . . . . . . 9  |-  ( ( y  e.  ZZ  /\  z  e.  ZZ )  ->  ( y  -  z
)  e.  ZZ )
2423adantl 466 . . . . . . . 8  |-  ( ( ( ( G  e. 
Grp  /\  A  e.  X )  /\  ( O `  A )  =  0 )  /\  ( y  e.  ZZ  /\  z  e.  ZZ ) )  ->  ( y  -  z )  e.  ZZ )
25 0dvds 13854 . . . . . . . 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 10858 . . . . . . . 8  |-  ( y  e.  ZZ  ->  y  e.  CC )
29 zcn 10858 . . . . . . . 8  |-  ( z  e.  ZZ  ->  z  e.  CC )
30 subeq0 9834 . . . . . . . 8  |-  ( ( y  e.  CC  /\  z  e.  CC )  ->  ( ( y  -  z )  =  0  <-> 
y  =  z ) )
3128, 29, 30syl2an 477 . . . . . . 7  |-  ( ( y  e.  ZZ  /\  z  e.  ZZ )  ->  ( ( y  -  z )  =  0  <-> 
y  =  z ) )
3231adantl 466 . . . . . 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 2878 . . 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 6145 . . 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 664 . 2  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  ( O `  A )  =  0 )  ->  F : ZZ
-1-1-> X )
381, 18, 2, 19odid 16351 . . . . . 6  |-  ( A  e.  X  ->  (
( O `  A
)  .x.  A )  =  ( 0g `  G ) )
391, 19, 2mulg0 15940 . . . . . 6  |-  ( A  e.  X  ->  (
0  .x.  A )  =  ( 0g `  G ) )
4038, 39eqtr4d 2504 . . . . 5  |-  ( A  e.  X  ->  (
( O `  A
)  .x.  A )  =  ( 0  .x. 
A ) )
4140ad2antlr 726 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  F : ZZ -1-1-> X )  ->  ( ( O `  A )  .x.  A )  =  ( 0  .x.  A ) )
421, 18odcl 16349 . . . . . . 7  |-  ( A  e.  X  ->  ( O `  A )  e.  NN0 )
4342ad2antlr 726 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  F : ZZ -1-1-> X )  ->  ( O `  A )  e.  NN0 )
4443nn0zd 10953 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  F : ZZ -1-1-> X )  ->  ( O `  A )  e.  ZZ )
45 oveq1 6282 . . . . . 6  |-  ( x  =  ( O `  A )  ->  (
x  .x.  A )  =  ( ( O `
 A )  .x.  A ) )
4645, 6, 10fvmpt3i 5945 . . . . 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 10865 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  F : ZZ -1-1-> X )  ->  0  e.  ZZ )
49 oveq1 6282 . . . . . 6  |-  ( x  =  0  ->  (
x  .x.  A )  =  ( 0  .x. 
A ) )
5049, 6, 10fvmpt3i 5945 . . . . 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 2511 . . 3  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  F : ZZ -1-1-> X )  ->  ( F `  ( O `  A
) )  =  ( F `  0 ) )
53 simpr 461 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  F : ZZ -1-1-> X )  ->  F : ZZ
-1-1-> X )
54 f1fveq 6149 . . . 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 1221 . . 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 829 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 369    = wceq 1374    e. wcel 1762   A.wral 2807   class class class wbr 4440    |-> cmpt 4498   -->wf 5575   -1-1->wf1 5576   ` cfv 5579  (class class class)co 6275   CCcc 9479   0cc0 9481    - cmin 9794   NN0cn0 10784   ZZcz 10853    || cdivides 13836   Basecbs 14479   0gc0g 14684   Grpcgrp 15716  .gcmg 15720   odcod 16338
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-inf2 8047  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-sup 7890  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-2 10583  df-3 10584  df-n0 10785  df-z 10854  df-uz 11072  df-rp 11210  df-fz 11662  df-fl 11886  df-mod 11953  df-seq 12064  df-exp 12123  df-cj 12882  df-re 12883  df-im 12884  df-sqr 13018  df-abs 13019  df-dvds 13837  df-0g 14686  df-mnd 15721  df-grp 15851  df-minusg 15852  df-sbg 15853  df-mulg 15854  df-od 16342
This theorem is referenced by:  odinf  16374  odcl2  16376  zrhchr  27579
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