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Theorem odf1 15190
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 14899 . . . . . . 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 5885 . . . 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 6080 . . . . . . . . 9  |-  ( x  =  y  ->  (
x  .x.  A )  =  ( y  .x.  A ) )
10 ovex 6098 . . . . . . . . 9  |-  ( x 
.x.  A )  e. 
_V
119, 6, 10fvmpt3i 5801 . . . . . . . 8  |-  ( y  e.  ZZ  ->  ( F `  y )  =  ( y  .x.  A ) )
12 oveq1 6080 . . . . . . . . 9  |-  ( x  =  z  ->  (
x  .x.  A )  =  ( z  .x.  A ) )
1312, 6, 10fvmpt3i 5801 . . . . . . . 8  |-  ( z  e.  ZZ  ->  ( F `  z )  =  ( z  .x.  A ) )
1411, 13eqeqan12d 2450 . . . . . . 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 4214 . . . . . . 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 2435 . . . . . . . . . 10  |-  ( 0g
`  G )  =  ( 0g `  G
)
201, 18, 2, 19odcong 15179 . . . . . . . . 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 10311 . . . . . . . . 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 12862 . . . . . . . 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 10279 . . . . . . . 8  |-  ( y  e.  ZZ  ->  y  e.  CC )
29 zcn 10279 . . . . . . . 8  |-  ( z  e.  ZZ  ->  z  e.  CC )
30 subeq0 9319 . . . . . . . 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 2790 . . 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 5996 . . 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 15168 . . . . . 6  |-  ( A  e.  X  ->  (
( O `  A
)  .x.  A )  =  ( 0g `  G ) )
391, 19, 2mulg0 14887 . . . . . 6  |-  ( A  e.  X  ->  (
0  .x.  A )  =  ( 0g `  G ) )
4038, 39eqtr4d 2470 . . . . 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 15166 . . . . . . 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 10365 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  F : ZZ -1-1-> X )  ->  ( O `  A )  e.  ZZ )
45 oveq1 6080 . . . . . 6  |-  ( x  =  ( O `  A )  ->  (
x  .x.  A )  =  ( ( O `
 A )  .x.  A ) )
4645, 6, 10fvmpt3i 5801 . . . . 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 10285 . . . . . 6  |-  0  e.  ZZ
4948a1i 11 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  F : ZZ -1-1-> X )  ->  0  e.  ZZ )
50 oveq1 6080 . . . . . 6  |-  ( x  =  0  ->  (
x  .x.  A )  =  ( 0  .x. 
A ) )
5150, 6, 10fvmpt3i 5801 . . . . 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 2477 . . 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 6000 . . . 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 1652    e. wcel 1725   A.wral 2697   class class class wbr 4204    e. cmpt 4258   -->wf 5442   -1-1->wf1 5443   ` cfv 5446  (class class class)co 6073   CCcc 8980   0cc0 8982    - cmin 9283   NN0cn0 10213   ZZcz 10274    || cdivides 12844   Basecbs 13461   0gc0g 13715   Grpcgrp 14677  .gcmg 14681   odcod 15155
This theorem is referenced by:  odinf  15191  odcl2  15193  zrhchr  24352
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-sup 7438  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-rp 10605  df-fz 11036  df-fl 11194  df-mod 11243  df-seq 11316  df-exp 11375  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-dvds 12845  df-0g 13719  df-mnd 14682  df-grp 14804  df-minusg 14805  df-sbg 14806  df-mulg 14807  df-od 15159
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