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Theorem odnncl 16419
Description: If a nonzero multiple of an element is zero, the element has positive order. (Contributed by Stefan O'Rear, 5-Sep-2015.) (Revised by Mario Carneiro, 22-Sep-2015.)
Hypotheses
Ref Expression
odcl.1  |-  X  =  ( Base `  G
)
odcl.2  |-  O  =  ( od `  G
)
odid.3  |-  .x.  =  (.g
`  G )
odid.4  |-  .0.  =  ( 0g `  G )
Assertion
Ref Expression
odnncl  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( N  =/=  0  /\  ( N  .x.  A
)  =  .0.  )
)  ->  ( O `  A )  e.  NN )

Proof of Theorem odnncl
StepHypRef Expression
1 simpl2 1000 . . 3  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( N  =/=  0  /\  ( N  .x.  A
)  =  .0.  )
)  ->  A  e.  X )
2 simprl 755 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( N  =/=  0  /\  ( N  .x.  A
)  =  .0.  )
)  ->  N  =/=  0 )
3 simpl3 1001 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( N  =/=  0  /\  ( N  .x.  A
)  =  .0.  )
)  ->  N  e.  ZZ )
43zcnd 10977 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( N  =/=  0  /\  ( N  .x.  A
)  =  .0.  )
)  ->  N  e.  CC )
5 abs00 13097 . . . . . . 7  |-  ( N  e.  CC  ->  (
( abs `  N
)  =  0  <->  N  =  0 ) )
65necon3bbid 2714 . . . . . 6  |-  ( N  e.  CC  ->  ( -.  ( abs `  N
)  =  0  <->  N  =/=  0 ) )
74, 6syl 16 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( N  =/=  0  /\  ( N  .x.  A
)  =  .0.  )
)  ->  ( -.  ( abs `  N )  =  0  <->  N  =/=  0 ) )
82, 7mpbird 232 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( N  =/=  0  /\  ( N  .x.  A
)  =  .0.  )
)  ->  -.  ( abs `  N )  =  0 )
9 nn0abscl 13120 . . . . . . 7  |-  ( N  e.  ZZ  ->  ( abs `  N )  e. 
NN0 )
103, 9syl 16 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( N  =/=  0  /\  ( N  .x.  A
)  =  .0.  )
)  ->  ( abs `  N )  e.  NN0 )
11 elnn0 10807 . . . . . 6  |-  ( ( abs `  N )  e.  NN0  <->  ( ( abs `  N )  e.  NN  \/  ( abs `  N
)  =  0 ) )
1210, 11sylib 196 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( N  =/=  0  /\  ( N  .x.  A
)  =  .0.  )
)  ->  ( ( abs `  N )  e.  NN  \/  ( abs `  N )  =  0 ) )
1312ord 377 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( N  =/=  0  /\  ( N  .x.  A
)  =  .0.  )
)  ->  ( -.  ( abs `  N )  e.  NN  ->  ( abs `  N )  =  0 ) )
148, 13mt3d 125 . . 3  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( N  =/=  0  /\  ( N  .x.  A
)  =  .0.  )
)  ->  ( abs `  N )  e.  NN )
15 simprr 756 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( N  =/=  0  /\  ( N  .x.  A
)  =  .0.  )
)  ->  ( N  .x.  A )  =  .0.  )
16 oveq1 6301 . . . . . 6  |-  ( ( abs `  N )  =  N  ->  (
( abs `  N
)  .x.  A )  =  ( N  .x.  A ) )
1716eqeq1d 2469 . . . . 5  |-  ( ( abs `  N )  =  N  ->  (
( ( abs `  N
)  .x.  A )  =  .0.  <->  ( N  .x.  A )  =  .0.  ) )
1815, 17syl5ibrcom 222 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( N  =/=  0  /\  ( N  .x.  A
)  =  .0.  )
)  ->  ( ( abs `  N )  =  N  ->  ( ( abs `  N )  .x.  A )  =  .0.  ) )
19 simpl1 999 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( N  =/=  0  /\  ( N  .x.  A
)  =  .0.  )
)  ->  G  e.  Grp )
20 odcl.1 . . . . . . . 8  |-  X  =  ( Base `  G
)
21 odid.3 . . . . . . . 8  |-  .x.  =  (.g
`  G )
22 eqid 2467 . . . . . . . 8  |-  ( invg `  G )  =  ( invg `  G )
2320, 21, 22mulgneg 16009 . . . . . . 7  |-  ( ( G  e.  Grp  /\  N  e.  ZZ  /\  A  e.  X )  ->  ( -u N  .x.  A )  =  ( ( invg `  G ) `
 ( N  .x.  A ) ) )
2419, 3, 1, 23syl3anc 1228 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( N  =/=  0  /\  ( N  .x.  A
)  =  .0.  )
)  ->  ( -u N  .x.  A )  =  ( ( invg `  G ) `  ( N  .x.  A ) ) )
2515fveq2d 5875 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( N  =/=  0  /\  ( N  .x.  A
)  =  .0.  )
)  ->  ( ( invg `  G ) `
 ( N  .x.  A ) )  =  ( ( invg `  G ) `  .0.  ) )
26 odid.4 . . . . . . . 8  |-  .0.  =  ( 0g `  G )
2726, 22grpinvid 15950 . . . . . . 7  |-  ( G  e.  Grp  ->  (
( invg `  G ) `  .0.  )  =  .0.  )
2819, 27syl 16 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( N  =/=  0  /\  ( N  .x.  A
)  =  .0.  )
)  ->  ( ( invg `  G ) `
 .0.  )  =  .0.  )
2924, 25, 283eqtrd 2512 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( N  =/=  0  /\  ( N  .x.  A
)  =  .0.  )
)  ->  ( -u N  .x.  A )  =  .0.  )
30 oveq1 6301 . . . . . 6  |-  ( ( abs `  N )  =  -u N  ->  (
( abs `  N
)  .x.  A )  =  ( -u N  .x.  A ) )
3130eqeq1d 2469 . . . . 5  |-  ( ( abs `  N )  =  -u N  ->  (
( ( abs `  N
)  .x.  A )  =  .0.  <->  ( -u N  .x.  A )  =  .0.  ) )
3229, 31syl5ibrcom 222 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( N  =/=  0  /\  ( N  .x.  A
)  =  .0.  )
)  ->  ( ( abs `  N )  = 
-u N  ->  (
( abs `  N
)  .x.  A )  =  .0.  ) )
333zred 10976 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( N  =/=  0  /\  ( N  .x.  A
)  =  .0.  )
)  ->  N  e.  RR )
3433absord 13222 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( N  =/=  0  /\  ( N  .x.  A
)  =  .0.  )
)  ->  ( ( abs `  N )  =  N  \/  ( abs `  N )  =  -u N ) )
3518, 32, 34mpjaod 381 . . 3  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( N  =/=  0  /\  ( N  .x.  A
)  =  .0.  )
)  ->  ( ( abs `  N )  .x.  A )  =  .0.  )
36 odcl.2 . . . 4  |-  O  =  ( od `  G
)
3720, 36, 21, 26odlem2 16413 . . 3  |-  ( ( A  e.  X  /\  ( abs `  N )  e.  NN  /\  (
( abs `  N
)  .x.  A )  =  .0.  )  ->  ( O `  A )  e.  ( 1 ... ( abs `  N ) ) )
381, 14, 35, 37syl3anc 1228 . 2  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( N  =/=  0  /\  ( N  .x.  A
)  =  .0.  )
)  ->  ( O `  A )  e.  ( 1 ... ( abs `  N ) ) )
39 elfznn 11724 . 2  |-  ( ( O `  A )  e.  ( 1 ... ( abs `  N
) )  ->  ( O `  A )  e.  NN )
4038, 39syl 16 1  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( N  =/=  0  /\  ( N  .x.  A
)  =  .0.  )
)  ->  ( O `  A )  e.  NN )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   ` cfv 5593  (class class class)co 6294   CCcc 9500   0cc0 9502   1c1 9503   -ucneg 9816   NNcn 10546   NN0cn0 10805   ZZcz 10874   ...cfz 11682   abscabs 13042   Basecbs 14502   0gc0g 14707   Grpcgrp 15902   invgcminusg 15903  .gcmg 15905   odcod 16399
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4563  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586  ax-inf2 8068  ax-cnex 9558  ax-resscn 9559  ax-1cn 9560  ax-icn 9561  ax-addcl 9562  ax-addrcl 9563  ax-mulcl 9564  ax-mulrcl 9565  ax-mulcom 9566  ax-addass 9567  ax-mulass 9568  ax-distr 9569  ax-i2m1 9570  ax-1ne0 9571  ax-1rid 9572  ax-rnegex 9573  ax-rrecex 9574  ax-cnre 9575  ax-pre-lttri 9576  ax-pre-lttrn 9577  ax-pre-ltadd 9578  ax-pre-mulgt0 9579  ax-pre-sup 9580
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4251  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-tr 4546  df-eprel 4796  df-id 4800  df-po 4805  df-so 4806  df-fr 4843  df-we 4845  df-ord 4886  df-on 4887  df-lim 4888  df-suc 4889  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-riota 6255  df-ov 6297  df-oprab 6298  df-mpt2 6299  df-om 6695  df-1st 6794  df-2nd 6795  df-recs 7052  df-rdg 7086  df-er 7321  df-en 7527  df-dom 7528  df-sdom 7529  df-sup 7911  df-pnf 9640  df-mnf 9641  df-xr 9642  df-ltxr 9643  df-le 9644  df-sub 9817  df-neg 9818  df-div 10217  df-nn 10547  df-2 10604  df-3 10605  df-n0 10806  df-z 10875  df-uz 11093  df-rp 11231  df-fz 11683  df-seq 12086  df-exp 12145  df-cj 12907  df-re 12908  df-im 12909  df-sqrt 13043  df-abs 13044  df-0g 14709  df-mgm 15741  df-sgrp 15764  df-mnd 15774  df-grp 15906  df-minusg 15907  df-mulg 15909  df-od 16403
This theorem is referenced by:  oddvds  16421
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