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Theorem odnncl 16041
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 987 . . 3  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( N  =/=  0  /\  ( N  .x.  A
)  =  .0.  )
)  ->  A  e.  X )
2 simprl 750 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( N  =/=  0  /\  ( N  .x.  A
)  =  .0.  )
)  ->  N  =/=  0 )
3 simpl3 988 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( N  =/=  0  /\  ( N  .x.  A
)  =  .0.  )
)  ->  N  e.  ZZ )
43zcnd 10744 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( N  =/=  0  /\  ( N  .x.  A
)  =  .0.  )
)  ->  N  e.  CC )
5 abs00 12774 . . . . . . 7  |-  ( N  e.  CC  ->  (
( abs `  N
)  =  0  <->  N  =  0 ) )
65necon3bbid 2640 . . . . . 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 12797 . . . . . . 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 10577 . . . . . 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 751 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( N  =/=  0  /\  ( N  .x.  A
)  =  .0.  )
)  ->  ( N  .x.  A )  =  .0.  )
16 oveq1 6097 . . . . . 6  |-  ( ( abs `  N )  =  N  ->  (
( abs `  N
)  .x.  A )  =  ( N  .x.  A ) )
1716eqeq1d 2449 . . . . 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 986 . . . . . . 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 2441 . . . . . . . 8  |-  ( invg `  G )  =  ( invg `  G )
2320, 21, 22mulgneg 15638 . . . . . . 7  |-  ( ( G  e.  Grp  /\  N  e.  ZZ  /\  A  e.  X )  ->  ( -u N  .x.  A )  =  ( ( invg `  G ) `
 ( N  .x.  A ) ) )
2419, 3, 1, 23syl3anc 1213 . . . . . 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 5692 . . . . . 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 15582 . . . . . . 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 2477 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( N  =/=  0  /\  ( N  .x.  A
)  =  .0.  )
)  ->  ( -u N  .x.  A )  =  .0.  )
30 oveq1 6097 . . . . . 6  |-  ( ( abs `  N )  =  -u N  ->  (
( abs `  N
)  .x.  A )  =  ( -u N  .x.  A ) )
3130eqeq1d 2449 . . . . 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 10743 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( N  =/=  0  /\  ( N  .x.  A
)  =  .0.  )
)  ->  N  e.  RR )
3433absord 12898 . . . 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 16035 . . 3  |-  ( ( A  e.  X  /\  ( abs `  N )  e.  NN  /\  (
( abs `  N
)  .x.  A )  =  .0.  )  ->  ( O `  A )  e.  ( 1 ... ( abs `  N ) ) )
381, 14, 35, 37syl3anc 1213 . 2  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( N  =/=  0  /\  ( N  .x.  A
)  =  .0.  )
)  ->  ( O `  A )  e.  ( 1 ... ( abs `  N ) ) )
39 elfznn 11474 . 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 960    = wceq 1364    e. wcel 1761    =/= wne 2604   ` cfv 5415  (class class class)co 6090   CCcc 9276   0cc0 9278   1c1 9279   -ucneg 9592   NNcn 10318   NN0cn0 10575   ZZcz 10642   ...cfz 11433   abscabs 12719   Basecbs 14170   0gc0g 14374   Grpcgrp 15406   invgcminusg 15407  .gcmg 15410   odcod 16021
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 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-sup 7687  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-n0 10576  df-z 10643  df-uz 10858  df-rp 10988  df-fz 11434  df-seq 11803  df-exp 11862  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-0g 14376  df-mnd 15411  df-grp 15538  df-minusg 15539  df-mulg 15541  df-od 16025
This theorem is referenced by:  oddvds  16043
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