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Theorem odnncl 16048
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 992 . . 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 993 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( N  =/=  0  /\  ( N  .x.  A
)  =  .0.  )
)  ->  N  e.  ZZ )
43zcnd 10748 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( N  =/=  0  /\  ( N  .x.  A
)  =  .0.  )
)  ->  N  e.  CC )
5 abs00 12778 . . . . . . 7  |-  ( N  e.  CC  ->  (
( abs `  N
)  =  0  <->  N  =  0 ) )
65necon3bbid 2642 . . . . . 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 12801 . . . . . . 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 10581 . . . . . 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 6098 . . . . . 6  |-  ( ( abs `  N )  =  N  ->  (
( abs `  N
)  .x.  A )  =  ( N  .x.  A ) )
1716eqeq1d 2451 . . . . 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 991 . . . . . . 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 2443 . . . . . . . 8  |-  ( invg `  G )  =  ( invg `  G )
2320, 21, 22mulgneg 15645 . . . . . . 7  |-  ( ( G  e.  Grp  /\  N  e.  ZZ  /\  A  e.  X )  ->  ( -u N  .x.  A )  =  ( ( invg `  G ) `
 ( N  .x.  A ) ) )
2419, 3, 1, 23syl3anc 1218 . . . . . 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 5695 . . . . . 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 15589 . . . . . . 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 2479 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( N  =/=  0  /\  ( N  .x.  A
)  =  .0.  )
)  ->  ( -u N  .x.  A )  =  .0.  )
30 oveq1 6098 . . . . . 6  |-  ( ( abs `  N )  =  -u N  ->  (
( abs `  N
)  .x.  A )  =  ( -u N  .x.  A ) )
3130eqeq1d 2451 . . . . 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 10747 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( N  =/=  0  /\  ( N  .x.  A
)  =  .0.  )
)  ->  N  e.  RR )
3433absord 12902 . . . 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 16042 . . 3  |-  ( ( A  e.  X  /\  ( abs `  N )  e.  NN  /\  (
( abs `  N
)  .x.  A )  =  .0.  )  ->  ( O `  A )  e.  ( 1 ... ( abs `  N ) ) )
381, 14, 35, 37syl3anc 1218 . 2  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( N  =/=  0  /\  ( N  .x.  A
)  =  .0.  )
)  ->  ( O `  A )  e.  ( 1 ... ( abs `  N ) ) )
39 elfznn 11478 . 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 965    = wceq 1369    e. wcel 1756    =/= wne 2606   ` cfv 5418  (class class class)co 6091   CCcc 9280   0cc0 9282   1c1 9283   -ucneg 9596   NNcn 10322   NN0cn0 10579   ZZcz 10646   ...cfz 11437   abscabs 12723   Basecbs 14174   0gc0g 14378   Grpcgrp 15410   invgcminusg 15411  .gcmg 15414   odcod 16028
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-inf2 7847  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-sup 7691  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-3 10381  df-n0 10580  df-z 10647  df-uz 10862  df-rp 10992  df-fz 11438  df-seq 11807  df-exp 11866  df-cj 12588  df-re 12589  df-im 12590  df-sqr 12724  df-abs 12725  df-0g 14380  df-mnd 15415  df-grp 15545  df-minusg 15546  df-mulg 15548  df-od 16032
This theorem is referenced by:  oddvds  16050
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