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Theorem odnncl 16443
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 1001 . . 3  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( N  =/=  0  /\  ( N  .x.  A
)  =  .0.  )
)  ->  A  e.  X )
2 simprl 756 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( N  =/=  0  /\  ( N  .x.  A
)  =  .0.  )
)  ->  N  =/=  0 )
3 simpl3 1002 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( N  =/=  0  /\  ( N  .x.  A
)  =  .0.  )
)  ->  N  e.  ZZ )
43zcnd 10975 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( N  =/=  0  /\  ( N  .x.  A
)  =  .0.  )
)  ->  N  e.  CC )
5 abs00 13101 . . . . . . 7  |-  ( N  e.  CC  ->  (
( abs `  N
)  =  0  <->  N  =  0 ) )
65necon3bbid 2690 . . . . . 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 13124 . . . . . . 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 10803 . . . . . 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 757 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( N  =/=  0  /\  ( N  .x.  A
)  =  .0.  )
)  ->  ( N  .x.  A )  =  .0.  )
16 oveq1 6288 . . . . . 6  |-  ( ( abs `  N )  =  N  ->  (
( abs `  N
)  .x.  A )  =  ( N  .x.  A ) )
1716eqeq1d 2445 . . . . 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 1000 . . . . . . 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 16034 . . . . . . 7  |-  ( ( G  e.  Grp  /\  N  e.  ZZ  /\  A  e.  X )  ->  ( -u N  .x.  A )  =  ( ( invg `  G ) `
 ( N  .x.  A ) ) )
2419, 3, 1, 23syl3anc 1229 . . . . . 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 5860 . . . . . 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 15975 . . . . . . 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 2488 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( N  =/=  0  /\  ( N  .x.  A
)  =  .0.  )
)  ->  ( -u N  .x.  A )  =  .0.  )
30 oveq1 6288 . . . . . 6  |-  ( ( abs `  N )  =  -u N  ->  (
( abs `  N
)  .x.  A )  =  ( -u N  .x.  A ) )
3130eqeq1d 2445 . . . . 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 10974 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( N  =/=  0  /\  ( N  .x.  A
)  =  .0.  )
)  ->  N  e.  RR )
3433absord 13226 . . . 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 16437 . . 3  |-  ( ( A  e.  X  /\  ( abs `  N )  e.  NN  /\  (
( abs `  N
)  .x.  A )  =  .0.  )  ->  ( O `  A )  e.  ( 1 ... ( abs `  N ) ) )
381, 14, 35, 37syl3anc 1229 . 2  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( N  =/=  0  /\  ( N  .x.  A
)  =  .0.  )
)  ->  ( O `  A )  e.  ( 1 ... ( abs `  N ) ) )
39 elfznn 11723 . 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 974    = wceq 1383    e. wcel 1804    =/= wne 2638   ` cfv 5578  (class class class)co 6281   CCcc 9493   0cc0 9495   1c1 9496   -ucneg 9811   NNcn 10542   NN0cn0 10801   ZZcz 10870   ...cfz 11681   abscabs 13046   Basecbs 14509   0gc0g 14714   Grpcgrp 15927   invgcminusg 15928  .gcmg 15930   odcod 16423
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-pre-sup 9573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-sup 7903  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10213  df-nn 10543  df-2 10600  df-3 10601  df-n0 10802  df-z 10871  df-uz 11091  df-rp 11230  df-fz 11682  df-seq 12087  df-exp 12146  df-cj 12911  df-re 12912  df-im 12913  df-sqrt 13047  df-abs 13048  df-0g 14716  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-grp 15931  df-minusg 15932  df-mulg 15934  df-od 16427
This theorem is referenced by:  oddvds  16445
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