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Theorem odeq 15143
Description: The oddvds 15140 property uniquely defines the group order. (Contributed by Stefan O'Rear, 6-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
odeq  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  NN0 )  -> 
( N  =  ( O `  A )  <->  A. y  e.  NN0  ( N  ||  y  <->  ( y  .x.  A )  =  .0.  ) ) )
Distinct variable groups:    y,  .0.    y, A    y, N    y, O    y,  .x.    y, G    y, X

Proof of Theorem odeq
StepHypRef Expression
1 nn0z 10260 . . . . . . 7  |-  ( y  e.  NN0  ->  y  e.  ZZ )
2 odcl.1 . . . . . . . 8  |-  X  =  ( Base `  G
)
3 odcl.2 . . . . . . . 8  |-  O  =  ( od `  G
)
4 odid.3 . . . . . . . 8  |-  .x.  =  (.g
`  G )
5 odid.4 . . . . . . . 8  |-  .0.  =  ( 0g `  G )
62, 3, 4, 5oddvds 15140 . . . . . . 7  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  y  e.  ZZ )  ->  ( ( O `  A )  ||  y  <->  ( y  .x.  A )  =  .0.  ) )
71, 6syl3an3 1219 . . . . . 6  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  y  e.  NN0 )  -> 
( ( O `  A )  ||  y  <->  ( y  .x.  A )  =  .0.  ) )
873expa 1153 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  y  e.  NN0 )  ->  ( ( O `
 A )  ||  y 
<->  ( y  .x.  A
)  =  .0.  )
)
98ralrimiva 2749 . . . 4  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  A. y  e.  NN0  ( ( O `  A )  ||  y  <->  ( y  .x.  A )  =  .0.  ) )
10 breq1 4175 . . . . . 6  |-  ( N  =  ( O `  A )  ->  ( N  ||  y  <->  ( O `  A )  ||  y
) )
1110bibi1d 311 . . . . 5  |-  ( N  =  ( O `  A )  ->  (
( N  ||  y  <->  ( y  .x.  A )  =  .0.  )  <->  ( ( O `  A )  ||  y  <->  ( y  .x.  A )  =  .0.  ) ) )
1211ralbidv 2686 . . . 4  |-  ( N  =  ( O `  A )  ->  ( A. y  e.  NN0  ( N  ||  y  <->  ( y  .x.  A )  =  .0.  )  <->  A. y  e.  NN0  ( ( O `  A )  ||  y  <->  ( y  .x.  A )  =  .0.  ) ) )
139, 12syl5ibrcom 214 . . 3  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( N  =  ( O `  A )  ->  A. y  e.  NN0  ( N  ||  y  <->  ( y  .x.  A )  =  .0.  ) ) )
14133adant3 977 . 2  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  NN0 )  -> 
( N  =  ( O `  A )  ->  A. y  e.  NN0  ( N  ||  y  <->  ( y  .x.  A )  =  .0.  ) ) )
15 simpl3 962 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  NN0 )  /\  A. y  e.  NN0  ( N  ||  y  <->  ( y  .x.  A )  =  .0.  ) )  ->  N  e.  NN0 )
16 simpl2 961 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  NN0 )  /\  A. y  e.  NN0  ( N  ||  y  <->  ( y  .x.  A )  =  .0.  ) )  ->  A  e.  X )
172, 3odcl 15129 . . . . 5  |-  ( A  e.  X  ->  ( O `  A )  e.  NN0 )
1816, 17syl 16 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  NN0 )  /\  A. y  e.  NN0  ( N  ||  y  <->  ( y  .x.  A )  =  .0.  ) )  ->  ( O `  A )  e.  NN0 )
192, 3, 4, 5odid 15131 . . . . . 6  |-  ( A  e.  X  ->  (
( O `  A
)  .x.  A )  =  .0.  )
2016, 19syl 16 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  NN0 )  /\  A. y  e.  NN0  ( N  ||  y  <->  ( y  .x.  A )  =  .0.  ) )  ->  (
( O `  A
)  .x.  A )  =  .0.  )
21173ad2ant2 979 . . . . . 6  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  NN0 )  -> 
( O `  A
)  e.  NN0 )
22 breq2 4176 . . . . . . . 8  |-  ( y  =  ( O `  A )  ->  ( N  ||  y  <->  N  ||  ( O `  A )
) )
23 oveq1 6047 . . . . . . . . 9  |-  ( y  =  ( O `  A )  ->  (
y  .x.  A )  =  ( ( O `
 A )  .x.  A ) )
2423eqeq1d 2412 . . . . . . . 8  |-  ( y  =  ( O `  A )  ->  (
( y  .x.  A
)  =  .0.  <->  ( ( O `  A )  .x.  A )  =  .0.  ) )
2522, 24bibi12d 313 . . . . . . 7  |-  ( y  =  ( O `  A )  ->  (
( N  ||  y  <->  ( y  .x.  A )  =  .0.  )  <->  ( N  ||  ( O `  A
)  <->  ( ( O `
 A )  .x.  A )  =  .0.  ) ) )
2625rspcva 3010 . . . . . 6  |-  ( ( ( O `  A
)  e.  NN0  /\  A. y  e.  NN0  ( N  ||  y  <->  ( y  .x.  A )  =  .0.  ) )  ->  ( N  ||  ( O `  A )  <->  ( ( O `  A )  .x.  A )  =  .0.  ) )
2721, 26sylan 458 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  NN0 )  /\  A. y  e.  NN0  ( N  ||  y  <->  ( y  .x.  A )  =  .0.  ) )  ->  ( N  ||  ( O `  A )  <->  ( ( O `  A )  .x.  A )  =  .0.  ) )
2820, 27mpbird 224 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  NN0 )  /\  A. y  e.  NN0  ( N  ||  y  <->  ( y  .x.  A )  =  .0.  ) )  ->  N  ||  ( O `  A
) )
29 nn0z 10260 . . . . . . 7  |-  ( N  e.  NN0  ->  N  e.  ZZ )
30 iddvds 12818 . . . . . . 7  |-  ( N  e.  ZZ  ->  N  ||  N )
3115, 29, 303syl 19 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  NN0 )  /\  A. y  e.  NN0  ( N  ||  y  <->  ( y  .x.  A )  =  .0.  ) )  ->  N  ||  N )
32 breq2 4176 . . . . . . . . 9  |-  ( y  =  N  ->  ( N  ||  y  <->  N  ||  N
) )
33 oveq1 6047 . . . . . . . . . 10  |-  ( y  =  N  ->  (
y  .x.  A )  =  ( N  .x.  A ) )
3433eqeq1d 2412 . . . . . . . . 9  |-  ( y  =  N  ->  (
( y  .x.  A
)  =  .0.  <->  ( N  .x.  A )  =  .0.  ) )
3532, 34bibi12d 313 . . . . . . . 8  |-  ( y  =  N  ->  (
( N  ||  y  <->  ( y  .x.  A )  =  .0.  )  <->  ( N  ||  N  <->  ( N  .x.  A )  =  .0.  ) ) )
3635rspcva 3010 . . . . . . 7  |-  ( ( N  e.  NN0  /\  A. y  e.  NN0  ( N  ||  y  <->  ( y  .x.  A )  =  .0.  ) )  ->  ( N  ||  N  <->  ( N  .x.  A )  =  .0.  ) )
37363ad2antl3 1121 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  NN0 )  /\  A. y  e.  NN0  ( N  ||  y  <->  ( y  .x.  A )  =  .0.  ) )  ->  ( N  ||  N  <->  ( N  .x.  A )  =  .0.  ) )
3831, 37mpbid 202 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  NN0 )  /\  A. y  e.  NN0  ( N  ||  y  <->  ( y  .x.  A )  =  .0.  ) )  ->  ( N  .x.  A )  =  .0.  )
392, 3, 4, 5oddvds 15140 . . . . . . 7  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  ->  ( ( O `  A )  ||  N  <->  ( N  .x.  A )  =  .0.  ) )
4029, 39syl3an3 1219 . . . . . 6  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  NN0 )  -> 
( ( O `  A )  ||  N  <->  ( N  .x.  A )  =  .0.  ) )
4140adantr 452 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  NN0 )  /\  A. y  e.  NN0  ( N  ||  y  <->  ( y  .x.  A )  =  .0.  ) )  ->  (
( O `  A
)  ||  N  <->  ( N  .x.  A )  =  .0.  ) )
4238, 41mpbird 224 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  NN0 )  /\  A. y  e.  NN0  ( N  ||  y  <->  ( y  .x.  A )  =  .0.  ) )  ->  ( O `  A )  ||  N )
43 dvdseq 12852 . . . 4  |-  ( ( ( N  e.  NN0  /\  ( O `  A
)  e.  NN0 )  /\  ( N  ||  ( O `  A )  /\  ( O `  A
)  ||  N )
)  ->  N  =  ( O `  A ) )
4415, 18, 28, 42, 43syl22anc 1185 . . 3  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  NN0 )  /\  A. y  e.  NN0  ( N  ||  y  <->  ( y  .x.  A )  =  .0.  ) )  ->  N  =  ( O `  A ) )
4544ex 424 . 2  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  NN0 )  -> 
( A. y  e. 
NN0  ( N  ||  y 
<->  ( y  .x.  A
)  =  .0.  )  ->  N  =  ( O `
 A ) ) )
4614, 45impbid 184 1  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  NN0 )  -> 
( N  =  ( O `  A )  <->  A. y  e.  NN0  ( N  ||  y  <->  ( y  .x.  A )  =  .0.  ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2666   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   NN0cn0 10177   ZZcz 10238    || cdivides 12807   Basecbs 13424   0gc0g 13678   Grpcgrp 14640  .gcmg 14644   odcod 15118
This theorem is referenced by:  odval2  15144  proot1ex  27388
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-fz 11000  df-fl 11157  df-mod 11206  df-seq 11279  df-exp 11338  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-dvds 12808  df-0g 13682  df-mnd 14645  df-grp 14767  df-minusg 14768  df-sbg 14769  df-mulg 14770  df-od 15122
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