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Theorem odeq 15193
Description: The oddvds 15190 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 10309 . . . . . . 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 15190 . . . . . . 7  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  y  e.  ZZ )  ->  ( ( O `  A )  ||  y  <->  ( y  .x.  A )  =  .0.  ) )
71, 6syl3an3 1220 . . . . . 6  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  y  e.  NN0 )  -> 
( ( O `  A )  ||  y  <->  ( y  .x.  A )  =  .0.  ) )
873expa 1154 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  y  e.  NN0 )  ->  ( ( O `
 A )  ||  y 
<->  ( y  .x.  A
)  =  .0.  )
)
98ralrimiva 2791 . . . 4  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  A. y  e.  NN0  ( ( O `  A )  ||  y  <->  ( y  .x.  A )  =  .0.  ) )
10 breq1 4218 . . . . . 6  |-  ( N  =  ( O `  A )  ->  ( N  ||  y  <->  ( O `  A )  ||  y
) )
1110bibi1d 312 . . . . 5  |-  ( N  =  ( O `  A )  ->  (
( N  ||  y  <->  ( y  .x.  A )  =  .0.  )  <->  ( ( O `  A )  ||  y  <->  ( y  .x.  A )  =  .0.  ) ) )
1211ralbidv 2727 . . . 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 215 . . 3  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( N  =  ( O `  A )  ->  A. y  e.  NN0  ( N  ||  y  <->  ( y  .x.  A )  =  .0.  ) ) )
14133adant3 978 . 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 963 . . . 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 962 . . . . 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 15179 . . . . 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 15181 . . . . . 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 980 . . . . . 6  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  NN0 )  -> 
( O `  A
)  e.  NN0 )
22 breq2 4219 . . . . . . . 8  |-  ( y  =  ( O `  A )  ->  ( N  ||  y  <->  N  ||  ( O `  A )
) )
23 oveq1 6091 . . . . . . . . 9  |-  ( y  =  ( O `  A )  ->  (
y  .x.  A )  =  ( ( O `
 A )  .x.  A ) )
2423eqeq1d 2446 . . . . . . . 8  |-  ( y  =  ( O `  A )  ->  (
( y  .x.  A
)  =  .0.  <->  ( ( O `  A )  .x.  A )  =  .0.  ) )
2522, 24bibi12d 314 . . . . . . 7  |-  ( y  =  ( O `  A )  ->  (
( N  ||  y  <->  ( y  .x.  A )  =  .0.  )  <->  ( N  ||  ( O `  A
)  <->  ( ( O `
 A )  .x.  A )  =  .0.  ) ) )
2625rspcva 3052 . . . . . 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 459 . . . . 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 225 . . . 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 10309 . . . . . . 7  |-  ( N  e.  NN0  ->  N  e.  ZZ )
30 iddvds 12868 . . . . . . 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 4219 . . . . . . . . 9  |-  ( y  =  N  ->  ( N  ||  y  <->  N  ||  N
) )
33 oveq1 6091 . . . . . . . . . 10  |-  ( y  =  N  ->  (
y  .x.  A )  =  ( N  .x.  A ) )
3433eqeq1d 2446 . . . . . . . . 9  |-  ( y  =  N  ->  (
( y  .x.  A
)  =  .0.  <->  ( N  .x.  A )  =  .0.  ) )
3532, 34bibi12d 314 . . . . . . . 8  |-  ( y  =  N  ->  (
( N  ||  y  <->  ( y  .x.  A )  =  .0.  )  <->  ( N  ||  N  <->  ( N  .x.  A )  =  .0.  ) ) )
3635rspcva 3052 . . . . . . 7  |-  ( ( N  e.  NN0  /\  A. y  e.  NN0  ( N  ||  y  <->  ( y  .x.  A )  =  .0.  ) )  ->  ( N  ||  N  <->  ( N  .x.  A )  =  .0.  ) )
37363ad2antl3 1122 . . . . . 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 203 . . . . 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 15190 . . . . . . 7  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  ->  ( ( O `  A )  ||  N  <->  ( N  .x.  A )  =  .0.  ) )
4029, 39syl3an3 1220 . . . . . 6  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  NN0 )  -> 
( ( O `  A )  ||  N  <->  ( N  .x.  A )  =  .0.  ) )
4140adantr 453 . . . . 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 225 . . . 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 12902 . . . 4  |-  ( ( ( N  e.  NN0  /\  ( O `  A
)  e.  NN0 )  /\  ( N  ||  ( O `  A )  /\  ( O `  A
)  ||  N )
)  ->  N  =  ( O `  A ) )
4415, 18, 28, 42, 43syl22anc 1186 . . 3  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  NN0 )  /\  A. y  e.  NN0  ( N  ||  y  <->  ( y  .x.  A )  =  .0.  ) )  ->  N  =  ( O `  A ) )
4544ex 425 . 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 185 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 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   A.wral 2707   class class class wbr 4215   ` cfv 5457  (class class class)co 6084   NN0cn0 10226   ZZcz 10287    || cdivides 12857   Basecbs 13474   0gc0g 13728   Grpcgrp 14690  .gcmg 14694   odcod 15168
This theorem is referenced by:  odval2  15194  proot1ex  27511
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-inf2 7599  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072  ax-pre-sup 9073
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-er 6908  df-en 7113  df-dom 7114  df-sdom 7115  df-sup 7449  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683  df-nn 10006  df-2 10063  df-3 10064  df-n0 10227  df-z 10288  df-uz 10494  df-rp 10618  df-fz 11049  df-fl 11207  df-mod 11256  df-seq 11329  df-exp 11388  df-cj 11909  df-re 11910  df-im 11911  df-sqr 12045  df-abs 12046  df-dvds 12858  df-0g 13732  df-mnd 14695  df-grp 14817  df-minusg 14818  df-sbg 14819  df-mulg 14820  df-od 15172
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