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Theorem odeq 16065
Description: The oddvds 16062 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 10681 . . . . . . 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 16062 . . . . . . 7  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  y  e.  ZZ )  ->  ( ( O `  A )  ||  y  <->  ( y  .x.  A )  =  .0.  ) )
71, 6syl3an3 1253 . . . . . 6  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  y  e.  NN0 )  -> 
( ( O `  A )  ||  y  <->  ( y  .x.  A )  =  .0.  ) )
873expa 1187 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  y  e.  NN0 )  ->  ( ( O `
 A )  ||  y 
<->  ( y  .x.  A
)  =  .0.  )
)
98ralrimiva 2811 . . . 4  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  A. y  e.  NN0  ( ( O `  A )  ||  y  <->  ( y  .x.  A )  =  .0.  ) )
10 breq1 4307 . . . . . 6  |-  ( N  =  ( O `  A )  ->  ( N  ||  y  <->  ( O `  A )  ||  y
) )
1110bibi1d 319 . . . . 5  |-  ( N  =  ( O `  A )  ->  (
( N  ||  y  <->  ( y  .x.  A )  =  .0.  )  <->  ( ( O `  A )  ||  y  <->  ( y  .x.  A )  =  .0.  ) ) )
1211ralbidv 2747 . . . 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 222 . . 3  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( N  =  ( O `  A )  ->  A. y  e.  NN0  ( N  ||  y  <->  ( y  .x.  A )  =  .0.  ) ) )
14133adant3 1008 . 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 993 . . . 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 992 . . . . 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 16051 . . . . 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 16053 . . . . . 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 1010 . . . . . 6  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  NN0 )  -> 
( O `  A
)  e.  NN0 )
22 breq2 4308 . . . . . . . 8  |-  ( y  =  ( O `  A )  ->  ( N  ||  y  <->  N  ||  ( O `  A )
) )
23 oveq1 6110 . . . . . . . . 9  |-  ( y  =  ( O `  A )  ->  (
y  .x.  A )  =  ( ( O `
 A )  .x.  A ) )
2423eqeq1d 2451 . . . . . . . 8  |-  ( y  =  ( O `  A )  ->  (
( y  .x.  A
)  =  .0.  <->  ( ( O `  A )  .x.  A )  =  .0.  ) )
2522, 24bibi12d 321 . . . . . . 7  |-  ( y  =  ( O `  A )  ->  (
( N  ||  y  <->  ( y  .x.  A )  =  .0.  )  <->  ( N  ||  ( O `  A
)  <->  ( ( O `
 A )  .x.  A )  =  .0.  ) ) )
2625rspcva 3083 . . . . . 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 471 . . . . 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 232 . . . 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 10681 . . . . . . 7  |-  ( N  e.  NN0  ->  N  e.  ZZ )
30 iddvds 13558 . . . . . . 7  |-  ( N  e.  ZZ  ->  N  ||  N )
3115, 29, 303syl 20 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  NN0 )  /\  A. y  e.  NN0  ( N  ||  y  <->  ( y  .x.  A )  =  .0.  ) )  ->  N  ||  N )
32 breq2 4308 . . . . . . . . 9  |-  ( y  =  N  ->  ( N  ||  y  <->  N  ||  N
) )
33 oveq1 6110 . . . . . . . . . 10  |-  ( y  =  N  ->  (
y  .x.  A )  =  ( N  .x.  A ) )
3433eqeq1d 2451 . . . . . . . . 9  |-  ( y  =  N  ->  (
( y  .x.  A
)  =  .0.  <->  ( N  .x.  A )  =  .0.  ) )
3532, 34bibi12d 321 . . . . . . . 8  |-  ( y  =  N  ->  (
( N  ||  y  <->  ( y  .x.  A )  =  .0.  )  <->  ( N  ||  N  <->  ( N  .x.  A )  =  .0.  ) ) )
3635rspcva 3083 . . . . . . 7  |-  ( ( N  e.  NN0  /\  A. y  e.  NN0  ( N  ||  y  <->  ( y  .x.  A )  =  .0.  ) )  ->  ( N  ||  N  <->  ( N  .x.  A )  =  .0.  ) )
37363ad2antl3 1152 . . . . . 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 210 . . . . 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 16062 . . . . . . 7  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  ->  ( ( O `  A )  ||  N  <->  ( N  .x.  A )  =  .0.  ) )
4029, 39syl3an3 1253 . . . . . 6  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  NN0 )  -> 
( ( O `  A )  ||  N  <->  ( N  .x.  A )  =  .0.  ) )
4140adantr 465 . . . . 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 232 . . . 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 13592 . . . 4  |-  ( ( ( N  e.  NN0  /\  ( O `  A
)  e.  NN0 )  /\  ( N  ||  ( O `  A )  /\  ( O `  A
)  ||  N )
)  ->  N  =  ( O `  A ) )
4415, 18, 28, 42, 43syl22anc 1219 . . 3  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  NN0 )  /\  A. y  e.  NN0  ( N  ||  y  <->  ( y  .x.  A )  =  .0.  ) )  ->  N  =  ( O `  A ) )
4544ex 434 . 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 191 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2727   class class class wbr 4304   ` cfv 5430  (class class class)co 6103   NN0cn0 10591   ZZcz 10658    || cdivides 13547   Basecbs 14186   0gc0g 14390   Grpcgrp 15422  .gcmg 15426   odcod 16040
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 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-inf2 7859  ax-cnex 9350  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-pre-mulgt0 9371  ax-pre-sup 9372
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 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-om 6489  df-1st 6589  df-2nd 6590  df-recs 6844  df-rdg 6878  df-er 7113  df-en 7323  df-dom 7324  df-sdom 7325  df-sup 7703  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610  df-div 10006  df-nn 10335  df-2 10392  df-3 10393  df-n0 10592  df-z 10659  df-uz 10874  df-rp 11004  df-fz 11450  df-fl 11654  df-mod 11721  df-seq 11819  df-exp 11878  df-cj 12600  df-re 12601  df-im 12602  df-sqr 12736  df-abs 12737  df-dvds 13548  df-0g 14392  df-mnd 15427  df-grp 15557  df-minusg 15558  df-sbg 15559  df-mulg 15560  df-od 16044
This theorem is referenced by:  odval2  16066  proot1ex  29581
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