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Theorem odeq 14865
Description: The oddvds 14862 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 10046 . . . . . . 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 14862 . . . . . . 7  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  y  e.  ZZ )  ->  ( ( O `  A )  ||  y  <->  ( y  .x.  A )  =  .0.  ) )
71, 6syl3an3 1217 . . . . . 6  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  y  e.  NN0 )  -> 
( ( O `  A )  ||  y  <->  ( y  .x.  A )  =  .0.  ) )
873expa 1151 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  y  e.  NN0 )  ->  ( ( O `
 A )  ||  y 
<->  ( y  .x.  A
)  =  .0.  )
)
98ralrimiva 2626 . . . 4  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  A. y  e.  NN0  ( ( O `  A )  ||  y  <->  ( y  .x.  A )  =  .0.  ) )
10 breq1 4026 . . . . . 6  |-  ( N  =  ( O `  A )  ->  ( N  ||  y  <->  ( O `  A )  ||  y
) )
1110bibi1d 310 . . . . 5  |-  ( N  =  ( O `  A )  ->  (
( N  ||  y  <->  ( y  .x.  A )  =  .0.  )  <->  ( ( O `  A )  ||  y  <->  ( y  .x.  A )  =  .0.  ) ) )
1211ralbidv 2563 . . . 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 213 . . 3  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( N  =  ( O `  A )  ->  A. y  e.  NN0  ( N  ||  y  <->  ( y  .x.  A )  =  .0.  ) ) )
14133adant3 975 . 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 960 . . . 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 959 . . . . 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 14851 . . . . 5  |-  ( A  e.  X  ->  ( O `  A )  e.  NN0 )
1816, 17syl 15 . . . 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 14853 . . . . . 6  |-  ( A  e.  X  ->  (
( O `  A
)  .x.  A )  =  .0.  )
2016, 19syl 15 . . . . 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 977 . . . . . 6  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  NN0 )  -> 
( O `  A
)  e.  NN0 )
22 breq2 4027 . . . . . . . 8  |-  ( y  =  ( O `  A )  ->  ( N  ||  y  <->  N  ||  ( O `  A )
) )
23 oveq1 5865 . . . . . . . . 9  |-  ( y  =  ( O `  A )  ->  (
y  .x.  A )  =  ( ( O `
 A )  .x.  A ) )
2423eqeq1d 2291 . . . . . . . 8  |-  ( y  =  ( O `  A )  ->  (
( y  .x.  A
)  =  .0.  <->  ( ( O `  A )  .x.  A )  =  .0.  ) )
2522, 24bibi12d 312 . . . . . . 7  |-  ( y  =  ( O `  A )  ->  (
( N  ||  y  <->  ( y  .x.  A )  =  .0.  )  <->  ( N  ||  ( O `  A
)  <->  ( ( O `
 A )  .x.  A )  =  .0.  ) ) )
2625rspcva 2882 . . . . . 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 457 . . . . 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 223 . . . 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 10046 . . . . . . 7  |-  ( N  e.  NN0  ->  N  e.  ZZ )
30 iddvds 12542 . . . . . . 7  |-  ( N  e.  ZZ  ->  N  ||  N )
3115, 29, 303syl 18 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  NN0 )  /\  A. y  e.  NN0  ( N  ||  y  <->  ( y  .x.  A )  =  .0.  ) )  ->  N  ||  N )
32 breq2 4027 . . . . . . . . 9  |-  ( y  =  N  ->  ( N  ||  y  <->  N  ||  N
) )
33 oveq1 5865 . . . . . . . . . 10  |-  ( y  =  N  ->  (
y  .x.  A )  =  ( N  .x.  A ) )
3433eqeq1d 2291 . . . . . . . . 9  |-  ( y  =  N  ->  (
( y  .x.  A
)  =  .0.  <->  ( N  .x.  A )  =  .0.  ) )
3532, 34bibi12d 312 . . . . . . . 8  |-  ( y  =  N  ->  (
( N  ||  y  <->  ( y  .x.  A )  =  .0.  )  <->  ( N  ||  N  <->  ( N  .x.  A )  =  .0.  ) ) )
3635rspcva 2882 . . . . . . 7  |-  ( ( N  e.  NN0  /\  A. y  e.  NN0  ( N  ||  y  <->  ( y  .x.  A )  =  .0.  ) )  ->  ( N  ||  N  <->  ( N  .x.  A )  =  .0.  ) )
37363ad2antl3 1119 . . . . . 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 201 . . . . 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 14862 . . . . . . 7  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  ->  ( ( O `  A )  ||  N  <->  ( N  .x.  A )  =  .0.  ) )
4029, 39syl3an3 1217 . . . . . 6  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  NN0 )  -> 
( ( O `  A )  ||  N  <->  ( N  .x.  A )  =  .0.  ) )
4140adantr 451 . . . . 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 223 . . . 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 12576 . . . 4  |-  ( ( ( N  e.  NN0  /\  ( O `  A
)  e.  NN0 )  /\  ( N  ||  ( O `  A )  /\  ( O `  A
)  ||  N )
)  ->  N  =  ( O `  A ) )
4415, 18, 28, 42, 43syl22anc 1183 . . 3  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  NN0 )  /\  A. y  e.  NN0  ( N  ||  y  <->  ( y  .x.  A )  =  .0.  ) )  ->  N  =  ( O `  A ) )
4544ex 423 . 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 183 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   NN0cn0 9965   ZZcz 10024    || cdivides 12531   Basecbs 13148   0gc0g 13400   Grpcgrp 14362  .gcmg 14366   odcod 14840
This theorem is referenced by:  odval2  14866  proot1ex  27520
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fz 10783  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-dvds 12532  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-sbg 14491  df-mulg 14492  df-od 14844
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