MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  odeq Unicode version

Theorem odeq 14964
Description: The oddvds 14961 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 10138 . . . . . . 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 14961 . . . . . . 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 2702 . . . 4  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  A. y  e.  NN0  ( ( O `  A )  ||  y  <->  ( y  .x.  A )  =  .0.  ) )
10 breq1 4107 . . . . . 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 2639 . . . 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 14950 . . . . 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 14952 . . . . . 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 4108 . . . . . . . 8  |-  ( y  =  ( O `  A )  ->  ( N  ||  y  <->  N  ||  ( O `  A )
) )
23 oveq1 5952 . . . . . . . . 9  |-  ( y  =  ( O `  A )  ->  (
y  .x.  A )  =  ( ( O `
 A )  .x.  A ) )
2423eqeq1d 2366 . . . . . . . 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 2958 . . . . . 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 10138 . . . . . . 7  |-  ( N  e.  NN0  ->  N  e.  ZZ )
30 iddvds 12639 . . . . . . 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 4108 . . . . . . . . 9  |-  ( y  =  N  ->  ( N  ||  y  <->  N  ||  N
) )
33 oveq1 5952 . . . . . . . . . 10  |-  ( y  =  N  ->  (
y  .x.  A )  =  ( N  .x.  A ) )
3433eqeq1d 2366 . . . . . . . . 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 2958 . . . . . . 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 14961 . . . . . . 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 12673 . . . 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 1642    e. wcel 1710   A.wral 2619   class class class wbr 4104   ` cfv 5337  (class class class)co 5945   NN0cn0 10057   ZZcz 10116    || cdivides 12628   Basecbs 13245   0gc0g 13499   Grpcgrp 14461  .gcmg 14465   odcod 14939
This theorem is referenced by:  odval2  14965  proot1ex  26843
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-inf2 7432  ax-cnex 8883  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-mulcom 8891  ax-addass 8892  ax-mulass 8893  ax-distr 8894  ax-i2m1 8895  ax-1ne0 8896  ax-1rid 8897  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902  ax-pre-ltadd 8903  ax-pre-mulgt0 8904  ax-pre-sup 8905
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-1st 6209  df-2nd 6210  df-riota 6391  df-recs 6475  df-rdg 6510  df-er 6747  df-en 6952  df-dom 6953  df-sdom 6954  df-sup 7284  df-pnf 8959  df-mnf 8960  df-xr 8961  df-ltxr 8962  df-le 8963  df-sub 9129  df-neg 9130  df-div 9514  df-nn 9837  df-2 9894  df-3 9895  df-n0 10058  df-z 10117  df-uz 10323  df-rp 10447  df-fz 10875  df-fl 11017  df-mod 11066  df-seq 11139  df-exp 11198  df-cj 11680  df-re 11681  df-im 11682  df-sqr 11816  df-abs 11817  df-dvds 12629  df-0g 13503  df-mnd 14466  df-grp 14588  df-minusg 14589  df-sbg 14590  df-mulg 14591  df-od 14943
  Copyright terms: Public domain W3C validator