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Theorem odlem1 14850
Description: The group element order is either zero or a nonzero multiplier that annihilates the element. (Contributed by Mario Carneiro, 14-Jan-2015.) (Revised by Stefan O'Rear, 5-Sep-2015.)
Hypotheses
Ref Expression
odval.1  |-  X  =  ( Base `  G
)
odval.2  |-  .x.  =  (.g
`  G )
odval.3  |-  .0.  =  ( 0g `  G )
odval.4  |-  O  =  ( od `  G
)
odval.i  |-  I  =  { y  e.  NN  |  ( y  .x.  A )  =  .0. 
}
Assertion
Ref Expression
odlem1  |-  ( A  e.  X  ->  (
( ( O `  A )  =  0  /\  I  =  (/) )  \/  ( O `  A )  e.  I
) )
Distinct variable groups:    y, A    y, G    y,  .x.    y,  .0.
Allowed substitution hints:    I( y)    O( y)    X( y)

Proof of Theorem odlem1
StepHypRef Expression
1 odval.1 . . 3  |-  X  =  ( Base `  G
)
2 odval.2 . . 3  |-  .x.  =  (.g
`  G )
3 odval.3 . . 3  |-  .0.  =  ( 0g `  G )
4 odval.4 . . 3  |-  O  =  ( od `  G
)
5 odval.i . . 3  |-  I  =  { y  e.  NN  |  ( y  .x.  A )  =  .0. 
}
61, 2, 3, 4, 5odval 14849 . 2  |-  ( A  e.  X  ->  ( O `  A )  =  if ( I  =  (/) ,  0 ,  sup ( I ,  RR ,  `'  <  ) ) )
7 eqeq2 2292 . . . 4  |-  ( 0  =  if ( I  =  (/) ,  0 ,  sup ( I ,  RR ,  `'  <  ) )  ->  ( ( O `  A )  =  0  <->  ( O `  A )  =  if ( I  =  (/) ,  0 ,  sup (
I ,  RR ,  `'  <  ) ) ) )
87imbi1d 308 . . 3  |-  ( 0  =  if ( I  =  (/) ,  0 ,  sup ( I ,  RR ,  `'  <  ) )  ->  ( (
( O `  A
)  =  0  -> 
( ( ( O `
 A )  =  0  /\  I  =  (/) )  \/  ( O `  A )  e.  I ) )  <->  ( ( O `  A )  =  if ( I  =  (/) ,  0 ,  sup ( I ,  RR ,  `'  <  ) )  ->  ( ( ( O `  A )  =  0  /\  I  =  (/) )  \/  ( O `  A )  e.  I ) ) ) )
9 eqeq2 2292 . . . 4  |-  ( sup ( I ,  RR ,  `'  <  )  =  if ( I  =  (/) ,  0 ,  sup ( I ,  RR ,  `'  <  ) )  ->  ( ( O `
 A )  =  sup ( I ,  RR ,  `'  <  )  <-> 
( O `  A
)  =  if ( I  =  (/) ,  0 ,  sup ( I ,  RR ,  `'  <  ) ) ) )
109imbi1d 308 . . 3  |-  ( sup ( I ,  RR ,  `'  <  )  =  if ( I  =  (/) ,  0 ,  sup ( I ,  RR ,  `'  <  ) )  ->  ( ( ( O `  A )  =  sup ( I ,  RR ,  `'  <  )  ->  ( (
( O `  A
)  =  0  /\  I  =  (/) )  \/  ( O `  A
)  e.  I ) )  <->  ( ( O `
 A )  =  if ( I  =  (/) ,  0 ,  sup ( I ,  RR ,  `'  <  ) )  ->  ( ( ( O `  A )  =  0  /\  I  =  (/) )  \/  ( O `  A )  e.  I ) ) ) )
11 orc 374 . . . . 5  |-  ( ( ( O `  A
)  =  0  /\  I  =  (/) )  -> 
( ( ( O `
 A )  =  0  /\  I  =  (/) )  \/  ( O `  A )  e.  I ) )
1211expcom 424 . . . 4  |-  ( I  =  (/)  ->  ( ( O `  A )  =  0  ->  (
( ( O `  A )  =  0  /\  I  =  (/) )  \/  ( O `  A )  e.  I
) ) )
1312adantl 452 . . 3  |-  ( ( A  e.  X  /\  I  =  (/) )  -> 
( ( O `  A )  =  0  ->  ( ( ( O `  A )  =  0  /\  I  =  (/) )  \/  ( O `  A )  e.  I ) ) )
14 ssrab2 3258 . . . . . . 7  |-  { y  e.  NN  |  ( y  .x.  A )  =  .0.  }  C_  NN
15 nnuz 10263 . . . . . . . 8  |-  NN  =  ( ZZ>= `  1 )
1615eqcomi 2287 . . . . . . 7  |-  ( ZZ>= ` 
1 )  =  NN
1714, 5, 163sstr4i 3217 . . . . . 6  |-  I  C_  ( ZZ>= `  1 )
18 df-ne 2448 . . . . . . . 8  |-  ( I  =/=  (/)  <->  -.  I  =  (/) )
1918biimpri 197 . . . . . . 7  |-  ( -.  I  =  (/)  ->  I  =/=  (/) )
2019adantl 452 . . . . . 6  |-  ( ( A  e.  X  /\  -.  I  =  (/) )  ->  I  =/=  (/) )
21 infmssuzcl 10301 . . . . . 6  |-  ( ( I  C_  ( ZZ>= ` 
1 )  /\  I  =/=  (/) )  ->  sup ( I ,  RR ,  `'  <  )  e.  I )
2217, 20, 21sylancr 644 . . . . 5  |-  ( ( A  e.  X  /\  -.  I  =  (/) )  ->  sup ( I ,  RR ,  `'  <  )  e.  I )
23 eleq1a 2352 . . . . 5  |-  ( sup ( I ,  RR ,  `'  <  )  e.  I  ->  ( ( O `  A )  =  sup ( I ,  RR ,  `'  <  )  ->  ( O `  A )  e.  I
) )
2422, 23syl 15 . . . 4  |-  ( ( A  e.  X  /\  -.  I  =  (/) )  -> 
( ( O `  A )  =  sup ( I ,  RR ,  `'  <  )  -> 
( O `  A
)  e.  I ) )
25 olc 373 . . . 4  |-  ( ( O `  A )  e.  I  ->  (
( ( O `  A )  =  0  /\  I  =  (/) )  \/  ( O `  A )  e.  I
) )
2624, 25syl6 29 . . 3  |-  ( ( A  e.  X  /\  -.  I  =  (/) )  -> 
( ( O `  A )  =  sup ( I ,  RR ,  `'  <  )  -> 
( ( ( O `
 A )  =  0  /\  I  =  (/) )  \/  ( O `  A )  e.  I ) ) )
278, 10, 13, 26ifbothda 3595 . 2  |-  ( A  e.  X  ->  (
( O `  A
)  =  if ( I  =  (/) ,  0 ,  sup ( I ,  RR ,  `'  <  ) )  ->  (
( ( O `  A )  =  0  /\  I  =  (/) )  \/  ( O `  A )  e.  I
) ) )
286, 27mpd 14 1  |-  ( A  e.  X  ->  (
( ( O `  A )  =  0  /\  I  =  (/) )  \/  ( O `  A )  e.  I
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   {crab 2547    C_ wss 3152   (/)c0 3455   ifcif 3565   `'ccnv 4688   ` cfv 5255  (class class class)co 5858   supcsup 7193   RRcr 8736   0cc0 8737   1c1 8738    < clt 8867   NNcn 9746   ZZ>=cuz 10230   Basecbs 13148   0gc0g 13400  .gcmg 14366   odcod 14840
This theorem is referenced by:  odcl  14851  odid  14853
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-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
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-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-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-od 14844
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