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Theorem odlem1 15173
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 15172 . 2  |-  ( A  e.  X  ->  ( O `  A )  =  if ( I  =  (/) ,  0 ,  sup ( I ,  RR ,  `'  <  ) ) )
7 eqeq2 2445 . . . 4  |-  ( 0  =  if ( I  =  (/) ,  0 ,  sup ( I ,  RR ,  `'  <  ) )  ->  ( ( O `  A )  =  0  <->  ( O `  A )  =  if ( I  =  (/) ,  0 ,  sup (
I ,  RR ,  `'  <  ) ) ) )
87imbi1d 309 . . 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 2445 . . . 4  |-  ( sup ( I ,  RR ,  `'  <  )  =  if ( I  =  (/) ,  0 ,  sup ( I ,  RR ,  `'  <  ) )  ->  ( ( O `
 A )  =  sup ( I ,  RR ,  `'  <  )  <-> 
( O `  A
)  =  if ( I  =  (/) ,  0 ,  sup ( I ,  RR ,  `'  <  ) ) ) )
109imbi1d 309 . . 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 375 . . . . 5  |-  ( ( ( O `  A
)  =  0  /\  I  =  (/) )  -> 
( ( ( O `
 A )  =  0  /\  I  =  (/) )  \/  ( O `  A )  e.  I ) )
1211expcom 425 . . . 4  |-  ( I  =  (/)  ->  ( ( O `  A )  =  0  ->  (
( ( O `  A )  =  0  /\  I  =  (/) )  \/  ( O `  A )  e.  I
) ) )
1312adantl 453 . . 3  |-  ( ( A  e.  X  /\  I  =  (/) )  -> 
( ( O `  A )  =  0  ->  ( ( ( O `  A )  =  0  /\  I  =  (/) )  \/  ( O `  A )  e.  I ) ) )
14 ssrab2 3428 . . . . . . 7  |-  { y  e.  NN  |  ( y  .x.  A )  =  .0.  }  C_  NN
15 nnuz 10521 . . . . . . . 8  |-  NN  =  ( ZZ>= `  1 )
1615eqcomi 2440 . . . . . . 7  |-  ( ZZ>= ` 
1 )  =  NN
1714, 5, 163sstr4i 3387 . . . . . 6  |-  I  C_  ( ZZ>= `  1 )
18 df-ne 2601 . . . . . . . 8  |-  ( I  =/=  (/)  <->  -.  I  =  (/) )
1918biimpri 198 . . . . . . 7  |-  ( -.  I  =  (/)  ->  I  =/=  (/) )
2019adantl 453 . . . . . 6  |-  ( ( A  e.  X  /\  -.  I  =  (/) )  ->  I  =/=  (/) )
21 infmssuzcl 10559 . . . . . 6  |-  ( ( I  C_  ( ZZ>= ` 
1 )  /\  I  =/=  (/) )  ->  sup ( I ,  RR ,  `'  <  )  e.  I )
2217, 20, 21sylancr 645 . . . . 5  |-  ( ( A  e.  X  /\  -.  I  =  (/) )  ->  sup ( I ,  RR ,  `'  <  )  e.  I )
23 eleq1a 2505 . . . . 5  |-  ( sup ( I ,  RR ,  `'  <  )  e.  I  ->  ( ( O `  A )  =  sup ( I ,  RR ,  `'  <  )  ->  ( O `  A )  e.  I
) )
2422, 23syl 16 . . . 4  |-  ( ( A  e.  X  /\  -.  I  =  (/) )  -> 
( ( O `  A )  =  sup ( I ,  RR ,  `'  <  )  -> 
( O `  A
)  e.  I ) )
25 olc 374 . . . 4  |-  ( ( O `  A )  e.  I  ->  (
( ( O `  A )  =  0  /\  I  =  (/) )  \/  ( O `  A )  e.  I
) )
2624, 25syl6 31 . . 3  |-  ( ( A  e.  X  /\  -.  I  =  (/) )  -> 
( ( O `  A )  =  sup ( I ,  RR ,  `'  <  )  -> 
( ( ( O `
 A )  =  0  /\  I  =  (/) )  \/  ( O `  A )  e.  I ) ) )
278, 10, 13, 26ifbothda 3769 . 2  |-  ( A  e.  X  ->  (
( O `  A
)  =  if ( I  =  (/) ,  0 ,  sup ( I ,  RR ,  `'  <  ) )  ->  (
( ( O `  A )  =  0  /\  I  =  (/) )  \/  ( O `  A )  e.  I
) ) )
286, 27mpd 15 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 358    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599   {crab 2709    C_ wss 3320   (/)c0 3628   ifcif 3739   `'ccnv 4877   ` cfv 5454  (class class class)co 6081   supcsup 7445   RRcr 8989   0cc0 8990   1c1 8991    < clt 9120   NNcn 10000   ZZ>=cuz 10488   Basecbs 13469   0gc0g 13723  .gcmg 14689   odcod 15163
This theorem is referenced by:  odcl  15174  odid  15176
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-sup 7446  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-n0 10222  df-z 10283  df-uz 10489  df-od 15167
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