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Theorem odlem1 16158
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 16157 . 2  |-  ( A  e.  X  ->  ( O `  A )  =  if ( I  =  (/) ,  0 ,  sup ( I ,  RR ,  `'  <  ) ) )
7 eqeq2 2469 . . . 4  |-  ( 0  =  if ( I  =  (/) ,  0 ,  sup ( I ,  RR ,  `'  <  ) )  ->  ( ( O `  A )  =  0  <->  ( O `  A )  =  if ( I  =  (/) ,  0 ,  sup (
I ,  RR ,  `'  <  ) ) ) )
87imbi1d 317 . . 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 2469 . . . 4  |-  ( sup ( I ,  RR ,  `'  <  )  =  if ( I  =  (/) ,  0 ,  sup ( I ,  RR ,  `'  <  ) )  ->  ( ( O `
 A )  =  sup ( I ,  RR ,  `'  <  )  <-> 
( O `  A
)  =  if ( I  =  (/) ,  0 ,  sup ( I ,  RR ,  `'  <  ) ) ) )
109imbi1d 317 . . 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 385 . . . . 5  |-  ( ( ( O `  A
)  =  0  /\  I  =  (/) )  -> 
( ( ( O `
 A )  =  0  /\  I  =  (/) )  \/  ( O `  A )  e.  I ) )
1211expcom 435 . . . 4  |-  ( I  =  (/)  ->  ( ( O `  A )  =  0  ->  (
( ( O `  A )  =  0  /\  I  =  (/) )  \/  ( O `  A )  e.  I
) ) )
1312adantl 466 . . 3  |-  ( ( A  e.  X  /\  I  =  (/) )  -> 
( ( O `  A )  =  0  ->  ( ( ( O `  A )  =  0  /\  I  =  (/) )  \/  ( O `  A )  e.  I ) ) )
14 ssrab2 3544 . . . . . . 7  |-  { y  e.  NN  |  ( y  .x.  A )  =  .0.  }  C_  NN
15 nnuz 11006 . . . . . . . 8  |-  NN  =  ( ZZ>= `  1 )
1615eqcomi 2467 . . . . . . 7  |-  ( ZZ>= ` 
1 )  =  NN
1714, 5, 163sstr4i 3502 . . . . . 6  |-  I  C_  ( ZZ>= `  1 )
18 df-ne 2649 . . . . . . . 8  |-  ( I  =/=  (/)  <->  -.  I  =  (/) )
1918biimpri 206 . . . . . . 7  |-  ( -.  I  =  (/)  ->  I  =/=  (/) )
2019adantl 466 . . . . . 6  |-  ( ( A  e.  X  /\  -.  I  =  (/) )  ->  I  =/=  (/) )
21 infmssuzcl 11048 . . . . . 6  |-  ( ( I  C_  ( ZZ>= ` 
1 )  /\  I  =/=  (/) )  ->  sup ( I ,  RR ,  `'  <  )  e.  I )
2217, 20, 21sylancr 663 . . . . 5  |-  ( ( A  e.  X  /\  -.  I  =  (/) )  ->  sup ( I ,  RR ,  `'  <  )  e.  I )
23 eleq1a 2537 . . . . 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 384 . . . 4  |-  ( ( O `  A )  e.  I  ->  (
( ( O `  A )  =  0  /\  I  =  (/) )  \/  ( O `  A )  e.  I
) )
2624, 25syl6 33 . . 3  |-  ( ( A  e.  X  /\  -.  I  =  (/) )  -> 
( ( O `  A )  =  sup ( I ,  RR ,  `'  <  )  -> 
( ( ( O `
 A )  =  0  /\  I  =  (/) )  \/  ( O `  A )  e.  I ) ) )
278, 10, 13, 26ifbothda 3931 . 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2647   {crab 2802    C_ wss 3435   (/)c0 3744   ifcif 3898   `'ccnv 4946   ` cfv 5525  (class class class)co 6199   supcsup 7800   RRcr 9391   0cc0 9392   1c1 9393    < clt 9528   NNcn 10432   ZZ>=cuz 10971   Basecbs 14291   0gc0g 14496  .gcmg 15532   odcod 16148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-cnex 9448  ax-resscn 9449  ax-1cn 9450  ax-icn 9451  ax-addcl 9452  ax-addrcl 9453  ax-mulcl 9454  ax-mulrcl 9455  ax-mulcom 9456  ax-addass 9457  ax-mulass 9458  ax-distr 9459  ax-i2m1 9460  ax-1ne0 9461  ax-1rid 9462  ax-rnegex 9463  ax-rrecex 9464  ax-cnre 9465  ax-pre-lttri 9466  ax-pre-lttrn 9467  ax-pre-ltadd 9468  ax-pre-mulgt0 9469
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-nel 2650  df-ral 2803  df-rex 2804  df-reu 2805  df-rmo 2806  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-pss 3451  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-tp 3989  df-op 3991  df-uni 4199  df-iun 4280  df-br 4400  df-opab 4458  df-mpt 4459  df-tr 4493  df-eprel 4739  df-id 4743  df-po 4748  df-so 4749  df-fr 4786  df-we 4788  df-ord 4829  df-on 4830  df-lim 4831  df-suc 4832  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-om 6586  df-recs 6941  df-rdg 6975  df-er 7210  df-en 7420  df-dom 7421  df-sdom 7422  df-sup 7801  df-pnf 9530  df-mnf 9531  df-xr 9532  df-ltxr 9533  df-le 9534  df-sub 9707  df-neg 9708  df-nn 10433  df-n0 10690  df-z 10757  df-uz 10972  df-od 16152
This theorem is referenced by:  odcl  16159  odid  16161
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