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Theorem odlem1 16885
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 16884 . 2  |-  ( A  e.  X  ->  ( O `  A )  =  if ( I  =  (/) ,  0 ,  sup ( I ,  RR ,  `'  <  ) ) )
7 eqeq2 2419 . . . 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 2419 . . . 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 3526 . . . . . . 7  |-  { y  e.  NN  |  ( y  .x.  A )  =  .0.  }  C_  NN
15 nnuz 11164 . . . . . . . 8  |-  NN  =  ( ZZ>= `  1 )
1615eqcomi 2417 . . . . . . 7  |-  ( ZZ>= ` 
1 )  =  NN
1714, 5, 163sstr4i 3483 . . . . . 6  |-  I  C_  ( ZZ>= `  1 )
18 df-ne 2602 . . . . . . . 8  |-  ( I  =/=  (/)  <->  -.  I  =  (/) )
1918biimpri 208 . . . . . . 7  |-  ( -.  I  =  (/)  ->  I  =/=  (/) )
2019adantl 466 . . . . . 6  |-  ( ( A  e.  X  /\  -.  I  =  (/) )  ->  I  =/=  (/) )
21 infmssuzcl 11212 . . . . . 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 2487 . . . . 5  |-  ( sup ( I ,  RR ,  `'  <  )  e.  I  ->  ( ( O `  A )  =  sup ( I ,  RR ,  `'  <  )  ->  ( O `  A )  e.  I
) )
2422, 23syl 17 . . . 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 3922 . 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 1407    e. wcel 1844    =/= wne 2600   {crab 2760    C_ wss 3416   (/)c0 3740   ifcif 3887   `'ccnv 4824   ` cfv 5571  (class class class)co 6280   supcsup 7936   RRcr 9523   0cc0 9524   1c1 9525    < clt 9660   NNcn 10578   ZZ>=cuz 11129   Basecbs 14843   0gc0g 15056  .gcmg 16382   odcod 16875
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576  ax-cnex 9580  ax-resscn 9581  ax-1cn 9582  ax-icn 9583  ax-addcl 9584  ax-addrcl 9585  ax-mulcl 9586  ax-mulrcl 9587  ax-mulcom 9588  ax-addass 9589  ax-mulass 9590  ax-distr 9591  ax-i2m1 9592  ax-1ne0 9593  ax-1rid 9594  ax-rnegex 9595  ax-rrecex 9596  ax-cnre 9597  ax-pre-lttri 9598  ax-pre-lttrn 9599  ax-pre-ltadd 9600  ax-pre-mulgt0 9601
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-nel 2603  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-tp 3979  df-op 3981  df-uni 4194  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-tr 4492  df-eprel 4736  df-id 4740  df-po 4746  df-so 4747  df-fr 4784  df-we 4786  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-pred 5369  df-ord 5415  df-on 5416  df-lim 5417  df-suc 5418  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-riota 6242  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-om 6686  df-wrecs 7015  df-recs 7077  df-rdg 7115  df-er 7350  df-en 7557  df-dom 7558  df-sdom 7559  df-sup 7937  df-pnf 9662  df-mnf 9663  df-xr 9664  df-ltxr 9665  df-le 9666  df-sub 9845  df-neg 9846  df-nn 10579  df-n0 10839  df-z 10908  df-uz 11130  df-od 16879
This theorem is referenced by:  odcl  16886  odid  16888
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