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Theorem odlem1 16355
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 16354 . 2  |-  ( A  e.  X  ->  ( O `  A )  =  if ( I  =  (/) ,  0 ,  sup ( I ,  RR ,  `'  <  ) ) )
7 eqeq2 2482 . . . 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 2482 . . . 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 3585 . . . . . . 7  |-  { y  e.  NN  |  ( y  .x.  A )  =  .0.  }  C_  NN
15 nnuz 11113 . . . . . . . 8  |-  NN  =  ( ZZ>= `  1 )
1615eqcomi 2480 . . . . . . 7  |-  ( ZZ>= ` 
1 )  =  NN
1714, 5, 163sstr4i 3543 . . . . . 6  |-  I  C_  ( ZZ>= `  1 )
18 df-ne 2664 . . . . . . . 8  |-  ( I  =/=  (/)  <->  -.  I  =  (/) )
1918biimpri 206 . . . . . . 7  |-  ( -.  I  =  (/)  ->  I  =/=  (/) )
2019adantl 466 . . . . . 6  |-  ( ( A  e.  X  /\  -.  I  =  (/) )  ->  I  =/=  (/) )
21 infmssuzcl 11161 . . . . . 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 2550 . . . . 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 3974 . 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 1379    e. wcel 1767    =/= wne 2662   {crab 2818    C_ wss 3476   (/)c0 3785   ifcif 3939   `'ccnv 4998   ` cfv 5586  (class class class)co 6282   supcsup 7896   RRcr 9487   0cc0 9488   1c1 9489    < clt 9624   NNcn 10532   ZZ>=cuz 11078   Basecbs 14486   0gc0g 14691  .gcmg 15727   odcod 16345
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-recs 7039  df-rdg 7073  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-sup 7897  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-n0 10792  df-z 10861  df-uz 11079  df-od 16349
This theorem is referenced by:  odcl  16356  odid  16358
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