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Theorem gexlem1 16390
Description: The group element order is either zero or a nonzero multiplier that annihilates the element. (Contributed by Mario Carneiro, 23-Apr-2016.)
Hypotheses
Ref Expression
gexval.1  |-  X  =  ( Base `  G
)
gexval.2  |-  .x.  =  (.g
`  G )
gexval.3  |-  .0.  =  ( 0g `  G )
gexval.4  |-  E  =  (gEx `  G )
gexval.i  |-  I  =  { y  e.  NN  |  A. x  e.  X  ( y  .x.  x
)  =  .0.  }
Assertion
Ref Expression
gexlem1  |-  ( G  e.  V  ->  (
( E  =  0  /\  I  =  (/) )  \/  E  e.  I ) )
Distinct variable groups:    x, y,  .0.    x, G, y    x, V, y    x,  .x. , y    x, X
Allowed substitution hints:    E( x, y)    I( x, y)    X( y)

Proof of Theorem gexlem1
StepHypRef Expression
1 gexval.1 . . 3  |-  X  =  ( Base `  G
)
2 gexval.2 . . 3  |-  .x.  =  (.g
`  G )
3 gexval.3 . . 3  |-  .0.  =  ( 0g `  G )
4 gexval.4 . . 3  |-  E  =  (gEx `  G )
5 gexval.i . . 3  |-  I  =  { y  e.  NN  |  A. x  e.  X  ( y  .x.  x
)  =  .0.  }
61, 2, 3, 4, 5gexval 16389 . 2  |-  ( G  e.  V  ->  E  =  if ( I  =  (/) ,  0 ,  sup ( I ,  RR ,  `'  <  ) ) )
7 eqeq2 2477 . . . 4  |-  ( 0  =  if ( I  =  (/) ,  0 ,  sup ( I ,  RR ,  `'  <  ) )  ->  ( E  =  0  <->  E  =  if ( I  =  (/) ,  0 ,  sup (
I ,  RR ,  `'  <  ) ) ) )
87imbi1d 317 . . 3  |-  ( 0  =  if ( I  =  (/) ,  0 ,  sup ( I ,  RR ,  `'  <  ) )  ->  ( ( E  =  0  ->  ( ( E  =  0  /\  I  =  (/) )  \/  E  e.  I ) )  <->  ( E  =  if ( I  =  (/) ,  0 ,  sup ( I ,  RR ,  `'  <  ) )  ->  ( ( E  =  0  /\  I  =  (/) )  \/  E  e.  I ) ) ) )
9 eqeq2 2477 . . . 4  |-  ( sup ( I ,  RR ,  `'  <  )  =  if ( I  =  (/) ,  0 ,  sup ( I ,  RR ,  `'  <  ) )  ->  ( E  =  sup ( I ,  RR ,  `'  <  )  <-> 
E  =  if ( I  =  (/) ,  0 ,  sup ( I ,  RR ,  `'  <  ) ) ) )
109imbi1d 317 . . 3  |-  ( sup ( I ,  RR ,  `'  <  )  =  if ( I  =  (/) ,  0 ,  sup ( I ,  RR ,  `'  <  ) )  ->  ( ( E  =  sup ( I ,  RR ,  `'  <  )  ->  ( ( E  =  0  /\  I  =  (/) )  \/  E  e.  I ) )  <->  ( E  =  if ( I  =  (/) ,  0 ,  sup ( I ,  RR ,  `'  <  ) )  ->  ( ( E  =  0  /\  I  =  (/) )  \/  E  e.  I ) ) ) )
11 orc 385 . . . . 5  |-  ( ( E  =  0  /\  I  =  (/) )  -> 
( ( E  =  0  /\  I  =  (/) )  \/  E  e.  I ) )
1211expcom 435 . . . 4  |-  ( I  =  (/)  ->  ( E  =  0  ->  (
( E  =  0  /\  I  =  (/) )  \/  E  e.  I ) ) )
1312adantl 466 . . 3  |-  ( ( G  e.  V  /\  I  =  (/) )  -> 
( E  =  0  ->  ( ( E  =  0  /\  I  =  (/) )  \/  E  e.  I ) ) )
14 ssrab2 3580 . . . . . . 7  |-  { y  e.  NN  |  A. x  e.  X  (
y  .x.  x )  =  .0.  }  C_  NN
15 nnuz 11108 . . . . . . . 8  |-  NN  =  ( ZZ>= `  1 )
1615eqcomi 2475 . . . . . . 7  |-  ( ZZ>= ` 
1 )  =  NN
1714, 5, 163sstr4i 3538 . . . . . 6  |-  I  C_  ( ZZ>= `  1 )
18 df-ne 2659 . . . . . . . 8  |-  ( I  =/=  (/)  <->  -.  I  =  (/) )
1918biimpri 206 . . . . . . 7  |-  ( -.  I  =  (/)  ->  I  =/=  (/) )
2019adantl 466 . . . . . 6  |-  ( ( G  e.  V  /\  -.  I  =  (/) )  ->  I  =/=  (/) )
21 infmssuzcl 11156 . . . . . 6  |-  ( ( I  C_  ( ZZ>= ` 
1 )  /\  I  =/=  (/) )  ->  sup ( I ,  RR ,  `'  <  )  e.  I )
2217, 20, 21sylancr 663 . . . . 5  |-  ( ( G  e.  V  /\  -.  I  =  (/) )  ->  sup ( I ,  RR ,  `'  <  )  e.  I )
23 eleq1a 2545 . . . . 5  |-  ( sup ( I ,  RR ,  `'  <  )  e.  I  ->  ( E  =  sup ( I ,  RR ,  `'  <  )  ->  E  e.  I
) )
2422, 23syl 16 . . . 4  |-  ( ( G  e.  V  /\  -.  I  =  (/) )  -> 
( E  =  sup ( I ,  RR ,  `'  <  )  ->  E  e.  I )
)
25 olc 384 . . . 4  |-  ( E  e.  I  ->  (
( E  =  0  /\  I  =  (/) )  \/  E  e.  I ) )
2624, 25syl6 33 . . 3  |-  ( ( G  e.  V  /\  -.  I  =  (/) )  -> 
( E  =  sup ( I ,  RR ,  `'  <  )  -> 
( ( E  =  0  /\  I  =  (/) )  \/  E  e.  I ) ) )
278, 10, 13, 26ifbothda 3969 . 2  |-  ( G  e.  V  ->  ( E  =  if (
I  =  (/) ,  0 ,  sup ( I ,  RR ,  `'  <  ) )  ->  (
( E  =  0  /\  I  =  (/) )  \/  E  e.  I ) ) )
286, 27mpd 15 1  |-  ( G  e.  V  ->  (
( E  =  0  /\  I  =  (/) )  \/  E  e.  I ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1374    e. wcel 1762    =/= wne 2657   A.wral 2809   {crab 2813    C_ wss 3471   (/)c0 3780   ifcif 3934   `'ccnv 4993   ` cfv 5581  (class class class)co 6277   supcsup 7891   RRcr 9482   0cc0 9483   1c1 9484    < clt 9619   NNcn 10527   ZZ>=cuz 11073   Basecbs 14481   0gc0g 14686  .gcmg 15722  gExcgex 16341
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6674  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7509  df-dom 7510  df-sdom 7511  df-sup 7892  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-nn 10528  df-n0 10787  df-z 10856  df-uz 11074  df-gex 16345
This theorem is referenced by:  gexcl  16391  gexid  16392  gexdvds  16395
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