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Theorem gexlem1 16716
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 16715 . 2  |-  ( G  e.  V  ->  E  =  if ( I  =  (/) ,  0 ,  sup ( I ,  RR ,  `'  <  ) ) )
7 eqeq2 2397 . . . 4  |-  ( 0  =  if ( I  =  (/) ,  0 ,  sup ( I ,  RR ,  `'  <  ) )  ->  ( E  =  0  <->  E  =  if ( I  =  (/) ,  0 ,  sup (
I ,  RR ,  `'  <  ) ) ) )
87imbi1d 315 . . 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 2397 . . . 4  |-  ( sup ( I ,  RR ,  `'  <  )  =  if ( I  =  (/) ,  0 ,  sup ( I ,  RR ,  `'  <  ) )  ->  ( E  =  sup ( I ,  RR ,  `'  <  )  <-> 
E  =  if ( I  =  (/) ,  0 ,  sup ( I ,  RR ,  `'  <  ) ) ) )
109imbi1d 315 . . 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 383 . . . . 5  |-  ( ( E  =  0  /\  I  =  (/) )  -> 
( ( E  =  0  /\  I  =  (/) )  \/  E  e.  I ) )
1211expcom 433 . . . 4  |-  ( I  =  (/)  ->  ( E  =  0  ->  (
( E  =  0  /\  I  =  (/) )  \/  E  e.  I ) ) )
1312adantl 464 . . 3  |-  ( ( G  e.  V  /\  I  =  (/) )  -> 
( E  =  0  ->  ( ( E  =  0  /\  I  =  (/) )  \/  E  e.  I ) ) )
14 ssrab2 3499 . . . . . . 7  |-  { y  e.  NN  |  A. x  e.  X  (
y  .x.  x )  =  .0.  }  C_  NN
15 nnuz 11036 . . . . . . . 8  |-  NN  =  ( ZZ>= `  1 )
1615eqcomi 2395 . . . . . . 7  |-  ( ZZ>= ` 
1 )  =  NN
1714, 5, 163sstr4i 3456 . . . . . 6  |-  I  C_  ( ZZ>= `  1 )
18 df-ne 2579 . . . . . . . 8  |-  ( I  =/=  (/)  <->  -.  I  =  (/) )
1918biimpri 206 . . . . . . 7  |-  ( -.  I  =  (/)  ->  I  =/=  (/) )
2019adantl 464 . . . . . 6  |-  ( ( G  e.  V  /\  -.  I  =  (/) )  ->  I  =/=  (/) )
21 infmssuzcl 11084 . . . . . 6  |-  ( ( I  C_  ( ZZ>= ` 
1 )  /\  I  =/=  (/) )  ->  sup ( I ,  RR ,  `'  <  )  e.  I )
2217, 20, 21sylancr 661 . . . . 5  |-  ( ( G  e.  V  /\  -.  I  =  (/) )  ->  sup ( I ,  RR ,  `'  <  )  e.  I )
23 eleq1a 2465 . . . . 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 382 . . . 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 3892 . 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 366    /\ wa 367    = wceq 1399    e. wcel 1826    =/= wne 2577   A.wral 2732   {crab 2736    C_ wss 3389   (/)c0 3711   ifcif 3857   `'ccnv 4912   ` cfv 5496  (class class class)co 6196   supcsup 7815   RRcr 9402   0cc0 9403   1c1 9404    < clt 9539   NNcn 10452   ZZ>=cuz 11001   Basecbs 14634   0gc0g 14847  .gcmg 16173  gExcgex 16667
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-recs 6960  df-rdg 6994  df-er 7229  df-en 7436  df-dom 7437  df-sdom 7438  df-sup 7816  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-nn 10453  df-n0 10713  df-z 10782  df-uz 11002  df-gex 16671
This theorem is referenced by:  gexcl  16717  gexid  16718  gexdvds  16721
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