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Theorem gexlem1 14906
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 14905 . 2  |-  ( G  e.  V  ->  E  =  if ( I  =  (/) ,  0 ,  sup ( I ,  RR ,  `'  <  ) ) )
7 eqeq2 2305 . . . 4  |-  ( 0  =  if ( I  =  (/) ,  0 ,  sup ( I ,  RR ,  `'  <  ) )  ->  ( E  =  0  <->  E  =  if ( I  =  (/) ,  0 ,  sup (
I ,  RR ,  `'  <  ) ) ) )
87imbi1d 308 . . 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 2305 . . . 4  |-  ( sup ( I ,  RR ,  `'  <  )  =  if ( I  =  (/) ,  0 ,  sup ( I ,  RR ,  `'  <  ) )  ->  ( E  =  sup ( I ,  RR ,  `'  <  )  <-> 
E  =  if ( I  =  (/) ,  0 ,  sup ( I ,  RR ,  `'  <  ) ) ) )
109imbi1d 308 . . 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 374 . . . . 5  |-  ( ( E  =  0  /\  I  =  (/) )  -> 
( ( E  =  0  /\  I  =  (/) )  \/  E  e.  I ) )
1211expcom 424 . . . 4  |-  ( I  =  (/)  ->  ( E  =  0  ->  (
( E  =  0  /\  I  =  (/) )  \/  E  e.  I ) ) )
1312adantl 452 . . 3  |-  ( ( G  e.  V  /\  I  =  (/) )  -> 
( E  =  0  ->  ( ( E  =  0  /\  I  =  (/) )  \/  E  e.  I ) ) )
14 ssrab2 3271 . . . . . . 7  |-  { y  e.  NN  |  A. x  e.  X  (
y  .x.  x )  =  .0.  }  C_  NN
15 nnuz 10279 . . . . . . . 8  |-  NN  =  ( ZZ>= `  1 )
1615eqcomi 2300 . . . . . . 7  |-  ( ZZ>= ` 
1 )  =  NN
1714, 5, 163sstr4i 3230 . . . . . 6  |-  I  C_  ( ZZ>= `  1 )
18 df-ne 2461 . . . . . . . 8  |-  ( I  =/=  (/)  <->  -.  I  =  (/) )
1918biimpri 197 . . . . . . 7  |-  ( -.  I  =  (/)  ->  I  =/=  (/) )
2019adantl 452 . . . . . 6  |-  ( ( G  e.  V  /\  -.  I  =  (/) )  ->  I  =/=  (/) )
21 infmssuzcl 10317 . . . . . 6  |-  ( ( I  C_  ( ZZ>= ` 
1 )  /\  I  =/=  (/) )  ->  sup ( I ,  RR ,  `'  <  )  e.  I )
2217, 20, 21sylancr 644 . . . . 5  |-  ( ( G  e.  V  /\  -.  I  =  (/) )  ->  sup ( I ,  RR ,  `'  <  )  e.  I )
23 eleq1a 2365 . . . . 5  |-  ( sup ( I ,  RR ,  `'  <  )  e.  I  ->  ( E  =  sup ( I ,  RR ,  `'  <  )  ->  E  e.  I
) )
2422, 23syl 15 . . . 4  |-  ( ( G  e.  V  /\  -.  I  =  (/) )  -> 
( E  =  sup ( I ,  RR ,  `'  <  )  ->  E  e.  I )
)
25 olc 373 . . . 4  |-  ( E  e.  I  ->  (
( E  =  0  /\  I  =  (/) )  \/  E  e.  I ) )
2624, 25syl6 29 . . 3  |-  ( ( G  e.  V  /\  -.  I  =  (/) )  -> 
( E  =  sup ( I ,  RR ,  `'  <  )  -> 
( ( E  =  0  /\  I  =  (/) )  \/  E  e.  I ) ) )
278, 10, 13, 26ifbothda 3608 . 2  |-  ( G  e.  V  ->  ( E  =  if (
I  =  (/) ,  0 ,  sup ( I ,  RR ,  `'  <  ) )  ->  (
( E  =  0  /\  I  =  (/) )  \/  E  e.  I ) ) )
286, 27mpd 14 1  |-  ( G  e.  V  ->  (
( E  =  0  /\  I  =  (/) )  \/  E  e.  I ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 357    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   {crab 2560    C_ wss 3165   (/)c0 3468   ifcif 3578   `'ccnv 4704   ` cfv 5271  (class class class)co 5874   supcsup 7209   RRcr 8752   0cc0 8753   1c1 8754    < clt 8883   NNcn 9762   ZZ>=cuz 10246   Basecbs 13164   0gc0g 13416  .gcmg 14382  gExcgex 14857
This theorem is referenced by:  gexcl  14907  gexid  14908  gexdvds  14911
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-n0 9982  df-z 10041  df-uz 10247  df-gex 14861
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