MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  gexlem1 Structured version   Unicode version

Theorem gexlem1 15214
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 15213 . 2  |-  ( G  e.  V  ->  E  =  if ( I  =  (/) ,  0 ,  sup ( I ,  RR ,  `'  <  ) ) )
7 eqeq2 2446 . . . 4  |-  ( 0  =  if ( I  =  (/) ,  0 ,  sup ( I ,  RR ,  `'  <  ) )  ->  ( E  =  0  <->  E  =  if ( I  =  (/) ,  0 ,  sup (
I ,  RR ,  `'  <  ) ) ) )
87imbi1d 310 . . 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 2446 . . . 4  |-  ( sup ( I ,  RR ,  `'  <  )  =  if ( I  =  (/) ,  0 ,  sup ( I ,  RR ,  `'  <  ) )  ->  ( E  =  sup ( I ,  RR ,  `'  <  )  <-> 
E  =  if ( I  =  (/) ,  0 ,  sup ( I ,  RR ,  `'  <  ) ) ) )
109imbi1d 310 . . 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 376 . . . . 5  |-  ( ( E  =  0  /\  I  =  (/) )  -> 
( ( E  =  0  /\  I  =  (/) )  \/  E  e.  I ) )
1211expcom 426 . . . 4  |-  ( I  =  (/)  ->  ( E  =  0  ->  (
( E  =  0  /\  I  =  (/) )  \/  E  e.  I ) ) )
1312adantl 454 . . 3  |-  ( ( G  e.  V  /\  I  =  (/) )  -> 
( E  =  0  ->  ( ( E  =  0  /\  I  =  (/) )  \/  E  e.  I ) ) )
14 ssrab2 3429 . . . . . . 7  |-  { y  e.  NN  |  A. x  e.  X  (
y  .x.  x )  =  .0.  }  C_  NN
15 nnuz 10522 . . . . . . . 8  |-  NN  =  ( ZZ>= `  1 )
1615eqcomi 2441 . . . . . . 7  |-  ( ZZ>= ` 
1 )  =  NN
1714, 5, 163sstr4i 3388 . . . . . 6  |-  I  C_  ( ZZ>= `  1 )
18 df-ne 2602 . . . . . . . 8  |-  ( I  =/=  (/)  <->  -.  I  =  (/) )
1918biimpri 199 . . . . . . 7  |-  ( -.  I  =  (/)  ->  I  =/=  (/) )
2019adantl 454 . . . . . 6  |-  ( ( G  e.  V  /\  -.  I  =  (/) )  ->  I  =/=  (/) )
21 infmssuzcl 10560 . . . . . 6  |-  ( ( I  C_  ( ZZ>= ` 
1 )  /\  I  =/=  (/) )  ->  sup ( I ,  RR ,  `'  <  )  e.  I )
2217, 20, 21sylancr 646 . . . . 5  |-  ( ( G  e.  V  /\  -.  I  =  (/) )  ->  sup ( I ,  RR ,  `'  <  )  e.  I )
23 eleq1a 2506 . . . . 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 375 . . . 4  |-  ( E  e.  I  ->  (
( E  =  0  /\  I  =  (/) )  \/  E  e.  I ) )
2624, 25syl6 32 . . 3  |-  ( ( G  e.  V  /\  -.  I  =  (/) )  -> 
( E  =  sup ( I ,  RR ,  `'  <  )  -> 
( ( E  =  0  /\  I  =  (/) )  \/  E  e.  I ) ) )
278, 10, 13, 26ifbothda 3770 . 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 set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 359    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2600   A.wral 2706   {crab 2710    C_ wss 3321   (/)c0 3629   ifcif 3740   `'ccnv 4878   ` cfv 5455  (class class class)co 6082   supcsup 7446   RRcr 8990   0cc0 8991   1c1 8992    < clt 9121   NNcn 10001   ZZ>=cuz 10489   Basecbs 13470   0gc0g 13724  .gcmg 14690  gExcgex 15165
This theorem is referenced by:  gexcl  15215  gexid  15216  gexdvds  15219
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702  ax-cnex 9047  ax-resscn 9048  ax-1cn 9049  ax-icn 9050  ax-addcl 9051  ax-addrcl 9052  ax-mulcl 9053  ax-mulrcl 9054  ax-mulcom 9055  ax-addass 9056  ax-mulass 9057  ax-distr 9058  ax-i2m1 9059  ax-1ne0 9060  ax-1rid 9061  ax-rnegex 9062  ax-rrecex 9063  ax-cnre 9064  ax-pre-lttri 9065  ax-pre-lttrn 9066  ax-pre-ltadd 9067  ax-pre-mulgt0 9068
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rmo 2714  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-riota 6550  df-recs 6634  df-rdg 6669  df-er 6906  df-en 7111  df-dom 7112  df-sdom 7113  df-sup 7447  df-pnf 9123  df-mnf 9124  df-xr 9125  df-ltxr 9126  df-le 9127  df-sub 9294  df-neg 9295  df-nn 10002  df-n0 10223  df-z 10284  df-uz 10490  df-gex 15169
  Copyright terms: Public domain W3C validator