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Theorem gexdvdsi 14894
Description: Any group element is annihilated by any multiple of the group exponent. (Contributed by Mario Carneiro, 24-Apr-2016.)
Hypotheses
Ref Expression
gexcl.1  |-  X  =  ( Base `  G
)
gexcl.2  |-  E  =  (gEx `  G )
gexid.3  |-  .x.  =  (.g
`  G )
gexid.4  |-  .0.  =  ( 0g `  G )
Assertion
Ref Expression
gexdvdsi  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  E  ||  N )  -> 
( N  .x.  A
)  =  .0.  )

Proof of Theorem gexdvdsi
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 simp3 957 . . . 4  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  E  ||  N )  ->  E  ||  N )
2 dvdszrcl 12536 . . . . 5  |-  ( E 
||  N  ->  ( E  e.  ZZ  /\  N  e.  ZZ ) )
3 divides 12533 . . . . 5  |-  ( ( E  e.  ZZ  /\  N  e.  ZZ )  ->  ( E  ||  N  <->  E. x  e.  ZZ  (
x  x.  E )  =  N ) )
42, 3biadan2 623 . . . 4  |-  ( E 
||  N  <->  ( ( E  e.  ZZ  /\  N  e.  ZZ )  /\  E. x  e.  ZZ  (
x  x.  E )  =  N ) )
51, 4sylib 188 . . 3  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  E  ||  N )  -> 
( ( E  e.  ZZ  /\  N  e.  ZZ )  /\  E. x  e.  ZZ  (
x  x.  E )  =  N ) )
65simprd 449 . 2  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  E  ||  N )  ->  E. x  e.  ZZ  ( x  x.  E
)  =  N )
7 simpl1 958 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  E  ||  N )  /\  x  e.  ZZ )  ->  G  e.  Grp )
8 simpr 447 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  E  ||  N )  /\  x  e.  ZZ )  ->  x  e.  ZZ )
95simpld 445 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  E  ||  N )  -> 
( E  e.  ZZ  /\  N  e.  ZZ ) )
109simpld 445 . . . . . . 7  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  E  ||  N )  ->  E  e.  ZZ )
1110adantr 451 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  E  ||  N )  /\  x  e.  ZZ )  ->  E  e.  ZZ )
12 simpl2 959 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  E  ||  N )  /\  x  e.  ZZ )  ->  A  e.  X )
13 gexcl.1 . . . . . . 7  |-  X  =  ( Base `  G
)
14 gexid.3 . . . . . . 7  |-  .x.  =  (.g
`  G )
1513, 14mulgass 14597 . . . . . 6  |-  ( ( G  e.  Grp  /\  ( x  e.  ZZ  /\  E  e.  ZZ  /\  A  e.  X )
)  ->  ( (
x  x.  E ) 
.x.  A )  =  ( x  .x.  ( E  .x.  A ) ) )
167, 8, 11, 12, 15syl13anc 1184 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  E  ||  N )  /\  x  e.  ZZ )  ->  ( ( x  x.  E )  .x.  A
)  =  ( x 
.x.  ( E  .x.  A ) ) )
17 gexcl.2 . . . . . . . 8  |-  E  =  (gEx `  G )
18 gexid.4 . . . . . . . 8  |-  .0.  =  ( 0g `  G )
1913, 17, 14, 18gexid 14892 . . . . . . 7  |-  ( A  e.  X  ->  ( E  .x.  A )  =  .0.  )
2012, 19syl 15 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  E  ||  N )  /\  x  e.  ZZ )  ->  ( E  .x.  A
)  =  .0.  )
2120oveq2d 5874 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  E  ||  N )  /\  x  e.  ZZ )  ->  ( x  .x.  ( E  .x.  A ) )  =  ( x  .x.  .0.  ) )
2213, 14, 18mulgz 14588 . . . . . 6  |-  ( ( G  e.  Grp  /\  x  e.  ZZ )  ->  ( x  .x.  .0.  )  =  .0.  )
23223ad2antl1 1117 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  E  ||  N )  /\  x  e.  ZZ )  ->  ( x  .x.  .0.  )  =  .0.  )
2416, 21, 233eqtrd 2319 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  E  ||  N )  /\  x  e.  ZZ )  ->  ( ( x  x.  E )  .x.  A
)  =  .0.  )
25 oveq1 5865 . . . . 5  |-  ( ( x  x.  E )  =  N  ->  (
( x  x.  E
)  .x.  A )  =  ( N  .x.  A ) )
2625eqeq1d 2291 . . . 4  |-  ( ( x  x.  E )  =  N  ->  (
( ( x  x.  E )  .x.  A
)  =  .0.  <->  ( N  .x.  A )  =  .0.  ) )
2724, 26syl5ibcom 211 . . 3  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  E  ||  N )  /\  x  e.  ZZ )  ->  ( ( x  x.  E )  =  N  ->  ( N  .x.  A )  =  .0.  ) )
2827rexlimdva 2667 . 2  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  E  ||  N )  -> 
( E. x  e.  ZZ  ( x  x.  E )  =  N  ->  ( N  .x.  A )  =  .0.  ) )
296, 28mpd 14 1  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  E  ||  N )  -> 
( N  .x.  A
)  =  .0.  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   E.wrex 2544   class class class wbr 4023   ` cfv 5255  (class class class)co 5858    x. cmul 8742   ZZcz 10024    || cdivides 12531   Basecbs 13148   0gc0g 13400   Grpcgrp 14362  .gcmg 14366  gExcgex 14841
This theorem is referenced by:  gexdvds  14895  gex2abl  15143
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-seq 11047  df-dvds 12532  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-mulg 14492  df-gex 14845
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