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Theorem gexdvdsi 14910
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 12552 . . . . 5  |-  ( E 
||  N  ->  ( E  e.  ZZ  /\  N  e.  ZZ ) )
3 divides 12549 . . . . 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 14613 . . . . . 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 14908 . . . . . . 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 5890 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  E  ||  N )  /\  x  e.  ZZ )  ->  ( x  .x.  ( E  .x.  A ) )  =  ( x  .x.  .0.  ) )
2213, 14, 18mulgz 14604 . . . . . 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 2332 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  E  ||  N )  /\  x  e.  ZZ )  ->  ( ( x  x.  E )  .x.  A
)  =  .0.  )
25 oveq1 5881 . . . . 5  |-  ( ( x  x.  E )  =  N  ->  (
( x  x.  E
)  .x.  A )  =  ( N  .x.  A ) )
2625eqeq1d 2304 . . . 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 2680 . 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 1632    e. wcel 1696   E.wrex 2557   class class class wbr 4039   ` cfv 5271  (class class class)co 5874    x. cmul 8758   ZZcz 10040    || cdivides 12547   Basecbs 13164   0gc0g 13416   Grpcgrp 14378  .gcmg 14382  gExcgex 14857
This theorem is referenced by:  gexdvds  14911  gex2abl  15159
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-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  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-1st 6138  df-2nd 6139  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-fz 10799  df-seq 11063  df-dvds 12548  df-0g 13420  df-mnd 14383  df-grp 14505  df-minusg 14506  df-mulg 14508  df-gex 14861
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