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Theorem gexdvdsi 15219
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 960 . . . 4  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  E  ||  N )  ->  E  ||  N )
2 dvdszrcl 12859 . . . . 5  |-  ( E 
||  N  ->  ( E  e.  ZZ  /\  N  e.  ZZ ) )
3 divides 12856 . . . . 5  |-  ( ( E  e.  ZZ  /\  N  e.  ZZ )  ->  ( E  ||  N  <->  E. x  e.  ZZ  (
x  x.  E )  =  N ) )
42, 3biadan2 625 . . . 4  |-  ( E 
||  N  <->  ( ( E  e.  ZZ  /\  N  e.  ZZ )  /\  E. x  e.  ZZ  (
x  x.  E )  =  N ) )
51, 4sylib 190 . . 3  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  E  ||  N )  -> 
( ( E  e.  ZZ  /\  N  e.  ZZ )  /\  E. x  e.  ZZ  (
x  x.  E )  =  N ) )
65simprd 451 . 2  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  E  ||  N )  ->  E. x  e.  ZZ  ( x  x.  E
)  =  N )
7 simpl1 961 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  E  ||  N )  /\  x  e.  ZZ )  ->  G  e.  Grp )
8 simpr 449 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  E  ||  N )  /\  x  e.  ZZ )  ->  x  e.  ZZ )
95simpld 447 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  E  ||  N )  -> 
( E  e.  ZZ  /\  N  e.  ZZ ) )
109simpld 447 . . . . . . 7  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  E  ||  N )  ->  E  e.  ZZ )
1110adantr 453 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  E  ||  N )  /\  x  e.  ZZ )  ->  E  e.  ZZ )
12 simpl2 962 . . . . . 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 14922 . . . . . 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 1187 . . . . 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 15217 . . . . . . 7  |-  ( A  e.  X  ->  ( E  .x.  A )  =  .0.  )
2012, 19syl 16 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  E  ||  N )  /\  x  e.  ZZ )  ->  ( E  .x.  A
)  =  .0.  )
2120oveq2d 6099 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  E  ||  N )  /\  x  e.  ZZ )  ->  ( x  .x.  ( E  .x.  A ) )  =  ( x  .x.  .0.  ) )
2213, 14, 18mulgz 14913 . . . . . 6  |-  ( ( G  e.  Grp  /\  x  e.  ZZ )  ->  ( x  .x.  .0.  )  =  .0.  )
23223ad2antl1 1120 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  E  ||  N )  /\  x  e.  ZZ )  ->  ( x  .x.  .0.  )  =  .0.  )
2416, 21, 233eqtrd 2474 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  E  ||  N )  /\  x  e.  ZZ )  ->  ( ( x  x.  E )  .x.  A
)  =  .0.  )
25 oveq1 6090 . . . . 5  |-  ( ( x  x.  E )  =  N  ->  (
( x  x.  E
)  .x.  A )  =  ( N  .x.  A ) )
2625eqeq1d 2446 . . . 4  |-  ( ( x  x.  E )  =  N  ->  (
( ( x  x.  E )  .x.  A
)  =  .0.  <->  ( N  .x.  A )  =  .0.  ) )
2724, 26syl5ibcom 213 . . 3  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  E  ||  N )  /\  x  e.  ZZ )  ->  ( ( x  x.  E )  =  N  ->  ( N  .x.  A )  =  .0.  ) )
2827rexlimdva 2832 . 2  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  E  ||  N )  -> 
( E. x  e.  ZZ  ( x  x.  E )  =  N  ->  ( N  .x.  A )  =  .0.  ) )
296, 28mpd 15 1  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  E  ||  N )  -> 
( N  .x.  A
)  =  .0.  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   E.wrex 2708   class class class wbr 4214   ` cfv 5456  (class class class)co 6083    x. cmul 8997   ZZcz 10284    || cdivides 12854   Basecbs 13471   0gc0g 13725   Grpcgrp 14687  .gcmg 14691  gExcgex 15166
This theorem is referenced by:  gexdvds  15220  gex2abl  15468
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 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-inf2 7598  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
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 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-sup 7448  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-nn 10003  df-n0 10224  df-z 10285  df-uz 10491  df-fz 11046  df-seq 11326  df-dvds 12855  df-0g 13729  df-mnd 14692  df-grp 14814  df-minusg 14815  df-mulg 14817  df-gex 15170
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