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Theorem gexdvdsi 17177
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 1007 . . . 4  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  E  ||  N )  ->  E  ||  N )
2 dvdszrcl 14253 . . . . 5  |-  ( E 
||  N  ->  ( E  e.  ZZ  /\  N  e.  ZZ ) )
3 divides 14250 . . . . 5  |-  ( ( E  e.  ZZ  /\  N  e.  ZZ )  ->  ( E  ||  N  <->  E. x  e.  ZZ  (
x  x.  E )  =  N ) )
42, 3biadan2 646 . . . 4  |-  ( E 
||  N  <->  ( ( E  e.  ZZ  /\  N  e.  ZZ )  /\  E. x  e.  ZZ  (
x  x.  E )  =  N ) )
51, 4sylib 199 . . 3  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  E  ||  N )  -> 
( ( E  e.  ZZ  /\  N  e.  ZZ )  /\  E. x  e.  ZZ  (
x  x.  E )  =  N ) )
65simprd 464 . 2  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  E  ||  N )  ->  E. x  e.  ZZ  ( x  x.  E
)  =  N )
7 simpl1 1008 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  E  ||  N )  /\  x  e.  ZZ )  ->  G  e.  Grp )
8 simpr 462 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  E  ||  N )  /\  x  e.  ZZ )  ->  x  e.  ZZ )
95simplld 759 . . . . . . 7  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  E  ||  N )  ->  E  e.  ZZ )
109adantr 466 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  E  ||  N )  /\  x  e.  ZZ )  ->  E  e.  ZZ )
11 simpl2 1009 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  E  ||  N )  /\  x  e.  ZZ )  ->  A  e.  X )
12 gexcl.1 . . . . . . 7  |-  X  =  ( Base `  G
)
13 gexid.3 . . . . . . 7  |-  .x.  =  (.g
`  G )
1412, 13mulgass 16731 . . . . . 6  |-  ( ( G  e.  Grp  /\  ( x  e.  ZZ  /\  E  e.  ZZ  /\  A  e.  X )
)  ->  ( (
x  x.  E ) 
.x.  A )  =  ( x  .x.  ( E  .x.  A ) ) )
157, 8, 10, 11, 14syl13anc 1266 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  E  ||  N )  /\  x  e.  ZZ )  ->  ( ( x  x.  E )  .x.  A
)  =  ( x 
.x.  ( E  .x.  A ) ) )
16 gexcl.2 . . . . . . . 8  |-  E  =  (gEx `  G )
17 gexid.4 . . . . . . . 8  |-  .0.  =  ( 0g `  G )
1812, 16, 13, 17gexid 17175 . . . . . . 7  |-  ( A  e.  X  ->  ( E  .x.  A )  =  .0.  )
1911, 18syl 17 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  E  ||  N )  /\  x  e.  ZZ )  ->  ( E  .x.  A
)  =  .0.  )
2019oveq2d 6265 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  E  ||  N )  /\  x  e.  ZZ )  ->  ( x  .x.  ( E  .x.  A ) )  =  ( x  .x.  .0.  ) )
2112, 13, 17mulgz 16722 . . . . . 6  |-  ( ( G  e.  Grp  /\  x  e.  ZZ )  ->  ( x  .x.  .0.  )  =  .0.  )
22213ad2antl1 1167 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  E  ||  N )  /\  x  e.  ZZ )  ->  ( x  .x.  .0.  )  =  .0.  )
2315, 20, 223eqtrd 2466 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  E  ||  N )  /\  x  e.  ZZ )  ->  ( ( x  x.  E )  .x.  A
)  =  .0.  )
24 oveq1 6256 . . . . 5  |-  ( ( x  x.  E )  =  N  ->  (
( x  x.  E
)  .x.  A )  =  ( N  .x.  A ) )
2524eqeq1d 2430 . . . 4  |-  ( ( x  x.  E )  =  N  ->  (
( ( x  x.  E )  .x.  A
)  =  .0.  <->  ( N  .x.  A )  =  .0.  ) )
2623, 25syl5ibcom 223 . . 3  |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  E  ||  N )  /\  x  e.  ZZ )  ->  ( ( x  x.  E )  =  N  ->  ( N  .x.  A )  =  .0.  ) )
2726rexlimdva 2856 . 2  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  E  ||  N )  -> 
( E. x  e.  ZZ  ( x  x.  E )  =  N  ->  ( N  .x.  A )  =  .0.  ) )
286, 27mpd 15 1  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  E  ||  N )  -> 
( N  .x.  A
)  =  .0.  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872   E.wrex 2715   class class class wbr 4366   ` cfv 5544  (class class class)co 6249    x. cmul 9495   ZZcz 10888    || cdvds 14248   Basecbs 15064   0gc0g 15281   Grpcgrp 16612  .gcmg 16615  gExcgex 17110
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-rep 4479  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541  ax-inf2 8099  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-nel 2602  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-pss 3395  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-tp 3946  df-op 3948  df-uni 4163  df-iun 4244  df-br 4367  df-opab 4426  df-mpt 4427  df-tr 4462  df-eprel 4707  df-id 4711  df-po 4717  df-so 4718  df-fr 4755  df-we 4757  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-pred 5342  df-ord 5388  df-on 5389  df-lim 5390  df-suc 5391  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-riota 6211  df-ov 6252  df-oprab 6253  df-mpt2 6254  df-om 6651  df-1st 6751  df-2nd 6752  df-wrecs 6983  df-recs 7045  df-rdg 7083  df-er 7318  df-en 7525  df-dom 7526  df-sdom 7527  df-sup 7909  df-inf 7910  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9813  df-neg 9814  df-nn 10561  df-n0 10821  df-z 10889  df-uz 11111  df-fz 11736  df-seq 12164  df-dvds 14249  df-0g 15283  df-mgm 16431  df-sgrp 16470  df-mnd 16480  df-grp 16616  df-minusg 16617  df-mulg 16619  df-gex 17117
This theorem is referenced by:  gexdvds  17178  gex2abl  17432
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