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Theorem pgpfi1 15234
Description: A finite group with order a power of a prime  P is a  P-group. (Contributed by Mario Carneiro, 16-Jan-2015.)
Hypothesis
Ref Expression
pgpfi1.1  |-  X  =  ( Base `  G
)
Assertion
Ref Expression
pgpfi1  |-  ( ( G  e.  Grp  /\  P  e.  Prime  /\  N  e.  NN0 )  ->  (
( # `  X )  =  ( P ^ N )  ->  P pGrp  G ) )

Proof of Theorem pgpfi1
Dummy variables  n  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl2 962 . . 3  |-  ( ( ( G  e.  Grp  /\  P  e.  Prime  /\  N  e.  NN0 )  /\  ( # `
 X )  =  ( P ^ N
) )  ->  P  e.  Prime )
2 simpl1 961 . . 3  |-  ( ( ( G  e.  Grp  /\  P  e.  Prime  /\  N  e.  NN0 )  /\  ( # `
 X )  =  ( P ^ N
) )  ->  G  e.  Grp )
3 simpll3 999 . . . . . 6  |-  ( ( ( ( G  e. 
Grp  /\  P  e.  Prime  /\  N  e.  NN0 )  /\  ( # `  X
)  =  ( P ^ N ) )  /\  x  e.  X
)  ->  N  e.  NN0 )
42adantr 453 . . . . . . . 8  |-  ( ( ( ( G  e. 
Grp  /\  P  e.  Prime  /\  N  e.  NN0 )  /\  ( # `  X
)  =  ( P ^ N ) )  /\  x  e.  X
)  ->  G  e.  Grp )
5 simplr 733 . . . . . . . . . 10  |-  ( ( ( ( G  e. 
Grp  /\  P  e.  Prime  /\  N  e.  NN0 )  /\  ( # `  X
)  =  ( P ^ N ) )  /\  x  e.  X
)  ->  ( # `  X
)  =  ( P ^ N ) )
61adantr 453 . . . . . . . . . . . . 13  |-  ( ( ( ( G  e. 
Grp  /\  P  e.  Prime  /\  N  e.  NN0 )  /\  ( # `  X
)  =  ( P ^ N ) )  /\  x  e.  X
)  ->  P  e.  Prime )
7 prmnn 13087 . . . . . . . . . . . . 13  |-  ( P  e.  Prime  ->  P  e.  NN )
86, 7syl 16 . . . . . . . . . . . 12  |-  ( ( ( ( G  e. 
Grp  /\  P  e.  Prime  /\  N  e.  NN0 )  /\  ( # `  X
)  =  ( P ^ N ) )  /\  x  e.  X
)  ->  P  e.  NN )
98, 3nnexpcld 11549 . . . . . . . . . . 11  |-  ( ( ( ( G  e. 
Grp  /\  P  e.  Prime  /\  N  e.  NN0 )  /\  ( # `  X
)  =  ( P ^ N ) )  /\  x  e.  X
)  ->  ( P ^ N )  e.  NN )
109nnnn0d 10279 . . . . . . . . . 10  |-  ( ( ( ( G  e. 
Grp  /\  P  e.  Prime  /\  N  e.  NN0 )  /\  ( # `  X
)  =  ( P ^ N ) )  /\  x  e.  X
)  ->  ( P ^ N )  e.  NN0 )
115, 10eqeltrd 2512 . . . . . . . . 9  |-  ( ( ( ( G  e. 
Grp  /\  P  e.  Prime  /\  N  e.  NN0 )  /\  ( # `  X
)  =  ( P ^ N ) )  /\  x  e.  X
)  ->  ( # `  X
)  e.  NN0 )
12 pgpfi1.1 . . . . . . . . . . 11  |-  X  =  ( Base `  G
)
13 fvex 5745 . . . . . . . . . . 11  |-  ( Base `  G )  e.  _V
1412, 13eqeltri 2508 . . . . . . . . . 10  |-  X  e. 
_V
15 hashclb 11646 . . . . . . . . . 10  |-  ( X  e.  _V  ->  ( X  e.  Fin  <->  ( # `  X
)  e.  NN0 )
)
1614, 15ax-mp 5 . . . . . . . . 9  |-  ( X  e.  Fin  <->  ( # `  X
)  e.  NN0 )
1711, 16sylibr 205 . . . . . . . 8  |-  ( ( ( ( G  e. 
Grp  /\  P  e.  Prime  /\  N  e.  NN0 )  /\  ( # `  X
)  =  ( P ^ N ) )  /\  x  e.  X
)  ->  X  e.  Fin )
18 simpr 449 . . . . . . . 8  |-  ( ( ( ( G  e. 
Grp  /\  P  e.  Prime  /\  N  e.  NN0 )  /\  ( # `  X
)  =  ( P ^ N ) )  /\  x  e.  X
)  ->  x  e.  X )
19 eqid 2438 . . . . . . . . 9  |-  ( od
`  G )  =  ( od `  G
)
2012, 19oddvds2 15207 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  X  e.  Fin  /\  x  e.  X )  ->  (
( od `  G
) `  x )  ||  ( # `  X
) )
214, 17, 18, 20syl3anc 1185 . . . . . . 7  |-  ( ( ( ( G  e. 
Grp  /\  P  e.  Prime  /\  N  e.  NN0 )  /\  ( # `  X
)  =  ( P ^ N ) )  /\  x  e.  X
)  ->  ( ( od `  G ) `  x )  ||  ( # `
 X ) )
2221, 5breqtrd 4239 . . . . . 6  |-  ( ( ( ( G  e. 
Grp  /\  P  e.  Prime  /\  N  e.  NN0 )  /\  ( # `  X
)  =  ( P ^ N ) )  /\  x  e.  X
)  ->  ( ( od `  G ) `  x )  ||  ( P ^ N ) )
23 oveq2 6092 . . . . . . . 8  |-  ( n  =  N  ->  ( P ^ n )  =  ( P ^ N
) )
2423breq2d 4227 . . . . . . 7  |-  ( n  =  N  ->  (
( ( od `  G ) `  x
)  ||  ( P ^ n )  <->  ( ( od `  G ) `  x )  ||  ( P ^ N ) ) )
2524rspcev 3054 . . . . . 6  |-  ( ( N  e.  NN0  /\  ( ( od `  G ) `  x
)  ||  ( P ^ N ) )  ->  E. n  e.  NN0  ( ( od `  G ) `  x
)  ||  ( P ^ n ) )
263, 22, 25syl2anc 644 . . . . 5  |-  ( ( ( ( G  e. 
Grp  /\  P  e.  Prime  /\  N  e.  NN0 )  /\  ( # `  X
)  =  ( P ^ N ) )  /\  x  e.  X
)  ->  E. n  e.  NN0  ( ( od
`  G ) `  x )  ||  ( P ^ n ) )
2712, 19odcl2 15206 . . . . . . 7  |-  ( ( G  e.  Grp  /\  X  e.  Fin  /\  x  e.  X )  ->  (
( od `  G
) `  x )  e.  NN )
284, 17, 18, 27syl3anc 1185 . . . . . 6  |-  ( ( ( ( G  e. 
Grp  /\  P  e.  Prime  /\  N  e.  NN0 )  /\  ( # `  X
)  =  ( P ^ N ) )  /\  x  e.  X
)  ->  ( ( od `  G ) `  x )  e.  NN )
29 pcprmpw2 13260 . . . . . . 7  |-  ( ( P  e.  Prime  /\  (
( od `  G
) `  x )  e.  NN )  ->  ( E. n  e.  NN0  ( ( od `  G ) `  x
)  ||  ( P ^ n )  <->  ( ( od `  G ) `  x )  =  ( P ^ ( P 
pCnt  ( ( od
`  G ) `  x ) ) ) ) )
30 pcprmpw 13261 . . . . . . 7  |-  ( ( P  e.  Prime  /\  (
( od `  G
) `  x )  e.  NN )  ->  ( E. n  e.  NN0  ( ( od `  G ) `  x
)  =  ( P ^ n )  <->  ( ( od `  G ) `  x )  =  ( P ^ ( P 
pCnt  ( ( od
`  G ) `  x ) ) ) ) )
3129, 30bitr4d 249 . . . . . 6  |-  ( ( P  e.  Prime  /\  (
( od `  G
) `  x )  e.  NN )  ->  ( E. n  e.  NN0  ( ( od `  G ) `  x
)  ||  ( P ^ n )  <->  E. n  e.  NN0  ( ( od
`  G ) `  x )  =  ( P ^ n ) ) )
326, 28, 31syl2anc 644 . . . . 5  |-  ( ( ( ( G  e. 
Grp  /\  P  e.  Prime  /\  N  e.  NN0 )  /\  ( # `  X
)  =  ( P ^ N ) )  /\  x  e.  X
)  ->  ( E. n  e.  NN0  ( ( od `  G ) `
 x )  ||  ( P ^ n )  <->  E. n  e.  NN0  ( ( od `  G ) `  x
)  =  ( P ^ n ) ) )
3326, 32mpbid 203 . . . 4  |-  ( ( ( ( G  e. 
Grp  /\  P  e.  Prime  /\  N  e.  NN0 )  /\  ( # `  X
)  =  ( P ^ N ) )  /\  x  e.  X
)  ->  E. n  e.  NN0  ( ( od
`  G ) `  x )  =  ( P ^ n ) )
3433ralrimiva 2791 . . 3  |-  ( ( ( G  e.  Grp  /\  P  e.  Prime  /\  N  e.  NN0 )  /\  ( # `
 X )  =  ( P ^ N
) )  ->  A. x  e.  X  E. n  e.  NN0  ( ( od
`  G ) `  x )  =  ( P ^ n ) )
3512, 19ispgp 15231 . . 3  |-  ( P pGrp 
G  <->  ( P  e. 
Prime  /\  G  e.  Grp  /\ 
A. x  e.  X  E. n  e.  NN0  ( ( od `  G ) `  x
)  =  ( P ^ n ) ) )
361, 2, 34, 35syl3anbrc 1139 . 2  |-  ( ( ( G  e.  Grp  /\  P  e.  Prime  /\  N  e.  NN0 )  /\  ( # `
 X )  =  ( P ^ N
) )  ->  P pGrp  G )
3736ex 425 1  |-  ( ( G  e.  Grp  /\  P  e.  Prime  /\  N  e.  NN0 )  ->  (
( # `  X )  =  ( P ^ N )  ->  P pGrp  G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   A.wral 2707   E.wrex 2708   _Vcvv 2958   class class class wbr 4215   ` cfv 5457  (class class class)co 6084   Fincfn 7112   NNcn 10005   NN0cn0 10226   ^cexp 11387   #chash 11623    || cdivides 12857   Primecprime 13084    pCnt cpc 13215   Basecbs 13474   Grpcgrp 14690   odcod 15168   pGrp cpgp 15170
This theorem is referenced by:  pgp0  15235  pgpfi  15244
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-inf2 7599  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072  ax-pre-sup 9073
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-int 4053  df-iun 4097  df-disj 4186  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-se 4545  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-isom 5466  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-2o 6728  df-oadd 6731  df-omul 6732  df-er 6908  df-ec 6910  df-qs 6914  df-map 7023  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-sup 7449  df-oi 7482  df-card 7831  df-acn 7834  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683  df-nn 10006  df-2 10063  df-3 10064  df-n0 10227  df-z 10288  df-uz 10494  df-q 10580  df-rp 10618  df-fz 11049  df-fzo 11141  df-fl 11207  df-mod 11256  df-seq 11329  df-exp 11388  df-hash 11624  df-cj 11909  df-re 11910  df-im 11911  df-sqr 12045  df-abs 12046  df-clim 12287  df-sum 12485  df-dvds 12858  df-gcd 13012  df-prm 13085  df-pc 13216  df-ndx 13477  df-slot 13478  df-base 13479  df-sets 13480  df-ress 13481  df-plusg 13547  df-0g 13732  df-mnd 14695  df-grp 14817  df-minusg 14818  df-sbg 14819  df-mulg 14820  df-subg 14946  df-eqg 14948  df-od 15172  df-pgp 15174
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