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Theorem pgpfi1 14906
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 959 . . 3  |-  ( ( ( G  e.  Grp  /\  P  e.  Prime  /\  N  e.  NN0 )  /\  ( # `
 X )  =  ( P ^ N
) )  ->  P  e.  Prime )
2 simpl1 958 . . 3  |-  ( ( ( G  e.  Grp  /\  P  e.  Prime  /\  N  e.  NN0 )  /\  ( # `
 X )  =  ( P ^ N
) )  ->  G  e.  Grp )
3 simpll3 996 . . . . . 6  |-  ( ( ( ( G  e. 
Grp  /\  P  e.  Prime  /\  N  e.  NN0 )  /\  ( # `  X
)  =  ( P ^ N ) )  /\  x  e.  X
)  ->  N  e.  NN0 )
42adantr 451 . . . . . . . 8  |-  ( ( ( ( G  e. 
Grp  /\  P  e.  Prime  /\  N  e.  NN0 )  /\  ( # `  X
)  =  ( P ^ N ) )  /\  x  e.  X
)  ->  G  e.  Grp )
5 simplr 731 . . . . . . . . . 10  |-  ( ( ( ( G  e. 
Grp  /\  P  e.  Prime  /\  N  e.  NN0 )  /\  ( # `  X
)  =  ( P ^ N ) )  /\  x  e.  X
)  ->  ( # `  X
)  =  ( P ^ N ) )
61adantr 451 . . . . . . . . . . . . 13  |-  ( ( ( ( G  e. 
Grp  /\  P  e.  Prime  /\  N  e.  NN0 )  /\  ( # `  X
)  =  ( P ^ N ) )  /\  x  e.  X
)  ->  P  e.  Prime )
7 prmnn 12761 . . . . . . . . . . . . 13  |-  ( P  e.  Prime  ->  P  e.  NN )
86, 7syl 15 . . . . . . . . . . . 12  |-  ( ( ( ( G  e. 
Grp  /\  P  e.  Prime  /\  N  e.  NN0 )  /\  ( # `  X
)  =  ( P ^ N ) )  /\  x  e.  X
)  ->  P  e.  NN )
98, 3nnexpcld 11266 . . . . . . . . . . 11  |-  ( ( ( ( G  e. 
Grp  /\  P  e.  Prime  /\  N  e.  NN0 )  /\  ( # `  X
)  =  ( P ^ N ) )  /\  x  e.  X
)  ->  ( P ^ N )  e.  NN )
109nnnn0d 10018 . . . . . . . . . 10  |-  ( ( ( ( G  e. 
Grp  /\  P  e.  Prime  /\  N  e.  NN0 )  /\  ( # `  X
)  =  ( P ^ N ) )  /\  x  e.  X
)  ->  ( P ^ N )  e.  NN0 )
115, 10eqeltrd 2357 . . . . . . . . 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 5539 . . . . . . . . . . 11  |-  ( Base `  G )  e.  _V
1412, 13eqeltri 2353 . . . . . . . . . 10  |-  X  e. 
_V
15 hashclb 11352 . . . . . . . . . 10  |-  ( X  e.  _V  ->  ( X  e.  Fin  <->  ( # `  X
)  e.  NN0 )
)
1614, 15ax-mp 8 . . . . . . . . 9  |-  ( X  e.  Fin  <->  ( # `  X
)  e.  NN0 )
1711, 16sylibr 203 . . . . . . . 8  |-  ( ( ( ( G  e. 
Grp  /\  P  e.  Prime  /\  N  e.  NN0 )  /\  ( # `  X
)  =  ( P ^ N ) )  /\  x  e.  X
)  ->  X  e.  Fin )
18 simpr 447 . . . . . . . 8  |-  ( ( ( ( G  e. 
Grp  /\  P  e.  Prime  /\  N  e.  NN0 )  /\  ( # `  X
)  =  ( P ^ N ) )  /\  x  e.  X
)  ->  x  e.  X )
19 eqid 2283 . . . . . . . . 9  |-  ( od
`  G )  =  ( od `  G
)
2012, 19oddvds2 14879 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  X  e.  Fin  /\  x  e.  X )  ->  (
( od `  G
) `  x )  ||  ( # `  X
) )
214, 17, 18, 20syl3anc 1182 . . . . . . 7  |-  ( ( ( ( G  e. 
Grp  /\  P  e.  Prime  /\  N  e.  NN0 )  /\  ( # `  X
)  =  ( P ^ N ) )  /\  x  e.  X
)  ->  ( ( od `  G ) `  x )  ||  ( # `
 X ) )
2221, 5breqtrd 4047 . . . . . 6  |-  ( ( ( ( G  e. 
Grp  /\  P  e.  Prime  /\  N  e.  NN0 )  /\  ( # `  X
)  =  ( P ^ N ) )  /\  x  e.  X
)  ->  ( ( od `  G ) `  x )  ||  ( P ^ N ) )
23 oveq2 5866 . . . . . . . 8  |-  ( n  =  N  ->  ( P ^ n )  =  ( P ^ N
) )
2423breq2d 4035 . . . . . . 7  |-  ( n  =  N  ->  (
( ( od `  G ) `  x
)  ||  ( P ^ n )  <->  ( ( od `  G ) `  x )  ||  ( P ^ N ) ) )
2524rspcev 2884 . . . . . 6  |-  ( ( N  e.  NN0  /\  ( ( od `  G ) `  x
)  ||  ( P ^ N ) )  ->  E. n  e.  NN0  ( ( od `  G ) `  x
)  ||  ( P ^ n ) )
263, 22, 25syl2anc 642 . . . . 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 14878 . . . . . . 7  |-  ( ( G  e.  Grp  /\  X  e.  Fin  /\  x  e.  X )  ->  (
( od `  G
) `  x )  e.  NN )
284, 17, 18, 27syl3anc 1182 . . . . . 6  |-  ( ( ( ( G  e. 
Grp  /\  P  e.  Prime  /\  N  e.  NN0 )  /\  ( # `  X
)  =  ( P ^ N ) )  /\  x  e.  X
)  ->  ( ( od `  G ) `  x )  e.  NN )
29 pcprmpw2 12934 . . . . . . 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 12935 . . . . . . 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 247 . . . . . 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 642 . . . . 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 201 . . . 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 2626 . . 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 14903 . . 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 1136 . 2  |-  ( ( ( G  e.  Grp  /\  P  e.  Prime  /\  N  e.  NN0 )  /\  ( # `
 X )  =  ( P ^ N
) )  ->  P pGrp  G )
3736ex 423 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   _Vcvv 2788   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Fincfn 6863   NNcn 9746   NN0cn0 9965   ^cexp 11104   #chash 11337    || cdivides 12531   Primecprime 12758    pCnt cpc 12889   Basecbs 13148   Grpcgrp 14362   odcod 14840   pGrp cpgp 14842
This theorem is referenced by:  pgp0  14907  pgpfi  14916
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  ax-pre-sup 8815
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-int 3863  df-iun 3907  df-disj 3994  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-se 4353  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-isom 5264  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-1o 6479  df-2o 6480  df-oadd 6483  df-omul 6484  df-er 6660  df-ec 6662  df-qs 6666  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-acn 7575  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317  df-rp 10355  df-fz 10783  df-fzo 10871  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-sum 12159  df-dvds 12532  df-gcd 12686  df-prm 12759  df-pc 12890  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-sbg 14491  df-mulg 14492  df-subg 14618  df-eqg 14620  df-od 14844  df-pgp 14846
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