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Theorem pgpfi1 16408
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 1000 . . 3  |-  ( ( ( G  e.  Grp  /\  P  e.  Prime  /\  N  e.  NN0 )  /\  ( # `
 X )  =  ( P ^ N
) )  ->  P  e.  Prime )
2 simpl1 999 . . 3  |-  ( ( ( G  e.  Grp  /\  P  e.  Prime  /\  N  e.  NN0 )  /\  ( # `
 X )  =  ( P ^ N
) )  ->  G  e.  Grp )
3 simpll3 1037 . . . . . 6  |-  ( ( ( ( G  e. 
Grp  /\  P  e.  Prime  /\  N  e.  NN0 )  /\  ( # `  X
)  =  ( P ^ N ) )  /\  x  e.  X
)  ->  N  e.  NN0 )
42adantr 465 . . . . . . . 8  |-  ( ( ( ( G  e. 
Grp  /\  P  e.  Prime  /\  N  e.  NN0 )  /\  ( # `  X
)  =  ( P ^ N ) )  /\  x  e.  X
)  ->  G  e.  Grp )
5 simplr 754 . . . . . . . . . 10  |-  ( ( ( ( G  e. 
Grp  /\  P  e.  Prime  /\  N  e.  NN0 )  /\  ( # `  X
)  =  ( P ^ N ) )  /\  x  e.  X
)  ->  ( # `  X
)  =  ( P ^ N ) )
61adantr 465 . . . . . . . . . . . . 13  |-  ( ( ( ( G  e. 
Grp  /\  P  e.  Prime  /\  N  e.  NN0 )  /\  ( # `  X
)  =  ( P ^ N ) )  /\  x  e.  X
)  ->  P  e.  Prime )
7 prmnn 14072 . . . . . . . . . . . . 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 12293 . . . . . . . . . . 11  |-  ( ( ( ( G  e. 
Grp  /\  P  e.  Prime  /\  N  e.  NN0 )  /\  ( # `  X
)  =  ( P ^ N ) )  /\  x  e.  X
)  ->  ( P ^ N )  e.  NN )
109nnnn0d 10848 . . . . . . . . . 10  |-  ( ( ( ( G  e. 
Grp  /\  P  e.  Prime  /\  N  e.  NN0 )  /\  ( # `  X
)  =  ( P ^ N ) )  /\  x  e.  X
)  ->  ( P ^ N )  e.  NN0 )
115, 10eqeltrd 2555 . . . . . . . . 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 5874 . . . . . . . . . . 11  |-  ( Base `  G )  e.  _V
1412, 13eqeltri 2551 . . . . . . . . . 10  |-  X  e. 
_V
15 hashclb 12392 . . . . . . . . . 10  |-  ( X  e.  _V  ->  ( X  e.  Fin  <->  ( # `  X
)  e.  NN0 )
)
1614, 15ax-mp 5 . . . . . . . . 9  |-  ( X  e.  Fin  <->  ( # `  X
)  e.  NN0 )
1711, 16sylibr 212 . . . . . . . 8  |-  ( ( ( ( G  e. 
Grp  /\  P  e.  Prime  /\  N  e.  NN0 )  /\  ( # `  X
)  =  ( P ^ N ) )  /\  x  e.  X
)  ->  X  e.  Fin )
18 simpr 461 . . . . . . . 8  |-  ( ( ( ( G  e. 
Grp  /\  P  e.  Prime  /\  N  e.  NN0 )  /\  ( # `  X
)  =  ( P ^ N ) )  /\  x  e.  X
)  ->  x  e.  X )
19 eqid 2467 . . . . . . . . 9  |-  ( od
`  G )  =  ( od `  G
)
2012, 19oddvds2 16381 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  X  e.  Fin  /\  x  e.  X )  ->  (
( od `  G
) `  x )  ||  ( # `  X
) )
214, 17, 18, 20syl3anc 1228 . . . . . . 7  |-  ( ( ( ( G  e. 
Grp  /\  P  e.  Prime  /\  N  e.  NN0 )  /\  ( # `  X
)  =  ( P ^ N ) )  /\  x  e.  X
)  ->  ( ( od `  G ) `  x )  ||  ( # `
 X ) )
2221, 5breqtrd 4471 . . . . . 6  |-  ( ( ( ( G  e. 
Grp  /\  P  e.  Prime  /\  N  e.  NN0 )  /\  ( # `  X
)  =  ( P ^ N ) )  /\  x  e.  X
)  ->  ( ( od `  G ) `  x )  ||  ( P ^ N ) )
23 oveq2 6290 . . . . . . . 8  |-  ( n  =  N  ->  ( P ^ n )  =  ( P ^ N
) )
2423breq2d 4459 . . . . . . 7  |-  ( n  =  N  ->  (
( ( od `  G ) `  x
)  ||  ( P ^ n )  <->  ( ( od `  G ) `  x )  ||  ( P ^ N ) ) )
2524rspcev 3214 . . . . . 6  |-  ( ( N  e.  NN0  /\  ( ( od `  G ) `  x
)  ||  ( P ^ N ) )  ->  E. n  e.  NN0  ( ( od `  G ) `  x
)  ||  ( P ^ n ) )
263, 22, 25syl2anc 661 . . . . 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 16380 . . . . . . 7  |-  ( ( G  e.  Grp  /\  X  e.  Fin  /\  x  e.  X )  ->  (
( od `  G
) `  x )  e.  NN )
284, 17, 18, 27syl3anc 1228 . . . . . 6  |-  ( ( ( ( G  e. 
Grp  /\  P  e.  Prime  /\  N  e.  NN0 )  /\  ( # `  X
)  =  ( P ^ N ) )  /\  x  e.  X
)  ->  ( ( od `  G ) `  x )  e.  NN )
29 pcprmpw2 14257 . . . . . . 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 14258 . . . . . . 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 256 . . . . . 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 661 . . . . 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 210 . . . 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 2878 . . 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 16405 . . 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 1180 . 2  |-  ( ( ( G  e.  Grp  /\  P  e.  Prime  /\  N  e.  NN0 )  /\  ( # `
 X )  =  ( P ^ N
) )  ->  P pGrp  G )
3736ex 434 1  |-  ( ( G  e.  Grp  /\  P  e.  Prime  /\  N  e.  NN0 )  ->  (
( # `  X )  =  ( P ^ N )  ->  P pGrp  G ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2814   E.wrex 2815   _Vcvv 3113   class class class wbr 4447   ` cfv 5586  (class class class)co 6282   Fincfn 7513   NNcn 10532   NN0cn0 10791   ^cexp 12129   #chash 12367    || cdivides 13840   Primecprime 14069    pCnt cpc 14212   Basecbs 14483   Grpcgrp 15720   odcod 16342   pGrp cpgp 16344
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-disj 4418  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-2o 7128  df-oadd 7131  df-omul 7132  df-er 7308  df-ec 7310  df-qs 7314  df-map 7419  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-sup 7897  df-oi 7931  df-card 8316  df-acn 8319  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11079  df-q 11179  df-rp 11217  df-fz 11669  df-fzo 11789  df-fl 11893  df-mod 11960  df-seq 12071  df-exp 12130  df-hash 12368  df-cj 12889  df-re 12890  df-im 12891  df-sqrt 13025  df-abs 13026  df-clim 13267  df-sum 13465  df-dvds 13841  df-gcd 13997  df-prm 14070  df-pc 14213  df-ndx 14486  df-slot 14487  df-base 14488  df-sets 14489  df-ress 14490  df-plusg 14561  df-0g 14690  df-mnd 15725  df-grp 15855  df-minusg 15856  df-sbg 15857  df-mulg 15858  df-subg 15990  df-eqg 15992  df-od 16346  df-pgp 16348
This theorem is referenced by:  pgp0  16409  pgpfi  16418
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