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Theorem pgpfi1 15188
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 961 . . 3  |-  ( ( ( G  e.  Grp  /\  P  e.  Prime  /\  N  e.  NN0 )  /\  ( # `
 X )  =  ( P ^ N
) )  ->  P  e.  Prime )
2 simpl1 960 . . 3  |-  ( ( ( G  e.  Grp  /\  P  e.  Prime  /\  N  e.  NN0 )  /\  ( # `
 X )  =  ( P ^ N
) )  ->  G  e.  Grp )
3 simpll3 998 . . . . . 6  |-  ( ( ( ( G  e. 
Grp  /\  P  e.  Prime  /\  N  e.  NN0 )  /\  ( # `  X
)  =  ( P ^ N ) )  /\  x  e.  X
)  ->  N  e.  NN0 )
42adantr 452 . . . . . . . 8  |-  ( ( ( ( G  e. 
Grp  /\  P  e.  Prime  /\  N  e.  NN0 )  /\  ( # `  X
)  =  ( P ^ N ) )  /\  x  e.  X
)  ->  G  e.  Grp )
5 simplr 732 . . . . . . . . . 10  |-  ( ( ( ( G  e. 
Grp  /\  P  e.  Prime  /\  N  e.  NN0 )  /\  ( # `  X
)  =  ( P ^ N ) )  /\  x  e.  X
)  ->  ( # `  X
)  =  ( P ^ N ) )
61adantr 452 . . . . . . . . . . . . 13  |-  ( ( ( ( G  e. 
Grp  /\  P  e.  Prime  /\  N  e.  NN0 )  /\  ( # `  X
)  =  ( P ^ N ) )  /\  x  e.  X
)  ->  P  e.  Prime )
7 prmnn 13041 . . . . . . . . . . . . 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 11503 . . . . . . . . . . 11  |-  ( ( ( ( G  e. 
Grp  /\  P  e.  Prime  /\  N  e.  NN0 )  /\  ( # `  X
)  =  ( P ^ N ) )  /\  x  e.  X
)  ->  ( P ^ N )  e.  NN )
109nnnn0d 10234 . . . . . . . . . 10  |-  ( ( ( ( G  e. 
Grp  /\  P  e.  Prime  /\  N  e.  NN0 )  /\  ( # `  X
)  =  ( P ^ N ) )  /\  x  e.  X
)  ->  ( P ^ N )  e.  NN0 )
115, 10eqeltrd 2482 . . . . . . . . 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 5705 . . . . . . . . . . 11  |-  ( Base `  G )  e.  _V
1412, 13eqeltri 2478 . . . . . . . . . 10  |-  X  e. 
_V
15 hashclb 11600 . . . . . . . . . 10  |-  ( X  e.  _V  ->  ( X  e.  Fin  <->  ( # `  X
)  e.  NN0 )
)
1614, 15ax-mp 8 . . . . . . . . 9  |-  ( X  e.  Fin  <->  ( # `  X
)  e.  NN0 )
1711, 16sylibr 204 . . . . . . . 8  |-  ( ( ( ( G  e. 
Grp  /\  P  e.  Prime  /\  N  e.  NN0 )  /\  ( # `  X
)  =  ( P ^ N ) )  /\  x  e.  X
)  ->  X  e.  Fin )
18 simpr 448 . . . . . . . 8  |-  ( ( ( ( G  e. 
Grp  /\  P  e.  Prime  /\  N  e.  NN0 )  /\  ( # `  X
)  =  ( P ^ N ) )  /\  x  e.  X
)  ->  x  e.  X )
19 eqid 2408 . . . . . . . . 9  |-  ( od
`  G )  =  ( od `  G
)
2012, 19oddvds2 15161 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  X  e.  Fin  /\  x  e.  X )  ->  (
( od `  G
) `  x )  ||  ( # `  X
) )
214, 17, 18, 20syl3anc 1184 . . . . . . 7  |-  ( ( ( ( G  e. 
Grp  /\  P  e.  Prime  /\  N  e.  NN0 )  /\  ( # `  X
)  =  ( P ^ N ) )  /\  x  e.  X
)  ->  ( ( od `  G ) `  x )  ||  ( # `
 X ) )
2221, 5breqtrd 4200 . . . . . 6  |-  ( ( ( ( G  e. 
Grp  /\  P  e.  Prime  /\  N  e.  NN0 )  /\  ( # `  X
)  =  ( P ^ N ) )  /\  x  e.  X
)  ->  ( ( od `  G ) `  x )  ||  ( P ^ N ) )
23 oveq2 6052 . . . . . . . 8  |-  ( n  =  N  ->  ( P ^ n )  =  ( P ^ N
) )
2423breq2d 4188 . . . . . . 7  |-  ( n  =  N  ->  (
( ( od `  G ) `  x
)  ||  ( P ^ n )  <->  ( ( od `  G ) `  x )  ||  ( P ^ N ) ) )
2524rspcev 3016 . . . . . 6  |-  ( ( N  e.  NN0  /\  ( ( od `  G ) `  x
)  ||  ( P ^ N ) )  ->  E. n  e.  NN0  ( ( od `  G ) `  x
)  ||  ( P ^ n ) )
263, 22, 25syl2anc 643 . . . . 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 15160 . . . . . . 7  |-  ( ( G  e.  Grp  /\  X  e.  Fin  /\  x  e.  X )  ->  (
( od `  G
) `  x )  e.  NN )
284, 17, 18, 27syl3anc 1184 . . . . . 6  |-  ( ( ( ( G  e. 
Grp  /\  P  e.  Prime  /\  N  e.  NN0 )  /\  ( # `  X
)  =  ( P ^ N ) )  /\  x  e.  X
)  ->  ( ( od `  G ) `  x )  e.  NN )
29 pcprmpw2 13214 . . . . . . 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 13215 . . . . . . 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 248 . . . . . 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 643 . . . . 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 202 . . . 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 2753 . . 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 15185 . . 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 1138 . 2  |-  ( ( ( G  e.  Grp  /\  P  e.  Prime  /\  N  e.  NN0 )  /\  ( # `
 X )  =  ( P ^ N
) )  ->  P pGrp  G )
3736ex 424 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 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2670   E.wrex 2671   _Vcvv 2920   class class class wbr 4176   ` cfv 5417  (class class class)co 6044   Fincfn 7072   NNcn 9960   NN0cn0 10181   ^cexp 11341   #chash 11577    || cdivides 12811   Primecprime 13038    pCnt cpc 13169   Basecbs 13428   Grpcgrp 14644   odcod 15122   pGrp cpgp 15124
This theorem is referenced by:  pgp0  15189  pgpfi  15198
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-inf2 7556  ax-cnex 9006  ax-resscn 9007  ax-1cn 9008  ax-icn 9009  ax-addcl 9010  ax-addrcl 9011  ax-mulcl 9012  ax-mulrcl 9013  ax-mulcom 9014  ax-addass 9015  ax-mulass 9016  ax-distr 9017  ax-i2m1 9018  ax-1ne0 9019  ax-1rid 9020  ax-rnegex 9021  ax-rrecex 9022  ax-cnre 9023  ax-pre-lttri 9024  ax-pre-lttrn 9025  ax-pre-ltadd 9026  ax-pre-mulgt0 9027  ax-pre-sup 9028
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-int 4015  df-iun 4059  df-disj 4147  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-se 4506  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-isom 5426  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1st 6312  df-2nd 6313  df-riota 6512  df-recs 6596  df-rdg 6631  df-1o 6687  df-2o 6688  df-oadd 6691  df-omul 6692  df-er 6868  df-ec 6870  df-qs 6874  df-map 6983  df-en 7073  df-dom 7074  df-sdom 7075  df-fin 7076  df-sup 7408  df-oi 7439  df-card 7786  df-acn 7789  df-pnf 9082  df-mnf 9083  df-xr 9084  df-ltxr 9085  df-le 9086  df-sub 9253  df-neg 9254  df-div 9638  df-nn 9961  df-2 10018  df-3 10019  df-n0 10182  df-z 10243  df-uz 10449  df-q 10535  df-rp 10573  df-fz 11004  df-fzo 11095  df-fl 11161  df-mod 11210  df-seq 11283  df-exp 11342  df-hash 11578  df-cj 11863  df-re 11864  df-im 11865  df-sqr 11999  df-abs 12000  df-clim 12241  df-sum 12439  df-dvds 12812  df-gcd 12966  df-prm 13039  df-pc 13170  df-ndx 13431  df-slot 13432  df-base 13433  df-sets 13434  df-ress 13435  df-plusg 13501  df-0g 13686  df-mnd 14649  df-grp 14771  df-minusg 14772  df-sbg 14773  df-mulg 14774  df-subg 14900  df-eqg 14902  df-od 15126  df-pgp 15128
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