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Theorem pgpfi1 15005
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 12858 . . . . . . . . . . . . 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 11359 . . . . . . . . . . 11  |-  ( ( ( ( G  e. 
Grp  /\  P  e.  Prime  /\  N  e.  NN0 )  /\  ( # `  X
)  =  ( P ^ N ) )  /\  x  e.  X
)  ->  ( P ^ N )  e.  NN )
109nnnn0d 10110 . . . . . . . . . 10  |-  ( ( ( ( G  e. 
Grp  /\  P  e.  Prime  /\  N  e.  NN0 )  /\  ( # `  X
)  =  ( P ^ N ) )  /\  x  e.  X
)  ->  ( P ^ N )  e.  NN0 )
115, 10eqeltrd 2432 . . . . . . . . 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 5622 . . . . . . . . . . 11  |-  ( Base `  G )  e.  _V
1412, 13eqeltri 2428 . . . . . . . . . 10  |-  X  e. 
_V
15 hashclb 11445 . . . . . . . . . 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 2358 . . . . . . . . 9  |-  ( od
`  G )  =  ( od `  G
)
2012, 19oddvds2 14978 . . . . . . . 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 4128 . . . . . 6  |-  ( ( ( ( G  e. 
Grp  /\  P  e.  Prime  /\  N  e.  NN0 )  /\  ( # `  X
)  =  ( P ^ N ) )  /\  x  e.  X
)  ->  ( ( od `  G ) `  x )  ||  ( P ^ N ) )
23 oveq2 5953 . . . . . . . 8  |-  ( n  =  N  ->  ( P ^ n )  =  ( P ^ N
) )
2423breq2d 4116 . . . . . . 7  |-  ( n  =  N  ->  (
( ( od `  G ) `  x
)  ||  ( P ^ n )  <->  ( ( od `  G ) `  x )  ||  ( P ^ N ) ) )
2524rspcev 2960 . . . . . 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 14977 . . . . . . 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 13031 . . . . . . 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 13032 . . . . . . 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 2702 . . 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 15002 . . 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 1642    e. wcel 1710   A.wral 2619   E.wrex 2620   _Vcvv 2864   class class class wbr 4104   ` cfv 5337  (class class class)co 5945   Fincfn 6951   NNcn 9836   NN0cn0 10057   ^cexp 11197   #chash 11430    || cdivides 12628   Primecprime 12855    pCnt cpc 12986   Basecbs 13245   Grpcgrp 14461   odcod 14939   pGrp cpgp 14941
This theorem is referenced by:  pgp0  15006  pgpfi  15015
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-inf2 7432  ax-cnex 8883  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-mulcom 8891  ax-addass 8892  ax-mulass 8893  ax-distr 8894  ax-i2m1 8895  ax-1ne0 8896  ax-1rid 8897  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902  ax-pre-ltadd 8903  ax-pre-mulgt0 8904  ax-pre-sup 8905
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-int 3944  df-iun 3988  df-disj 4075  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-se 4435  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-isom 5346  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-1st 6209  df-2nd 6210  df-riota 6391  df-recs 6475  df-rdg 6510  df-1o 6566  df-2o 6567  df-oadd 6570  df-omul 6571  df-er 6747  df-ec 6749  df-qs 6753  df-map 6862  df-en 6952  df-dom 6953  df-sdom 6954  df-fin 6955  df-sup 7284  df-oi 7315  df-card 7662  df-acn 7665  df-pnf 8959  df-mnf 8960  df-xr 8961  df-ltxr 8962  df-le 8963  df-sub 9129  df-neg 9130  df-div 9514  df-nn 9837  df-2 9894  df-3 9895  df-n0 10058  df-z 10117  df-uz 10323  df-q 10409  df-rp 10447  df-fz 10875  df-fzo 10963  df-fl 11017  df-mod 11066  df-seq 11139  df-exp 11198  df-hash 11431  df-cj 11680  df-re 11681  df-im 11682  df-sqr 11816  df-abs 11817  df-clim 12058  df-sum 12256  df-dvds 12629  df-gcd 12783  df-prm 12856  df-pc 12987  df-ndx 13248  df-slot 13249  df-base 13250  df-sets 13251  df-ress 13252  df-plusg 13318  df-0g 13503  df-mnd 14466  df-grp 14588  df-minusg 14589  df-sbg 14590  df-mulg 14591  df-subg 14717  df-eqg 14719  df-od 14943  df-pgp 14945
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