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Theorem subgpgp 14908
Description: A subgroup of a p-group is a p-group. (Contributed by Mario Carneiro, 27-Apr-2016.)
Assertion
Ref Expression
subgpgp  |-  ( ( P pGrp  G  /\  S  e.  (SubGrp `  G )
)  ->  P pGrp  ( Gs  S ) )

Proof of Theorem subgpgp
Dummy variables  x  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pgpprm 14904 . . 3  |-  ( P pGrp 
G  ->  P  e.  Prime )
21adantr 451 . 2  |-  ( ( P pGrp  G  /\  S  e.  (SubGrp `  G )
)  ->  P  e.  Prime )
3 eqid 2283 . . . 4  |-  ( Gs  S )  =  ( Gs  S )
43subggrp 14624 . . 3  |-  ( S  e.  (SubGrp `  G
)  ->  ( Gs  S
)  e.  Grp )
54adantl 452 . 2  |-  ( ( P pGrp  G  /\  S  e.  (SubGrp `  G )
)  ->  ( Gs  S
)  e.  Grp )
6 eqid 2283 . . . . . . 7  |-  ( Base `  G )  =  (
Base `  G )
7 eqid 2283 . . . . . . 7  |-  ( od
`  G )  =  ( od `  G
)
86, 7ispgp 14903 . . . . . 6  |-  ( P pGrp 
G  <->  ( P  e. 
Prime  /\  G  e.  Grp  /\ 
A. x  e.  (
Base `  G ) E. n  e.  NN0  ( ( od `  G ) `  x
)  =  ( P ^ n ) ) )
98simp3bi 972 . . . . 5  |-  ( P pGrp 
G  ->  A. x  e.  ( Base `  G
) E. n  e. 
NN0  ( ( od
`  G ) `  x )  =  ( P ^ n ) )
109adantr 451 . . . 4  |-  ( ( P pGrp  G  /\  S  e.  (SubGrp `  G )
)  ->  A. x  e.  ( Base `  G
) E. n  e. 
NN0  ( ( od
`  G ) `  x )  =  ( P ^ n ) )
116subgss 14622 . . . . . . 7  |-  ( S  e.  (SubGrp `  G
)  ->  S  C_  ( Base `  G ) )
1211adantl 452 . . . . . 6  |-  ( ( P pGrp  G  /\  S  e.  (SubGrp `  G )
)  ->  S  C_  ( Base `  G ) )
13 ssralv 3237 . . . . . 6  |-  ( S 
C_  ( Base `  G
)  ->  ( A. x  e.  ( Base `  G ) E. n  e.  NN0  ( ( od
`  G ) `  x )  =  ( P ^ n )  ->  A. x  e.  S  E. n  e.  NN0  ( ( od `  G ) `  x
)  =  ( P ^ n ) ) )
1412, 13syl 15 . . . . 5  |-  ( ( P pGrp  G  /\  S  e.  (SubGrp `  G )
)  ->  ( A. x  e.  ( Base `  G ) E. n  e.  NN0  ( ( od
`  G ) `  x )  =  ( P ^ n )  ->  A. x  e.  S  E. n  e.  NN0  ( ( od `  G ) `  x
)  =  ( P ^ n ) ) )
15 eqid 2283 . . . . . . . . . 10  |-  ( od
`  ( Gs  S ) )  =  ( od
`  ( Gs  S ) )
163, 7, 15subgod 14881 . . . . . . . . 9  |-  ( ( S  e.  (SubGrp `  G )  /\  x  e.  S )  ->  (
( od `  G
) `  x )  =  ( ( od
`  ( Gs  S ) ) `  x ) )
1716adantll 694 . . . . . . . 8  |-  ( ( ( P pGrp  G  /\  S  e.  (SubGrp `  G
) )  /\  x  e.  S )  ->  (
( od `  G
) `  x )  =  ( ( od
`  ( Gs  S ) ) `  x ) )
1817eqeq1d 2291 . . . . . . 7  |-  ( ( ( P pGrp  G  /\  S  e.  (SubGrp `  G
) )  /\  x  e.  S )  ->  (
( ( od `  G ) `  x
)  =  ( P ^ n )  <->  ( ( od `  ( Gs  S ) ) `  x )  =  ( P ^
n ) ) )
1918rexbidv 2564 . . . . . 6  |-  ( ( ( P pGrp  G  /\  S  e.  (SubGrp `  G
) )  /\  x  e.  S )  ->  ( E. n  e.  NN0  ( ( od `  G ) `  x
)  =  ( P ^ n )  <->  E. n  e.  NN0  ( ( od
`  ( Gs  S ) ) `  x )  =  ( P ^
n ) ) )
2019ralbidva 2559 . . . . 5  |-  ( ( P pGrp  G  /\  S  e.  (SubGrp `  G )
)  ->  ( A. x  e.  S  E. n  e.  NN0  ( ( od `  G ) `
 x )  =  ( P ^ n
)  <->  A. x  e.  S  E. n  e.  NN0  ( ( od `  ( Gs  S ) ) `  x )  =  ( P ^ n ) ) )
2114, 20sylibd 205 . . . 4  |-  ( ( P pGrp  G  /\  S  e.  (SubGrp `  G )
)  ->  ( A. x  e.  ( Base `  G ) E. n  e.  NN0  ( ( od
`  G ) `  x )  =  ( P ^ n )  ->  A. x  e.  S  E. n  e.  NN0  ( ( od `  ( Gs  S ) ) `  x )  =  ( P ^ n ) ) )
2210, 21mpd 14 . . 3  |-  ( ( P pGrp  G  /\  S  e.  (SubGrp `  G )
)  ->  A. x  e.  S  E. n  e.  NN0  ( ( od
`  ( Gs  S ) ) `  x )  =  ( P ^
n ) )
233subgbas 14625 . . . . 5  |-  ( S  e.  (SubGrp `  G
)  ->  S  =  ( Base `  ( Gs  S
) ) )
2423adantl 452 . . . 4  |-  ( ( P pGrp  G  /\  S  e.  (SubGrp `  G )
)  ->  S  =  ( Base `  ( Gs  S
) ) )
2524raleqdv 2742 . . 3  |-  ( ( P pGrp  G  /\  S  e.  (SubGrp `  G )
)  ->  ( A. x  e.  S  E. n  e.  NN0  ( ( od `  ( Gs  S ) ) `  x
)  =  ( P ^ n )  <->  A. x  e.  ( Base `  ( Gs  S ) ) E. n  e.  NN0  (
( od `  ( Gs  S ) ) `  x )  =  ( P ^ n ) ) )
2622, 25mpbid 201 . 2  |-  ( ( P pGrp  G  /\  S  e.  (SubGrp `  G )
)  ->  A. x  e.  ( Base `  ( Gs  S ) ) E. n  e.  NN0  (
( od `  ( Gs  S ) ) `  x )  =  ( P ^ n ) )
27 eqid 2283 . . 3  |-  ( Base `  ( Gs  S ) )  =  ( Base `  ( Gs  S ) )
2827, 15ispgp 14903 . 2  |-  ( P pGrp  ( Gs  S )  <->  ( P  e.  Prime  /\  ( Gs  S
)  e.  Grp  /\  A. x  e.  ( Base `  ( Gs  S ) ) E. n  e.  NN0  (
( od `  ( Gs  S ) ) `  x )  =  ( P ^ n ) ) )
292, 5, 26, 28syl3anbrc 1136 1  |-  ( ( P pGrp  G  /\  S  e.  (SubGrp `  G )
)  ->  P pGrp  ( Gs  S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544    C_ wss 3152   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   NN0cn0 9965   ^cexp 11104   Primecprime 12758   Basecbs 13148   ↾s cress 13149   Grpcgrp 14362  SubGrpcsubg 14615   odcod 14840   pGrp cpgp 14842
This theorem is referenced by:  pgpfaclem1  15316  pgpfaclem3  15318
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
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-iun 3907  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-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-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-n0 9966  df-z 10025  df-seq 11047  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-submnd 14416  df-grp 14489  df-minusg 14490  df-mulg 14492  df-subg 14618  df-od 14844  df-pgp 14846
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