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Theorem subgpgp 14924
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 14920 . . 3  |-  ( P pGrp 
G  ->  P  e.  Prime )
21adantr 451 . 2  |-  ( ( P pGrp  G  /\  S  e.  (SubGrp `  G )
)  ->  P  e.  Prime )
3 eqid 2296 . . . 4  |-  ( Gs  S )  =  ( Gs  S )
43subggrp 14640 . . 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 2296 . . . . . . 7  |-  ( Base `  G )  =  (
Base `  G )
7 eqid 2296 . . . . . . 7  |-  ( od
`  G )  =  ( od `  G
)
86, 7ispgp 14919 . . . . . 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 14638 . . . . . . 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 3250 . . . . . 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 2296 . . . . . . . . . 10  |-  ( od
`  ( Gs  S ) )  =  ( od
`  ( Gs  S ) )
163, 7, 15subgod 14897 . . . . . . . . 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 2304 . . . . . . 7  |-  ( ( ( P pGrp  G  /\  S  e.  (SubGrp `  G
) )  /\  x  e.  S )  ->  (
( ( od `  G ) `  x
)  =  ( P ^ n )  <->  ( ( od `  ( Gs  S ) ) `  x )  =  ( P ^
n ) ) )
1918rexbidv 2577 . . . . . 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 2572 . . . . 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 14641 . . . . 5  |-  ( S  e.  (SubGrp `  G
)  ->  S  =  ( Base `  ( Gs  S
) ) )
2423adantl 452 . . . 4  |-  ( ( P pGrp  G  /\  S  e.  (SubGrp `  G )
)  ->  S  =  ( Base `  ( Gs  S
) ) )
2524raleqdv 2755 . . 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 2296 . . 3  |-  ( Base `  ( Gs  S ) )  =  ( Base `  ( Gs  S ) )
2827, 15ispgp 14919 . 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 1632    e. wcel 1696   A.wral 2556   E.wrex 2557    C_ wss 3165   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   NN0cn0 9981   ^cexp 11120   Primecprime 12774   Basecbs 13164   ↾s cress 13165   Grpcgrp 14378  SubGrpcsubg 14631   odcod 14856   pGrp cpgp 14858
This theorem is referenced by:  pgpfaclem1  15332  pgpfaclem3  15334
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-n0 9982  df-z 10041  df-seq 11063  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-0g 13420  df-mnd 14383  df-submnd 14432  df-grp 14505  df-minusg 14506  df-mulg 14508  df-subg 14634  df-od 14860  df-pgp 14862
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