Theorem ispgp 18007

Description: A group is a P-group if every element has some power of P as its order. (Contributed by Mario Carneiro, 15-Jan-2015.)

Hypotheses

Ref	Expression
ispgp.1	|- X = ( Base ` G )
ispgp.2	|- O = ( od ` G )

Assertion

Ref	Expression
ispgp	|- ( P pGrp G <-> ( P e. Prime /\ G e. Grp /\ A. x e. X E. n e. NN0 ( O ` x ) = ( P ^ n ) ) )

Distinct variable groups:   x, n, G   P, n, x   x, X

Allowed substitution hints:   O( x, n)   X( n)

Proof of Theorem ispgp

Dummy variables g p are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 477	. . . . . 6 |- ( ( p = P /\ g = G ) -> g = G )
2	1	fveq2d 6195	. . . . 5 |- ( ( p = P /\ g = G ) -> ( Base ` g ) = ( Base ` G ) )
3		ispgp.1	. . . . 5 |- X = ( Base ` G )
4	2, 3	syl6eqr 2674	. . . 4 |- ( ( p = P /\ g = G ) -> ( Base ` g ) = X )
5	1	fveq2d 6195	. . . . . . . 8 |- ( ( p = P /\ g = G ) -> ( od ` g ) = ( od ` G ) )
6		ispgp.2	. . . . . . . 8 |- O = ( od ` G )
7	5, 6	syl6eqr 2674	. . . . . . 7 |- ( ( p = P /\ g = G ) -> ( od ` g ) = O )
8	7	fveq1d 6193	. . . . . 6 |- ( ( p = P /\ g = G ) -> ( ( od ` g ) ` x ) = ( O ` x ) )
9		simpl 473	. . . . . . 7 |- ( ( p = P /\ g = G ) -> p = P )
10	9	oveq1d 6665	. . . . . 6 |- ( ( p = P /\ g = G ) -> ( p ^ n ) = ( P ^ n ) )
11	8, 10	eqeq12d 2637	. . . . 5 |- ( ( p = P /\ g = G ) -> ( ( ( od ` g ) ` x ) = ( p ^ n ) <-> ( O ` x ) = ( P ^ n ) ) )
12	11	rexbidv 3052	. . . 4 |- ( ( p = P /\ g = G ) -> ( E. n e. NN0 ( ( od ` g ) ` x ) = ( p ^ n ) <-> E. n e. NN0 ( O ` x ) = ( P ^ n ) ) )
13	4, 12	raleqbidv 3152	. . 3 |- ( ( p = P /\ g = G ) -> ( A. x e. ( Base ` g ) E. n e. NN0 ( ( od ` g ) ` x ) = ( p ^ n ) <-> A. x e. X E. n e. NN0 ( O ` x ) = ( P ^ n ) ) )
14		df-pgp 17950	. . 3 |- pGrp = { <. p , g >. | ( ( p e. Prime /\ g e. Grp ) /\ A. x e. ( Base ` g ) E. n e. NN0 ( ( od ` g ) ` x ) = ( p ^ n ) ) }
15	13, 14	brab2a 5194	. 2 |- ( P pGrp G <-> ( ( P e. Prime /\ G e. Grp ) /\ A. x e. X E. n e. NN0 ( O ` x ) = ( P ^ n ) ) )
16		df-3an 1039	. 2 |- ( ( P e. Prime /\ G e. Grp /\ A. x e. X E. n e. NN0 ( O ` x ) = ( P ^ n ) ) <-> ( ( P e. Prime /\ G e. Grp ) /\ A. x e. X E. n e. NN0 ( O ` x ) = ( P ^ n ) ) )
17	15, 16	bitr4i 267	1 |- ( P pGrp G <-> ( P e. Prime /\ G e. Grp /\ A. x e. X E. n e. NN0 ( O ` x ) = ( P ^ n ) ) )

Syntax hints:   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   NN0cn0 11292   ^cexp 12860   Primecprime 15385   Basecbs 15857   Grpcgrp 17422   odcod 17944   pGrp cpgp 17946

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-iota 5851  df-fv 5896  df-ov 6653  df-pgp 17950

This theorem is referenced by:  pgpprm  18008  pgpgrp  18009  pgpfi1  18010  subgpgp  18012  pgpfi  18020