Theorem pgpfi1 18010

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 x n are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl2 1065	. . 3 |- ( ( ( G e. Grp /\ P e. Prime /\ N e. NN0 ) /\ ( # ` X ) = ( P ^ N ) ) -> P e. Prime )
2		simpl1 1064	. . 3 |- ( ( ( G e. Grp /\ P e. Prime /\ N e. NN0 ) /\ ( # ` X ) = ( P ^ N ) ) -> G e. Grp )
3		simpll3 1102	. . . . . 6 |- ( ( ( ( G e. Grp /\ P e. Prime /\ N e. NN0 ) /\ ( # ` X ) = ( P ^ N ) ) /\ x e. X ) -> N e. NN0 )
4	2	adantr 481	. . . . . . . 8 |- ( ( ( ( G e. Grp /\ P e. Prime /\ N e. NN0 ) /\ ( # ` X ) = ( P ^ N ) ) /\ x e. X ) -> G e. Grp )
5		simplr 792	. . . . . . . . . 10 |- ( ( ( ( G e. Grp /\ P e. Prime /\ N e. NN0 ) /\ ( # ` X ) = ( P ^ N ) ) /\ x e. X ) -> ( # ` X ) = ( P ^ N ) )
6	1	adantr 481	. . . . . . . . . . . . 13 |- ( ( ( ( G e. Grp /\ P e. Prime /\ N e. NN0 ) /\ ( # ` X ) = ( P ^ N ) ) /\ x e. X ) -> P e. Prime )
7		prmnn 15388	. . . . . . . . . . . . 13 |- ( P e. Prime -> P e. NN )
8	6, 7	syl 17	. . . . . . . . . . . 12 |- ( ( ( ( G e. Grp /\ P e. Prime /\ N e. NN0 ) /\ ( # ` X ) = ( P ^ N ) ) /\ x e. X ) -> P e. NN )
9	8, 3	nnexpcld 13030	. . . . . . . . . . 11 |- ( ( ( ( G e. Grp /\ P e. Prime /\ N e. NN0 ) /\ ( # ` X ) = ( P ^ N ) ) /\ x e. X ) -> ( P ^ N ) e. NN )
10	9	nnnn0d 11351	. . . . . . . . . 10 |- ( ( ( ( G e. Grp /\ P e. Prime /\ N e. NN0 ) /\ ( # ` X ) = ( P ^ N ) ) /\ x e. X ) -> ( P ^ N ) e. NN0 )
11	5, 10	eqeltrd 2701	. . . . . . . . 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 6201	. . . . . . . . . . 11 |- ( Base ` G ) e. _V
14	12, 13	eqeltri 2697	. . . . . . . . . 10 |- X e. _V
15		hashclb 13149	. . . . . . . . . 10 |- ( X e. _V -> ( X e. Fin <-> ( # ` X ) e. NN0 ) )
16	14, 15	ax-mp 5	. . . . . . . . 9 |- ( X e. Fin <-> ( # ` X ) e. NN0 )
17	11, 16	sylibr 224	. . . . . . . 8 |- ( ( ( ( G e. Grp /\ P e. Prime /\ N e. NN0 ) /\ ( # ` X ) = ( P ^ N ) ) /\ x e. X ) -> X e. Fin )
18		simpr 477	. . . . . . . 8 |- ( ( ( ( G e. Grp /\ P e. Prime /\ N e. NN0 ) /\ ( # ` X ) = ( P ^ N ) ) /\ x e. X ) -> x e. X )
19		eqid 2622	. . . . . . . . 9 |- ( od ` G ) = ( od ` G )
20	12, 19	oddvds2 17983	. . . . . . . 8 |- ( ( G e. Grp /\ X e. Fin /\ x e. X ) -> ( ( od ` G ) ` x ) || ( # ` X ) )
21	4, 17, 18, 20	syl3anc 1326	. . . . . . 7 |- ( ( ( ( G e. Grp /\ P e. Prime /\ N e. NN0 ) /\ ( # ` X ) = ( P ^ N ) ) /\ x e. X ) -> ( ( od ` G ) ` x ) || ( # ` X ) )
22	21, 5	breqtrd 4679	. . . . . 6 |- ( ( ( ( G e. Grp /\ P e. Prime /\ N e. NN0 ) /\ ( # ` X ) = ( P ^ N ) ) /\ x e. X ) -> ( ( od ` G ) ` x ) || ( P ^ N ) )
23		oveq2 6658	. . . . . . . 8 |- ( n = N -> ( P ^ n ) = ( P ^ N ) )
24	23	breq2d 4665	. . . . . . 7 |- ( n = N -> ( ( ( od ` G ) ` x ) || ( P ^ n ) <-> ( ( od ` G ) ` x ) || ( P ^ N ) ) )
25	24	rspcev 3309	. . . . . 6 |- ( ( N e. NN0 /\ ( ( od ` G ) ` x ) || ( P ^ N ) ) -> E. n e. NN0 ( ( od ` G ) ` x ) || ( P ^ n ) )
26	3, 22, 25	syl2anc 693	. . . . 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 ) )
27	12, 19	odcl2 17982	. . . . . . 7 |- ( ( G e. Grp /\ X e. Fin /\ x e. X ) -> ( ( od ` G ) ` x ) e. NN )
28	4, 17, 18, 27	syl3anc 1326	. . . . . 6 |- ( ( ( ( G e. Grp /\ P e. Prime /\ N e. NN0 ) /\ ( # ` X ) = ( P ^ N ) ) /\ x e. X ) -> ( ( od ` G ) ` x ) e. NN )
29		pcprmpw2 15586	. . . . . . 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 15587	. . . . . . 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 ) ) ) ) )
31	29, 30	bitr4d 271	. . . . . 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 ) ) )
32	6, 28, 31	syl2anc 693	. . . . 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 ) ) )
33	26, 32	mpbid 222	. . . 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 ) )
34	33	ralrimiva 2966	. . 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 ) )
35	12, 19	ispgp 18007	. . 3 |- ( P pGrp G <-> ( P e. Prime /\ G e. Grp /\ A. x e. X E. n e. NN0 ( ( od ` G ) ` x ) = ( P ^ n ) ) )
36	1, 2, 34, 35	syl3anbrc 1246	. 2 |- ( ( ( G e. Grp /\ P e. Prime /\ N e. NN0 ) /\ ( # ` X ) = ( P ^ N ) ) -> P pGrp G )
37	36	ex 450	1 |- ( ( G e. Grp /\ P e. Prime /\ N e. NN0 ) -> ( ( # ` X ) = ( P ^ N ) -> P pGrp G ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   _Vcvv 3200   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Fincfn 7955   NNcn 11020   NN0cn0 11292   ^cexp 12860   #chash 13117   || cdvds 14983   Primecprime 15385   pCnt cpc 15541   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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  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-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-disj 4621  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-omul 7565  df-er 7742  df-ec 7744  df-qs 7748  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-acn 8768  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417  df-dvds 14984  df-gcd 15217  df-prm 15386  df-pc 15542  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-sbg 17427  df-mulg 17541  df-subg 17591  df-eqg 17593  df-od 17948  df-pgp 17950

This theorem is referenced by:  pgp0  18011  pgpfi  18020