Theorem sylow3 16112

Description: Sylow's third theorem. The number of Sylow subgroups is a divisor of | G | / d, where d is the common order of a Sylow subgroup, and is equivalent to 1 mod P. (Contributed by Mario Carneiro, 19-Jan-2015.)

Hypotheses

Ref	Expression
sylow3.x	|- X = ( Base ` G )
sylow3.g	|- ( ph -> G e. Grp )
sylow3.xf	|- ( ph -> X e. Fin )
sylow3.p	|- ( ph -> P e. Prime )
sylow3.n	|- N = ( # ` ( P pSyl G ) )

Assertion

Ref	Expression
sylow3	|- ( ph -> ( N || ( ( # ` X ) / ( P ^ ( P pCnt ( # ` X ) ) ) ) /\ ( N mod P ) = 1 ) )

Proof of Theorem sylow3

Dummy variables a b c u x y z s k are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sylow3.g	. . . 4 |- ( ph -> G e. Grp )
2		sylow3.xf	. . . 4 |- ( ph -> X e. Fin )
3		sylow3.p	. . . 4 |- ( ph -> P e. Prime )
4		sylow3.x	. . . . 5 |- X = ( Base ` G )
5	4	slwn0 16094	. . . 4 |- ( ( G e. Grp /\ X e. Fin /\ P e. Prime ) -> ( P pSyl G ) =/= (/) )
6	1, 2, 3, 5	syl3anc 1211	. . 3 |- ( ph -> ( P pSyl G ) =/= (/) )
7		n0 3634	. . 3 |- ( ( P pSyl G ) =/= (/) <-> E. k k e. ( P pSyl G ) )
8	6, 7	sylib 196	. 2 |- ( ph -> E. k k e. ( P pSyl G ) )
9		sylow3.n	. . . 4 |- N = ( # ` ( P pSyl G ) )
10	1	adantr 462	. . . . 5 |- ( ( ph /\ k e. ( P pSyl G ) ) -> G e. Grp )
11	2	adantr 462	. . . . 5 |- ( ( ph /\ k e. ( P pSyl G ) ) -> X e. Fin )
12	3	adantr 462	. . . . 5 |- ( ( ph /\ k e. ( P pSyl G ) ) -> P e. Prime )
13		eqid 2433	. . . . 5 |- ( +g ` G ) = ( +g ` G )
14		eqid 2433	. . . . 5 |- ( -g ` G ) = ( -g ` G )
15		oveq2 6088	. . . . . . . . . 10 |- ( c = z -> ( a ( +g ` G ) c ) = ( a ( +g ` G ) z ) )
16	15	oveq1d 6095	. . . . . . . . 9 |- ( c = z -> ( ( a ( +g ` G ) c ) ( -g ` G ) a ) = ( ( a ( +g ` G ) z ) ( -g ` G ) a ) )
17	16	cbvmptv 4371	. . . . . . . 8 |- ( c e. b |-> ( ( a ( +g ` G ) c ) ( -g ` G ) a ) ) = ( z e. b |-> ( ( a ( +g ` G ) z ) ( -g ` G ) a ) )
18		oveq1 6087	. . . . . . . . . 10 |- ( a = x -> ( a ( +g ` G ) z ) = ( x ( +g ` G ) z ) )
19		id 22	. . . . . . . . . 10 |- ( a = x -> a = x )
20	18, 19	oveq12d 6098	. . . . . . . . 9 |- ( a = x -> ( ( a ( +g ` G ) z ) ( -g ` G ) a ) = ( ( x ( +g ` G ) z ) ( -g ` G ) x ) )
21	20	mpteq2dv 4367	. . . . . . . 8 |- ( a = x -> ( z e. b |-> ( ( a ( +g ` G ) z ) ( -g ` G ) a ) ) = ( z e. b |-> ( ( x ( +g ` G ) z ) ( -g ` G ) x ) ) )
22	17, 21	syl5eq 2477	. . . . . . 7 |- ( a = x -> ( c e. b |-> ( ( a ( +g ` G ) c ) ( -g ` G ) a ) ) = ( z e. b |-> ( ( x ( +g ` G ) z ) ( -g ` G ) x ) ) )
23	22	rneqd 5054	. . . . . 6 |- ( a = x -> ran ( c e. b |-> ( ( a ( +g ` G ) c ) ( -g ` G ) a ) ) = ran ( z e. b |-> ( ( x ( +g ` G ) z ) ( -g ` G ) x ) ) )
24		mpteq1 4360	. . . . . . 7 |- ( b = y -> ( z e. b |-> ( ( x ( +g ` G ) z ) ( -g ` G ) x ) ) = ( z e. y |-> ( ( x ( +g ` G ) z ) ( -g ` G ) x ) ) )
25	24	rneqd 5054	. . . . . 6 |- ( b = y -> ran ( z e. b |-> ( ( x ( +g ` G ) z ) ( -g ` G ) x ) ) = ran ( z e. y |-> ( ( x ( +g ` G ) z ) ( -g ` G ) x ) ) )
26	23, 25	cbvmpt2v 6155	. . . . 5 |- ( a e. X , b e. ( P pSyl G ) |-> ran ( c e. b |-> ( ( a ( +g ` G ) c ) ( -g ` G ) a ) ) ) = ( x e. X , y e. ( P pSyl G ) |-> ran ( z e. y |-> ( ( x ( +g ` G ) z ) ( -g ` G ) x ) ) )
27		simpr 458	. . . . 5 |- ( ( ph /\ k e. ( P pSyl G ) ) -> k e. ( P pSyl G ) )
28		eqid 2433	. . . . 5 |- { u e. X | ( u ( a e. X , b e. ( P pSyl G ) |-> ran ( c e. b |-> ( ( a ( +g ` G ) c ) ( -g ` G ) a ) ) ) k ) = k } = { u e. X | ( u ( a e. X , b e. ( P pSyl G ) |-> ran ( c e. b |-> ( ( a ( +g ` G ) c ) ( -g ` G ) a ) ) ) k ) = k }
29		eqid 2433	. . . . 5 |- { x e. X | A. y e. X ( ( x ( +g ` G ) y ) e. k <-> ( y ( +g ` G ) x ) e. k ) } = { x e. X | A. y e. X ( ( x ( +g ` G ) y ) e. k <-> ( y ( +g ` G ) x ) e. k ) }
30	4, 10, 11, 12, 13, 14, 26, 27, 28, 29	sylow3lem4 16109	. . . 4 |- ( ( ph /\ k e. ( P pSyl G ) ) -> ( # ` ( P pSyl G ) ) || ( ( # ` X ) / ( P ^ ( P pCnt ( # ` X ) ) ) ) )
31	9, 30	syl5eqbr 4313	. . 3 |- ( ( ph /\ k e. ( P pSyl G ) ) -> N || ( ( # ` X ) / ( P ^ ( P pCnt ( # ` X ) ) ) ) )
32	9	oveq1i 6090	. . . 4 |- ( N mod P ) = ( ( # ` ( P pSyl G ) ) mod P )
33	23, 25	cbvmpt2v 6155	. . . . 5 |- ( a e. k , b e. ( P pSyl G ) |-> ran ( c e. b |-> ( ( a ( +g ` G ) c ) ( -g ` G ) a ) ) ) = ( x e. k , y e. ( P pSyl G ) |-> ran ( z e. y |-> ( ( x ( +g ` G ) z ) ( -g ` G ) x ) ) )
34		eqid 2433	. . . . 5 |- { x e. X | A. y e. X ( ( x ( +g ` G ) y ) e. s <-> ( y ( +g ` G ) x ) e. s ) } = { x e. X | A. y e. X ( ( x ( +g ` G ) y ) e. s <-> ( y ( +g ` G ) x ) e. s ) }
35	4, 10, 11, 12, 13, 14, 27, 33, 34	sylow3lem6 16111	. . . 4 |- ( ( ph /\ k e. ( P pSyl G ) ) -> ( ( # ` ( P pSyl G ) ) mod P ) = 1 )
36	32, 35	syl5eq 2477	. . 3 |- ( ( ph /\ k e. ( P pSyl G ) ) -> ( N mod P ) = 1 )
37	31, 36	jca 529	. 2 |- ( ( ph /\ k e. ( P pSyl G ) ) -> ( N || ( ( # ` X ) / ( P ^ ( P pCnt ( # ` X ) ) ) ) /\ ( N mod P ) = 1 ) )
38	8, 37	exlimddv 1691	1 |- ( ph -> ( N || ( ( # ` X ) / ( P ^ ( P pCnt ( # ` X ) ) ) ) /\ ( N mod P ) = 1 ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1362   E.wex 1589   e. wcel 1755   =/= wne 2596   A.wral 2705   {crab 2709   (/)c0 3625   class class class wbr 4280   e. cmpt 4338   ran crn 4828   ` cfv 5406  (class class class)co 6080   e. cmpt2 6082   Fincfn 7298   1c1 9271   / cdiv 9981   mod cmo 11692   ^cexp 11849   #chash 12087   || cdivides 13518   Primecprime 13746   pCnt cpc 13886   Basecbs 14157   +g cplusg 14221   Grpcgrp 15393   -gcsg 15396   pSyl cslw 16011

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-inf2 7835  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347  ax-pre-sup 9348

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-fal 1368  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-disj 4251  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-se 4667  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-isom 5415  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-om 6466  df-1st 6566  df-2nd 6567  df-recs 6818  df-rdg 6852  df-1o 6908  df-2o 6909  df-oadd 6912  df-omul 6913  df-er 7089  df-ec 7091  df-qs 7095  df-map 7204  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-sup 7679  df-oi 7712  df-card 8097  df-acn 8100  df-cda 8325  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-div 9982  df-nn 10311  df-2 10368  df-3 10369  df-n0 10568  df-z 10635  df-uz 10850  df-q 10942  df-rp 10980  df-fz 11425  df-fzo 11533  df-fl 11626  df-mod 11693  df-seq 11791  df-exp 11850  df-fac 12036  df-bc 12063  df-hash 12088  df-cj 12572  df-re 12573  df-im 12574  df-sqr 12708  df-abs 12709  df-clim 12950  df-sum 13148  df-dvds 13519  df-gcd 13674  df-prm 13747  df-pc 13887  df-ndx 14160  df-slot 14161  df-base 14162  df-sets 14163  df-ress 14164  df-plusg 14234  df-0g 14363  df-mnd 15398  df-submnd 15448  df-grp 15525  df-minusg 15526  df-sbg 15527  df-mulg 15528  df-subg 15658  df-nsg 15659  df-eqg 15660  df-ghm 15725  df-ga 15788  df-od 16012  df-pgp 16014  df-slw 16015

This theorem is referenced by: (None)