Theorem sylow3 16252

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. This is part of Metamath 100 proof #72. (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 16234	. . . 4 |- ( ( G e. Grp /\ X e. Fin /\ P e. Prime ) -> ( P pSyl G ) =/= (/) )
6	1, 2, 3, 5	syl3anc 1219	. . 3 |- ( ph -> ( P pSyl G ) =/= (/) )
7		n0 3753	. . 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 465	. . . . 5 |- ( ( ph /\ k e. ( P pSyl G ) ) -> G e. Grp )
11	2	adantr 465	. . . . 5 |- ( ( ph /\ k e. ( P pSyl G ) ) -> X e. Fin )
12	3	adantr 465	. . . . 5 |- ( ( ph /\ k e. ( P pSyl G ) ) -> P e. Prime )
13		eqid 2454	. . . . 5 |- ( +g ` G ) = ( +g ` G )
14		eqid 2454	. . . . 5 |- ( -g ` G ) = ( -g ` G )
15		oveq2 6207	. . . . . . . . . 10 |- ( c = z -> ( a ( +g ` G ) c ) = ( a ( +g ` G ) z ) )
16	15	oveq1d 6214	. . . . . . . . 9 |- ( c = z -> ( ( a ( +g ` G ) c ) ( -g ` G ) a ) = ( ( a ( +g ` G ) z ) ( -g ` G ) a ) )
17	16	cbvmptv 4490	. . . . . . . 8 |- ( c e. b |-> ( ( a ( +g ` G ) c ) ( -g ` G ) a ) ) = ( z e. b |-> ( ( a ( +g ` G ) z ) ( -g ` G ) a ) )
18		oveq1 6206	. . . . . . . . . 10 |- ( a = x -> ( a ( +g ` G ) z ) = ( x ( +g ` G ) z ) )
19		id 22	. . . . . . . . . 10 |- ( a = x -> a = x )
20	18, 19	oveq12d 6217	. . . . . . . . 9 |- ( a = x -> ( ( a ( +g ` G ) z ) ( -g ` G ) a ) = ( ( x ( +g ` G ) z ) ( -g ` G ) x ) )
21	20	mpteq2dv 4486	. . . . . . . 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 2507	. . . . . . 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 5174	. . . . . 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 4479	. . . . . . 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 5174	. . . . . 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 6274	. . . . 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 461	. . . . 5 |- ( ( ph /\ k e. ( P pSyl G ) ) -> k e. ( P pSyl G ) )
28		eqid 2454	. . . . 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 2454	. . . . 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 16249	. . . 4 |- ( ( ph /\ k e. ( P pSyl G ) ) -> ( # ` ( P pSyl G ) ) || ( ( # ` X ) / ( P ^ ( P pCnt ( # ` X ) ) ) ) )
31	9, 30	syl5eqbr 4432	. . 3 |- ( ( ph /\ k e. ( P pSyl G ) ) -> N || ( ( # ` X ) / ( P ^ ( P pCnt ( # ` X ) ) ) ) )
32	9	oveq1i 6209	. . . 4 |- ( N mod P ) = ( ( # ` ( P pSyl G ) ) mod P )
33	23, 25	cbvmpt2v 6274	. . . . 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 2454	. . . . 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 16251	. . . 4 |- ( ( ph /\ k e. ( P pSyl G ) ) -> ( ( # ` ( P pSyl G ) ) mod P ) = 1 )
36	32, 35	syl5eq 2507	. . 3 |- ( ( ph /\ k e. ( P pSyl G ) ) -> ( N mod P ) = 1 )
37	31, 36	jca 532	. 2 |- ( ( ph /\ k e. ( P pSyl G ) ) -> ( N || ( ( # ` X ) / ( P ^ ( P pCnt ( # ` X ) ) ) ) /\ ( N mod P ) = 1 ) )
38	8, 37	exlimddv 1693	1 |- ( ph -> ( N || ( ( # ` X ) / ( P ^ ( P pCnt ( # ` X ) ) ) ) /\ ( N mod P ) = 1 ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1370   E.wex 1587   e. wcel 1758   =/= wne 2647   A.wral 2798   {crab 2802   (/)c0 3744   class class class wbr 4399   |-> cmpt 4457   ran crn 4948   ` cfv 5525  (class class class)co 6199   |-> cmpt2 6201   Fincfn 7419   1c1 9393   / cdiv 10103   mod cmo 11824   ^cexp 11981   #chash 12219   || cdivides 13652   Primecprime 13880   pCnt cpc 14020   Basecbs 14291   +g cplusg 14356   Grpcgrp 15528   -gcsg 15531   pSyl cslw 16151

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-inf2 7957  ax-cnex 9448  ax-resscn 9449  ax-1cn 9450  ax-icn 9451  ax-addcl 9452  ax-addrcl 9453  ax-mulcl 9454  ax-mulrcl 9455  ax-mulcom 9456  ax-addass 9457  ax-mulass 9458  ax-distr 9459  ax-i2m1 9460  ax-1ne0 9461  ax-1rid 9462  ax-rnegex 9463  ax-rrecex 9464  ax-cnre 9465  ax-pre-lttri 9466  ax-pre-lttrn 9467  ax-pre-ltadd 9468  ax-pre-mulgt0 9469  ax-pre-sup 9470

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-nel 2650  df-ral 2803  df-rex 2804  df-reu 2805  df-rmo 2806  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-pss 3451  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-tp 3989  df-op 3991  df-uni 4199  df-int 4236  df-iun 4280  df-disj 4370  df-br 4400  df-opab 4458  df-mpt 4459  df-tr 4493  df-eprel 4739  df-id 4743  df-po 4748  df-so 4749  df-fr 4786  df-se 4787  df-we 4788  df-ord 4829  df-on 4830  df-lim 4831  df-suc 4832  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-isom 5534  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-om 6586  df-1st 6686  df-2nd 6687  df-recs 6941  df-rdg 6975  df-1o 7029  df-2o 7030  df-oadd 7033  df-omul 7034  df-er 7210  df-ec 7212  df-qs 7216  df-map 7325  df-en 7420  df-dom 7421  df-sdom 7422  df-fin 7423  df-sup 7801  df-oi 7834  df-card 8219  df-acn 8222  df-cda 8447  df-pnf 9530  df-mnf 9531  df-xr 9532  df-ltxr 9533  df-le 9534  df-sub 9707  df-neg 9708  df-div 10104  df-nn 10433  df-2 10490  df-3 10491  df-n0 10690  df-z 10757  df-uz 10972  df-q 11064  df-rp 11102  df-fz 11554  df-fzo 11665  df-fl 11758  df-mod 11825  df-seq 11923  df-exp 11982  df-fac 12168  df-bc 12195  df-hash 12220  df-cj 12705  df-re 12706  df-im 12707  df-sqr 12841  df-abs 12842  df-clim 13083  df-sum 13281  df-dvds 13653  df-gcd 13808  df-prm 13881  df-pc 14021  df-ndx 14294  df-slot 14295  df-base 14296  df-sets 14297  df-ress 14298  df-plusg 14369  df-0g 14498  df-mnd 15533  df-submnd 15583  df-grp 15663  df-minusg 15664  df-sbg 15665  df-mulg 15666  df-subg 15796  df-nsg 15797  df-eqg 15798  df-ghm 15863  df-ga 15926  df-od 16152  df-pgp 16154  df-slw 16155

This theorem is referenced by: (None)