Theorem sylow3 16977

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 16959	. . . 4 |- ( ( G e. Grp /\ X e. Fin /\ P e. Prime ) -> ( P pSyl G ) =/= (/) )
6	1, 2, 3, 5	syl3anc 1230	. . 3 |- ( ph -> ( P pSyl G ) =/= (/) )
7		n0 3748	. . 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 463	. . . . 5 |- ( ( ph /\ k e. ( P pSyl G ) ) -> G e. Grp )
11	2	adantr 463	. . . . 5 |- ( ( ph /\ k e. ( P pSyl G ) ) -> X e. Fin )
12	3	adantr 463	. . . . 5 |- ( ( ph /\ k e. ( P pSyl G ) ) -> P e. Prime )
13		eqid 2402	. . . . 5 |- ( +g ` G ) = ( +g ` G )
14		eqid 2402	. . . . 5 |- ( -g ` G ) = ( -g ` G )
15		oveq2 6286	. . . . . . . . . 10 |- ( c = z -> ( a ( +g ` G ) c ) = ( a ( +g ` G ) z ) )
16	15	oveq1d 6293	. . . . . . . . 9 |- ( c = z -> ( ( a ( +g ` G ) c ) ( -g ` G ) a ) = ( ( a ( +g ` G ) z ) ( -g ` G ) a ) )
17	16	cbvmptv 4487	. . . . . . . 8 |- ( c e. b |-> ( ( a ( +g ` G ) c ) ( -g ` G ) a ) ) = ( z e. b |-> ( ( a ( +g ` G ) z ) ( -g ` G ) a ) )
18		oveq1 6285	. . . . . . . . . 10 |- ( a = x -> ( a ( +g ` G ) z ) = ( x ( +g ` G ) z ) )
19		id 22	. . . . . . . . . 10 |- ( a = x -> a = x )
20	18, 19	oveq12d 6296	. . . . . . . . 9 |- ( a = x -> ( ( a ( +g ` G ) z ) ( -g ` G ) a ) = ( ( x ( +g ` G ) z ) ( -g ` G ) x ) )
21	20	mpteq2dv 4482	. . . . . . . 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 2455	. . . . . . 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 5051	. . . . . 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 4475	. . . . . . 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 5051	. . . . . 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 6358	. . . . 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 459	. . . . 5 |- ( ( ph /\ k e. ( P pSyl G ) ) -> k e. ( P pSyl G ) )
28		eqid 2402	. . . . 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 2402	. . . . 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 16974	. . . 4 |- ( ( ph /\ k e. ( P pSyl G ) ) -> ( # ` ( P pSyl G ) ) || ( ( # ` X ) / ( P ^ ( P pCnt ( # ` X ) ) ) ) )
31	9, 30	syl5eqbr 4428	. . 3 |- ( ( ph /\ k e. ( P pSyl G ) ) -> N || ( ( # ` X ) / ( P ^ ( P pCnt ( # ` X ) ) ) ) )
32	9	oveq1i 6288	. . . 4 |- ( N mod P ) = ( ( # ` ( P pSyl G ) ) mod P )
33	23, 25	cbvmpt2v 6358	. . . . 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 2402	. . . . 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 16976	. . . 4 |- ( ( ph /\ k e. ( P pSyl G ) ) -> ( ( # ` ( P pSyl G ) ) mod P ) = 1 )
36	32, 35	syl5eq 2455	. . 3 |- ( ( ph /\ k e. ( P pSyl G ) ) -> ( N mod P ) = 1 )
37	31, 36	jca 530	. 2 |- ( ( ph /\ k e. ( P pSyl G ) ) -> ( N || ( ( # ` X ) / ( P ^ ( P pCnt ( # ` X ) ) ) ) /\ ( N mod P ) = 1 ) )
38	8, 37	exlimddv 1747	1 |- ( ph -> ( N || ( ( # ` X ) / ( P ^ ( P pCnt ( # ` X ) ) ) ) /\ ( N mod P ) = 1 ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 367   = wceq 1405   E.wex 1633   e. wcel 1842   =/= wne 2598   A.wral 2754   {crab 2758   (/)c0 3738   class class class wbr 4395   |-> cmpt 4453   ran crn 4824   ` cfv 5569  (class class class)co 6278   |-> cmpt2 6280   Fincfn 7554   1c1 9523   / cdiv 10247   mod cmo 12034   ^cexp 12210   #chash 12452   || cdvds 14195   Primecprime 14426   pCnt cpc 14569   Basecbs 14841   +g cplusg 14909   Grpcgrp 16377   -gcsg 16379   pSyl cslw 16876

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-inf2 8091  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599  ax-pre-sup 9600

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-disj 4367  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-isom 5578  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-1st 6784  df-2nd 6785  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-2o 7168  df-oadd 7171  df-omul 7172  df-er 7348  df-ec 7350  df-qs 7354  df-map 7459  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-sup 7935  df-oi 7969  df-card 8352  df-acn 8355  df-cda 8580  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-div 10248  df-nn 10577  df-2 10635  df-3 10636  df-n0 10837  df-z 10906  df-uz 11128  df-q 11228  df-rp 11266  df-fz 11727  df-fzo 11855  df-fl 11966  df-mod 12035  df-seq 12152  df-exp 12211  df-fac 12398  df-bc 12425  df-hash 12453  df-cj 13081  df-re 13082  df-im 13083  df-sqrt 13217  df-abs 13218  df-clim 13460  df-sum 13658  df-dvds 14196  df-gcd 14354  df-prm 14427  df-pc 14570  df-ndx 14844  df-slot 14845  df-base 14846  df-sets 14847  df-ress 14848  df-plusg 14922  df-0g 15056  df-mgm 16196  df-sgrp 16235  df-mnd 16245  df-submnd 16291  df-grp 16381  df-minusg 16382  df-sbg 16383  df-mulg 16384  df-subg 16522  df-nsg 16523  df-eqg 16524  df-ghm 16589  df-ga 16652  df-od 16877  df-pgp 16879  df-slw 16880

This theorem is referenced by: (None)