Theorem sylow3 16446

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 16428	. . . 4 |- ( ( G e. Grp /\ X e. Fin /\ P e. Prime ) -> ( P pSyl G ) =/= (/) )
6	1, 2, 3, 5	syl3anc 1228	. . 3 |- ( ph -> ( P pSyl G ) =/= (/) )
7		n0 3794	. . 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 2467	. . . . 5 |- ( +g ` G ) = ( +g ` G )
14		eqid 2467	. . . . 5 |- ( -g ` G ) = ( -g ` G )
15		oveq2 6290	. . . . . . . . . 10 |- ( c = z -> ( a ( +g ` G ) c ) = ( a ( +g ` G ) z ) )
16	15	oveq1d 6297	. . . . . . . . 9 |- ( c = z -> ( ( a ( +g ` G ) c ) ( -g ` G ) a ) = ( ( a ( +g ` G ) z ) ( -g ` G ) a ) )
17	16	cbvmptv 4538	. . . . . . . 8 |- ( c e. b |-> ( ( a ( +g ` G ) c ) ( -g ` G ) a ) ) = ( z e. b |-> ( ( a ( +g ` G ) z ) ( -g ` G ) a ) )
18		oveq1 6289	. . . . . . . . . 10 |- ( a = x -> ( a ( +g ` G ) z ) = ( x ( +g ` G ) z ) )
19		id 22	. . . . . . . . . 10 |- ( a = x -> a = x )
20	18, 19	oveq12d 6300	. . . . . . . . 9 |- ( a = x -> ( ( a ( +g ` G ) z ) ( -g ` G ) a ) = ( ( x ( +g ` G ) z ) ( -g ` G ) x ) )
21	20	mpteq2dv 4534	. . . . . . . 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 2520	. . . . . . 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 5228	. . . . . 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 4527	. . . . . . 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 5228	. . . . . 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 6359	. . . . 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 2467	. . . . 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 2467	. . . . 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 16443	. . . 4 |- ( ( ph /\ k e. ( P pSyl G ) ) -> ( # ` ( P pSyl G ) ) || ( ( # ` X ) / ( P ^ ( P pCnt ( # ` X ) ) ) ) )
31	9, 30	syl5eqbr 4480	. . 3 |- ( ( ph /\ k e. ( P pSyl G ) ) -> N || ( ( # ` X ) / ( P ^ ( P pCnt ( # ` X ) ) ) ) )
32	9	oveq1i 6292	. . . 4 |- ( N mod P ) = ( ( # ` ( P pSyl G ) ) mod P )
33	23, 25	cbvmpt2v 6359	. . . . 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 2467	. . . . 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 16445	. . . 4 |- ( ( ph /\ k e. ( P pSyl G ) ) -> ( ( # ` ( P pSyl G ) ) mod P ) = 1 )
36	32, 35	syl5eq 2520	. . 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 1702	1 |- ( ph -> ( N || ( ( # ` X ) / ( P ^ ( P pCnt ( # ` X ) ) ) ) /\ ( N mod P ) = 1 ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1379   E.wex 1596   e. wcel 1767   =/= wne 2662   A.wral 2814   {crab 2818   (/)c0 3785   class class class wbr 4447   |-> cmpt 4505   ran crn 5000   ` cfv 5586  (class class class)co 6282   |-> cmpt2 6284   Fincfn 7513   1c1 9489   / cdiv 10202   mod cmo 11959   ^cexp 12129   #chash 12367   || cdivides 13840   Primecprime 14069   pCnt cpc 14212   Basecbs 14483   +g cplusg 14548   Grpcgrp 15720   -gcsg 15723   pSyl cslw 16345

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-disj 4418  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-2o 7128  df-oadd 7131  df-omul 7132  df-er 7308  df-ec 7310  df-qs 7314  df-map 7419  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-sup 7897  df-oi 7931  df-card 8316  df-acn 8319  df-cda 8544  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11079  df-q 11179  df-rp 11217  df-fz 11669  df-fzo 11789  df-fl 11893  df-mod 11960  df-seq 12071  df-exp 12130  df-fac 12316  df-bc 12343  df-hash 12368  df-cj 12889  df-re 12890  df-im 12891  df-sqrt 13025  df-abs 13026  df-clim 13267  df-sum 13465  df-dvds 13841  df-gcd 13997  df-prm 14070  df-pc 14213  df-ndx 14486  df-slot 14487  df-base 14488  df-sets 14489  df-ress 14490  df-plusg 14561  df-0g 14690  df-mnd 15725  df-submnd 15775  df-grp 15855  df-minusg 15856  df-sbg 15857  df-mulg 15858  df-subg 15990  df-nsg 15991  df-eqg 15992  df-ghm 16057  df-ga 16120  df-od 16346  df-pgp 16348  df-slw 16349

This theorem is referenced by: (None)