Theorem sylow3lem3 16140

Description: Lemma for sylow3 16144, first part. The number of Sylow subgroups is the same as the index (number of cosets) of the normalizer of the Sylow subgroup K. (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 )
sylow3lem1.a	|- .+ = ( +g ` G )
sylow3lem1.d	|- .- = ( -g ` G )
sylow3lem1.m	|- .(+) = ( x e. X , y e. ( P pSyl G ) |-> ran ( z e. y |-> ( ( x .+ z ) .- x ) ) )
sylow3lem2.k	|- ( ph -> K e. ( P pSyl G ) )
sylow3lem2.h	|- H = { u e. X | ( u .(+) K ) = K }
sylow3lem2.n	|- N = { x e. X | A. y e. X ( ( x .+ y ) e. K <-> ( y .+ x ) e. K ) }

Assertion

Ref	Expression
sylow3lem3	|- ( ph -> ( # ` ( P pSyl G ) ) = ( # ` ( X /. ( G ~QG N ) ) ) )

Distinct variable groups:   x, u, y, z, .-   u, .(+) , x, y, z   x, H, y   u, K, x, y, z   u, N, z   u, X, x, y, z   u, G, x, y, z   ph, u, x, y, z   u, .+ , x, y, z   u, P, x, y, z

Allowed substitution hints:   H( z, u)   N( x, y)

Proof of Theorem sylow3lem3

Dummy variables g h are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sylow3.xf	. . . . . 6 |- ( ph -> X e. Fin )
2		pwfi 7618	. . . . . 6 |- ( X e. Fin <-> ~P X e. Fin )
3	1, 2	sylib 196	. . . . 5 |- ( ph -> ~P X e. Fin )
4		slwsubg 16121	. . . . . . . 8 |- ( x e. ( P pSyl G ) -> x e. (SubGrp ` G ) )
5		sylow3.x	. . . . . . . . 9 |- X = ( Base ` G )
6	5	subgss 15694	. . . . . . . 8 |- ( x e. (SubGrp ` G ) -> x C_ X )
7	4, 6	syl 16	. . . . . . 7 |- ( x e. ( P pSyl G ) -> x C_ X )
8		selpw 3879	. . . . . . 7 |- ( x e. ~P X <-> x C_ X )
9	7, 8	sylibr 212	. . . . . 6 |- ( x e. ( P pSyl G ) -> x e. ~P X )
10	9	ssriv 3372	. . . . 5 |- ( P pSyl G ) C_ ~P X
11		ssfi 7545	. . . . 5 |- ( ( ~P X e. Fin /\ ( P pSyl G ) C_ ~P X ) -> ( P pSyl G ) e. Fin )
12	3, 10, 11	sylancl 662	. . . 4 |- ( ph -> ( P pSyl G ) e. Fin )
13		hashcl 12138	. . . 4 |- ( ( P pSyl G ) e. Fin -> ( # ` ( P pSyl G ) ) e. NN0 )
14	12, 13	syl 16	. . 3 |- ( ph -> ( # ` ( P pSyl G ) ) e. NN0 )
15	14	nn0cnd 10650	. 2 |- ( ph -> ( # ` ( P pSyl G ) ) e. CC )
16		sylow3.g	. . . . . . . 8 |- ( ph -> G e. Grp )
17		sylow3lem2.n	. . . . . . . . 9 |- N = { x e. X | A. y e. X ( ( x .+ y ) e. K <-> ( y .+ x ) e. K ) }
18		sylow3lem1.a	. . . . . . . . 9 |- .+ = ( +g ` G )
19	17, 5, 18	nmzsubg 15734	. . . . . . . 8 |- ( G e. Grp -> N e. (SubGrp ` G ) )
20	16, 19	syl 16	. . . . . . 7 |- ( ph -> N e. (SubGrp ` G ) )
21		eqid 2443	. . . . . . . 8 |- ( G ~QG N ) = ( G ~QG N )
22	5, 21	eqger 15743	. . . . . . 7 |- ( N e. (SubGrp ` G ) -> ( G ~QG N ) Er X )
23	20, 22	syl 16	. . . . . 6 |- ( ph -> ( G ~QG N ) Er X )
24	23	qsss 7173	. . . . 5 |- ( ph -> ( X /. ( G ~QG N ) ) C_ ~P X )
25		ssfi 7545	. . . . 5 |- ( ( ~P X e. Fin /\ ( X /. ( G ~QG N ) ) C_ ~P X ) -> ( X /. ( G ~QG N ) ) e. Fin )
26	3, 24, 25	syl2anc 661	. . . 4 |- ( ph -> ( X /. ( G ~QG N ) ) e. Fin )
27		hashcl 12138	. . . 4 |- ( ( X /. ( G ~QG N ) ) e. Fin -> ( # ` ( X /. ( G ~QG N ) ) ) e. NN0 )
28	26, 27	syl 16	. . 3 |- ( ph -> ( # ` ( X /. ( G ~QG N ) ) ) e. NN0 )
29	28	nn0cnd 10650	. 2 |- ( ph -> ( # ` ( X /. ( G ~QG N ) ) ) e. CC )
30		eqid 2443	. . . . . . 7 |- ( 0g ` G ) = ( 0g ` G )
31	30	subg0cl 15701	. . . . . 6 |- ( N e. (SubGrp ` G ) -> ( 0g ` G ) e. N )
32	20, 31	syl 16	. . . . 5 |- ( ph -> ( 0g ` G ) e. N )
33		ne0i 3655	. . . . 5 |- ( ( 0g ` G ) e. N -> N =/= (/) )
34	32, 33	syl 16	. . . 4 |- ( ph -> N =/= (/) )
35	5	subgss 15694	. . . . . . 7 |- ( N e. (SubGrp ` G ) -> N C_ X )
36	20, 35	syl 16	. . . . . 6 |- ( ph -> N C_ X )
37		ssfi 7545	. . . . . 6 |- ( ( X e. Fin /\ N C_ X ) -> N e. Fin )
38	1, 36, 37	syl2anc 661	. . . . 5 |- ( ph -> N e. Fin )
39		hashnncl 12146	. . . . 5 |- ( N e. Fin -> ( ( # ` N ) e. NN <-> N =/= (/) ) )
40	38, 39	syl 16	. . . 4 |- ( ph -> ( ( # ` N ) e. NN <-> N =/= (/) ) )
41	34, 40	mpbird 232	. . 3 |- ( ph -> ( # ` N ) e. NN )
42	41	nncnd 10350	. 2 |- ( ph -> ( # ` N ) e. CC )
43	41	nnne0d 10378	. 2 |- ( ph -> ( # ` N ) =/= 0 )
44		sylow3.p	. . . . 5 |- ( ph -> P e. Prime )
45		sylow3lem1.d	. . . . 5 |- .- = ( -g ` G )
46		sylow3lem1.m	. . . . 5 |- .(+) = ( x e. X , y e. ( P pSyl G ) |-> ran ( z e. y |-> ( ( x .+ z ) .- x ) ) )
47	5, 16, 1, 44, 18, 45, 46	sylow3lem1 16138	. . . 4 |- ( ph -> .(+) e. ( G GrpAct ( P pSyl G ) ) )
48		sylow3lem2.k	. . . 4 |- ( ph -> K e. ( P pSyl G ) )
49		sylow3lem2.h	. . . . 5 |- H = { u e. X | ( u .(+) K ) = K }
50		eqid 2443	. . . . 5 |- ( G ~QG H ) = ( G ~QG H )
51		eqid 2443	. . . . 5 |- { <. x , y >. | ( { x , y } C_ ( P pSyl G ) /\ E. g e. X ( g .(+) x ) = y ) } = { <. x , y >. | ( { x , y } C_ ( P pSyl G ) /\ E. g e. X ( g .(+) x ) = y ) }
52	5, 49, 50, 51	orbsta2 15844	. . . 4 |- ( ( ( .(+) e. ( G GrpAct ( P pSyl G ) ) /\ K e. ( P pSyl G ) ) /\ X e. Fin ) -> ( # ` X ) = ( ( # ` [ K ] { <. x , y >. | ( { x , y } C_ ( P pSyl G ) /\ E. g e. X ( g .(+) x ) = y ) } ) x. ( # ` H ) ) )
53	47, 48, 1, 52	syl21anc 1217	. . 3 |- ( ph -> ( # ` X ) = ( ( # ` [ K ] { <. x , y >. | ( { x , y } C_ ( P pSyl G ) /\ E. g e. X ( g .(+) x ) = y ) } ) x. ( # ` H ) ) )
54	5, 21, 20, 1	lagsubg2 15754	. . 3 |- ( ph -> ( # ` X ) = ( ( # ` ( X /. ( G ~QG N ) ) ) x. ( # ` N ) ) )
55	51, 5	gaorber 15838	. . . . . . . 8 |- ( .(+) e. ( G GrpAct ( P pSyl G ) ) -> { <. x , y >. | ( { x , y } C_ ( P pSyl G ) /\ E. g e. X ( g .(+) x ) = y ) } Er ( P pSyl G ) )
56	47, 55	syl 16	. . . . . . 7 |- ( ph -> { <. x , y >. | ( { x , y } C_ ( P pSyl G ) /\ E. g e. X ( g .(+) x ) = y ) } Er ( P pSyl G ) )
57	56	ecss 7154	. . . . . 6 |- ( ph -> [ K ] { <. x , y >. | ( { x , y } C_ ( P pSyl G ) /\ E. g e. X ( g .(+) x ) = y ) } C_ ( P pSyl G ) )
58	48	adantr 465	. . . . . . . . . 10 |- ( ( ph /\ h e. ( P pSyl G ) ) -> K e. ( P pSyl G ) )
59		simpr 461	. . . . . . . . . 10 |- ( ( ph /\ h e. ( P pSyl G ) ) -> h e. ( P pSyl G ) )
60	1	adantr 465	. . . . . . . . . . . 12 |- ( ( ph /\ h e. ( P pSyl G ) ) -> X e. Fin )
61	5, 60, 59, 58, 18, 45	sylow2 16137	. . . . . . . . . . 11 |- ( ( ph /\ h e. ( P pSyl G ) ) -> E. u e. X h = ran ( z e. K |-> ( ( u .+ z ) .- u ) ) )
62		eqcom 2445	. . . . . . . . . . . . 13 |- ( ( u .(+) K ) = h <-> h = ( u .(+) K ) )
63		simpr 461	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ h e. ( P pSyl G ) ) /\ u e. X ) -> u e. X )
64	58	adantr 465	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ h e. ( P pSyl G ) ) /\ u e. X ) -> K e. ( P pSyl G ) )
65		mptexg 5959	. . . . . . . . . . . . . . . . 17 |- ( K e. ( P pSyl G ) -> ( z e. K |-> ( ( u .+ z ) .- u ) ) e. _V )
66	64, 65	syl 16	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ h e. ( P pSyl G ) ) /\ u e. X ) -> ( z e. K |-> ( ( u .+ z ) .- u ) ) e. _V )
67		rnexg 6522	. . . . . . . . . . . . . . . 16 |- ( ( z e. K |-> ( ( u .+ z ) .- u ) ) e. _V -> ran ( z e. K |-> ( ( u .+ z ) .- u ) ) e. _V )
68	66, 67	syl 16	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ h e. ( P pSyl G ) ) /\ u e. X ) -> ran ( z e. K |-> ( ( u .+ z ) .- u ) ) e. _V )
69		simpr 461	. . . . . . . . . . . . . . . . . 18 |- ( ( x = u /\ y = K ) -> y = K )
70		simpl 457	. . . . . . . . . . . . . . . . . . . 20 |- ( ( x = u /\ y = K ) -> x = u )
71	70	oveq1d 6118	. . . . . . . . . . . . . . . . . . 19 |- ( ( x = u /\ y = K ) -> ( x .+ z ) = ( u .+ z ) )
72	71, 70	oveq12d 6121	. . . . . . . . . . . . . . . . . 18 |- ( ( x = u /\ y = K ) -> ( ( x .+ z ) .- x ) = ( ( u .+ z ) .- u ) )
73	69, 72	mpteq12dv 4382	. . . . . . . . . . . . . . . . 17 |- ( ( x = u /\ y = K ) -> ( z e. y |-> ( ( x .+ z ) .- x ) ) = ( z e. K |-> ( ( u .+ z ) .- u ) ) )
74	73	rneqd 5079	. . . . . . . . . . . . . . . 16 |- ( ( x = u /\ y = K ) -> ran ( z e. y |-> ( ( x .+ z ) .- x ) ) = ran ( z e. K |-> ( ( u .+ z ) .- u ) ) )
75	74, 46	ovmpt2ga 6232	. . . . . . . . . . . . . . 15 |- ( ( u e. X /\ K e. ( P pSyl G ) /\ ran ( z e. K |-> ( ( u .+ z ) .- u ) ) e. _V ) -> ( u .(+) K ) = ran ( z e. K |-> ( ( u .+ z ) .- u ) ) )
76	63, 64, 68, 75	syl3anc 1218	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ h e. ( P pSyl G ) ) /\ u e. X ) -> ( u .(+) K ) = ran ( z e. K |-> ( ( u .+ z ) .- u ) ) )
77	76	eqeq2d 2454	. . . . . . . . . . . . 13 |- ( ( ( ph /\ h e. ( P pSyl G ) ) /\ u e. X ) -> ( h = ( u .(+) K ) <-> h = ran ( z e. K |-> ( ( u .+ z ) .- u ) ) ) )
78	62, 77	syl5bb 257	. . . . . . . . . . . 12 |- ( ( ( ph /\ h e. ( P pSyl G ) ) /\ u e. X ) -> ( ( u .(+) K ) = h <-> h = ran ( z e. K |-> ( ( u .+ z ) .- u ) ) ) )
79	78	rexbidva 2744	. . . . . . . . . . 11 |- ( ( ph /\ h e. ( P pSyl G ) ) -> ( E. u e. X ( u .(+) K ) = h <-> E. u e. X h = ran ( z e. K |-> ( ( u .+ z ) .- u ) ) ) )
80	61, 79	mpbird 232	. . . . . . . . . 10 |- ( ( ph /\ h e. ( P pSyl G ) ) -> E. u e. X ( u .(+) K ) = h )
81	51	gaorb 15837	. . . . . . . . . 10 |- ( K { <. x , y >. | ( { x , y } C_ ( P pSyl G ) /\ E. g e. X ( g .(+) x ) = y ) } h <-> ( K e. ( P pSyl G ) /\ h e. ( P pSyl G ) /\ E. u e. X ( u .(+) K ) = h ) )
82	58, 59, 80, 81	syl3anbrc 1172	. . . . . . . . 9 |- ( ( ph /\ h e. ( P pSyl G ) ) -> K { <. x , y >. | ( { x , y } C_ ( P pSyl G ) /\ E. g e. X ( g .(+) x ) = y ) } h )
83		elecg 7151	. . . . . . . . . 10 |- ( ( h e. ( P pSyl G ) /\ K e. ( P pSyl G ) ) -> ( h e. [ K ] { <. x , y >. | ( { x , y } C_ ( P pSyl G ) /\ E. g e. X ( g .(+) x ) = y ) } <-> K { <. x , y >. | ( { x , y } C_ ( P pSyl G ) /\ E. g e. X ( g .(+) x ) = y ) } h ) )
84	59, 58, 83	syl2anc 661	. . . . . . . . 9 |- ( ( ph /\ h e. ( P pSyl G ) ) -> ( h e. [ K ] { <. x , y >. | ( { x , y } C_ ( P pSyl G ) /\ E. g e. X ( g .(+) x ) = y ) } <-> K { <. x , y >. | ( { x , y } C_ ( P pSyl G ) /\ E. g e. X ( g .(+) x ) = y ) } h ) )
85	82, 84	mpbird 232	. . . . . . . 8 |- ( ( ph /\ h e. ( P pSyl G ) ) -> h e. [ K ] { <. x , y >. | ( { x , y } C_ ( P pSyl G ) /\ E. g e. X ( g .(+) x ) = y ) } )
86	85	ex 434	. . . . . . 7 |- ( ph -> ( h e. ( P pSyl G ) -> h e. [ K ] { <. x , y >. | ( { x , y } C_ ( P pSyl G ) /\ E. g e. X ( g .(+) x ) = y ) } ) )
87	86	ssrdv 3374	. . . . . 6 |- ( ph -> ( P pSyl G ) C_ [ K ] { <. x , y >. | ( { x , y } C_ ( P pSyl G ) /\ E. g e. X ( g .(+) x ) = y ) } )
88	57, 87	eqssd 3385	. . . . 5 |- ( ph -> [ K ] { <. x , y >. | ( { x , y } C_ ( P pSyl G ) /\ E. g e. X ( g .(+) x ) = y ) } = ( P pSyl G ) )
89	88	fveq2d 5707	. . . 4 |- ( ph -> ( # ` [ K ] { <. x , y >. | ( { x , y } C_ ( P pSyl G ) /\ E. g e. X ( g .(+) x ) = y ) } ) = ( # ` ( P pSyl G ) ) )
90	5, 16, 1, 44, 18, 45, 46, 48, 49, 17	sylow3lem2 16139	. . . . 5 |- ( ph -> H = N )
91	90	fveq2d 5707	. . . 4 |- ( ph -> ( # ` H ) = ( # ` N ) )
92	89, 91	oveq12d 6121	. . 3 |- ( ph -> ( ( # ` [ K ] { <. x , y >. | ( { x , y } C_ ( P pSyl G ) /\ E. g e. X ( g .(+) x ) = y ) } ) x. ( # ` H ) ) = ( ( # ` ( P pSyl G ) ) x. ( # ` N ) ) )
93	53, 54, 92	3eqtr3rd 2484	. 2 |- ( ph -> ( ( # ` ( P pSyl G ) ) x. ( # ` N ) ) = ( ( # ` ( X /. ( G ~QG N ) ) ) x. ( # ` N ) ) )
94	15, 29, 42, 43, 93	mulcan2ad 9984	1 |- ( ph -> ( # ` ( P pSyl G ) ) = ( # ` ( X /. ( G ~QG N ) ) ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1369   e. wcel 1756   =/= wne 2618   A.wral 2727   E.wrex 2728   {crab 2731   _Vcvv 2984   C_ wss 3340   (/)c0 3649   ~Pcpw 3872   {cpr 3891   class class class wbr 4304   {copab 4361   e. cmpt 4362   ran crn 4853   ` cfv 5430  (class class class)co 6103   e. cmpt2 6105   Er wer 7110   [cec 7111   /.cqs 7112   Fincfn 7322   x. cmul 9299   NNcn 10334   NN0cn0 10591   #chash 12115   Primecprime 13775   Basecbs 14186   +g cplusg 14250   0gc0g 14390   Grpcgrp 15422   -gcsg 15425  SubGrpcsubg 15687   ~_QG cqg 15689   GrpAct cga 15819   pSyl cslw 16043

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-inf2 7859  ax-cnex 9350  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-pre-mulgt0 9371  ax-pre-sup 9372

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-int 4141  df-iun 4185  df-disj 4275  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-se 4692  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-isom 5439  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-om 6489  df-1st 6589  df-2nd 6590  df-recs 6844  df-rdg 6878  df-1o 6932  df-2o 6933  df-oadd 6936  df-omul 6937  df-er 7113  df-ec 7115  df-qs 7119  df-map 7228  df-en 7323  df-dom 7324  df-sdom 7325  df-fin 7326  df-sup 7703  df-oi 7736  df-card 8121  df-acn 8124  df-cda 8349  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610  df-div 10006  df-nn 10335  df-2 10392  df-3 10393  df-n0 10592  df-z 10659  df-uz 10874  df-q 10966  df-rp 11004  df-fz 11450  df-fzo 11561  df-fl 11654  df-mod 11721  df-seq 11819  df-exp 11878  df-fac 12064  df-bc 12091  df-hash 12116  df-cj 12600  df-re 12601  df-im 12602  df-sqr 12736  df-abs 12737  df-clim 12978  df-sum 13176  df-dvds 13548  df-gcd 13703  df-prm 13776  df-pc 13916  df-ndx 14189  df-slot 14190  df-base 14191  df-sets 14192  df-ress 14193  df-plusg 14263  df-0g 14392  df-mnd 15427  df-submnd 15477  df-grp 15557  df-minusg 15558  df-sbg 15559  df-mulg 15560  df-subg 15690  df-eqg 15692  df-ghm 15757  df-ga 15820  df-od 16044  df-pgp 16046  df-slw 16047

This theorem is referenced by:  sylow3lem4  16141