Theorem sylow3lem3 16452

Description: Lemma for sylow3 16456, 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 7814	. . . . . 6 |- ( X e. Fin <-> ~P X e. Fin )
3	1, 2	sylib 196	. . . . 5 |- ( ph -> ~P X e. Fin )
4		slwsubg 16433	. . . . . . . 8 |- ( x e. ( P pSyl G ) -> x e. (SubGrp ` G ) )
5		sylow3.x	. . . . . . . . 9 |- X = ( Base ` G )
6	5	subgss 16004	. . . . . . . 8 |- ( x e. (SubGrp ` G ) -> x C_ X )
7	4, 6	syl 16	. . . . . . 7 |- ( x e. ( P pSyl G ) -> x C_ X )
8		selpw 4017	. . . . . . 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 3508	. . . . 5 |- ( P pSyl G ) C_ ~P X
11		ssfi 7740	. . . . 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 12395	. . . 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 10853	. 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 16044	. . . . . . . 8 |- ( G e. Grp -> N e. (SubGrp ` G ) )
20	16, 19	syl 16	. . . . . . 7 |- ( ph -> N e. (SubGrp ` G ) )
21		eqid 2467	. . . . . . . 8 |- ( G ~QG N ) = ( G ~QG N )
22	5, 21	eqger 16053	. . . . . . 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 7372	. . . . 5 |- ( ph -> ( X /. ( G ~QG N ) ) C_ ~P X )
25		ssfi 7740	. . . . 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 12395	. . . 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 10853	. 2 |- ( ph -> ( # ` ( X /. ( G ~QG N ) ) ) e. CC )
30		eqid 2467	. . . . . . 7 |- ( 0g ` G ) = ( 0g ` G )
31	30	subg0cl 16011	. . . . . 6 |- ( N e. (SubGrp ` G ) -> ( 0g ` G ) e. N )
32	20, 31	syl 16	. . . . 5 |- ( ph -> ( 0g ` G ) e. N )
33		ne0i 3791	. . . . 5 |- ( ( 0g ` G ) e. N -> N =/= (/) )
34	32, 33	syl 16	. . . 4 |- ( ph -> N =/= (/) )
35	5	subgss 16004	. . . . . . 7 |- ( N e. (SubGrp ` G ) -> N C_ X )
36	20, 35	syl 16	. . . . . 6 |- ( ph -> N C_ X )
37		ssfi 7740	. . . . . 6 |- ( ( X e. Fin /\ N C_ X ) -> N e. Fin )
38	1, 36, 37	syl2anc 661	. . . . 5 |- ( ph -> N e. Fin )
39		hashnncl 12403	. . . . 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 10551	. 2 |- ( ph -> ( # ` N ) e. CC )
43	41	nnne0d 10579	. 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 16450	. . . 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 2467	. . . . 5 |- ( G ~QG H ) = ( G ~QG H )
51		eqid 2467	. . . . 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 16154	. . . 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 1227	. . 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 16064	. . 3 |- ( ph -> ( # ` X ) = ( ( # ` ( X /. ( G ~QG N ) ) ) x. ( # ` N ) ) )
55	51, 5	gaorber 16148	. . . . . . . 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 7353	. . . . . 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 16449	. . . . . . . . . . 11 |- ( ( ph /\ h e. ( P pSyl G ) ) -> E. u e. X h = ran ( z e. K |-> ( ( u .+ z ) .- u ) ) )
62		eqcom 2476	. . . . . . . . . . . . 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 6129	. . . . . . . . . . . . . . . . 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 6716	. . . . . . . . . . . . . . . 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 6298	. . . . . . . . . . . . . . . . . . 19 |- ( ( x = u /\ y = K ) -> ( x .+ z ) = ( u .+ z ) )
72	71, 70	oveq12d 6301	. . . . . . . . . . . . . . . . . 18 |- ( ( x = u /\ y = K ) -> ( ( x .+ z ) .- x ) = ( ( u .+ z ) .- u ) )
73	69, 72	mpteq12dv 4525	. . . . . . . . . . . . . . . . 17 |- ( ( x = u /\ y = K ) -> ( z e. y |-> ( ( x .+ z ) .- x ) ) = ( z e. K |-> ( ( u .+ z ) .- u ) ) )
74	73	rneqd 5229	. . . . . . . . . . . . . . . 16 |- ( ( x = u /\ y = K ) -> ran ( z e. y |-> ( ( x .+ z ) .- x ) ) = ran ( z e. K |-> ( ( u .+ z ) .- u ) ) )
75	74, 46	ovmpt2ga 6415	. . . . . . . . . . . . . . 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 1228	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ h e. ( P pSyl G ) ) /\ u e. X ) -> ( u .(+) K ) = ran ( z e. K |-> ( ( u .+ z ) .- u ) ) )
77	76	eqeq2d 2481	. . . . . . . . . . . . 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 2970	. . . . . . . . . . 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 16147	. . . . . . . . . 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 1180	. . . . . . . . 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 7350	. . . . . . . . . 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 3510	. . . . . 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 3521	. . . . 5 |- ( ph -> [ K ] { <. x , y >. | ( { x , y } C_ ( P pSyl G ) /\ E. g e. X ( g .(+) x ) = y ) } = ( P pSyl G ) )
89	88	fveq2d 5869	. . . 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 16451	. . . . 5 |- ( ph -> H = N )
91	90	fveq2d 5869	. . . 4 |- ( ph -> ( # ` H ) = ( # ` N ) )
92	89, 91	oveq12d 6301	. . 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 2517	. 2 |- ( ph -> ( ( # ` ( P pSyl G ) ) x. ( # ` N ) ) = ( ( # ` ( X /. ( G ~QG N ) ) ) x. ( # ` N ) ) )
94	15, 29, 42, 43, 93	mulcan2ad 10184	1 |- ( ph -> ( # ` ( P pSyl G ) ) = ( # ` ( X /. ( G ~QG N ) ) ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1379   e. wcel 1767   =/= wne 2662   A.wral 2814   E.wrex 2815   {crab 2818   _Vcvv 3113   C_ wss 3476   (/)c0 3785   ~Pcpw 4010   {cpr 4029   class class class wbr 4447   {copab 4504   |-> cmpt 4505   ran crn 5000   ` cfv 5587  (class class class)co 6283   |-> cmpt2 6285   Er wer 7308   [cec 7309   /.cqs 7310   Fincfn 7516   x. cmul 9496   NNcn 10535   NN0cn0 10794   #chash 12372   Primecprime 14075   Basecbs 14489   +g cplusg 14554   0gc0g 14694   Grpcgrp 15726   -gcsg 15729  SubGrpcsubg 15997   ~_QG cqg 15999   GrpAct cga 16129   pSyl cslw 16355

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 6575  ax-inf2 8057  ax-cnex 9547  ax-resscn 9548  ax-1cn 9549  ax-icn 9550  ax-addcl 9551  ax-addrcl 9552  ax-mulcl 9553  ax-mulrcl 9554  ax-mulcom 9555  ax-addass 9556  ax-mulass 9557  ax-distr 9558  ax-i2m1 9559  ax-1ne0 9560  ax-1rid 9561  ax-rnegex 9562  ax-rrecex 9563  ax-cnre 9564  ax-pre-lttri 9565  ax-pre-lttrn 9566  ax-pre-ltadd 9567  ax-pre-mulgt0 9568  ax-pre-sup 9569

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 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-isom 5596  df-riota 6244  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-om 6680  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-omul 7135  df-er 7311  df-ec 7313  df-qs 7317  df-map 7422  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-sup 7900  df-oi 7934  df-card 8319  df-acn 8322  df-cda 8547  df-pnf 9629  df-mnf 9630  df-xr 9631  df-ltxr 9632  df-le 9633  df-sub 9806  df-neg 9807  df-div 10206  df-nn 10536  df-2 10593  df-3 10594  df-n0 10795  df-z 10864  df-uz 11082  df-q 11182  df-rp 11220  df-fz 11672  df-fzo 11792  df-fl 11896  df-mod 11964  df-seq 12075  df-exp 12134  df-fac 12321  df-bc 12348  df-hash 12373  df-cj 12894  df-re 12895  df-im 12896  df-sqrt 13030  df-abs 13031  df-clim 13273  df-sum 13471  df-dvds 13847  df-gcd 14003  df-prm 14076  df-pc 14219  df-ndx 14492  df-slot 14493  df-base 14494  df-sets 14495  df-ress 14496  df-plusg 14567  df-0g 14696  df-mnd 15731  df-submnd 15784  df-grp 15864  df-minusg 15865  df-sbg 15866  df-mulg 15867  df-subg 16000  df-eqg 16002  df-ghm 16067  df-ga 16130  df-od 16356  df-pgp 16358  df-slw 16359

This theorem is referenced by:  sylow3lem4  16453