Theorem sylow3lem3 17281

Description: Lemma for sylow3 17285, 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 7869	. . . . . 6 |- ( X e. Fin <-> ~P X e. Fin )
3	1, 2	sylib 200	. . . . 5 |- ( ph -> ~P X e. Fin )
4		slwsubg 17262	. . . . . . . 8 |- ( x e. ( P pSyl G ) -> x e. (SubGrp ` G ) )
5		sylow3.x	. . . . . . . . 9 |- X = ( Base ` G )
6	5	subgss 16818	. . . . . . . 8 |- ( x e. (SubGrp ` G ) -> x C_ X )
7	4, 6	syl 17	. . . . . . 7 |- ( x e. ( P pSyl G ) -> x C_ X )
8		selpw 3958	. . . . . . 7 |- ( x e. ~P X <-> x C_ X )
9	7, 8	sylibr 216	. . . . . 6 |- ( x e. ( P pSyl G ) -> x e. ~P X )
10	9	ssriv 3436	. . . . 5 |- ( P pSyl G ) C_ ~P X
11		ssfi 7792	. . . . 5 |- ( ( ~P X e. Fin /\ ( P pSyl G ) C_ ~P X ) -> ( P pSyl G ) e. Fin )
12	3, 10, 11	sylancl 668	. . . 4 |- ( ph -> ( P pSyl G ) e. Fin )
13		hashcl 12538	. . . 4 |- ( ( P pSyl G ) e. Fin -> ( # ` ( P pSyl G ) ) e. NN0 )
14	12, 13	syl 17	. . 3 |- ( ph -> ( # ` ( P pSyl G ) ) e. NN0 )
15	14	nn0cnd 10927	. 2 |- ( ph -> ( # ` ( P pSyl G ) ) e. CC )
16		sylow3.g	. . . . . . 7 |- ( ph -> G e. Grp )
17		sylow3lem2.n	. . . . . . . 8 |- N = { x e. X | A. y e. X ( ( x .+ y ) e. K <-> ( y .+ x ) e. K ) }
18		sylow3lem1.a	. . . . . . . 8 |- .+ = ( +g ` G )
19	17, 5, 18	nmzsubg 16858	. . . . . . 7 |- ( G e. Grp -> N e. (SubGrp ` G ) )
20		eqid 2451	. . . . . . . 8 |- ( G ~QG N ) = ( G ~QG N )
21	5, 20	eqger 16867	. . . . . . 7 |- ( N e. (SubGrp ` G ) -> ( G ~QG N ) Er X )
22	16, 19, 21	3syl 18	. . . . . 6 |- ( ph -> ( G ~QG N ) Er X )
23	22	qsss 7424	. . . . 5 |- ( ph -> ( X /. ( G ~QG N ) ) C_ ~P X )
24		ssfi 7792	. . . . 5 |- ( ( ~P X e. Fin /\ ( X /. ( G ~QG N ) ) C_ ~P X ) -> ( X /. ( G ~QG N ) ) e. Fin )
25	3, 23, 24	syl2anc 667	. . . 4 |- ( ph -> ( X /. ( G ~QG N ) ) e. Fin )
26		hashcl 12538	. . . 4 |- ( ( X /. ( G ~QG N ) ) e. Fin -> ( # ` ( X /. ( G ~QG N ) ) ) e. NN0 )
27	25, 26	syl 17	. . 3 |- ( ph -> ( # ` ( X /. ( G ~QG N ) ) ) e. NN0 )
28	27	nn0cnd 10927	. 2 |- ( ph -> ( # ` ( X /. ( G ~QG N ) ) ) e. CC )
29	16, 19	syl 17	. . . . 5 |- ( ph -> N e. (SubGrp ` G ) )
30		eqid 2451	. . . . . 6 |- ( 0g ` G ) = ( 0g ` G )
31	30	subg0cl 16825	. . . . 5 |- ( N e. (SubGrp ` G ) -> ( 0g ` G ) e. N )
32		ne0i 3737	. . . . 5 |- ( ( 0g ` G ) e. N -> N =/= (/) )
33	29, 31, 32	3syl 18	. . . 4 |- ( ph -> N =/= (/) )
34	5	subgss 16818	. . . . . . 7 |- ( N e. (SubGrp ` G ) -> N C_ X )
35	16, 19, 34	3syl 18	. . . . . 6 |- ( ph -> N C_ X )
36		ssfi 7792	. . . . . 6 |- ( ( X e. Fin /\ N C_ X ) -> N e. Fin )
37	1, 35, 36	syl2anc 667	. . . . 5 |- ( ph -> N e. Fin )
38		hashnncl 12547	. . . . 5 |- ( N e. Fin -> ( ( # ` N ) e. NN <-> N =/= (/) ) )
39	37, 38	syl 17	. . . 4 |- ( ph -> ( ( # ` N ) e. NN <-> N =/= (/) ) )
40	33, 39	mpbird 236	. . 3 |- ( ph -> ( # ` N ) e. NN )
41	40	nncnd 10625	. 2 |- ( ph -> ( # ` N ) e. CC )
42	40	nnne0d 10654	. 2 |- ( ph -> ( # ` N ) =/= 0 )
43		sylow3.p	. . . . 5 |- ( ph -> P e. Prime )
44		sylow3lem1.d	. . . . 5 |- .- = ( -g ` G )
45		sylow3lem1.m	. . . . 5 |- .(+) = ( x e. X , y e. ( P pSyl G ) |-> ran ( z e. y |-> ( ( x .+ z ) .- x ) ) )
46	5, 16, 1, 43, 18, 44, 45	sylow3lem1 17279	. . . 4 |- ( ph -> .(+) e. ( G GrpAct ( P pSyl G ) ) )
47		sylow3lem2.k	. . . 4 |- ( ph -> K e. ( P pSyl G ) )
48		sylow3lem2.h	. . . . 5 |- H = { u e. X | ( u .(+) K ) = K }
49		eqid 2451	. . . . 5 |- ( G ~QG H ) = ( G ~QG H )
50		eqid 2451	. . . . 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 ) }
51	5, 48, 49, 50	orbsta2 16968	. . . 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 ) ) )
52	46, 47, 1, 51	syl21anc 1267	. . 3 |- ( ph -> ( # ` X ) = ( ( # ` [ K ] { <. x , y >. | ( { x , y } C_ ( P pSyl G ) /\ E. g e. X ( g .(+) x ) = y ) } ) x. ( # ` H ) ) )
53	5, 20, 29, 1	lagsubg2 16878	. . 3 |- ( ph -> ( # ` X ) = ( ( # ` ( X /. ( G ~QG N ) ) ) x. ( # ` N ) ) )
54	50, 5	gaorber 16962	. . . . . . . 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 ) )
55	46, 54	syl 17	. . . . . . 7 |- ( ph -> { <. x , y >. | ( { x , y } C_ ( P pSyl G ) /\ E. g e. X ( g .(+) x ) = y ) } Er ( P pSyl G ) )
56	55	ecss 7405	. . . . . 6 |- ( ph -> [ K ] { <. x , y >. | ( { x , y } C_ ( P pSyl G ) /\ E. g e. X ( g .(+) x ) = y ) } C_ ( P pSyl G ) )
57	47	adantr 467	. . . . . . . . . 10 |- ( ( ph /\ h e. ( P pSyl G ) ) -> K e. ( P pSyl G ) )
58		simpr 463	. . . . . . . . . 10 |- ( ( ph /\ h e. ( P pSyl G ) ) -> h e. ( P pSyl G ) )
59	1	adantr 467	. . . . . . . . . . . 12 |- ( ( ph /\ h e. ( P pSyl G ) ) -> X e. Fin )
60	5, 59, 58, 57, 18, 44	sylow2 17278	. . . . . . . . . . 11 |- ( ( ph /\ h e. ( P pSyl G ) ) -> E. u e. X h = ran ( z e. K |-> ( ( u .+ z ) .- u ) ) )
61		eqcom 2458	. . . . . . . . . . . . 13 |- ( ( u .(+) K ) = h <-> h = ( u .(+) K ) )
62		simpr 463	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ h e. ( P pSyl G ) ) /\ u e. X ) -> u e. X )
63	57	adantr 467	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ h e. ( P pSyl G ) ) /\ u e. X ) -> K e. ( P pSyl G ) )
64		mptexg 6135	. . . . . . . . . . . . . . . 16 |- ( K e. ( P pSyl G ) -> ( z e. K |-> ( ( u .+ z ) .- u ) ) e. _V )
65		rnexg 6725	. . . . . . . . . . . . . . . 16 |- ( ( z e. K |-> ( ( u .+ z ) .- u ) ) e. _V -> ran ( z e. K |-> ( ( u .+ z ) .- u ) ) e. _V )
66	63, 64, 65	3syl 18	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ h e. ( P pSyl G ) ) /\ u e. X ) -> ran ( z e. K |-> ( ( u .+ z ) .- u ) ) e. _V )
67		simpr 463	. . . . . . . . . . . . . . . . . 18 |- ( ( x = u /\ y = K ) -> y = K )
68		simpl 459	. . . . . . . . . . . . . . . . . . . 20 |- ( ( x = u /\ y = K ) -> x = u )
69	68	oveq1d 6305	. . . . . . . . . . . . . . . . . . 19 |- ( ( x = u /\ y = K ) -> ( x .+ z ) = ( u .+ z ) )
70	69, 68	oveq12d 6308	. . . . . . . . . . . . . . . . . 18 |- ( ( x = u /\ y = K ) -> ( ( x .+ z ) .- x ) = ( ( u .+ z ) .- u ) )
71	67, 70	mpteq12dv 4481	. . . . . . . . . . . . . . . . 17 |- ( ( x = u /\ y = K ) -> ( z e. y |-> ( ( x .+ z ) .- x ) ) = ( z e. K |-> ( ( u .+ z ) .- u ) ) )
72	71	rneqd 5062	. . . . . . . . . . . . . . . 16 |- ( ( x = u /\ y = K ) -> ran ( z e. y |-> ( ( x .+ z ) .- x ) ) = ran ( z e. K |-> ( ( u .+ z ) .- u ) ) )
73	72, 45	ovmpt2ga 6426	. . . . . . . . . . . . . . 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 ) ) )
74	62, 63, 66, 73	syl3anc 1268	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ h e. ( P pSyl G ) ) /\ u e. X ) -> ( u .(+) K ) = ran ( z e. K |-> ( ( u .+ z ) .- u ) ) )
75	74	eqeq2d 2461	. . . . . . . . . . . . 13 |- ( ( ( ph /\ h e. ( P pSyl G ) ) /\ u e. X ) -> ( h = ( u .(+) K ) <-> h = ran ( z e. K |-> ( ( u .+ z ) .- u ) ) ) )
76	61, 75	syl5bb 261	. . . . . . . . . . . 12 |- ( ( ( ph /\ h e. ( P pSyl G ) ) /\ u e. X ) -> ( ( u .(+) K ) = h <-> h = ran ( z e. K |-> ( ( u .+ z ) .- u ) ) ) )
77	76	rexbidva 2898	. . . . . . . . . . 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 ) ) ) )
78	60, 77	mpbird 236	. . . . . . . . . 10 |- ( ( ph /\ h e. ( P pSyl G ) ) -> E. u e. X ( u .(+) K ) = h )
79	50	gaorb 16961	. . . . . . . . . 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 ) )
80	57, 58, 78, 79	syl3anbrc 1192	. . . . . . . . 9 |- ( ( ph /\ h e. ( P pSyl G ) ) -> K { <. x , y >. | ( { x , y } C_ ( P pSyl G ) /\ E. g e. X ( g .(+) x ) = y ) } h )
81		elecg 7402	. . . . . . . . . 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 ) )
82	58, 57, 81	syl2anc 667	. . . . . . . . 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 ) )
83	80, 82	mpbird 236	. . . . . . . 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 ) } )
84	83	ex 436	. . . . . . 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 ) } ) )
85	84	ssrdv 3438	. . . . . 6 |- ( ph -> ( P pSyl G ) C_ [ K ] { <. x , y >. | ( { x , y } C_ ( P pSyl G ) /\ E. g e. X ( g .(+) x ) = y ) } )
86	56, 85	eqssd 3449	. . . . 5 |- ( ph -> [ K ] { <. x , y >. | ( { x , y } C_ ( P pSyl G ) /\ E. g e. X ( g .(+) x ) = y ) } = ( P pSyl G ) )
87	86	fveq2d 5869	. . . 4 |- ( ph -> ( # ` [ K ] { <. x , y >. | ( { x , y } C_ ( P pSyl G ) /\ E. g e. X ( g .(+) x ) = y ) } ) = ( # ` ( P pSyl G ) ) )
88	5, 16, 1, 43, 18, 44, 45, 47, 48, 17	sylow3lem2 17280	. . . . 5 |- ( ph -> H = N )
89	88	fveq2d 5869	. . . 4 |- ( ph -> ( # ` H ) = ( # ` N ) )
90	87, 89	oveq12d 6308	. . 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 ) ) )
91	52, 53, 90	3eqtr3rd 2494	. 2 |- ( ph -> ( ( # ` ( P pSyl G ) ) x. ( # ` N ) ) = ( ( # ` ( X /. ( G ~QG N ) ) ) x. ( # ` N ) ) )
92	15, 28, 41, 42, 91	mulcan2ad 10248	1 |- ( ph -> ( # ` ( P pSyl G ) ) = ( # ` ( X /. ( G ~QG N ) ) ) )

Syntax hints:   -> wi 4   <-> wb 188   /\ wa 371   = wceq 1444   e. wcel 1887   =/= wne 2622   A.wral 2737   E.wrex 2738   {crab 2741   _Vcvv 3045   C_ wss 3404   (/)c0 3731   ~Pcpw 3951   {cpr 3970   class class class wbr 4402   {copab 4460   |-> cmpt 4461   ran crn 4835   ` cfv 5582  (class class class)co 6290   |-> cmpt2 6292   Er wer 7360   [cec 7361   /.cqs 7362   Fincfn 7569   x. cmul 9544   NNcn 10609   NN0cn0 10869   #chash 12515   Primecprime 14622   Basecbs 15121   +g cplusg 15190   0gc0g 15338   Grpcgrp 16669   -gcsg 16671  SubGrpcsubg 16811   ~_QG cqg 16813   GrpAct cga 16943   pSyl cslw 17171

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-disj 4374  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-2o 7183  df-oadd 7186  df-omul 7187  df-er 7363  df-ec 7365  df-qs 7369  df-map 7474  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-sup 7956  df-inf 7957  df-oi 8025  df-card 8373  df-acn 8376  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-q 11265  df-rp 11303  df-fz 11785  df-fzo 11916  df-fl 12028  df-mod 12097  df-seq 12214  df-exp 12273  df-fac 12460  df-bc 12488  df-hash 12516  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-clim 13552  df-sum 13753  df-dvds 14306  df-gcd 14469  df-prm 14623  df-pc 14787  df-ndx 15124  df-slot 15125  df-base 15126  df-sets 15127  df-ress 15128  df-plusg 15203  df-0g 15340  df-mgm 16488  df-sgrp 16527  df-mnd 16537  df-submnd 16583  df-grp 16673  df-minusg 16674  df-sbg 16675  df-mulg 16676  df-subg 16814  df-eqg 16816  df-ghm 16881  df-ga 16944  df-od 17172  df-pgp 17176  df-slw 17178

This theorem is referenced by:  sylow3lem4  17282