Theorem sylow3lem3 18044

Description: Lemma for sylow3 18048, 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 8261	. . . . . 6 |- ( X e. Fin <-> ~P X e. Fin )
3	1, 2	sylib 208	. . . . 5 |- ( ph -> ~P X e. Fin )
4		slwsubg 18025	. . . . . . . 8 |- ( x e. ( P pSyl G ) -> x e. (SubGrp ` G ) )
5		sylow3.x	. . . . . . . . 9 |- X = ( Base ` G )
6	5	subgss 17595	. . . . . . . 8 |- ( x e. (SubGrp ` G ) -> x C_ X )
7	4, 6	syl 17	. . . . . . 7 |- ( x e. ( P pSyl G ) -> x C_ X )
8		selpw 4165	. . . . . . 7 |- ( x e. ~P X <-> x C_ X )
9	7, 8	sylibr 224	. . . . . 6 |- ( x e. ( P pSyl G ) -> x e. ~P X )
10	9	ssriv 3607	. . . . 5 |- ( P pSyl G ) C_ ~P X
11		ssfi 8180	. . . . 5 |- ( ( ~P X e. Fin /\ ( P pSyl G ) C_ ~P X ) -> ( P pSyl G ) e. Fin )
12	3, 10, 11	sylancl 694	. . . 4 |- ( ph -> ( P pSyl G ) e. Fin )
13		hashcl 13147	. . . 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 11353	. 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 17635	. . . . . . 7 |- ( G e. Grp -> N e. (SubGrp ` G ) )
20		eqid 2622	. . . . . . . 8 |- ( G ~QG N ) = ( G ~QG N )
21	5, 20	eqger 17644	. . . . . . 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 7808	. . . . 5 |- ( ph -> ( X /. ( G ~QG N ) ) C_ ~P X )
24		ssfi 8180	. . . . 5 |- ( ( ~P X e. Fin /\ ( X /. ( G ~QG N ) ) C_ ~P X ) -> ( X /. ( G ~QG N ) ) e. Fin )
25	3, 23, 24	syl2anc 693	. . . 4 |- ( ph -> ( X /. ( G ~QG N ) ) e. Fin )
26		hashcl 13147	. . . 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 11353	. 2 |- ( ph -> ( # ` ( X /. ( G ~QG N ) ) ) e. CC )
29	16, 19	syl 17	. . . . 5 |- ( ph -> N e. (SubGrp ` G ) )
30		eqid 2622	. . . . . 6 |- ( 0g ` G ) = ( 0g ` G )
31	30	subg0cl 17602	. . . . 5 |- ( N e. (SubGrp ` G ) -> ( 0g ` G ) e. N )
32		ne0i 3921	. . . . 5 |- ( ( 0g ` G ) e. N -> N =/= (/) )
33	29, 31, 32	3syl 18	. . . 4 |- ( ph -> N =/= (/) )
34	5	subgss 17595	. . . . . . 7 |- ( N e. (SubGrp ` G ) -> N C_ X )
35	16, 19, 34	3syl 18	. . . . . 6 |- ( ph -> N C_ X )
36		ssfi 8180	. . . . . 6 |- ( ( X e. Fin /\ N C_ X ) -> N e. Fin )
37	1, 35, 36	syl2anc 693	. . . . 5 |- ( ph -> N e. Fin )
38		hashnncl 13157	. . . . 5 |- ( N e. Fin -> ( ( # ` N ) e. NN <-> N =/= (/) ) )
39	37, 38	syl 17	. . . 4 |- ( ph -> ( ( # ` N ) e. NN <-> N =/= (/) ) )
40	33, 39	mpbird 247	. . 3 |- ( ph -> ( # ` N ) e. NN )
41	40	nncnd 11036	. 2 |- ( ph -> ( # ` N ) e. CC )
42	40	nnne0d 11065	. 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 18042	. . . 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 2622	. . . . 5 |- ( G ~QG H ) = ( G ~QG H )
50		eqid 2622	. . . . 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 17747	. . . 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 1325	. . 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 17655	. . 3 |- ( ph -> ( # ` X ) = ( ( # ` ( X /. ( G ~QG N ) ) ) x. ( # ` N ) ) )
54	50, 5	gaorber 17741	. . . . . . . 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 7788	. . . . . 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 481	. . . . . . . . . 10 |- ( ( ph /\ h e. ( P pSyl G ) ) -> K e. ( P pSyl G ) )
58		simpr 477	. . . . . . . . . 10 |- ( ( ph /\ h e. ( P pSyl G ) ) -> h e. ( P pSyl G ) )
59	1	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ h e. ( P pSyl G ) ) -> X e. Fin )
60	5, 59, 58, 57, 18, 44	sylow2 18041	. . . . . . . . . . 11 |- ( ( ph /\ h e. ( P pSyl G ) ) -> E. u e. X h = ran ( z e. K |-> ( ( u .+ z ) .- u ) ) )
61		eqcom 2629	. . . . . . . . . . . . 13 |- ( ( u .(+) K ) = h <-> h = ( u .(+) K ) )
62		simpr 477	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ h e. ( P pSyl G ) ) /\ u e. X ) -> u e. X )
63	57	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ h e. ( P pSyl G ) ) /\ u e. X ) -> K e. ( P pSyl G ) )
64		mptexg 6484	. . . . . . . . . . . . . . . 16 |- ( K e. ( P pSyl G ) -> ( z e. K |-> ( ( u .+ z ) .- u ) ) e. _V )
65		rnexg 7098	. . . . . . . . . . . . . . . 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 477	. . . . . . . . . . . . . . . . . 18 |- ( ( x = u /\ y = K ) -> y = K )
68		simpl 473	. . . . . . . . . . . . . . . . . . . 20 |- ( ( x = u /\ y = K ) -> x = u )
69	68	oveq1d 6665	. . . . . . . . . . . . . . . . . . 19 |- ( ( x = u /\ y = K ) -> ( x .+ z ) = ( u .+ z ) )
70	69, 68	oveq12d 6668	. . . . . . . . . . . . . . . . . 18 |- ( ( x = u /\ y = K ) -> ( ( x .+ z ) .- x ) = ( ( u .+ z ) .- u ) )
71	67, 70	mpteq12dv 4733	. . . . . . . . . . . . . . . . 17 |- ( ( x = u /\ y = K ) -> ( z e. y |-> ( ( x .+ z ) .- x ) ) = ( z e. K |-> ( ( u .+ z ) .- u ) ) )
72	71	rneqd 5353	. . . . . . . . . . . . . . . 16 |- ( ( x = u /\ y = K ) -> ran ( z e. y |-> ( ( x .+ z ) .- x ) ) = ran ( z e. K |-> ( ( u .+ z ) .- u ) ) )
73	72, 45	ovmpt2ga 6790	. . . . . . . . . . . . . . 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 1326	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ h e. ( P pSyl G ) ) /\ u e. X ) -> ( u .(+) K ) = ran ( z e. K |-> ( ( u .+ z ) .- u ) ) )
75	74	eqeq2d 2632	. . . . . . . . . . . . 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 272	. . . . . . . . . . . 12 |- ( ( ( ph /\ h e. ( P pSyl G ) ) /\ u e. X ) -> ( ( u .(+) K ) = h <-> h = ran ( z e. K |-> ( ( u .+ z ) .- u ) ) ) )
77	76	rexbidva 3049	. . . . . . . . . . 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 247	. . . . . . . . . 10 |- ( ( ph /\ h e. ( P pSyl G ) ) -> E. u e. X ( u .(+) K ) = h )
79	50	gaorb 17740	. . . . . . . . . 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 1246	. . . . . . . . 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 7785	. . . . . . . . . 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 693	. . . . . . . . 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 247	. . . . . . . 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 450	. . . . . . 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 3609	. . . . . 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 3620	. . . . 5 |- ( ph -> [ K ] { <. x , y >. | ( { x , y } C_ ( P pSyl G ) /\ E. g e. X ( g .(+) x ) = y ) } = ( P pSyl G ) )
87	86	fveq2d 6195	. . . 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 18043	. . . . 5 |- ( ph -> H = N )
89	88	fveq2d 6195	. . . 4 |- ( ph -> ( # ` H ) = ( # ` N ) )
90	87, 89	oveq12d 6668	. . 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 2665	. 2 |- ( ph -> ( ( # ` ( P pSyl G ) ) x. ( # ` N ) ) = ( ( # ` ( X /. ( G ~QG N ) ) ) x. ( # ` N ) ) )
92	15, 28, 41, 42, 91	mulcan2ad 10663	1 |- ( ph -> ( # ` ( P pSyl G ) ) = ( # ` ( X /. ( G ~QG N ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   {crab 2916   _Vcvv 3200   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   {cpr 4179   class class class wbr 4653   {copab 4712   |-> cmpt 4729   ran crn 5115   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   Er wer 7739   [cec 7740   /.cqs 7741   Fincfn 7955   x. cmul 9941   NNcn 11020   NN0cn0 11292   #chash 13117   Primecprime 15385   Basecbs 15857   +g cplusg 15941   0gc0g 16100   Grpcgrp 17422   -gcsg 17424  SubGrpcsubg 17588   ~_QG cqg 17590   GrpAct cga 17722   pSyl cslw 17947

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-disj 4621  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-omul 7565  df-er 7742  df-ec 7744  df-qs 7748  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-acn 8768  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417  df-dvds 14984  df-gcd 15217  df-prm 15386  df-pc 15542  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-grp 17425  df-minusg 17426  df-sbg 17427  df-mulg 17541  df-subg 17591  df-eqg 17593  df-ghm 17658  df-ga 17723  df-od 17948  df-pgp 17950  df-slw 17951

This theorem is referenced by:  sylow3lem4  18045