Theorem sylow3lem3 16848

Description: Lemma for sylow3 16852, 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 7807	. . . . . 6 |- ( X e. Fin <-> ~P X e. Fin )
3	1, 2	sylib 196	. . . . 5 |- ( ph -> ~P X e. Fin )
4		slwsubg 16829	. . . . . . . 8 |- ( x e. ( P pSyl G ) -> x e. (SubGrp ` G ) )
5		sylow3.x	. . . . . . . . 9 |- X = ( Base ` G )
6	5	subgss 16401	. . . . . . . 8 |- ( x e. (SubGrp ` G ) -> x C_ X )
7	4, 6	syl 16	. . . . . . 7 |- ( x e. ( P pSyl G ) -> x C_ X )
8		selpw 4006	. . . . . . 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 3493	. . . . 5 |- ( P pSyl G ) C_ ~P X
11		ssfi 7733	. . . . 5 |- ( ( ~P X e. Fin /\ ( P pSyl G ) C_ ~P X ) -> ( P pSyl G ) e. Fin )
12	3, 10, 11	sylancl 660	. . . 4 |- ( ph -> ( P pSyl G ) e. Fin )
13		hashcl 12410	. . . 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 10850	. 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 16441	. . . . . . 7 |- ( G e. Grp -> N e. (SubGrp ` G ) )
20		eqid 2454	. . . . . . . 8 |- ( G ~QG N ) = ( G ~QG N )
21	5, 20	eqger 16450	. . . . . . 7 |- ( N e. (SubGrp ` G ) -> ( G ~QG N ) Er X )
22	16, 19, 21	3syl 20	. . . . . 6 |- ( ph -> ( G ~QG N ) Er X )
23	22	qsss 7364	. . . . 5 |- ( ph -> ( X /. ( G ~QG N ) ) C_ ~P X )
24		ssfi 7733	. . . . 5 |- ( ( ~P X e. Fin /\ ( X /. ( G ~QG N ) ) C_ ~P X ) -> ( X /. ( G ~QG N ) ) e. Fin )
25	3, 23, 24	syl2anc 659	. . . 4 |- ( ph -> ( X /. ( G ~QG N ) ) e. Fin )
26		hashcl 12410	. . . 4 |- ( ( X /. ( G ~QG N ) ) e. Fin -> ( # ` ( X /. ( G ~QG N ) ) ) e. NN0 )
27	25, 26	syl 16	. . 3 |- ( ph -> ( # ` ( X /. ( G ~QG N ) ) ) e. NN0 )
28	27	nn0cnd 10850	. 2 |- ( ph -> ( # ` ( X /. ( G ~QG N ) ) ) e. CC )
29	16, 19	syl 16	. . . . 5 |- ( ph -> N e. (SubGrp ` G ) )
30		eqid 2454	. . . . . 6 |- ( 0g ` G ) = ( 0g ` G )
31	30	subg0cl 16408	. . . . 5 |- ( N e. (SubGrp ` G ) -> ( 0g ` G ) e. N )
32		ne0i 3789	. . . . 5 |- ( ( 0g ` G ) e. N -> N =/= (/) )
33	29, 31, 32	3syl 20	. . . 4 |- ( ph -> N =/= (/) )
34	5	subgss 16401	. . . . . . 7 |- ( N e. (SubGrp ` G ) -> N C_ X )
35	16, 19, 34	3syl 20	. . . . . 6 |- ( ph -> N C_ X )
36		ssfi 7733	. . . . . 6 |- ( ( X e. Fin /\ N C_ X ) -> N e. Fin )
37	1, 35, 36	syl2anc 659	. . . . 5 |- ( ph -> N e. Fin )
38		hashnncl 12419	. . . . 5 |- ( N e. Fin -> ( ( # ` N ) e. NN <-> N =/= (/) ) )
39	37, 38	syl 16	. . . 4 |- ( ph -> ( ( # ` N ) e. NN <-> N =/= (/) ) )
40	33, 39	mpbird 232	. . 3 |- ( ph -> ( # ` N ) e. NN )
41	40	nncnd 10547	. 2 |- ( ph -> ( # ` N ) e. CC )
42	40	nnne0d 10576	. 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 16846	. . . 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 2454	. . . . 5 |- ( G ~QG H ) = ( G ~QG H )
50		eqid 2454	. . . . 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 16551	. . . 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 1225	. . 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 16461	. . 3 |- ( ph -> ( # ` X ) = ( ( # ` ( X /. ( G ~QG N ) ) ) x. ( # ` N ) ) )
54	50, 5	gaorber 16545	. . . . . . . 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 16	. . . . . . 7 |- ( ph -> { <. x , y >. | ( { x , y } C_ ( P pSyl G ) /\ E. g e. X ( g .(+) x ) = y ) } Er ( P pSyl G ) )
56	55	ecss 7345	. . . . . 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 463	. . . . . . . . . 10 |- ( ( ph /\ h e. ( P pSyl G ) ) -> K e. ( P pSyl G ) )
58		simpr 459	. . . . . . . . . 10 |- ( ( ph /\ h e. ( P pSyl G ) ) -> h e. ( P pSyl G ) )
59	1	adantr 463	. . . . . . . . . . . 12 |- ( ( ph /\ h e. ( P pSyl G ) ) -> X e. Fin )
60	5, 59, 58, 57, 18, 44	sylow2 16845	. . . . . . . . . . 11 |- ( ( ph /\ h e. ( P pSyl G ) ) -> E. u e. X h = ran ( z e. K |-> ( ( u .+ z ) .- u ) ) )
61		eqcom 2463	. . . . . . . . . . . . 13 |- ( ( u .(+) K ) = h <-> h = ( u .(+) K ) )
62		simpr 459	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ h e. ( P pSyl G ) ) /\ u e. X ) -> u e. X )
63	57	adantr 463	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ h e. ( P pSyl G ) ) /\ u e. X ) -> K e. ( P pSyl G ) )
64		mptexg 6117	. . . . . . . . . . . . . . . 16 |- ( K e. ( P pSyl G ) -> ( z e. K |-> ( ( u .+ z ) .- u ) ) e. _V )
65		rnexg 6705	. . . . . . . . . . . . . . . 16 |- ( ( z e. K |-> ( ( u .+ z ) .- u ) ) e. _V -> ran ( z e. K |-> ( ( u .+ z ) .- u ) ) e. _V )
66	63, 64, 65	3syl 20	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ h e. ( P pSyl G ) ) /\ u e. X ) -> ran ( z e. K |-> ( ( u .+ z ) .- u ) ) e. _V )
67		simpr 459	. . . . . . . . . . . . . . . . . 18 |- ( ( x = u /\ y = K ) -> y = K )
68		simpl 455	. . . . . . . . . . . . . . . . . . . 20 |- ( ( x = u /\ y = K ) -> x = u )
69	68	oveq1d 6285	. . . . . . . . . . . . . . . . . . 19 |- ( ( x = u /\ y = K ) -> ( x .+ z ) = ( u .+ z ) )
70	69, 68	oveq12d 6288	. . . . . . . . . . . . . . . . . 18 |- ( ( x = u /\ y = K ) -> ( ( x .+ z ) .- x ) = ( ( u .+ z ) .- u ) )
71	67, 70	mpteq12dv 4517	. . . . . . . . . . . . . . . . 17 |- ( ( x = u /\ y = K ) -> ( z e. y |-> ( ( x .+ z ) .- x ) ) = ( z e. K |-> ( ( u .+ z ) .- u ) ) )
72	71	rneqd 5219	. . . . . . . . . . . . . . . 16 |- ( ( x = u /\ y = K ) -> ran ( z e. y |-> ( ( x .+ z ) .- x ) ) = ran ( z e. K |-> ( ( u .+ z ) .- u ) ) )
73	72, 45	ovmpt2ga 6405	. . . . . . . . . . . . . . 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 1226	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ h e. ( P pSyl G ) ) /\ u e. X ) -> ( u .(+) K ) = ran ( z e. K |-> ( ( u .+ z ) .- u ) ) )
75	74	eqeq2d 2468	. . . . . . . . . . . . 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 257	. . . . . . . . . . . 12 |- ( ( ( ph /\ h e. ( P pSyl G ) ) /\ u e. X ) -> ( ( u .(+) K ) = h <-> h = ran ( z e. K |-> ( ( u .+ z ) .- u ) ) ) )
77	76	rexbidva 2962	. . . . . . . . . . 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 232	. . . . . . . . . 10 |- ( ( ph /\ h e. ( P pSyl G ) ) -> E. u e. X ( u .(+) K ) = h )
79	50	gaorb 16544	. . . . . . . . . 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 1178	. . . . . . . . 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 7342	. . . . . . . . . 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 659	. . . . . . . . 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 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 ) } )
84	83	ex 432	. . . . . . 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 3495	. . . . . 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 3506	. . . . 5 |- ( ph -> [ K ] { <. x , y >. | ( { x , y } C_ ( P pSyl G ) /\ E. g e. X ( g .(+) x ) = y ) } = ( P pSyl G ) )
87	86	fveq2d 5852	. . . 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 16847	. . . . 5 |- ( ph -> H = N )
89	88	fveq2d 5852	. . . 4 |- ( ph -> ( # ` H ) = ( # ` N ) )
90	87, 89	oveq12d 6288	. . 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 2504	. 2 |- ( ph -> ( ( # ` ( P pSyl G ) ) x. ( # ` N ) ) = ( ( # ` ( X /. ( G ~QG N ) ) ) x. ( # ` N ) ) )
92	15, 28, 41, 42, 91	mulcan2ad 10181	1 |- ( ph -> ( # ` ( P pSyl G ) ) = ( # ` ( X /. ( G ~QG N ) ) ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 367   = wceq 1398   e. wcel 1823   =/= wne 2649   A.wral 2804   E.wrex 2805   {crab 2808   _Vcvv 3106   C_ wss 3461   (/)c0 3783   ~Pcpw 3999   {cpr 4018   class class class wbr 4439   {copab 4496   |-> cmpt 4497   ran crn 4989   ` cfv 5570  (class class class)co 6270   |-> cmpt2 6272   Er wer 7300   [cec 7301   /.cqs 7302   Fincfn 7509   x. cmul 9486   NNcn 10531   NN0cn0 10791   #chash 12387   Primecprime 14301   Basecbs 14716   +g cplusg 14784   0gc0g 14929   Grpcgrp 16252   -gcsg 16254  SubGrpcsubg 16394   ~_QG cqg 16396   GrpAct cga 16526   pSyl cslw 16751

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-disj 4411  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-oadd 7126  df-omul 7127  df-er 7303  df-ec 7305  df-qs 7309  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-sup 7893  df-oi 7927  df-card 8311  df-acn 8314  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11083  df-q 11184  df-rp 11222  df-fz 11676  df-fzo 11800  df-fl 11910  df-mod 11979  df-seq 12090  df-exp 12149  df-fac 12336  df-bc 12363  df-hash 12388  df-cj 13014  df-re 13015  df-im 13016  df-sqrt 13150  df-abs 13151  df-clim 13393  df-sum 13591  df-dvds 14071  df-gcd 14229  df-prm 14302  df-pc 14445  df-ndx 14719  df-slot 14720  df-base 14721  df-sets 14722  df-ress 14723  df-plusg 14797  df-0g 14931  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-submnd 16166  df-grp 16256  df-minusg 16257  df-sbg 16258  df-mulg 16259  df-subg 16397  df-eqg 16399  df-ghm 16464  df-ga 16527  df-od 16752  df-pgp 16754  df-slw 16755

This theorem is referenced by:  sylow3lem4  16849