Theorem sylow3lem4 18045

Description: Lemma for sylow3 18048, first part. The number of Sylow subgroups is a divisor of the size of G reduced by the size of a Sylow subgroup of G. (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
sylow3lem4	|- ( ph -> ( # ` ( P pSyl G ) ) || ( ( # ` X ) / ( P ^ ( P pCnt ( # ` X ) ) ) ) )

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 sylow3lem4

Step	Hyp	Ref	Expression
1		sylow3.x	. . 3 |- X = ( Base ` G )
2		sylow3.g	. . 3 |- ( ph -> G e. Grp )
3		sylow3.xf	. . 3 |- ( ph -> X e. Fin )
4		sylow3.p	. . 3 |- ( ph -> P e. Prime )
5		sylow3lem1.a	. . 3 |- .+ = ( +g ` G )
6		sylow3lem1.d	. . 3 |- .- = ( -g ` G )
7		sylow3lem1.m	. . 3 |- .(+) = ( x e. X , y e. ( P pSyl G ) |-> ran ( z e. y |-> ( ( x .+ z ) .- x ) ) )
8		sylow3lem2.k	. . 3 |- ( ph -> K e. ( P pSyl G ) )
9		sylow3lem2.h	. . 3 |- H = { u e. X | ( u .(+) K ) = K }
10		sylow3lem2.n	. . 3 |- N = { x e. X | A. y e. X ( ( x .+ y ) e. K <-> ( y .+ x ) e. K ) }
11	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	sylow3lem3 18044	. 2 |- ( ph -> ( # ` ( P pSyl G ) ) = ( # ` ( X /. ( G ~QG N ) ) ) )
12		slwsubg 18025	. . . . . . . . . 10 |- ( K e. ( P pSyl G ) -> K e. (SubGrp ` G ) )
13	8, 12	syl 17	. . . . . . . . 9 |- ( ph -> K e. (SubGrp ` G ) )
14		eqid 2622	. . . . . . . . . . 11 |- ( G ↾s N ) = ( G ↾s N )
15	10, 1, 5, 14	nmznsg 17638	. . . . . . . . . 10 |- ( K e. (SubGrp ` G ) -> K e. (NrmSGrp ` ( G ↾s N ) ) )
16		nsgsubg 17626	. . . . . . . . . 10 |- ( K e. (NrmSGrp ` ( G ↾s N ) ) -> K e. (SubGrp ` ( G ↾s N ) ) )
17	15, 16	syl 17	. . . . . . . . 9 |- ( K e. (SubGrp ` G ) -> K e. (SubGrp ` ( G ↾s N ) ) )
18	13, 17	syl 17	. . . . . . . 8 |- ( ph -> K e. (SubGrp ` ( G ↾s N ) ) )
19	10, 1, 5	nmzsubg 17635	. . . . . . . . . . 11 |- ( G e. Grp -> N e. (SubGrp ` G ) )
20	2, 19	syl 17	. . . . . . . . . 10 |- ( ph -> N e. (SubGrp ` G ) )
21	14	subgbas 17598	. . . . . . . . . 10 |- ( N e. (SubGrp ` G ) -> N = ( Base ` ( G ↾s N ) ) )
22	20, 21	syl 17	. . . . . . . . 9 |- ( ph -> N = ( Base ` ( G ↾s N ) ) )
23	1	subgss 17595	. . . . . . . . . . 11 |- ( N e. (SubGrp ` G ) -> N C_ X )
24	20, 23	syl 17	. . . . . . . . . 10 |- ( ph -> N C_ X )
25		ssfi 8180	. . . . . . . . . 10 |- ( ( X e. Fin /\ N C_ X ) -> N e. Fin )
26	3, 24, 25	syl2anc 693	. . . . . . . . 9 |- ( ph -> N e. Fin )
27	22, 26	eqeltrrd 2702	. . . . . . . 8 |- ( ph -> ( Base ` ( G ↾s N ) ) e. Fin )
28		eqid 2622	. . . . . . . . 9 |- ( Base ` ( G ↾s N ) ) = ( Base ` ( G ↾s N ) )
29	28	lagsubg 17656	. . . . . . . 8 |- ( ( K e. (SubGrp ` ( G ↾s N ) ) /\ ( Base ` ( G ↾s N ) ) e. Fin ) -> ( # ` K ) || ( # ` ( Base ` ( G ↾s N ) ) ) )
30	18, 27, 29	syl2anc 693	. . . . . . 7 |- ( ph -> ( # ` K ) || ( # ` ( Base ` ( G ↾s N ) ) ) )
31	22	fveq2d 6195	. . . . . . 7 |- ( ph -> ( # ` N ) = ( # ` ( Base ` ( G ↾s N ) ) ) )
32	30, 31	breqtrrd 4681	. . . . . 6 |- ( ph -> ( # ` K ) || ( # ` N ) )
33		eqid 2622	. . . . . . . . . . . 12 |- ( 0g ` G ) = ( 0g ` G )
34	33	subg0cl 17602	. . . . . . . . . . 11 |- ( K e. (SubGrp ` G ) -> ( 0g ` G ) e. K )
35	13, 34	syl 17	. . . . . . . . . 10 |- ( ph -> ( 0g ` G ) e. K )
36		ne0i 3921	. . . . . . . . . 10 |- ( ( 0g ` G ) e. K -> K =/= (/) )
37	35, 36	syl 17	. . . . . . . . 9 |- ( ph -> K =/= (/) )
38	1	subgss 17595	. . . . . . . . . . . 12 |- ( K e. (SubGrp ` G ) -> K C_ X )
39	13, 38	syl 17	. . . . . . . . . . 11 |- ( ph -> K C_ X )
40		ssfi 8180	. . . . . . . . . . 11 |- ( ( X e. Fin /\ K C_ X ) -> K e. Fin )
41	3, 39, 40	syl2anc 693	. . . . . . . . . 10 |- ( ph -> K e. Fin )
42		hashnncl 13157	. . . . . . . . . 10 |- ( K e. Fin -> ( ( # ` K ) e. NN <-> K =/= (/) ) )
43	41, 42	syl 17	. . . . . . . . 9 |- ( ph -> ( ( # ` K ) e. NN <-> K =/= (/) ) )
44	37, 43	mpbird 247	. . . . . . . 8 |- ( ph -> ( # ` K ) e. NN )
45	44	nnzd 11481	. . . . . . 7 |- ( ph -> ( # ` K ) e. ZZ )
46		hashcl 13147	. . . . . . . . 9 |- ( N e. Fin -> ( # ` N ) e. NN0 )
47	26, 46	syl 17	. . . . . . . 8 |- ( ph -> ( # ` N ) e. NN0 )
48	47	nn0zd 11480	. . . . . . 7 |- ( ph -> ( # ` N ) e. ZZ )
49		pwfi 8261	. . . . . . . . . . 11 |- ( X e. Fin <-> ~P X e. Fin )
50	3, 49	sylib 208	. . . . . . . . . 10 |- ( ph -> ~P X e. Fin )
51		eqid 2622	. . . . . . . . . . . . 13 |- ( G ~QG N ) = ( G ~QG N )
52	1, 51	eqger 17644	. . . . . . . . . . . 12 |- ( N e. (SubGrp ` G ) -> ( G ~QG N ) Er X )
53	20, 52	syl 17	. . . . . . . . . . 11 |- ( ph -> ( G ~QG N ) Er X )
54	53	qsss 7808	. . . . . . . . . 10 |- ( ph -> ( X /. ( G ~QG N ) ) C_ ~P X )
55		ssfi 8180	. . . . . . . . . 10 |- ( ( ~P X e. Fin /\ ( X /. ( G ~QG N ) ) C_ ~P X ) -> ( X /. ( G ~QG N ) ) e. Fin )
56	50, 54, 55	syl2anc 693	. . . . . . . . 9 |- ( ph -> ( X /. ( G ~QG N ) ) e. Fin )
57		hashcl 13147	. . . . . . . . 9 |- ( ( X /. ( G ~QG N ) ) e. Fin -> ( # ` ( X /. ( G ~QG N ) ) ) e. NN0 )
58	56, 57	syl 17	. . . . . . . 8 |- ( ph -> ( # ` ( X /. ( G ~QG N ) ) ) e. NN0 )
59	58	nn0zd 11480	. . . . . . 7 |- ( ph -> ( # ` ( X /. ( G ~QG N ) ) ) e. ZZ )
60		dvdscmul 15008	. . . . . . 7 |- ( ( ( # ` K ) e. ZZ /\ ( # ` N ) e. ZZ /\ ( # ` ( X /. ( G ~QG N ) ) ) e. ZZ ) -> ( ( # ` K ) || ( # ` N ) -> ( ( # ` ( X /. ( G ~QG N ) ) ) x. ( # ` K ) ) || ( ( # ` ( X /. ( G ~QG N ) ) ) x. ( # ` N ) ) ) )
61	45, 48, 59, 60	syl3anc 1326	. . . . . 6 |- ( ph -> ( ( # ` K ) || ( # ` N ) -> ( ( # ` ( X /. ( G ~QG N ) ) ) x. ( # ` K ) ) || ( ( # ` ( X /. ( G ~QG N ) ) ) x. ( # ` N ) ) ) )
62	32, 61	mpd 15	. . . . 5 |- ( ph -> ( ( # ` ( X /. ( G ~QG N ) ) ) x. ( # ` K ) ) || ( ( # ` ( X /. ( G ~QG N ) ) ) x. ( # ` N ) ) )
63		hashcl 13147	. . . . . . . . 9 |- ( X e. Fin -> ( # ` X ) e. NN0 )
64	3, 63	syl 17	. . . . . . . 8 |- ( ph -> ( # ` X ) e. NN0 )
65	64	nn0cnd 11353	. . . . . . 7 |- ( ph -> ( # ` X ) e. CC )
66	44	nncnd 11036	. . . . . . 7 |- ( ph -> ( # ` K ) e. CC )
67	44	nnne0d 11065	. . . . . . 7 |- ( ph -> ( # ` K ) =/= 0 )
68	65, 66, 67	divcan1d 10802	. . . . . 6 |- ( ph -> ( ( ( # ` X ) / ( # ` K ) ) x. ( # ` K ) ) = ( # ` X ) )
69	1, 51, 20, 3	lagsubg2 17655	. . . . . 6 |- ( ph -> ( # ` X ) = ( ( # ` ( X /. ( G ~QG N ) ) ) x. ( # ` N ) ) )
70	68, 69	eqtrd 2656	. . . . 5 |- ( ph -> ( ( ( # ` X ) / ( # ` K ) ) x. ( # ` K ) ) = ( ( # ` ( X /. ( G ~QG N ) ) ) x. ( # ` N ) ) )
71	62, 70	breqtrrd 4681	. . . 4 |- ( ph -> ( ( # ` ( X /. ( G ~QG N ) ) ) x. ( # ` K ) ) || ( ( ( # ` X ) / ( # ` K ) ) x. ( # ` K ) ) )
72	1	lagsubg 17656	. . . . . . 7 |- ( ( K e. (SubGrp ` G ) /\ X e. Fin ) -> ( # ` K ) || ( # ` X ) )
73	13, 3, 72	syl2anc 693	. . . . . 6 |- ( ph -> ( # ` K ) || ( # ` X ) )
74	64	nn0zd 11480	. . . . . . 7 |- ( ph -> ( # ` X ) e. ZZ )
75		dvdsval2 14986	. . . . . . 7 |- ( ( ( # ` K ) e. ZZ /\ ( # ` K ) =/= 0 /\ ( # ` X ) e. ZZ ) -> ( ( # ` K ) || ( # ` X ) <-> ( ( # ` X ) / ( # ` K ) ) e. ZZ ) )
76	45, 67, 74, 75	syl3anc 1326	. . . . . 6 |- ( ph -> ( ( # ` K ) || ( # ` X ) <-> ( ( # ` X ) / ( # ` K ) ) e. ZZ ) )
77	73, 76	mpbid 222	. . . . 5 |- ( ph -> ( ( # ` X ) / ( # ` K ) ) e. ZZ )
78		dvdsmulcr 15011	. . . . 5 |- ( ( ( # ` ( X /. ( G ~QG N ) ) ) e. ZZ /\ ( ( # ` X ) / ( # ` K ) ) e. ZZ /\ ( ( # ` K ) e. ZZ /\ ( # ` K ) =/= 0 ) ) -> ( ( ( # ` ( X /. ( G ~QG N ) ) ) x. ( # ` K ) ) || ( ( ( # ` X ) / ( # ` K ) ) x. ( # ` K ) ) <-> ( # ` ( X /. ( G ~QG N ) ) ) || ( ( # ` X ) / ( # ` K ) ) ) )
79	59, 77, 45, 67, 78	syl112anc 1330	. . . 4 |- ( ph -> ( ( ( # ` ( X /. ( G ~QG N ) ) ) x. ( # ` K ) ) || ( ( ( # ` X ) / ( # ` K ) ) x. ( # ` K ) ) <-> ( # ` ( X /. ( G ~QG N ) ) ) || ( ( # ` X ) / ( # ` K ) ) ) )
80	71, 79	mpbid 222	. . 3 |- ( ph -> ( # ` ( X /. ( G ~QG N ) ) ) || ( ( # ` X ) / ( # ` K ) ) )
81	1, 3, 8	slwhash 18039	. . . 4 |- ( ph -> ( # ` K ) = ( P ^ ( P pCnt ( # ` X ) ) ) )
82	81	oveq2d 6666	. . 3 |- ( ph -> ( ( # ` X ) / ( # ` K ) ) = ( ( # ` X ) / ( P ^ ( P pCnt ( # ` X ) ) ) ) )
83	80, 82	breqtrd 4679	. 2 |- ( ph -> ( # ` ( X /. ( G ~QG N ) ) ) || ( ( # ` X ) / ( P ^ ( P pCnt ( # ` X ) ) ) ) )
84	11, 83	eqbrtrd 4675	1 |- ( ph -> ( # ` ( P pSyl G ) ) || ( ( # ` X ) / ( P ^ ( P pCnt ( # ` X ) ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   {crab 2916   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   class class class wbr 4653   |-> cmpt 4729   ran crn 5115   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   Er wer 7739   /.cqs 7741   Fincfn 7955   0cc0 9936   x. cmul 9941   / cdiv 10684   NNcn 11020   NN0cn0 11292   ZZcz 11377   ^cexp 12860   #chash 13117   || cdvds 14983   Primecprime 15385   pCnt cpc 15541   Basecbs 15857   ↾_s cress 15858   +g cplusg 15941   0gc0g 16100   Grpcgrp 17422   -gcsg 17424  SubGrpcsubg 17588  NrmSGrpcnsg 17589   ~_QG cqg 17590   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-nsg 17592  df-eqg 17593  df-ghm 17658  df-ga 17723  df-od 17948  df-pgp 17950  df-slw 17951

This theorem is referenced by:  sylow3  18048