Theorem sylow1lem5 18017

Description: Lemma for sylow1 18018. Using Lagrange's theorem and the orbit-stabilizer theorem, show that there is a subgroup with size exactly P ^ N. (Contributed by Mario Carneiro, 16-Jan-2015.)

Hypotheses

Ref	Expression
sylow1.x	|- X = ( Base ` G )
sylow1.g	|- ( ph -> G e. Grp )
sylow1.f	|- ( ph -> X e. Fin )
sylow1.p	|- ( ph -> P e. Prime )
sylow1.n	|- ( ph -> N e. NN0 )
sylow1.d	|- ( ph -> ( P ^ N ) || ( # ` X ) )
sylow1lem.a	|- .+ = ( +g ` G )
sylow1lem.s	|- S = { s e. ~P X | ( # ` s ) = ( P ^ N ) }
sylow1lem.m	|- .(+) = ( x e. X , y e. S |-> ran ( z e. y |-> ( x .+ z ) ) )
sylow1lem3.1	|- .~ = { <. x , y >. | ( { x , y } C_ S /\ E. g e. X ( g .(+) x ) = y ) }
sylow1lem4.b	|- ( ph -> B e. S )
sylow1lem4.h	|- H = { u e. X | ( u .(+) B ) = B }
sylow1lem5.l	|- ( ph -> ( P pCnt ( # ` [ B ] .~ ) ) <_ ( ( P pCnt ( # ` X ) ) - N ) )

Assertion

Ref	Expression
sylow1lem5	|- ( ph -> E. h e. (SubGrp ` G ) ( # ` h ) = ( P ^ N ) )

Distinct variable groups:   g, s, u, x, y, z, B   g, h, H, x, y   S, g, u, x, y, z   g, N   h, s, u, z, N, x, y   g, X, h, s, u, x, y, z   .+ , s, u, x, y, z   z, .~   .(+) , g, u, x, y, z   g, G, h, s, u, x, y, z   P, g, h, s, u, x, y, z   ph, u, x, y, z

Allowed substitution hints:   ph( g, h, s)   B( h)   .+ ( g, h)   .(+) ( h, s)   .~ ( x, y, u, g, h, s)   S( h, s)   H( z, u, s)

Proof of Theorem sylow1lem5

Step	Hyp	Ref	Expression
1		sylow1.x	. . . 4 |- X = ( Base ` G )
2		sylow1.g	. . . 4 |- ( ph -> G e. Grp )
3		sylow1.f	. . . 4 |- ( ph -> X e. Fin )
4		sylow1.p	. . . 4 |- ( ph -> P e. Prime )
5		sylow1.n	. . . 4 |- ( ph -> N e. NN0 )
6		sylow1.d	. . . 4 |- ( ph -> ( P ^ N ) || ( # ` X ) )
7		sylow1lem.a	. . . 4 |- .+ = ( +g ` G )
8		sylow1lem.s	. . . 4 |- S = { s e. ~P X | ( # ` s ) = ( P ^ N ) }
9		sylow1lem.m	. . . 4 |- .(+) = ( x e. X , y e. S |-> ran ( z e. y |-> ( x .+ z ) ) )
10	1, 2, 3, 4, 5, 6, 7, 8, 9	sylow1lem2 18014	. . 3 |- ( ph -> .(+) e. ( G GrpAct S ) )
11		sylow1lem4.b	. . 3 |- ( ph -> B e. S )
12		sylow1lem4.h	. . . 4 |- H = { u e. X | ( u .(+) B ) = B }
13	1, 12	gastacl 17742	. . 3 |- ( ( .(+) e. ( G GrpAct S ) /\ B e. S ) -> H e. (SubGrp ` G ) )
14	10, 11, 13	syl2anc 693	. 2 |- ( ph -> H e. (SubGrp ` G ) )
15		sylow1lem3.1	. . . 4 |- .~ = { <. x , y >. | ( { x , y } C_ S /\ E. g e. X ( g .(+) x ) = y ) }
16	1, 2, 3, 4, 5, 6, 7, 8, 9, 15, 11, 12	sylow1lem4 18016	. . 3 |- ( ph -> ( # ` H ) <_ ( P ^ N ) )
17		sylow1lem5.l	. . . . . . . 8 |- ( ph -> ( P pCnt ( # ` [ B ] .~ ) ) <_ ( ( P pCnt ( # ` X ) ) - N ) )
18	15, 1	gaorber 17741	. . . . . . . . . . . . . . . 16 |- ( .(+) e. ( G GrpAct S ) -> .~ Er S )
19	10, 18	syl 17	. . . . . . . . . . . . . . 15 |- ( ph -> .~ Er S )
20		erdm 7752	. . . . . . . . . . . . . . 15 |- ( .~ Er S -> dom .~ = S )
21	19, 20	syl 17	. . . . . . . . . . . . . 14 |- ( ph -> dom .~ = S )
22	11, 21	eleqtrrd 2704	. . . . . . . . . . . . 13 |- ( ph -> B e. dom .~ )
23		ecdmn0 7789	. . . . . . . . . . . . 13 |- ( B e. dom .~ <-> [ B ] .~ =/= (/) )
24	22, 23	sylib 208	. . . . . . . . . . . 12 |- ( ph -> [ B ] .~ =/= (/) )
25		pwfi 8261	. . . . . . . . . . . . . . . 16 |- ( X e. Fin <-> ~P X e. Fin )
26	3, 25	sylib 208	. . . . . . . . . . . . . . 15 |- ( ph -> ~P X e. Fin )
27		ssrab2 3687	. . . . . . . . . . . . . . . 16 |- { s e. ~P X | ( # ` s ) = ( P ^ N ) } C_ ~P X
28	8, 27	eqsstri 3635	. . . . . . . . . . . . . . 15 |- S C_ ~P X
29		ssfi 8180	. . . . . . . . . . . . . . 15 |- ( ( ~P X e. Fin /\ S C_ ~P X ) -> S e. Fin )
30	26, 28, 29	sylancl 694	. . . . . . . . . . . . . 14 |- ( ph -> S e. Fin )
31	19	ecss 7788	. . . . . . . . . . . . . 14 |- ( ph -> [ B ] .~ C_ S )
32		ssfi 8180	. . . . . . . . . . . . . 14 |- ( ( S e. Fin /\ [ B ] .~ C_ S ) -> [ B ] .~ e. Fin )
33	30, 31, 32	syl2anc 693	. . . . . . . . . . . . 13 |- ( ph -> [ B ] .~ e. Fin )
34		hashnncl 13157	. . . . . . . . . . . . 13 |- ( [ B ] .~ e. Fin -> ( ( # ` [ B ] .~ ) e. NN <-> [ B ] .~ =/= (/) ) )
35	33, 34	syl 17	. . . . . . . . . . . 12 |- ( ph -> ( ( # ` [ B ] .~ ) e. NN <-> [ B ] .~ =/= (/) ) )
36	24, 35	mpbird 247	. . . . . . . . . . 11 |- ( ph -> ( # ` [ B ] .~ ) e. NN )
37	4, 36	pccld 15555	. . . . . . . . . 10 |- ( ph -> ( P pCnt ( # ` [ B ] .~ ) ) e. NN0 )
38	37	nn0red 11352	. . . . . . . . 9 |- ( ph -> ( P pCnt ( # ` [ B ] .~ ) ) e. RR )
39	5	nn0red 11352	. . . . . . . . 9 |- ( ph -> N e. RR )
40	1	grpbn0 17451	. . . . . . . . . . . . 13 |- ( G e. Grp -> X =/= (/) )
41	2, 40	syl 17	. . . . . . . . . . . 12 |- ( ph -> X =/= (/) )
42		hashnncl 13157	. . . . . . . . . . . . 13 |- ( X e. Fin -> ( ( # ` X ) e. NN <-> X =/= (/) ) )
43	3, 42	syl 17	. . . . . . . . . . . 12 |- ( ph -> ( ( # ` X ) e. NN <-> X =/= (/) ) )
44	41, 43	mpbird 247	. . . . . . . . . . 11 |- ( ph -> ( # ` X ) e. NN )
45	4, 44	pccld 15555	. . . . . . . . . 10 |- ( ph -> ( P pCnt ( # ` X ) ) e. NN0 )
46	45	nn0red 11352	. . . . . . . . 9 |- ( ph -> ( P pCnt ( # ` X ) ) e. RR )
47		leaddsub 10504	. . . . . . . . 9 |- ( ( ( P pCnt ( # ` [ B ] .~ ) ) e. RR /\ N e. RR /\ ( P pCnt ( # ` X ) ) e. RR ) -> ( ( ( P pCnt ( # ` [ B ] .~ ) ) + N ) <_ ( P pCnt ( # ` X ) ) <-> ( P pCnt ( # ` [ B ] .~ ) ) <_ ( ( P pCnt ( # ` X ) ) - N ) ) )
48	38, 39, 46, 47	syl3anc 1326	. . . . . . . 8 |- ( ph -> ( ( ( P pCnt ( # ` [ B ] .~ ) ) + N ) <_ ( P pCnt ( # ` X ) ) <-> ( P pCnt ( # ` [ B ] .~ ) ) <_ ( ( P pCnt ( # ` X ) ) - N ) ) )
49	17, 48	mpbird 247	. . . . . . 7 |- ( ph -> ( ( P pCnt ( # ` [ B ] .~ ) ) + N ) <_ ( P pCnt ( # ` X ) ) )
50		eqid 2622	. . . . . . . . . . 11 |- ( G ~QG H ) = ( G ~QG H )
51	1, 12, 50, 15	orbsta2 17747	. . . . . . . . . 10 |- ( ( ( .(+) e. ( G GrpAct S ) /\ B e. S ) /\ X e. Fin ) -> ( # ` X ) = ( ( # ` [ B ] .~ ) x. ( # ` H ) ) )
52	10, 11, 3, 51	syl21anc 1325	. . . . . . . . 9 |- ( ph -> ( # ` X ) = ( ( # ` [ B ] .~ ) x. ( # ` H ) ) )
53	52	oveq2d 6666	. . . . . . . 8 |- ( ph -> ( P pCnt ( # ` X ) ) = ( P pCnt ( ( # ` [ B ] .~ ) x. ( # ` H ) ) ) )
54	36	nnzd 11481	. . . . . . . . 9 |- ( ph -> ( # ` [ B ] .~ ) e. ZZ )
55	36	nnne0d 11065	. . . . . . . . 9 |- ( ph -> ( # ` [ B ] .~ ) =/= 0 )
56		eqid 2622	. . . . . . . . . . . . . 14 |- ( 0g ` G ) = ( 0g ` G )
57	56	subg0cl 17602	. . . . . . . . . . . . 13 |- ( H e. (SubGrp ` G ) -> ( 0g ` G ) e. H )
58	14, 57	syl 17	. . . . . . . . . . . 12 |- ( ph -> ( 0g ` G ) e. H )
59		ne0i 3921	. . . . . . . . . . . 12 |- ( ( 0g ` G ) e. H -> H =/= (/) )
60	58, 59	syl 17	. . . . . . . . . . 11 |- ( ph -> H =/= (/) )
61		ssrab2 3687	. . . . . . . . . . . . . 14 |- { u e. X | ( u .(+) B ) = B } C_ X
62	12, 61	eqsstri 3635	. . . . . . . . . . . . 13 |- H C_ X
63		ssfi 8180	. . . . . . . . . . . . 13 |- ( ( X e. Fin /\ H C_ X ) -> H e. Fin )
64	3, 62, 63	sylancl 694	. . . . . . . . . . . 12 |- ( ph -> H e. Fin )
65		hashnncl 13157	. . . . . . . . . . . 12 |- ( H e. Fin -> ( ( # ` H ) e. NN <-> H =/= (/) ) )
66	64, 65	syl 17	. . . . . . . . . . 11 |- ( ph -> ( ( # ` H ) e. NN <-> H =/= (/) ) )
67	60, 66	mpbird 247	. . . . . . . . . 10 |- ( ph -> ( # ` H ) e. NN )
68	67	nnzd 11481	. . . . . . . . 9 |- ( ph -> ( # ` H ) e. ZZ )
69	67	nnne0d 11065	. . . . . . . . 9 |- ( ph -> ( # ` H ) =/= 0 )
70		pcmul 15556	. . . . . . . . 9 |- ( ( P e. Prime /\ ( ( # ` [ B ] .~ ) e. ZZ /\ ( # ` [ B ] .~ ) =/= 0 ) /\ ( ( # ` H ) e. ZZ /\ ( # ` H ) =/= 0 ) ) -> ( P pCnt ( ( # ` [ B ] .~ ) x. ( # ` H ) ) ) = ( ( P pCnt ( # ` [ B ] .~ ) ) + ( P pCnt ( # ` H ) ) ) )
71	4, 54, 55, 68, 69, 70	syl122anc 1335	. . . . . . . 8 |- ( ph -> ( P pCnt ( ( # ` [ B ] .~ ) x. ( # ` H ) ) ) = ( ( P pCnt ( # ` [ B ] .~ ) ) + ( P pCnt ( # ` H ) ) ) )
72	53, 71	eqtrd 2656	. . . . . . 7 |- ( ph -> ( P pCnt ( # ` X ) ) = ( ( P pCnt ( # ` [ B ] .~ ) ) + ( P pCnt ( # ` H ) ) ) )
73	49, 72	breqtrd 4679	. . . . . 6 |- ( ph -> ( ( P pCnt ( # ` [ B ] .~ ) ) + N ) <_ ( ( P pCnt ( # ` [ B ] .~ ) ) + ( P pCnt ( # ` H ) ) ) )
74	4, 67	pccld 15555	. . . . . . . 8 |- ( ph -> ( P pCnt ( # ` H ) ) e. NN0 )
75	74	nn0red 11352	. . . . . . 7 |- ( ph -> ( P pCnt ( # ` H ) ) e. RR )
76	39, 75, 38	leadd2d 10622	. . . . . 6 |- ( ph -> ( N <_ ( P pCnt ( # ` H ) ) <-> ( ( P pCnt ( # ` [ B ] .~ ) ) + N ) <_ ( ( P pCnt ( # ` [ B ] .~ ) ) + ( P pCnt ( # ` H ) ) ) ) )
77	73, 76	mpbird 247	. . . . 5 |- ( ph -> N <_ ( P pCnt ( # ` H ) ) )
78		pcdvdsb 15573	. . . . . 6 |- ( ( P e. Prime /\ ( # ` H ) e. ZZ /\ N e. NN0 ) -> ( N <_ ( P pCnt ( # ` H ) ) <-> ( P ^ N ) || ( # ` H ) ) )
79	4, 68, 5, 78	syl3anc 1326	. . . . 5 |- ( ph -> ( N <_ ( P pCnt ( # ` H ) ) <-> ( P ^ N ) || ( # ` H ) ) )
80	77, 79	mpbid 222	. . . 4 |- ( ph -> ( P ^ N ) || ( # ` H ) )
81		prmnn 15388	. . . . . . . 8 |- ( P e. Prime -> P e. NN )
82	4, 81	syl 17	. . . . . . 7 |- ( ph -> P e. NN )
83	82, 5	nnexpcld 13030	. . . . . 6 |- ( ph -> ( P ^ N ) e. NN )
84	83	nnzd 11481	. . . . 5 |- ( ph -> ( P ^ N ) e. ZZ )
85		dvdsle 15032	. . . . 5 |- ( ( ( P ^ N ) e. ZZ /\ ( # ` H ) e. NN ) -> ( ( P ^ N ) || ( # ` H ) -> ( P ^ N ) <_ ( # ` H ) ) )
86	84, 67, 85	syl2anc 693	. . . 4 |- ( ph -> ( ( P ^ N ) || ( # ` H ) -> ( P ^ N ) <_ ( # ` H ) ) )
87	80, 86	mpd 15	. . 3 |- ( ph -> ( P ^ N ) <_ ( # ` H ) )
88		hashcl 13147	. . . . . 6 |- ( H e. Fin -> ( # ` H ) e. NN0 )
89	64, 88	syl 17	. . . . 5 |- ( ph -> ( # ` H ) e. NN0 )
90	89	nn0red 11352	. . . 4 |- ( ph -> ( # ` H ) e. RR )
91	83	nnred 11035	. . . 4 |- ( ph -> ( P ^ N ) e. RR )
92	90, 91	letri3d 10179	. . 3 |- ( ph -> ( ( # ` H ) = ( P ^ N ) <-> ( ( # ` H ) <_ ( P ^ N ) /\ ( P ^ N ) <_ ( # ` H ) ) ) )
93	16, 87, 92	mpbir2and 957	. 2 |- ( ph -> ( # ` H ) = ( P ^ N ) )
94		fveq2 6191	. . . 4 |- ( h = H -> ( # ` h ) = ( # ` H ) )
95	94	eqeq1d 2624	. . 3 |- ( h = H -> ( ( # ` h ) = ( P ^ N ) <-> ( # ` H ) = ( P ^ N ) ) )
96	95	rspcev 3309	. 2 |- ( ( H e. (SubGrp ` G ) /\ ( # ` H ) = ( P ^ N ) ) -> E. h e. (SubGrp ` G ) ( # ` h ) = ( P ^ N ) )
97	14, 93, 96	syl2anc 693	1 |- ( ph -> E. h e. (SubGrp ` G ) ( # ` h ) = ( P ^ N ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   {crab 2916   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   {cpr 4179   class class class wbr 4653   {copab 4712   |-> cmpt 4729   dom cdm 5114   ran crn 5115   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   Er wer 7739   [cec 7740   Fincfn 7955   RRcr 9935   0cc0 9936   + caddc 9939   x. cmul 9941   <_ cle 10075   - cmin 10266   NNcn 11020   NN0cn0 11292   ZZcz 11377   ^cexp 12860   #chash 13117   || cdvds 14983   Primecprime 15385   pCnt cpc 15541   Basecbs 15857   +g cplusg 15941   0gc0g 16100   Grpcgrp 17422  SubGrpcsubg 17588   ~_QG cqg 17590   GrpAct cga 17722

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-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-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-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-grp 17425  df-minusg 17426  df-subg 17591  df-eqg 17593  df-ga 17723

This theorem is referenced by:  sylow1  18018