Theorem sylow3 17363

Description: Sylow's third theorem. The number of Sylow subgroups is a divisor of | G | / d, where d is the common order of a Sylow subgroup, and is equivalent to 1 mod P. This is part of Metamath 100 proof #72. (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 )
sylow3.n	|- N = ( # ` ( P pSyl G ) )

Assertion

Ref	Expression
sylow3	|- ( ph -> ( N || ( ( # ` X ) / ( P ^ ( P pCnt ( # ` X ) ) ) ) /\ ( N mod P ) = 1 ) )

Proof of Theorem sylow3

Dummy variables a b c u x y z s k are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sylow3.g	. . . 4 |- ( ph -> G e. Grp )
2		sylow3.xf	. . . 4 |- ( ph -> X e. Fin )
3		sylow3.p	. . . 4 |- ( ph -> P e. Prime )
4		sylow3.x	. . . . 5 |- X = ( Base ` G )
5	4	slwn0 17345	. . . 4 |- ( ( G e. Grp /\ X e. Fin /\ P e. Prime ) -> ( P pSyl G ) =/= (/) )
6	1, 2, 3, 5	syl3anc 1292	. . 3 |- ( ph -> ( P pSyl G ) =/= (/) )
7		n0 3732	. . 3 |- ( ( P pSyl G ) =/= (/) <-> E. k k e. ( P pSyl G ) )
8	6, 7	sylib 201	. 2 |- ( ph -> E. k k e. ( P pSyl G ) )
9		sylow3.n	. . . 4 |- N = ( # ` ( P pSyl G ) )
10	1	adantr 472	. . . . 5 |- ( ( ph /\ k e. ( P pSyl G ) ) -> G e. Grp )
11	2	adantr 472	. . . . 5 |- ( ( ph /\ k e. ( P pSyl G ) ) -> X e. Fin )
12	3	adantr 472	. . . . 5 |- ( ( ph /\ k e. ( P pSyl G ) ) -> P e. Prime )
13		eqid 2471	. . . . 5 |- ( +g ` G ) = ( +g ` G )
14		eqid 2471	. . . . 5 |- ( -g ` G ) = ( -g ` G )
15		oveq2 6316	. . . . . . . . . 10 |- ( c = z -> ( a ( +g ` G ) c ) = ( a ( +g ` G ) z ) )
16	15	oveq1d 6323	. . . . . . . . 9 |- ( c = z -> ( ( a ( +g ` G ) c ) ( -g ` G ) a ) = ( ( a ( +g ` G ) z ) ( -g ` G ) a ) )
17	16	cbvmptv 4488	. . . . . . . 8 |- ( c e. b |-> ( ( a ( +g ` G ) c ) ( -g ` G ) a ) ) = ( z e. b |-> ( ( a ( +g ` G ) z ) ( -g ` G ) a ) )
18		oveq1 6315	. . . . . . . . . 10 |- ( a = x -> ( a ( +g ` G ) z ) = ( x ( +g ` G ) z ) )
19		id 22	. . . . . . . . . 10 |- ( a = x -> a = x )
20	18, 19	oveq12d 6326	. . . . . . . . 9 |- ( a = x -> ( ( a ( +g ` G ) z ) ( -g ` G ) a ) = ( ( x ( +g ` G ) z ) ( -g ` G ) x ) )
21	20	mpteq2dv 4483	. . . . . . . 8 |- ( a = x -> ( z e. b |-> ( ( a ( +g ` G ) z ) ( -g ` G ) a ) ) = ( z e. b |-> ( ( x ( +g ` G ) z ) ( -g ` G ) x ) ) )
22	17, 21	syl5eq 2517	. . . . . . 7 |- ( a = x -> ( c e. b |-> ( ( a ( +g ` G ) c ) ( -g ` G ) a ) ) = ( z e. b |-> ( ( x ( +g ` G ) z ) ( -g ` G ) x ) ) )
23	22	rneqd 5068	. . . . . 6 |- ( a = x -> ran ( c e. b |-> ( ( a ( +g ` G ) c ) ( -g ` G ) a ) ) = ran ( z e. b |-> ( ( x ( +g ` G ) z ) ( -g ` G ) x ) ) )
24		mpteq1 4476	. . . . . . 7 |- ( b = y -> ( z e. b |-> ( ( x ( +g ` G ) z ) ( -g ` G ) x ) ) = ( z e. y |-> ( ( x ( +g ` G ) z ) ( -g ` G ) x ) ) )
25	24	rneqd 5068	. . . . . 6 |- ( b = y -> ran ( z e. b |-> ( ( x ( +g ` G ) z ) ( -g ` G ) x ) ) = ran ( z e. y |-> ( ( x ( +g ` G ) z ) ( -g ` G ) x ) ) )
26	23, 25	cbvmpt2v 6390	. . . . 5 |- ( a e. X , b e. ( P pSyl G ) |-> ran ( c e. b |-> ( ( a ( +g ` G ) c ) ( -g ` G ) a ) ) ) = ( x e. X , y e. ( P pSyl G ) |-> ran ( z e. y |-> ( ( x ( +g ` G ) z ) ( -g ` G ) x ) ) )
27		simpr 468	. . . . 5 |- ( ( ph /\ k e. ( P pSyl G ) ) -> k e. ( P pSyl G ) )
28		eqid 2471	. . . . 5 |- { u e. X | ( u ( a e. X , b e. ( P pSyl G ) |-> ran ( c e. b |-> ( ( a ( +g ` G ) c ) ( -g ` G ) a ) ) ) k ) = k } = { u e. X | ( u ( a e. X , b e. ( P pSyl G ) |-> ran ( c e. b |-> ( ( a ( +g ` G ) c ) ( -g ` G ) a ) ) ) k ) = k }
29		eqid 2471	. . . . 5 |- { x e. X | A. y e. X ( ( x ( +g ` G ) y ) e. k <-> ( y ( +g ` G ) x ) e. k ) } = { x e. X | A. y e. X ( ( x ( +g ` G ) y ) e. k <-> ( y ( +g ` G ) x ) e. k ) }
30	4, 10, 11, 12, 13, 14, 26, 27, 28, 29	sylow3lem4 17360	. . . 4 |- ( ( ph /\ k e. ( P pSyl G ) ) -> ( # ` ( P pSyl G ) ) || ( ( # ` X ) / ( P ^ ( P pCnt ( # ` X ) ) ) ) )
31	9, 30	syl5eqbr 4429	. . 3 |- ( ( ph /\ k e. ( P pSyl G ) ) -> N || ( ( # ` X ) / ( P ^ ( P pCnt ( # ` X ) ) ) ) )
32	9	oveq1i 6318	. . . 4 |- ( N mod P ) = ( ( # ` ( P pSyl G ) ) mod P )
33	23, 25	cbvmpt2v 6390	. . . . 5 |- ( a e. k , b e. ( P pSyl G ) |-> ran ( c e. b |-> ( ( a ( +g ` G ) c ) ( -g ` G ) a ) ) ) = ( x e. k , y e. ( P pSyl G ) |-> ran ( z e. y |-> ( ( x ( +g ` G ) z ) ( -g ` G ) x ) ) )
34		eqid 2471	. . . . 5 |- { x e. X | A. y e. X ( ( x ( +g ` G ) y ) e. s <-> ( y ( +g ` G ) x ) e. s ) } = { x e. X | A. y e. X ( ( x ( +g ` G ) y ) e. s <-> ( y ( +g ` G ) x ) e. s ) }
35	4, 10, 11, 12, 13, 14, 27, 33, 34	sylow3lem6 17362	. . . 4 |- ( ( ph /\ k e. ( P pSyl G ) ) -> ( ( # ` ( P pSyl G ) ) mod P ) = 1 )
36	32, 35	syl5eq 2517	. . 3 |- ( ( ph /\ k e. ( P pSyl G ) ) -> ( N mod P ) = 1 )
37	31, 36	jca 541	. 2 |- ( ( ph /\ k e. ( P pSyl G ) ) -> ( N || ( ( # ` X ) / ( P ^ ( P pCnt ( # ` X ) ) ) ) /\ ( N mod P ) = 1 ) )
38	8, 37	exlimddv 1789	1 |- ( ph -> ( N || ( ( # ` X ) / ( P ^ ( P pCnt ( # ` X ) ) ) ) /\ ( N mod P ) = 1 ) )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 376   = wceq 1452   E.wex 1671   e. wcel 1904   =/= wne 2641   A.wral 2756   {crab 2760   (/)c0 3722   class class class wbr 4395   |-> cmpt 4454   ran crn 4840   ` cfv 5589  (class class class)co 6308   |-> cmpt2 6310   Fincfn 7587   1c1 9558   / cdiv 10291   mod cmo 12129   ^cexp 12310   #chash 12553   || cdvds 14382   Primecprime 14701   pCnt cpc 14865   Basecbs 15199   +g cplusg 15268   Grpcgrp 16747   -gcsg 16749   pSyl cslw 17249

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-disj 4367  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-omul 7205  df-er 7381  df-ec 7383  df-qs 7387  df-map 7492  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-acn 8394  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-n0 10894  df-z 10962  df-uz 11183  df-q 11288  df-rp 11326  df-fz 11811  df-fzo 11943  df-fl 12061  df-mod 12130  df-seq 12252  df-exp 12311  df-fac 12498  df-bc 12526  df-hash 12554  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-clim 13629  df-sum 13830  df-dvds 14383  df-gcd 14548  df-prm 14702  df-pc 14866  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-0g 15418  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-submnd 16661  df-grp 16751  df-minusg 16752  df-sbg 16753  df-mulg 16754  df-subg 16892  df-nsg 16893  df-eqg 16894  df-ghm 16959  df-ga 17022  df-od 17250  df-pgp 17254  df-slw 17256

This theorem is referenced by: (None)