Theorem isslw 18023

Description: The property of being a Sylow subgroup. A Sylow P-subgroup is a P-group which has no proper supersets that are also P-groups. (Contributed by Mario Carneiro, 16-Jan-2015.)

Assertion

Ref	Expression
isslw	|- ( H e. ( P pSyl G ) <-> ( P e. Prime /\ H e. (SubGrp ` G ) /\ A. k e. (SubGrp ` G ) ( ( H C_ k /\ P pGrp ( G ↾s k ) ) <-> H = k ) ) )

Distinct variable groups:   k, G   k, H   P, k

Proof of Theorem isslw

Dummy variables g h p are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-slw 17951	. . 3 |- pSyl = ( p e. Prime , g e. Grp |-> { h e. (SubGrp ` g ) | A. k e. (SubGrp ` g ) ( ( h C_ k /\ p pGrp ( g ↾s k ) ) <-> h = k ) } )
2	1	elmpt2cl 6876	. 2 |- ( H e. ( P pSyl G ) -> ( P e. Prime /\ G e. Grp ) )
3		simp1 1061	. . 3 |- ( ( P e. Prime /\ H e. (SubGrp ` G ) /\ A. k e. (SubGrp ` G ) ( ( H C_ k /\ P pGrp ( G ↾s k ) ) <-> H = k ) ) -> P e. Prime )
4		subgrcl 17599	. . . 4 |- ( H e. (SubGrp ` G ) -> G e. Grp )
5	4	3ad2ant2 1083	. . 3 |- ( ( P e. Prime /\ H e. (SubGrp ` G ) /\ A. k e. (SubGrp ` G ) ( ( H C_ k /\ P pGrp ( G ↾s k ) ) <-> H = k ) ) -> G e. Grp )
6	3, 5	jca 554	. 2 |- ( ( P e. Prime /\ H e. (SubGrp ` G ) /\ A. k e. (SubGrp ` G ) ( ( H C_ k /\ P pGrp ( G ↾s k ) ) <-> H = k ) ) -> ( P e. Prime /\ G e. Grp ) )
7		simpr 477	. . . . . . . . 9 |- ( ( p = P /\ g = G ) -> g = G )
8	7	fveq2d 6195	. . . . . . . 8 |- ( ( p = P /\ g = G ) -> (SubGrp ` g ) = (SubGrp ` G ) )
9		simpl 473	. . . . . . . . . . . 12 |- ( ( p = P /\ g = G ) -> p = P )
10	7	oveq1d 6665	. . . . . . . . . . . 12 |- ( ( p = P /\ g = G ) -> ( g ↾s k ) = ( G ↾s k ) )
11	9, 10	breq12d 4666	. . . . . . . . . . 11 |- ( ( p = P /\ g = G ) -> ( p pGrp ( g ↾s k ) <-> P pGrp ( G ↾s k ) ) )
12	11	anbi2d 740	. . . . . . . . . 10 |- ( ( p = P /\ g = G ) -> ( ( h C_ k /\ p pGrp ( g ↾s k ) ) <-> ( h C_ k /\ P pGrp ( G ↾s k ) ) ) )
13	12	bibi1d 333	. . . . . . . . 9 |- ( ( p = P /\ g = G ) -> ( ( ( h C_ k /\ p pGrp ( g ↾s k ) ) <-> h = k ) <-> ( ( h C_ k /\ P pGrp ( G ↾s k ) ) <-> h = k ) ) )
14	8, 13	raleqbidv 3152	. . . . . . . 8 |- ( ( p = P /\ g = G ) -> ( A. k e. (SubGrp ` g ) ( ( h C_ k /\ p pGrp ( g ↾s k ) ) <-> h = k ) <-> A. k e. (SubGrp ` G ) ( ( h C_ k /\ P pGrp ( G ↾s k ) ) <-> h = k ) ) )
15	8, 14	rabeqbidv 3195	. . . . . . 7 |- ( ( p = P /\ g = G ) -> { h e. (SubGrp ` g ) | A. k e. (SubGrp ` g ) ( ( h C_ k /\ p pGrp ( g ↾s k ) ) <-> h = k ) } = { h e. (SubGrp ` G ) | A. k e. (SubGrp ` G ) ( ( h C_ k /\ P pGrp ( G ↾s k ) ) <-> h = k ) } )
16		fvex 6201	. . . . . . . 8 |- (SubGrp ` G ) e. _V
17	16	rabex 4813	. . . . . . 7 |- { h e. (SubGrp ` G ) | A. k e. (SubGrp ` G ) ( ( h C_ k /\ P pGrp ( G ↾s k ) ) <-> h = k ) } e. _V
18	15, 1, 17	ovmpt2a 6791	. . . . . 6 |- ( ( P e. Prime /\ G e. Grp ) -> ( P pSyl G ) = { h e. (SubGrp ` G ) | A. k e. (SubGrp ` G ) ( ( h C_ k /\ P pGrp ( G ↾s k ) ) <-> h = k ) } )
19	18	eleq2d 2687	. . . . 5 |- ( ( P e. Prime /\ G e. Grp ) -> ( H e. ( P pSyl G ) <-> H e. { h e. (SubGrp ` G ) | A. k e. (SubGrp ` G ) ( ( h C_ k /\ P pGrp ( G ↾s k ) ) <-> h = k ) } ) )
20		sseq1 3626	. . . . . . . . 9 |- ( h = H -> ( h C_ k <-> H C_ k ) )
21	20	anbi1d 741	. . . . . . . 8 |- ( h = H -> ( ( h C_ k /\ P pGrp ( G ↾s k ) ) <-> ( H C_ k /\ P pGrp ( G ↾s k ) ) ) )
22		eqeq1 2626	. . . . . . . 8 |- ( h = H -> ( h = k <-> H = k ) )
23	21, 22	bibi12d 335	. . . . . . 7 |- ( h = H -> ( ( ( h C_ k /\ P pGrp ( G ↾s k ) ) <-> h = k ) <-> ( ( H C_ k /\ P pGrp ( G ↾s k ) ) <-> H = k ) ) )
24	23	ralbidv 2986	. . . . . 6 |- ( h = H -> ( A. k e. (SubGrp ` G ) ( ( h C_ k /\ P pGrp ( G ↾s k ) ) <-> h = k ) <-> A. k e. (SubGrp ` G ) ( ( H C_ k /\ P pGrp ( G ↾s k ) ) <-> H = k ) ) )
25	24	elrab 3363	. . . . 5 |- ( H e. { h e. (SubGrp ` G ) | A. k e. (SubGrp ` G ) ( ( h C_ k /\ P pGrp ( G ↾s k ) ) <-> h = k ) } <-> ( H e. (SubGrp ` G ) /\ A. k e. (SubGrp ` G ) ( ( H C_ k /\ P pGrp ( G ↾s k ) ) <-> H = k ) ) )
26	19, 25	syl6bb 276	. . . 4 |- ( ( P e. Prime /\ G e. Grp ) -> ( H e. ( P pSyl G ) <-> ( H e. (SubGrp ` G ) /\ A. k e. (SubGrp ` G ) ( ( H C_ k /\ P pGrp ( G ↾s k ) ) <-> H = k ) ) ) )
27		simpl 473	. . . . 5 |- ( ( P e. Prime /\ G e. Grp ) -> P e. Prime )
28	27	biantrurd 529	. . . 4 |- ( ( P e. Prime /\ G e. Grp ) -> ( ( H e. (SubGrp ` G ) /\ A. k e. (SubGrp ` G ) ( ( H C_ k /\ P pGrp ( G ↾s k ) ) <-> H = k ) ) <-> ( P e. Prime /\ ( H e. (SubGrp ` G ) /\ A. k e. (SubGrp ` G ) ( ( H C_ k /\ P pGrp ( G ↾s k ) ) <-> H = k ) ) ) ) )
29	26, 28	bitrd 268	. . 3 |- ( ( P e. Prime /\ G e. Grp ) -> ( H e. ( P pSyl G ) <-> ( P e. Prime /\ ( H e. (SubGrp ` G ) /\ A. k e. (SubGrp ` G ) ( ( H C_ k /\ P pGrp ( G ↾s k ) ) <-> H = k ) ) ) ) )
30		3anass 1042	. . 3 |- ( ( P e. Prime /\ H e. (SubGrp ` G ) /\ A. k e. (SubGrp ` G ) ( ( H C_ k /\ P pGrp ( G ↾s k ) ) <-> H = k ) ) <-> ( P e. Prime /\ ( H e. (SubGrp ` G ) /\ A. k e. (SubGrp ` G ) ( ( H C_ k /\ P pGrp ( G ↾s k ) ) <-> H = k ) ) ) )
31	29, 30	syl6bbr 278	. 2 |- ( ( P e. Prime /\ G e. Grp ) -> ( H e. ( P pSyl G ) <-> ( P e. Prime /\ H e. (SubGrp ` G ) /\ A. k e. (SubGrp ` G ) ( ( H C_ k /\ P pGrp ( G ↾s k ) ) <-> H = k ) ) ) )
32	2, 6, 31	pm5.21nii 368	1 |- ( H e. ( P pSyl G ) <-> ( P e. Prime /\ H e. (SubGrp ` G ) /\ A. k e. (SubGrp ` G ) ( ( H C_ k /\ P pGrp ( G ↾s k ) ) <-> H = k ) ) )

Syntax hints:   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   C_ wss 3574   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Primecprime 15385   ↾_s cress 15858   Grpcgrp 17422  SubGrpcsubg 17588   pGrp cpgp 17946   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-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  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-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-subg 17591  df-slw 17951

This theorem is referenced by:  slwprm  18024  slwsubg  18025  slwispgp  18026  pgpssslw  18029  subgslw  18031  fislw  18040