Theorem slwispgp 18026

Description: Defining property of a Sylow P-subgroup. (Contributed by Mario Carneiro, 16-Jan-2015.)

Hypothesis

Ref	Expression
slwispgp.1	|- S = ( G ↾s K )

Assertion

Ref	Expression
slwispgp	|- ( ( H e. ( P pSyl G ) /\ K e. (SubGrp ` G ) ) -> ( ( H C_ K /\ P pGrp S ) <-> H = K ) )

Proof of Theorem slwispgp

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		isslw 18023	. . 3 |- ( 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 ) ) )
2	1	simp3bi 1078	. 2 |- ( H e. ( P pSyl G ) -> A. k e. (SubGrp ` G ) ( ( H C_ k /\ P pGrp ( G ↾s k ) ) <-> H = k ) )
3		sseq2 3627	. . . . 5 |- ( k = K -> ( H C_ k <-> H C_ K ) )
4		oveq2 6658	. . . . . . 7 |- ( k = K -> ( G ↾s k ) = ( G ↾s K ) )
5		slwispgp.1	. . . . . . 7 |- S = ( G ↾s K )
6	4, 5	syl6eqr 2674	. . . . . 6 |- ( k = K -> ( G ↾s k ) = S )
7	6	breq2d 4665	. . . . 5 |- ( k = K -> ( P pGrp ( G ↾s k ) <-> P pGrp S ) )
8	3, 7	anbi12d 747	. . . 4 |- ( k = K -> ( ( H C_ k /\ P pGrp ( G ↾s k ) ) <-> ( H C_ K /\ P pGrp S ) ) )
9		eqeq2 2633	. . . 4 |- ( k = K -> ( H = k <-> H = K ) )
10	8, 9	bibi12d 335	. . 3 |- ( k = K -> ( ( ( H C_ k /\ P pGrp ( G ↾s k ) ) <-> H = k ) <-> ( ( H C_ K /\ P pGrp S ) <-> H = K ) ) )
11	10	rspccva 3308	. 2 |- ( ( A. k e. (SubGrp ` G ) ( ( H C_ k /\ P pGrp ( G ↾s k ) ) <-> H = k ) /\ K e. (SubGrp ` G ) ) -> ( ( H C_ K /\ P pGrp S ) <-> H = K ) )
12	2, 11	sylan 488	1 |- ( ( H e. ( P pSyl G ) /\ K e. (SubGrp ` G ) ) -> ( ( H C_ K /\ P pGrp S ) <-> H = K ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   C_ wss 3574   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Primecprime 15385   ↾_s cress 15858  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:  slwpss  18027  slwpgp  18028  subgslw  18031  slwhash  18039