Theorem subgslw 18031

Description: A Sylow subgroup that is contained in a larger subgroup is also Sylow with respect to the subgroup. (The converse need not be true.) (Contributed by Mario Carneiro, 19-Jan-2015.)

Hypothesis

Ref	Expression
subgslw.1	|- H = ( G ↾s S )

Assertion

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

Proof of Theorem subgslw

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		slwprm 18024	. . 3 |- ( K e. ( P pSyl G ) -> P e. Prime )
2	1	3ad2ant2 1083	. 2 |- ( ( S e. (SubGrp ` G ) /\ K e. ( P pSyl G ) /\ K C_ S ) -> P e. Prime )
3		slwsubg 18025	. . . 4 |- ( K e. ( P pSyl G ) -> K e. (SubGrp ` G ) )
4	3	3ad2ant2 1083	. . 3 |- ( ( S e. (SubGrp ` G ) /\ K e. ( P pSyl G ) /\ K C_ S ) -> K e. (SubGrp ` G ) )
5		simp3 1063	. . 3 |- ( ( S e. (SubGrp ` G ) /\ K e. ( P pSyl G ) /\ K C_ S ) -> K C_ S )
6		subgslw.1	. . . . 5 |- H = ( G ↾s S )
7	6	subsubg 17617	. . . 4 |- ( S e. (SubGrp ` G ) -> ( K e. (SubGrp ` H ) <-> ( K e. (SubGrp ` G ) /\ K C_ S ) ) )
8	7	3ad2ant1 1082	. . 3 |- ( ( S e. (SubGrp ` G ) /\ K e. ( P pSyl G ) /\ K C_ S ) -> ( K e. (SubGrp ` H ) <-> ( K e. (SubGrp ` G ) /\ K C_ S ) ) )
9	4, 5, 8	mpbir2and 957	. 2 |- ( ( S e. (SubGrp ` G ) /\ K e. ( P pSyl G ) /\ K C_ S ) -> K e. (SubGrp ` H ) )
10	6	oveq1i 6660	. . . . . . 7 |- ( H ↾s x ) = ( ( G ↾s S ) ↾s x )
11		simpl1 1064	. . . . . . . 8 |- ( ( ( S e. (SubGrp ` G ) /\ K e. ( P pSyl G ) /\ K C_ S ) /\ x e. (SubGrp ` H ) ) -> S e. (SubGrp ` G ) )
12	6	subsubg 17617	. . . . . . . . . 10 |- ( S e. (SubGrp ` G ) -> ( x e. (SubGrp ` H ) <-> ( x e. (SubGrp ` G ) /\ x C_ S ) ) )
13	12	3ad2ant1 1082	. . . . . . . . 9 |- ( ( S e. (SubGrp ` G ) /\ K e. ( P pSyl G ) /\ K C_ S ) -> ( x e. (SubGrp ` H ) <-> ( x e. (SubGrp ` G ) /\ x C_ S ) ) )
14	13	simplbda 654	. . . . . . . 8 |- ( ( ( S e. (SubGrp ` G ) /\ K e. ( P pSyl G ) /\ K C_ S ) /\ x e. (SubGrp ` H ) ) -> x C_ S )
15		ressabs 15939	. . . . . . . 8 |- ( ( S e. (SubGrp ` G ) /\ x C_ S ) -> ( ( G ↾s S ) ↾s x ) = ( G ↾s x ) )
16	11, 14, 15	syl2anc 693	. . . . . . 7 |- ( ( ( S e. (SubGrp ` G ) /\ K e. ( P pSyl G ) /\ K C_ S ) /\ x e. (SubGrp ` H ) ) -> ( ( G ↾s S ) ↾s x ) = ( G ↾s x ) )
17	10, 16	syl5eq 2668	. . . . . 6 |- ( ( ( S e. (SubGrp ` G ) /\ K e. ( P pSyl G ) /\ K C_ S ) /\ x e. (SubGrp ` H ) ) -> ( H ↾s x ) = ( G ↾s x ) )
18	17	breq2d 4665	. . . . 5 |- ( ( ( S e. (SubGrp ` G ) /\ K e. ( P pSyl G ) /\ K C_ S ) /\ x e. (SubGrp ` H ) ) -> ( P pGrp ( H ↾s x ) <-> P pGrp ( G ↾s x ) ) )
19	18	anbi2d 740	. . . 4 |- ( ( ( S e. (SubGrp ` G ) /\ K e. ( P pSyl G ) /\ K C_ S ) /\ x e. (SubGrp ` H ) ) -> ( ( K C_ x /\ P pGrp ( H ↾s x ) ) <-> ( K C_ x /\ P pGrp ( G ↾s x ) ) ) )
20		simpl2 1065	. . . . 5 |- ( ( ( S e. (SubGrp ` G ) /\ K e. ( P pSyl G ) /\ K C_ S ) /\ x e. (SubGrp ` H ) ) -> K e. ( P pSyl G ) )
21	13	simprbda 653	. . . . 5 |- ( ( ( S e. (SubGrp ` G ) /\ K e. ( P pSyl G ) /\ K C_ S ) /\ x e. (SubGrp ` H ) ) -> x e. (SubGrp ` G ) )
22		eqid 2622	. . . . . 6 |- ( G ↾s x ) = ( G ↾s x )
23	22	slwispgp 18026	. . . . 5 |- ( ( K e. ( P pSyl G ) /\ x e. (SubGrp ` G ) ) -> ( ( K C_ x /\ P pGrp ( G ↾s x ) ) <-> K = x ) )
24	20, 21, 23	syl2anc 693	. . . 4 |- ( ( ( S e. (SubGrp ` G ) /\ K e. ( P pSyl G ) /\ K C_ S ) /\ x e. (SubGrp ` H ) ) -> ( ( K C_ x /\ P pGrp ( G ↾s x ) ) <-> K = x ) )
25	19, 24	bitrd 268	. . 3 |- ( ( ( S e. (SubGrp ` G ) /\ K e. ( P pSyl G ) /\ K C_ S ) /\ x e. (SubGrp ` H ) ) -> ( ( K C_ x /\ P pGrp ( H ↾s x ) ) <-> K = x ) )
26	25	ralrimiva 2966	. 2 |- ( ( S e. (SubGrp ` G ) /\ K e. ( P pSyl G ) /\ K C_ S ) -> A. x e. (SubGrp ` H ) ( ( K C_ x /\ P pGrp ( H ↾s x ) ) <-> K = x ) )
27		isslw 18023	. 2 |- ( K e. ( P pSyl H ) <-> ( P e. Prime /\ K e. (SubGrp ` H ) /\ A. x e. (SubGrp ` H ) ( ( K C_ x /\ P pGrp ( H ↾s x ) ) <-> K = x ) ) )
28	2, 9, 26, 27	syl3anbrc 1246	1 |- ( ( S e. (SubGrp ` G ) /\ K e. ( P pSyl G ) /\ K C_ S ) -> K e. ( P pSyl H ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = 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  ax-un 6949  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-i2m1 10004  ax-1ne0 10005  ax-rrecex 10008  ax-cnre 10009

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-reu 2919  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-iun 4522  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-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-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-nn 11021  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-subg 17591  df-slw 17951

This theorem is referenced by:  sylow3lem6  18047