Theorem subsubg 13327

Description: A subgroup of a subgroup is a subgroup. (Contributed by Mario Carneiro, 19-Jan-2015.)

Hypothesis

Ref	Expression
subsubg.h	|- H = ( G ↾s S )

Assertion

Ref	Expression
subsubg	|- ( S e. (SubGrp ` G ) -> ( A e. (SubGrp ` H ) <-> ( A e. (SubGrp ` G ) /\ A C_ S ) ) )

Proof of Theorem subsubg

Step	Hyp	Ref	Expression
1		subgrcl 13309	. . . . 5 |- ( S e. (SubGrp ` G ) -> G e. Grp )
2	1	adantr 276	. . . 4 |- ( ( S e. (SubGrp ` G ) /\ A e. (SubGrp ` H ) ) -> G e. Grp )
3		eqid 2196	. . . . . . . 8 |- ( Base ` H ) = ( Base ` H )
4	3	subgss 13304	. . . . . . 7 |- ( A e. (SubGrp ` H ) -> A C_ ( Base ` H ) )
5	4	adantl 277	. . . . . 6 |- ( ( S e. (SubGrp ` G ) /\ A e. (SubGrp ` H ) ) -> A C_ ( Base ` H ) )
6		subsubg.h	. . . . . . . 8 |- H = ( G ↾s S )
7	6	subgbas 13308	. . . . . . 7 |- ( S e. (SubGrp ` G ) -> S = ( Base ` H ) )
8	7	adantr 276	. . . . . 6 |- ( ( S e. (SubGrp ` G ) /\ A e. (SubGrp ` H ) ) -> S = ( Base ` H ) )
9	5, 8	sseqtrrd 3222	. . . . 5 |- ( ( S e. (SubGrp ` G ) /\ A e. (SubGrp ` H ) ) -> A C_ S )
10		eqid 2196	. . . . . . 7 |- ( Base ` G ) = ( Base ` G )
11	10	subgss 13304	. . . . . 6 |- ( S e. (SubGrp ` G ) -> S C_ ( Base ` G ) )
12	11	adantr 276	. . . . 5 |- ( ( S e. (SubGrp ` G ) /\ A e. (SubGrp ` H ) ) -> S C_ ( Base ` G ) )
13	9, 12	sstrd 3193	. . . 4 |- ( ( S e. (SubGrp ` G ) /\ A e. (SubGrp ` H ) ) -> A C_ ( Base ` G ) )
14	6	oveq1i 5932	. . . . . . 7 |- ( H ↾s A ) = ( ( G ↾s S ) ↾s A )
15	1	adantr 276	. . . . . . . 8 |- ( ( S e. (SubGrp ` G ) /\ A C_ S ) -> G e. Grp )
16		ressabsg 12754	. . . . . . . 8 |- ( ( S e. (SubGrp ` G ) /\ A C_ S /\ G e. Grp ) -> ( ( G ↾s S ) ↾s A ) = ( G ↾s A ) )
17	15, 16	mpd3an3 1349	. . . . . . 7 |- ( ( S e. (SubGrp ` G ) /\ A C_ S ) -> ( ( G ↾s S ) ↾s A ) = ( G ↾s A ) )
18	14, 17	eqtrid 2241	. . . . . 6 |- ( ( S e. (SubGrp ` G ) /\ A C_ S ) -> ( H ↾s A ) = ( G ↾s A ) )
19	9, 18	syldan 282	. . . . 5 |- ( ( S e. (SubGrp ` G ) /\ A e. (SubGrp ` H ) ) -> ( H ↾s A ) = ( G ↾s A ) )
20		eqid 2196	. . . . . . 7 |- ( H ↾s A ) = ( H ↾s A )
21	20	subggrp 13307	. . . . . 6 |- ( A e. (SubGrp ` H ) -> ( H ↾s A ) e. Grp )
22	21	adantl 277	. . . . 5 |- ( ( S e. (SubGrp ` G ) /\ A e. (SubGrp ` H ) ) -> ( H ↾s A ) e. Grp )
23	19, 22	eqeltrrd 2274	. . . 4 |- ( ( S e. (SubGrp ` G ) /\ A e. (SubGrp ` H ) ) -> ( G ↾s A ) e. Grp )
24	10	issubg 13303	. . . 4 |- ( A e. (SubGrp ` G ) <-> ( G e. Grp /\ A C_ ( Base ` G ) /\ ( G ↾s A ) e. Grp ) )
25	2, 13, 23, 24	syl3anbrc 1183	. . 3 |- ( ( S e. (SubGrp ` G ) /\ A e. (SubGrp ` H ) ) -> A e. (SubGrp ` G ) )
26	25, 9	jca 306	. 2 |- ( ( S e. (SubGrp ` G ) /\ A e. (SubGrp ` H ) ) -> ( A e. (SubGrp ` G ) /\ A C_ S ) )
27	6	subggrp 13307	. . . 4 |- ( S e. (SubGrp ` G ) -> H e. Grp )
28	27	adantr 276	. . 3 |- ( ( S e. (SubGrp ` G ) /\ ( A e. (SubGrp ` G ) /\ A C_ S ) ) -> H e. Grp )
29		simprr 531	. . . 4 |- ( ( S e. (SubGrp ` G ) /\ ( A e. (SubGrp ` G ) /\ A C_ S ) ) -> A C_ S )
30	7	adantr 276	. . . 4 |- ( ( S e. (SubGrp ` G ) /\ ( A e. (SubGrp ` G ) /\ A C_ S ) ) -> S = ( Base ` H ) )
31	29, 30	sseqtrd 3221	. . 3 |- ( ( S e. (SubGrp ` G ) /\ ( A e. (SubGrp ` G ) /\ A C_ S ) ) -> A C_ ( Base ` H ) )
32	18	adantrl 478	. . . 4 |- ( ( S e. (SubGrp ` G ) /\ ( A e. (SubGrp ` G ) /\ A C_ S ) ) -> ( H ↾s A ) = ( G ↾s A ) )
33		eqid 2196	. . . . . 6 |- ( G ↾s A ) = ( G ↾s A )
34	33	subggrp 13307	. . . . 5 |- ( A e. (SubGrp ` G ) -> ( G ↾s A ) e. Grp )
35	34	ad2antrl 490	. . . 4 |- ( ( S e. (SubGrp ` G ) /\ ( A e. (SubGrp ` G ) /\ A C_ S ) ) -> ( G ↾s A ) e. Grp )
36	32, 35	eqeltrd 2273	. . 3 |- ( ( S e. (SubGrp ` G ) /\ ( A e. (SubGrp ` G ) /\ A C_ S ) ) -> ( H ↾s A ) e. Grp )
37	3	issubg 13303	. . 3 |- ( A e. (SubGrp ` H ) <-> ( H e. Grp /\ A C_ ( Base ` H ) /\ ( H ↾s A ) e. Grp ) )
38	28, 31, 36, 37	syl3anbrc 1183	. 2 |- ( ( S e. (SubGrp ` G ) /\ ( A e. (SubGrp ` G ) /\ A C_ S ) ) -> A e. (SubGrp ` H ) )
39	26, 38	impbida 596	1 |- ( S e. (SubGrp ` G ) -> ( A e. (SubGrp ` H ) <-> ( A e. (SubGrp ` G ) /\ A C_ S ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2167   C_ wss 3157   ` cfv 5258  (class class class)co 5922   Basecbs 12678   ↾_s cress 12679   Grpcgrp 13132  SubGrpcsubg 13297

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1re 7973  ax-addrcl 7976

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-fv 5266  df-ov 5925  df-oprab 5926  df-mpo 5927  df-inn 8991  df-ndx 12681  df-slot 12682  df-base 12684  df-sets 12685  df-iress 12686  df-subg 13300

This theorem is referenced by:  nmznsg  13343