Theorem subsubg 13057

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 13039	. . . . 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 2177	. . . . . . . 8 |- ( Base ` H ) = ( Base ` H )
4	3	subgss 13034	. . . . . . 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 13038	. . . . . . 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 3195	. . . . 5 |- ( ( S e. (SubGrp ` G ) /\ A e. (SubGrp ` H ) ) -> A C_ S )
10		eqid 2177	. . . . . . 7 |- ( Base ` G ) = ( Base ` G )
11	10	subgss 13034	. . . . . 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 3166	. . . 4 |- ( ( S e. (SubGrp ` G ) /\ A e. (SubGrp ` H ) ) -> A C_ ( Base ` G ) )
14	6	oveq1i 5885	. . . . . . 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 12535	. . . . . . . 8 |- ( ( S e. (SubGrp ` G ) /\ A C_ S /\ G e. Grp ) -> ( ( G ↾s S ) ↾s A ) = ( G ↾s A ) )
17	15, 16	mpd3an3 1338	. . . . . . 7 |- ( ( S e. (SubGrp ` G ) /\ A C_ S ) -> ( ( G ↾s S ) ↾s A ) = ( G ↾s A ) )
18	14, 17	eqtrid 2222	. . . . . 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 2177	. . . . . . 7 |- ( H ↾s A ) = ( H ↾s A )
21	20	subggrp 13037	. . . . . 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 2255	. . . 4 |- ( ( S e. (SubGrp ` G ) /\ A e. (SubGrp ` H ) ) -> ( G ↾s A ) e. Grp )
24	10	issubg 13033	. . . 4 |- ( A e. (SubGrp ` G ) <-> ( G e. Grp /\ A C_ ( Base ` G ) /\ ( G ↾s A ) e. Grp ) )
25	2, 13, 23, 24	syl3anbrc 1181	. . 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 13037	. . . 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 3194	. . 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 2177	. . . . . 6 |- ( G ↾s A ) = ( G ↾s A )
34	33	subggrp 13037	. . . . 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 2254	. . 3 |- ( ( S e. (SubGrp ` G ) /\ ( A e. (SubGrp ` G ) /\ A C_ S ) ) -> ( H ↾s A ) e. Grp )
37	3	issubg 13033	. . 3 |- ( A e. (SubGrp ` H ) <-> ( H e. Grp /\ A C_ ( Base ` H ) /\ ( H ↾s A ) e. Grp ) )
38	28, 31, 36, 37	syl3anbrc 1181	. 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 1353   e. wcel 2148   C_ wss 3130   ` cfv 5217  (class class class)co 5875   Basecbs 12462   ↾_s cress 12463   Grpcgrp 12877  SubGrpcsubg 13027

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-cnex 7902  ax-resscn 7903  ax-1re 7905  ax-addrcl 7908

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-fv 5225  df-ov 5878  df-oprab 5879  df-mpo 5880  df-inn 8920  df-ndx 12465  df-slot 12466  df-base 12468  df-sets 12469  df-iress 12470  df-subg 13030

This theorem is referenced by:  nmznsg  13073