Theorem subsubg 13774

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 13756	. . . . 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 2229	. . . . . . . 8 |- ( Base ` H ) = ( Base ` H )
4	3	subgss 13751	. . . . . . 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 13755	. . . . . . 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 3264	. . . . 5 |- ( ( S e. (SubGrp ` G ) /\ A e. (SubGrp ` H ) ) -> A C_ S )
10		eqid 2229	. . . . . . 7 |- ( Base ` G ) = ( Base ` G )
11	10	subgss 13751	. . . . . 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 3235	. . . 4 |- ( ( S e. (SubGrp ` G ) /\ A e. (SubGrp ` H ) ) -> A C_ ( Base ` G ) )
14	6	oveq1i 6023	. . . . . . 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 13149	. . . . . . . 8 |- ( ( S e. (SubGrp ` G ) /\ A C_ S /\ G e. Grp ) -> ( ( G ↾s S ) ↾s A ) = ( G ↾s A ) )
17	15, 16	mpd3an3 1372	. . . . . . 7 |- ( ( S e. (SubGrp ` G ) /\ A C_ S ) -> ( ( G ↾s S ) ↾s A ) = ( G ↾s A ) )
18	14, 17	eqtrid 2274	. . . . . 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 2229	. . . . . . 7 |- ( H ↾s A ) = ( H ↾s A )
21	20	subggrp 13754	. . . . . 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 2307	. . . 4 |- ( ( S e. (SubGrp ` G ) /\ A e. (SubGrp ` H ) ) -> ( G ↾s A ) e. Grp )
24	10	issubg 13750	. . . 4 |- ( A e. (SubGrp ` G ) <-> ( G e. Grp /\ A C_ ( Base ` G ) /\ ( G ↾s A ) e. Grp ) )
25	2, 13, 23, 24	syl3anbrc 1205	. . 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 13754	. . . 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 3263	. . 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 2229	. . . . . 6 |- ( G ↾s A ) = ( G ↾s A )
34	33	subggrp 13754	. . . . 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 2306	. . 3 |- ( ( S e. (SubGrp ` G ) /\ ( A e. (SubGrp ` G ) /\ A C_ S ) ) -> ( H ↾s A ) e. Grp )
37	3	issubg 13750	. . 3 |- ( A e. (SubGrp ` H ) <-> ( H e. Grp /\ A C_ ( Base ` H ) /\ ( H ↾s A ) e. Grp ) )
38	28, 31, 36, 37	syl3anbrc 1205	. 2 |- ( ( S e. (SubGrp ` G ) /\ ( A e. (SubGrp ` G ) /\ A C_ S ) ) -> A e. (SubGrp ` H ) )
39	26, 38	impbida 598	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 1395   e. wcel 2200   C_ wss 3198   ` cfv 5324  (class class class)co 6013   Basecbs 13072   ↾_s cress 13073   Grpcgrp 13573  SubGrpcsubg 13744

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8113  ax-resscn 8114  ax-1re 8116  ax-addrcl 8119

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-fv 5332  df-ov 6016  df-oprab 6017  df-mpo 6018  df-inn 9134  df-ndx 13075  df-slot 13076  df-base 13078  df-sets 13079  df-iress 13080  df-subg 13747

This theorem is referenced by:  nmznsg  13790