Theorem subsubg 13533

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 13515	. . . . 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 2205	. . . . . . . 8 |- ( Base ` H ) = ( Base ` H )
4	3	subgss 13510	. . . . . . 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 13514	. . . . . . 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 3232	. . . . 5 |- ( ( S e. (SubGrp ` G ) /\ A e. (SubGrp ` H ) ) -> A C_ S )
10		eqid 2205	. . . . . . 7 |- ( Base ` G ) = ( Base ` G )
11	10	subgss 13510	. . . . . 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 3203	. . . 4 |- ( ( S e. (SubGrp ` G ) /\ A e. (SubGrp ` H ) ) -> A C_ ( Base ` G ) )
14	6	oveq1i 5954	. . . . . . 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 12908	. . . . . . . 8 |- ( ( S e. (SubGrp ` G ) /\ A C_ S /\ G e. Grp ) -> ( ( G ↾s S ) ↾s A ) = ( G ↾s A ) )
17	15, 16	mpd3an3 1351	. . . . . . 7 |- ( ( S e. (SubGrp ` G ) /\ A C_ S ) -> ( ( G ↾s S ) ↾s A ) = ( G ↾s A ) )
18	14, 17	eqtrid 2250	. . . . . 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 2205	. . . . . . 7 |- ( H ↾s A ) = ( H ↾s A )
21	20	subggrp 13513	. . . . . 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 2283	. . . 4 |- ( ( S e. (SubGrp ` G ) /\ A e. (SubGrp ` H ) ) -> ( G ↾s A ) e. Grp )
24	10	issubg 13509	. . . 4 |- ( A e. (SubGrp ` G ) <-> ( G e. Grp /\ A C_ ( Base ` G ) /\ ( G ↾s A ) e. Grp ) )
25	2, 13, 23, 24	syl3anbrc 1184	. . 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 13513	. . . 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 3231	. . 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 2205	. . . . . 6 |- ( G ↾s A ) = ( G ↾s A )
34	33	subggrp 13513	. . . . 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 2282	. . 3 |- ( ( S e. (SubGrp ` G ) /\ ( A e. (SubGrp ` G ) /\ A C_ S ) ) -> ( H ↾s A ) e. Grp )
37	3	issubg 13509	. . 3 |- ( A e. (SubGrp ` H ) <-> ( H e. Grp /\ A C_ ( Base ` H ) /\ ( H ↾s A ) e. Grp ) )
38	28, 31, 36, 37	syl3anbrc 1184	. 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 1373   e. wcel 2176   C_ wss 3166   ` cfv 5271  (class class class)co 5944   Basecbs 12832   ↾_s cress 12833   Grpcgrp 13332  SubGrpcsubg 13503

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-cnex 8016  ax-resscn 8017  ax-1re 8019  ax-addrcl 8022

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-fv 5279  df-ov 5947  df-oprab 5948  df-mpo 5949  df-inn 9037  df-ndx 12835  df-slot 12836  df-base 12838  df-sets 12839  df-iress 12840  df-subg 13506

This theorem is referenced by:  nmznsg  13549