Theorem subsubg 13977

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 13959	. . . . 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 2238	. . . . . . . 8 |- ( Base ` H ) = ( Base ` H )
4	3	subgss 13954	. . . . . . 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 13958	. . . . . . 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 3287	. . . . 5 |- ( ( S e. (SubGrp ` G ) /\ A e. (SubGrp ` H ) ) -> A C_ S )
10		eqid 2238	. . . . . . 7 |- ( Base ` G ) = ( Base ` G )
11	10	subgss 13954	. . . . . 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 3258	. . . 4 |- ( ( S e. (SubGrp ` G ) /\ A e. (SubGrp ` H ) ) -> A C_ ( Base ` G ) )
14	6	oveq1i 6085	. . . . . . 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 13407	. . . . . . . 8 |- ( ( S e. (SubGrp ` G ) /\ A C_ S /\ G e. Grp ) -> ( ( G ↾s S ) ↾s A ) = ( G ↾s A ) )
17	15, 16	mpd3an3 1379	. . . . . . 7 |- ( ( S e. (SubGrp ` G ) /\ A C_ S ) -> ( ( G ↾s S ) ↾s A ) = ( G ↾s A ) )
18	14, 17	eqtrid 2283	. . . . . 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 2238	. . . . . . 7 |- ( H ↾s A ) = ( H ↾s A )
21	20	subggrp 13957	. . . . . 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 2316	. . . 4 |- ( ( S e. (SubGrp ` G ) /\ A e. (SubGrp ` H ) ) -> ( G ↾s A ) e. Grp )
24	10	issubg 13953	. . . 4 |- ( A e. (SubGrp ` G ) <-> ( G e. Grp /\ A C_ ( Base ` G ) /\ ( G ↾s A ) e. Grp ) )
25	2, 13, 23, 24	syl3anbrc 1212	. . 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 13957	. . . 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 537	. . . 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 3286	. . 3 |- ( ( S e. (SubGrp ` G ) /\ ( A e. (SubGrp ` G ) /\ A C_ S ) ) -> A C_ ( Base ` H ) )
32	18	adantrl 482	. . . 4 |- ( ( S e. (SubGrp ` G ) /\ ( A e. (SubGrp ` G ) /\ A C_ S ) ) -> ( H ↾s A ) = ( G ↾s A ) )
33		eqid 2238	. . . . . 6 |- ( G ↾s A ) = ( G ↾s A )
34	33	subggrp 13957	. . . . 5 |- ( A e. (SubGrp ` G ) -> ( G ↾s A ) e. Grp )
35	34	ad2antrl 494	. . . 4 |- ( ( S e. (SubGrp ` G ) /\ ( A e. (SubGrp ` G ) /\ A C_ S ) ) -> ( G ↾s A ) e. Grp )
36	32, 35	eqeltrd 2315	. . 3 |- ( ( S e. (SubGrp ` G ) /\ ( A e. (SubGrp ` G ) /\ A C_ S ) ) -> ( H ↾s A ) e. Grp )
37	3	issubg 13953	. . 3 |- ( A e. (SubGrp ` H ) <-> ( H e. Grp /\ A C_ ( Base ` H ) /\ ( H ↾s A ) e. Grp ) )
38	28, 31, 36, 37	syl3anbrc 1212	. 2 |- ( ( S e. (SubGrp ` G ) /\ ( A e. (SubGrp ` G ) /\ A C_ S ) ) -> A e. (SubGrp ` H ) )
39	26, 38	impbida 604	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 1402   e. wcel 2209   C_ wss 3220   ` cfv 5372  (class class class)co 6075   Basecbs 13330   ↾_s cress 13331   Grpcgrp 13782  SubGrpcsubg 13947

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1re 8263  ax-addrcl 8266

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-inn 9284  df-ndx 13333  df-slot 13334  df-base 13336  df-sets 13337  df-iress 13338  df-subg 13950

This theorem is referenced by:  nmznsg  13993