Theorem subsubg 13403

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 ∈ (SubGrp‘𝐺) → (𝐴 ∈ (SubGrp‘𝐻) ↔ (𝐴 ∈ (SubGrp‘𝐺) ∧ 𝐴 ⊆ 𝑆)))

Proof of Theorem subsubg

Step	Hyp	Ref	Expression
1		subgrcl 13385	. . . . 5 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
2	1	adantr 276	. . . 4 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ (SubGrp‘𝐻)) → 𝐺 ∈ Grp)
3		eqid 2196	. . . . . . . 8 ⊢ (Base‘𝐻) = (Base‘𝐻)
4	3	subgss 13380	. . . . . . 7 ⊢ (𝐴 ∈ (SubGrp‘𝐻) → 𝐴 ⊆ (Base‘𝐻))
5	4	adantl 277	. . . . . 6 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ (SubGrp‘𝐻)) → 𝐴 ⊆ (Base‘𝐻))
6		subsubg.h	. . . . . . . 8 ⊢ 𝐻 = (𝐺 ↾s 𝑆)
7	6	subgbas 13384	. . . . . . 7 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 = (Base‘𝐻))
8	7	adantr 276	. . . . . 6 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ (SubGrp‘𝐻)) → 𝑆 = (Base‘𝐻))
9	5, 8	sseqtrrd 3223	. . . . 5 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ (SubGrp‘𝐻)) → 𝐴 ⊆ 𝑆)
10		eqid 2196	. . . . . . 7 ⊢ (Base‘𝐺) = (Base‘𝐺)
11	10	subgss 13380	. . . . . 6 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ (Base‘𝐺))
12	11	adantr 276	. . . . 5 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ (SubGrp‘𝐻)) → 𝑆 ⊆ (Base‘𝐺))
13	9, 12	sstrd 3194	. . . 4 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ (SubGrp‘𝐻)) → 𝐴 ⊆ (Base‘𝐺))
14	6	oveq1i 5935	. . . . . . 7 ⊢ (𝐻 ↾s 𝐴) = ((𝐺 ↾s 𝑆) ↾s 𝐴)
15	1	adantr 276	. . . . . . . 8 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ⊆ 𝑆) → 𝐺 ∈ Grp)
16		ressabsg 12779	. . . . . . . 8 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ⊆ 𝑆 ∧ 𝐺 ∈ Grp) → ((𝐺 ↾s 𝑆) ↾s 𝐴) = (𝐺 ↾s 𝐴))
17	15, 16	mpd3an3 1349	. . . . . . 7 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ⊆ 𝑆) → ((𝐺 ↾s 𝑆) ↾s 𝐴) = (𝐺 ↾s 𝐴))
18	14, 17	eqtrid 2241	. . . . . 6 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ⊆ 𝑆) → (𝐻 ↾s 𝐴) = (𝐺 ↾s 𝐴))
19	9, 18	syldan 282	. . . . 5 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ (SubGrp‘𝐻)) → (𝐻 ↾s 𝐴) = (𝐺 ↾s 𝐴))
20		eqid 2196	. . . . . . 7 ⊢ (𝐻 ↾s 𝐴) = (𝐻 ↾s 𝐴)
21	20	subggrp 13383	. . . . . 6 ⊢ (𝐴 ∈ (SubGrp‘𝐻) → (𝐻 ↾s 𝐴) ∈ Grp)
22	21	adantl 277	. . . . 5 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ (SubGrp‘𝐻)) → (𝐻 ↾s 𝐴) ∈ Grp)
23	19, 22	eqeltrrd 2274	. . . 4 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ (SubGrp‘𝐻)) → (𝐺 ↾s 𝐴) ∈ Grp)
24	10	issubg 13379	. . . 4 ⊢ (𝐴 ∈ (SubGrp‘𝐺) ↔ (𝐺 ∈ Grp ∧ 𝐴 ⊆ (Base‘𝐺) ∧ (𝐺 ↾s 𝐴) ∈ Grp))
25	2, 13, 23, 24	syl3anbrc 1183	. . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ (SubGrp‘𝐻)) → 𝐴 ∈ (SubGrp‘𝐺))
26	25, 9	jca 306	. 2 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ (SubGrp‘𝐻)) → (𝐴 ∈ (SubGrp‘𝐺) ∧ 𝐴 ⊆ 𝑆))
27	6	subggrp 13383	. . . 4 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐻 ∈ Grp)
28	27	adantr 276	. . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴 ∈ (SubGrp‘𝐺) ∧ 𝐴 ⊆ 𝑆)) → 𝐻 ∈ Grp)
29		simprr 531	. . . 4 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴 ∈ (SubGrp‘𝐺) ∧ 𝐴 ⊆ 𝑆)) → 𝐴 ⊆ 𝑆)
30	7	adantr 276	. . . 4 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴 ∈ (SubGrp‘𝐺) ∧ 𝐴 ⊆ 𝑆)) → 𝑆 = (Base‘𝐻))
31	29, 30	sseqtrd 3222	. . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴 ∈ (SubGrp‘𝐺) ∧ 𝐴 ⊆ 𝑆)) → 𝐴 ⊆ (Base‘𝐻))
32	18	adantrl 478	. . . 4 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴 ∈ (SubGrp‘𝐺) ∧ 𝐴 ⊆ 𝑆)) → (𝐻 ↾s 𝐴) = (𝐺 ↾s 𝐴))
33		eqid 2196	. . . . . 6 ⊢ (𝐺 ↾s 𝐴) = (𝐺 ↾s 𝐴)
34	33	subggrp 13383	. . . . 5 ⊢ (𝐴 ∈ (SubGrp‘𝐺) → (𝐺 ↾s 𝐴) ∈ Grp)
35	34	ad2antrl 490	. . . 4 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴 ∈ (SubGrp‘𝐺) ∧ 𝐴 ⊆ 𝑆)) → (𝐺 ↾s 𝐴) ∈ Grp)
36	32, 35	eqeltrd 2273	. . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴 ∈ (SubGrp‘𝐺) ∧ 𝐴 ⊆ 𝑆)) → (𝐻 ↾s 𝐴) ∈ Grp)
37	3	issubg 13379	. . 3 ⊢ (𝐴 ∈ (SubGrp‘𝐻) ↔ (𝐻 ∈ Grp ∧ 𝐴 ⊆ (Base‘𝐻) ∧ (𝐻 ↾s 𝐴) ∈ Grp))
38	28, 31, 36, 37	syl3anbrc 1183	. 2 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴 ∈ (SubGrp‘𝐺) ∧ 𝐴 ⊆ 𝑆)) → 𝐴 ∈ (SubGrp‘𝐻))
39	26, 38	impbida 596	1 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (𝐴 ∈ (SubGrp‘𝐻) ↔ (𝐴 ∈ (SubGrp‘𝐺) ∧ 𝐴 ⊆ 𝑆)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364   ∈ wcel 2167   ⊆ wss 3157  ‘cfv 5259  (class class class)co 5925  Basecbs 12703   ↾_s cress 12704  Grpcgrp 13202  SubGrpcsubg 13373

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 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-1re 7990  ax-addrcl 7993

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 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-fv 5267  df-ov 5928  df-oprab 5929  df-mpo 5930  df-inn 9008  df-ndx 12706  df-slot 12707  df-base 12709  df-sets 12710  df-iress 12711  df-subg 13376

This theorem is referenced by:  nmznsg  13419