Theorem subsubg 19059

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 19041	. . . . 5 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
2	1	adantr 480	. . . 4 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ (SubGrp‘𝐻)) → 𝐺 ∈ Grp)
3		eqid 2731	. . . . . . . 8 ⊢ (Base‘𝐻) = (Base‘𝐻)
4	3	subgss 19037	. . . . . . 7 ⊢ (𝐴 ∈ (SubGrp‘𝐻) → 𝐴 ⊆ (Base‘𝐻))
5	4	adantl 481	. . . . . 6 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ (SubGrp‘𝐻)) → 𝐴 ⊆ (Base‘𝐻))
6		subsubg.h	. . . . . . . 8 ⊢ 𝐻 = (𝐺 ↾s 𝑆)
7	6	subgbas 19040	. . . . . . 7 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 = (Base‘𝐻))
8	7	adantr 480	. . . . . 6 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ (SubGrp‘𝐻)) → 𝑆 = (Base‘𝐻))
9	5, 8	sseqtrrd 3972	. . . . 5 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ (SubGrp‘𝐻)) → 𝐴 ⊆ 𝑆)
10		eqid 2731	. . . . . . 7 ⊢ (Base‘𝐺) = (Base‘𝐺)
11	10	subgss 19037	. . . . . 6 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ (Base‘𝐺))
12	11	adantr 480	. . . . 5 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ (SubGrp‘𝐻)) → 𝑆 ⊆ (Base‘𝐺))
13	9, 12	sstrd 3945	. . . 4 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ (SubGrp‘𝐻)) → 𝐴 ⊆ (Base‘𝐺))
14	6	oveq1i 7356	. . . . . . 7 ⊢ (𝐻 ↾s 𝐴) = ((𝐺 ↾s 𝑆) ↾s 𝐴)
15		ressabs 17156	. . . . . . 7 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ⊆ 𝑆) → ((𝐺 ↾s 𝑆) ↾s 𝐴) = (𝐺 ↾s 𝐴))
16	14, 15	eqtrid 2778	. . . . . 6 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ⊆ 𝑆) → (𝐻 ↾s 𝐴) = (𝐺 ↾s 𝐴))
17	9, 16	syldan 591	. . . . 5 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ (SubGrp‘𝐻)) → (𝐻 ↾s 𝐴) = (𝐺 ↾s 𝐴))
18		eqid 2731	. . . . . . 7 ⊢ (𝐻 ↾s 𝐴) = (𝐻 ↾s 𝐴)
19	18	subggrp 19039	. . . . . 6 ⊢ (𝐴 ∈ (SubGrp‘𝐻) → (𝐻 ↾s 𝐴) ∈ Grp)
20	19	adantl 481	. . . . 5 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ (SubGrp‘𝐻)) → (𝐻 ↾s 𝐴) ∈ Grp)
21	17, 20	eqeltrrd 2832	. . . 4 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ (SubGrp‘𝐻)) → (𝐺 ↾s 𝐴) ∈ Grp)
22	10	issubg 19036	. . . 4 ⊢ (𝐴 ∈ (SubGrp‘𝐺) ↔ (𝐺 ∈ Grp ∧ 𝐴 ⊆ (Base‘𝐺) ∧ (𝐺 ↾s 𝐴) ∈ Grp))
23	2, 13, 21, 22	syl3anbrc 1344	. . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ (SubGrp‘𝐻)) → 𝐴 ∈ (SubGrp‘𝐺))
24	23, 9	jca 511	. 2 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ (SubGrp‘𝐻)) → (𝐴 ∈ (SubGrp‘𝐺) ∧ 𝐴 ⊆ 𝑆))
25	6	subggrp 19039	. . . 4 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐻 ∈ Grp)
26	25	adantr 480	. . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴 ∈ (SubGrp‘𝐺) ∧ 𝐴 ⊆ 𝑆)) → 𝐻 ∈ Grp)
27		simprr 772	. . . 4 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴 ∈ (SubGrp‘𝐺) ∧ 𝐴 ⊆ 𝑆)) → 𝐴 ⊆ 𝑆)
28	7	adantr 480	. . . 4 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴 ∈ (SubGrp‘𝐺) ∧ 𝐴 ⊆ 𝑆)) → 𝑆 = (Base‘𝐻))
29	27, 28	sseqtrd 3971	. . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴 ∈ (SubGrp‘𝐺) ∧ 𝐴 ⊆ 𝑆)) → 𝐴 ⊆ (Base‘𝐻))
30	16	adantrl 716	. . . 4 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴 ∈ (SubGrp‘𝐺) ∧ 𝐴 ⊆ 𝑆)) → (𝐻 ↾s 𝐴) = (𝐺 ↾s 𝐴))
31		eqid 2731	. . . . . 6 ⊢ (𝐺 ↾s 𝐴) = (𝐺 ↾s 𝐴)
32	31	subggrp 19039	. . . . 5 ⊢ (𝐴 ∈ (SubGrp‘𝐺) → (𝐺 ↾s 𝐴) ∈ Grp)
33	32	ad2antrl 728	. . . 4 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴 ∈ (SubGrp‘𝐺) ∧ 𝐴 ⊆ 𝑆)) → (𝐺 ↾s 𝐴) ∈ Grp)
34	30, 33	eqeltrd 2831	. . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴 ∈ (SubGrp‘𝐺) ∧ 𝐴 ⊆ 𝑆)) → (𝐻 ↾s 𝐴) ∈ Grp)
35	3	issubg 19036	. . 3 ⊢ (𝐴 ∈ (SubGrp‘𝐻) ↔ (𝐻 ∈ Grp ∧ 𝐴 ⊆ (Base‘𝐻) ∧ (𝐻 ↾s 𝐴) ∈ Grp))
36	26, 29, 34, 35	syl3anbrc 1344	. 2 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ (𝐴 ∈ (SubGrp‘𝐺) ∧ 𝐴 ⊆ 𝑆)) → 𝐴 ∈ (SubGrp‘𝐻))
37	24, 36	impbida 800	1 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (𝐴 ∈ (SubGrp‘𝐻) ↔ (𝐴 ∈ (SubGrp‘𝐺) ∧ 𝐴 ⊆ 𝑆)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111   ⊆ wss 3902  ‘cfv 6481  (class class class)co 7346  Basecbs 17117   ↾_s cress 17138  Grpcgrp 18843  SubGrpcsubg 19030

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668  ax-cnex 11059  ax-1cn 11061  ax-addcl 11063

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-tr 5199  df-id 5511  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-we 5571  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-nn 12123  df-sets 17072  df-slot 17090  df-ndx 17102  df-base 17118  df-ress 17139  df-subg 19033

This theorem is referenced by:  nmznsg  19078  subgslw  19526  subgdmdprd  19946  subgdprd  19947  ablfac1c  19983  pgpfaclem1  19993  pgpfaclem2  19994  ablfaclem3  19999