Theorem subgtgp 24051

Description: A subgroup of a topological group is a topological group. (Contributed by Mario Carneiro, 17-Sep-2015.)

Hypothesis

Ref	Expression
subgtgp.h	⊢ 𝐻 = (𝐺 ↾s 𝑆)

Assertion

Ref	Expression
subgtgp	⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐻 ∈ TopGrp)

Proof of Theorem subgtgp

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		subgtgp.h	. . . 4 ⊢ 𝐻 = (𝐺 ↾s 𝑆)
2	1	subggrp 19061	. . 3 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐻 ∈ Grp)
3	2	adantl 481	. 2 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐻 ∈ Grp)
4		tgptmd 24025	. . 3 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ TopMnd)
5		subgsubm 19080	. . 3 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ∈ (SubMnd‘𝐺))
6	1	submtmd 24050	. . 3 ⊢ ((𝐺 ∈ TopMnd ∧ 𝑆 ∈ (SubMnd‘𝐺)) → 𝐻 ∈ TopMnd)
7	4, 5, 6	syl2an 597	. 2 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐻 ∈ TopMnd)
8	1	subgbas 19062	. . . . . . 7 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 = (Base‘𝐻))
9	8	adantl 481	. . . . . 6 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝑆 = (Base‘𝐻))
10	9	mpteq1d 5187	. . . . 5 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝑥 ∈ 𝑆 ↦ ((invg‘𝐻)‘𝑥)) = (𝑥 ∈ (Base‘𝐻) ↦ ((invg‘𝐻)‘𝑥)))
11		eqid 2735	. . . . . . . 8 ⊢ (invg‘𝐺) = (invg‘𝐺)
12		eqid 2735	. . . . . . . 8 ⊢ (invg‘𝐻) = (invg‘𝐻)
13	1, 11, 12	subginv 19065	. . . . . . 7 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑆) → ((invg‘𝐺)‘𝑥) = ((invg‘𝐻)‘𝑥))
14	13	adantll 715	. . . . . 6 ⊢ (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥 ∈ 𝑆) → ((invg‘𝐺)‘𝑥) = ((invg‘𝐻)‘𝑥))
15	14	mpteq2dva 5190	. . . . 5 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝑥 ∈ 𝑆 ↦ ((invg‘𝐺)‘𝑥)) = (𝑥 ∈ 𝑆 ↦ ((invg‘𝐻)‘𝑥)))
16		eqid 2735	. . . . . . . 8 ⊢ (Base‘𝐻) = (Base‘𝐻)
17	16, 12	grpinvf 18918	. . . . . . 7 ⊢ (𝐻 ∈ Grp → (invg‘𝐻):(Base‘𝐻)⟶(Base‘𝐻))
18	3, 17	syl 17	. . . . . 6 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (invg‘𝐻):(Base‘𝐻)⟶(Base‘𝐻))
19	18	feqmptd 6901	. . . . 5 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (invg‘𝐻) = (𝑥 ∈ (Base‘𝐻) ↦ ((invg‘𝐻)‘𝑥)))
20	10, 15, 19	3eqtr4rd 2781	. . . 4 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (invg‘𝐻) = (𝑥 ∈ 𝑆 ↦ ((invg‘𝐺)‘𝑥)))
21		eqid 2735	. . . . 5 ⊢ ((TopOpen‘𝐺) ↾t 𝑆) = ((TopOpen‘𝐺) ↾t 𝑆)
22		eqid 2735	. . . . . . 7 ⊢ (TopOpen‘𝐺) = (TopOpen‘𝐺)
23		eqid 2735	. . . . . . 7 ⊢ (Base‘𝐺) = (Base‘𝐺)
24	22, 23	tgptopon 24028	. . . . . 6 ⊢ (𝐺 ∈ TopGrp → (TopOpen‘𝐺) ∈ (TopOn‘(Base‘𝐺)))
25	24	adantr 480	. . . . 5 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (TopOpen‘𝐺) ∈ (TopOn‘(Base‘𝐺)))
26	23	subgss 19059	. . . . . 6 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ (Base‘𝐺))
27	26	adantl 481	. . . . 5 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝑆 ⊆ (Base‘𝐺))
28		tgpgrp 24024	. . . . . . . . 9 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
29	28	adantr 480	. . . . . . . 8 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐺 ∈ Grp)
30	23, 11	grpinvf 18918	. . . . . . . 8 ⊢ (𝐺 ∈ Grp → (invg‘𝐺):(Base‘𝐺)⟶(Base‘𝐺))
31	29, 30	syl 17	. . . . . . 7 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (invg‘𝐺):(Base‘𝐺)⟶(Base‘𝐺))
32	31	feqmptd 6901	. . . . . 6 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (invg‘𝐺) = (𝑥 ∈ (Base‘𝐺) ↦ ((invg‘𝐺)‘𝑥)))
33	22, 11	tgpinv 24031	. . . . . . 7 ⊢ (𝐺 ∈ TopGrp → (invg‘𝐺) ∈ ((TopOpen‘𝐺) Cn (TopOpen‘𝐺)))
34	33	adantr 480	. . . . . 6 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (invg‘𝐺) ∈ ((TopOpen‘𝐺) Cn (TopOpen‘𝐺)))
35	32, 34	eqeltrrd 2836	. . . . 5 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝑥 ∈ (Base‘𝐺) ↦ ((invg‘𝐺)‘𝑥)) ∈ ((TopOpen‘𝐺) Cn (TopOpen‘𝐺)))
36	21, 25, 27, 35	cnmpt1res 23622	. . . 4 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝑥 ∈ 𝑆 ↦ ((invg‘𝐺)‘𝑥)) ∈ (((TopOpen‘𝐺) ↾t 𝑆) Cn (TopOpen‘𝐺)))
37	20, 36	eqeltrd 2835	. . 3 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (invg‘𝐻) ∈ (((TopOpen‘𝐺) ↾t 𝑆) Cn (TopOpen‘𝐺)))
38	18	frnd 6669	. . . . 5 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ran (invg‘𝐻) ⊆ (Base‘𝐻))
39	38, 9	sseqtrrd 3970	. . . 4 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ran (invg‘𝐻) ⊆ 𝑆)
40		cnrest2 23232	. . . 4 ⊢ (((TopOpen‘𝐺) ∈ (TopOn‘(Base‘𝐺)) ∧ ran (invg‘𝐻) ⊆ 𝑆 ∧ 𝑆 ⊆ (Base‘𝐺)) → ((invg‘𝐻) ∈ (((TopOpen‘𝐺) ↾t 𝑆) Cn (TopOpen‘𝐺)) ↔ (invg‘𝐻) ∈ (((TopOpen‘𝐺) ↾t 𝑆) Cn ((TopOpen‘𝐺) ↾t 𝑆))))
41	25, 39, 27, 40	syl3anc 1374	. . 3 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((invg‘𝐻) ∈ (((TopOpen‘𝐺) ↾t 𝑆) Cn (TopOpen‘𝐺)) ↔ (invg‘𝐻) ∈ (((TopOpen‘𝐺) ↾t 𝑆) Cn ((TopOpen‘𝐺) ↾t 𝑆))))
42	37, 41	mpbid 232	. 2 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (invg‘𝐻) ∈ (((TopOpen‘𝐺) ↾t 𝑆) Cn ((TopOpen‘𝐺) ↾t 𝑆)))
43	1, 22	resstopn 23132	. . 3 ⊢ ((TopOpen‘𝐺) ↾t 𝑆) = (TopOpen‘𝐻)
44	43, 12	istgp 24023	. 2 ⊢ (𝐻 ∈ TopGrp ↔ (𝐻 ∈ Grp ∧ 𝐻 ∈ TopMnd ∧ (invg‘𝐻) ∈ (((TopOpen‘𝐺) ↾t 𝑆) Cn ((TopOpen‘𝐺) ↾t 𝑆))))
45	3, 7, 42, 44	syl3anbrc 1345	1 ⊢ ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐻 ∈ TopGrp)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ⊆ wss 3900   ↦ cmpt 5178  ran crn 5624  ⟶wf 6487  ‘cfv 6491  (class class class)co 7358  Basecbs 17138   ↾_s cress 17159   ↾_t crest 17342  TopOpenctopn 17343  SubMndcsubmnd 18709  Grpcgrp 18865  inv_gcminusg 18866  SubGrpcsubg 19052  TopOnctopon 22856   Cn ccn 23170  TopMndctmd 24016  TopGrpctgp 24017

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2183  ax-ext 2707  ax-rep 5223  ax-sep 5240  ax-nul 5250  ax-pow 5309  ax-pr 5376  ax-un 7680  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2932  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3349  df-reu 3350  df-rab 3399  df-v 3441  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4285  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-int 4902  df-iun 4947  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6258  df-ord 6319  df-on 6320  df-lim 6321  df-suc 6322  df-iota 6447  df-fun 6493  df-fn 6494  df-f 6495  df-f1 6496  df-fo 6497  df-f1o 6498  df-fv 6499  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-er 8635  df-map 8767  df-en 8886  df-dom 8887  df-sdom 8888  df-fin 8889  df-fi 9316  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11368  df-neg 11369  df-nn 12148  df-2 12210  df-3 12211  df-4 12212  df-5 12213  df-6 12214  df-7 12215  df-8 12216  df-9 12217  df-sets 17093  df-slot 17111  df-ndx 17123  df-base 17139  df-ress 17160  df-plusg 17192  df-tset 17198  df-rest 17344  df-topn 17345  df-0g 17363  df-topgen 17365  df-plusf 18566  df-mgm 18567  df-sgrp 18646  df-mnd 18662  df-submnd 18711  df-grp 18868  df-minusg 18869  df-subg 19055  df-top 22840  df-topon 22857  df-topsp 22879  df-bases 22892  df-cn 23173  df-tx 23508  df-tmd 24018  df-tgp 24019

This theorem is referenced by:  qqhcn  34127