Theorem ressxms 23129

Description: The restriction of a metric space is a metric space. (Contributed by Mario Carneiro, 24-Aug-2015.)

Assertion

Ref	Expression
ressxms	⊢ ((𝐾 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑉) → (𝐾 ↾s 𝐴) ∈ ∞MetSp)

Proof of Theorem ressxms

Step	Hyp	Ref	Expression
1		eqid 2821	. . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2821	. . . . . 6 ⊢ ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))
3	1, 2	xmsxmet 23060	. . . . 5 ⊢ (𝐾 ∈ ∞MetSp → ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ∈ (∞Met‘(Base‘𝐾)))
4	3	adantr 483	. . . 4 ⊢ ((𝐾 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑉) → ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ∈ (∞Met‘(Base‘𝐾)))
5		xmetres 22968	. . . 4 ⊢ (((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ∈ (∞Met‘(Base‘𝐾)) → (((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ↾ (𝐴 × 𝐴)) ∈ (∞Met‘((Base‘𝐾) ∩ 𝐴)))
6	4, 5	syl 17	. . 3 ⊢ ((𝐾 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑉) → (((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ↾ (𝐴 × 𝐴)) ∈ (∞Met‘((Base‘𝐾) ∩ 𝐴)))
7		resres 5860	. . . . 5 ⊢ (((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ↾ (𝐴 × 𝐴)) = ((dist‘𝐾) ↾ (((Base‘𝐾) × (Base‘𝐾)) ∩ (𝐴 × 𝐴)))
8		inxp 5697	. . . . . 6 ⊢ (((Base‘𝐾) × (Base‘𝐾)) ∩ (𝐴 × 𝐴)) = (((Base‘𝐾) ∩ 𝐴) × ((Base‘𝐾) ∩ 𝐴))
9	8	reseq2i 5844	. . . . 5 ⊢ ((dist‘𝐾) ↾ (((Base‘𝐾) × (Base‘𝐾)) ∩ (𝐴 × 𝐴))) = ((dist‘𝐾) ↾ (((Base‘𝐾) ∩ 𝐴) × ((Base‘𝐾) ∩ 𝐴)))
10	7, 9	eqtri 2844	. . . 4 ⊢ (((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ↾ (𝐴 × 𝐴)) = ((dist‘𝐾) ↾ (((Base‘𝐾) ∩ 𝐴) × ((Base‘𝐾) ∩ 𝐴)))
11		eqid 2821	. . . . . . 7 ⊢ (𝐾 ↾s 𝐴) = (𝐾 ↾s 𝐴)
12		eqid 2821	. . . . . . 7 ⊢ (dist‘𝐾) = (dist‘𝐾)
13	11, 12	ressds 16680	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (dist‘𝐾) = (dist‘(𝐾 ↾s 𝐴)))
14	13	adantl 484	. . . . 5 ⊢ ((𝐾 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑉) → (dist‘𝐾) = (dist‘(𝐾 ↾s 𝐴)))
15		incom 4177	. . . . . . 7 ⊢ ((Base‘𝐾) ∩ 𝐴) = (𝐴 ∩ (Base‘𝐾))
16	11, 1	ressbas 16548	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ (Base‘𝐾)) = (Base‘(𝐾 ↾s 𝐴)))
17	16	adantl 484	. . . . . . 7 ⊢ ((𝐾 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑉) → (𝐴 ∩ (Base‘𝐾)) = (Base‘(𝐾 ↾s 𝐴)))
18	15, 17	syl5eq 2868	. . . . . 6 ⊢ ((𝐾 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑉) → ((Base‘𝐾) ∩ 𝐴) = (Base‘(𝐾 ↾s 𝐴)))
19	18	sqxpeqd 5581	. . . . 5 ⊢ ((𝐾 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑉) → (((Base‘𝐾) ∩ 𝐴) × ((Base‘𝐾) ∩ 𝐴)) = ((Base‘(𝐾 ↾s 𝐴)) × (Base‘(𝐾 ↾s 𝐴))))
20	14, 19	reseq12d 5848	. . . 4 ⊢ ((𝐾 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑉) → ((dist‘𝐾) ↾ (((Base‘𝐾) ∩ 𝐴) × ((Base‘𝐾) ∩ 𝐴))) = ((dist‘(𝐾 ↾s 𝐴)) ↾ ((Base‘(𝐾 ↾s 𝐴)) × (Base‘(𝐾 ↾s 𝐴)))))
21	10, 20	syl5eq 2868	. . 3 ⊢ ((𝐾 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑉) → (((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ↾ (𝐴 × 𝐴)) = ((dist‘(𝐾 ↾s 𝐴)) ↾ ((Base‘(𝐾 ↾s 𝐴)) × (Base‘(𝐾 ↾s 𝐴)))))
22	18	fveq2d 6668	. . 3 ⊢ ((𝐾 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑉) → (∞Met‘((Base‘𝐾) ∩ 𝐴)) = (∞Met‘(Base‘(𝐾 ↾s 𝐴))))
23	6, 21, 22	3eltr3d 2927	. 2 ⊢ ((𝐾 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑉) → ((dist‘(𝐾 ↾s 𝐴)) ↾ ((Base‘(𝐾 ↾s 𝐴)) × (Base‘(𝐾 ↾s 𝐴)))) ∈ (∞Met‘(Base‘(𝐾 ↾s 𝐴))))
24		eqid 2821	. . . . . . 7 ⊢ (TopOpen‘𝐾) = (TopOpen‘𝐾)
25	24, 1, 2	xmstopn 23055	. . . . . 6 ⊢ (𝐾 ∈ ∞MetSp → (TopOpen‘𝐾) = (MetOpen‘((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))))
26	25	adantr 483	. . . . 5 ⊢ ((𝐾 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑉) → (TopOpen‘𝐾) = (MetOpen‘((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))))
27	26	oveq1d 7165	. . . 4 ⊢ ((𝐾 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑉) → ((TopOpen‘𝐾) ↾t ((Base‘𝐾) ∩ 𝐴)) = ((MetOpen‘((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))) ↾t ((Base‘𝐾) ∩ 𝐴)))
28		inss1 4204	. . . . 5 ⊢ ((Base‘𝐾) ∩ 𝐴) ⊆ (Base‘𝐾)
29		xpss12 5564	. . . . . . . . 9 ⊢ ((((Base‘𝐾) ∩ 𝐴) ⊆ (Base‘𝐾) ∧ ((Base‘𝐾) ∩ 𝐴) ⊆ (Base‘𝐾)) → (((Base‘𝐾) ∩ 𝐴) × ((Base‘𝐾) ∩ 𝐴)) ⊆ ((Base‘𝐾) × (Base‘𝐾)))
30	28, 28, 29	mp2an 690	. . . . . . . 8 ⊢ (((Base‘𝐾) ∩ 𝐴) × ((Base‘𝐾) ∩ 𝐴)) ⊆ ((Base‘𝐾) × (Base‘𝐾))
31		resabs1 5877	. . . . . . . 8 ⊢ ((((Base‘𝐾) ∩ 𝐴) × ((Base‘𝐾) ∩ 𝐴)) ⊆ ((Base‘𝐾) × (Base‘𝐾)) → (((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ↾ (((Base‘𝐾) ∩ 𝐴) × ((Base‘𝐾) ∩ 𝐴))) = ((dist‘𝐾) ↾ (((Base‘𝐾) ∩ 𝐴) × ((Base‘𝐾) ∩ 𝐴))))
32	30, 31	ax-mp 5	. . . . . . 7 ⊢ (((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ↾ (((Base‘𝐾) ∩ 𝐴) × ((Base‘𝐾) ∩ 𝐴))) = ((dist‘𝐾) ↾ (((Base‘𝐾) ∩ 𝐴) × ((Base‘𝐾) ∩ 𝐴)))
33	10, 32	eqtr4i 2847	. . . . . 6 ⊢ (((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ↾ (𝐴 × 𝐴)) = (((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ↾ (((Base‘𝐾) ∩ 𝐴) × ((Base‘𝐾) ∩ 𝐴)))
34		eqid 2821	. . . . . 6 ⊢ (MetOpen‘((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))) = (MetOpen‘((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))))
35		eqid 2821	. . . . . 6 ⊢ (MetOpen‘(((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ↾ (𝐴 × 𝐴))) = (MetOpen‘(((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ↾ (𝐴 × 𝐴)))
36	33, 34, 35	metrest 23128	. . . . 5 ⊢ ((((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ∈ (∞Met‘(Base‘𝐾)) ∧ ((Base‘𝐾) ∩ 𝐴) ⊆ (Base‘𝐾)) → ((MetOpen‘((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))) ↾t ((Base‘𝐾) ∩ 𝐴)) = (MetOpen‘(((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ↾ (𝐴 × 𝐴))))
37	4, 28, 36	sylancl 588	. . . 4 ⊢ ((𝐾 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑉) → ((MetOpen‘((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))) ↾t ((Base‘𝐾) ∩ 𝐴)) = (MetOpen‘(((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ↾ (𝐴 × 𝐴))))
38	27, 37	eqtrd 2856	. . 3 ⊢ ((𝐾 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑉) → ((TopOpen‘𝐾) ↾t ((Base‘𝐾) ∩ 𝐴)) = (MetOpen‘(((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ↾ (𝐴 × 𝐴))))
39		xmstps 23057	. . . . . . . . 9 ⊢ (𝐾 ∈ ∞MetSp → 𝐾 ∈ TopSp)
40	1, 24	tpsuni 21538	. . . . . . . . 9 ⊢ (𝐾 ∈ TopSp → (Base‘𝐾) = ∪ (TopOpen‘𝐾))
41	39, 40	syl 17	. . . . . . . 8 ⊢ (𝐾 ∈ ∞MetSp → (Base‘𝐾) = ∪ (TopOpen‘𝐾))
42	41	adantr 483	. . . . . . 7 ⊢ ((𝐾 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑉) → (Base‘𝐾) = ∪ (TopOpen‘𝐾))
43	42	ineq2d 4188	. . . . . 6 ⊢ ((𝐾 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑉) → (𝐴 ∩ (Base‘𝐾)) = (𝐴 ∩ ∪ (TopOpen‘𝐾)))
44	15, 43	syl5eq 2868	. . . . 5 ⊢ ((𝐾 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑉) → ((Base‘𝐾) ∩ 𝐴) = (𝐴 ∩ ∪ (TopOpen‘𝐾)))
45	44	oveq2d 7166	. . . 4 ⊢ ((𝐾 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑉) → ((TopOpen‘𝐾) ↾t ((Base‘𝐾) ∩ 𝐴)) = ((TopOpen‘𝐾) ↾t (𝐴 ∩ ∪ (TopOpen‘𝐾))))
46	1, 24	istps 21536	. . . . . 6 ⊢ (𝐾 ∈ TopSp ↔ (TopOpen‘𝐾) ∈ (TopOn‘(Base‘𝐾)))
47	39, 46	sylib 220	. . . . 5 ⊢ (𝐾 ∈ ∞MetSp → (TopOpen‘𝐾) ∈ (TopOn‘(Base‘𝐾)))
48		eqid 2821	. . . . . 6 ⊢ ∪ (TopOpen‘𝐾) = ∪ (TopOpen‘𝐾)
49	48	restin 21768	. . . . 5 ⊢ (((TopOpen‘𝐾) ∈ (TopOn‘(Base‘𝐾)) ∧ 𝐴 ∈ 𝑉) → ((TopOpen‘𝐾) ↾t 𝐴) = ((TopOpen‘𝐾) ↾t (𝐴 ∩ ∪ (TopOpen‘𝐾))))
50	47, 49	sylan 582	. . . 4 ⊢ ((𝐾 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑉) → ((TopOpen‘𝐾) ↾t 𝐴) = ((TopOpen‘𝐾) ↾t (𝐴 ∩ ∪ (TopOpen‘𝐾))))
51	45, 50	eqtr4d 2859	. . 3 ⊢ ((𝐾 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑉) → ((TopOpen‘𝐾) ↾t ((Base‘𝐾) ∩ 𝐴)) = ((TopOpen‘𝐾) ↾t 𝐴))
52	21	fveq2d 6668	. . 3 ⊢ ((𝐾 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑉) → (MetOpen‘(((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ↾ (𝐴 × 𝐴))) = (MetOpen‘((dist‘(𝐾 ↾s 𝐴)) ↾ ((Base‘(𝐾 ↾s 𝐴)) × (Base‘(𝐾 ↾s 𝐴))))))
53	38, 51, 52	3eqtr3d 2864	. 2 ⊢ ((𝐾 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑉) → ((TopOpen‘𝐾) ↾t 𝐴) = (MetOpen‘((dist‘(𝐾 ↾s 𝐴)) ↾ ((Base‘(𝐾 ↾s 𝐴)) × (Base‘(𝐾 ↾s 𝐴))))))
54	11, 24	resstopn 21788	. . 3 ⊢ ((TopOpen‘𝐾) ↾t 𝐴) = (TopOpen‘(𝐾 ↾s 𝐴))
55		eqid 2821	. . 3 ⊢ (Base‘(𝐾 ↾s 𝐴)) = (Base‘(𝐾 ↾s 𝐴))
56		eqid 2821	. . 3 ⊢ ((dist‘(𝐾 ↾s 𝐴)) ↾ ((Base‘(𝐾 ↾s 𝐴)) × (Base‘(𝐾 ↾s 𝐴)))) = ((dist‘(𝐾 ↾s 𝐴)) ↾ ((Base‘(𝐾 ↾s 𝐴)) × (Base‘(𝐾 ↾s 𝐴))))
57	54, 55, 56	isxms2 23052	. 2 ⊢ ((𝐾 ↾s 𝐴) ∈ ∞MetSp ↔ (((dist‘(𝐾 ↾s 𝐴)) ↾ ((Base‘(𝐾 ↾s 𝐴)) × (Base‘(𝐾 ↾s 𝐴)))) ∈ (∞Met‘(Base‘(𝐾 ↾s 𝐴))) ∧ ((TopOpen‘𝐾) ↾t 𝐴) = (MetOpen‘((dist‘(𝐾 ↾s 𝐴)) ↾ ((Base‘(𝐾 ↾s 𝐴)) × (Base‘(𝐾 ↾s 𝐴)))))))
58	23, 53, 57	sylanbrc 585	1 ⊢ ((𝐾 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑉) → (𝐾 ↾s 𝐴) ∈ ∞MetSp)

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1533   ∈ wcel 2110   ∩ cin 3934   ⊆ wss 3935  ∪ cuni 4831   × cxp 5547   ↾ cres 5551  ‘cfv 6349  (class class class)co 7150  Basecbs 16477   ↾_s cress 16478  distcds 16568   ↾_t crest 16688  TopOpenctopn 16689  ∞Metcxmet 20524  MetOpencmopn 20529  TopOnctopon 21512  TopSpctps 21534  ∞MetSpcxms 22921

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608  ax-pre-sup 10609

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-tp 4565  df-op 4567  df-uni 4832  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7575  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-er 8283  df-map 8402  df-en 8504  df-dom 8505  df-sdom 8506  df-sup 8900  df-inf 8901  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-div 11292  df-nn 11633  df-2 11694  df-3 11695  df-4 11696  df-5 11697  df-6 11698  df-7 11699  df-8 11700  df-9 11701  df-n0 11892  df-z 11976  df-dec 12093  df-uz 12238  df-q 12343  df-rp 12384  df-xneg 12501  df-xadd 12502  df-xmul 12503  df-ndx 16480  df-slot 16481  df-base 16483  df-sets 16484  df-ress 16485  df-tset 16578  df-ds 16581  df-rest 16690  df-topn 16691  df-topgen 16711  df-psmet 20531  df-xmet 20532  df-bl 20534  df-mopn 20535  df-top 21496  df-topon 21513  df-topsp 21535  df-bases 21548  df-xms 22924

This theorem is referenced by:  ressms  23130  qqhcn  31227  qqhucn  31228