Theorem ressxms 22140

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 2610	. . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2610	. . . . . 6 ⊢ ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))
3	1, 2	xmsxmet 22071	. . . . 5 ⊢ (𝐾 ∈ ∞MetSp → ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ∈ (∞Met‘(Base‘𝐾)))
4	3	adantr 480	. . . 4 ⊢ ((𝐾 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑉) → ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ∈ (∞Met‘(Base‘𝐾)))
5		xmetres 21979	. . . 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 5329	. . . . 5 ⊢ (((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ↾ (𝐴 × 𝐴)) = ((dist‘𝐾) ↾ (((Base‘𝐾) × (Base‘𝐾)) ∩ (𝐴 × 𝐴)))
8		inxp 5176	. . . . . 6 ⊢ (((Base‘𝐾) × (Base‘𝐾)) ∩ (𝐴 × 𝐴)) = (((Base‘𝐾) ∩ 𝐴) × ((Base‘𝐾) ∩ 𝐴))
9	8	reseq2i 5314	. . . . 5 ⊢ ((dist‘𝐾) ↾ (((Base‘𝐾) × (Base‘𝐾)) ∩ (𝐴 × 𝐴))) = ((dist‘𝐾) ↾ (((Base‘𝐾) ∩ 𝐴) × ((Base‘𝐾) ∩ 𝐴)))
10	7, 9	eqtri 2632	. . . 4 ⊢ (((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ↾ (𝐴 × 𝐴)) = ((dist‘𝐾) ↾ (((Base‘𝐾) ∩ 𝐴) × ((Base‘𝐾) ∩ 𝐴)))
11		eqid 2610	. . . . . . 7 ⊢ (𝐾 ↾s 𝐴) = (𝐾 ↾s 𝐴)
12		eqid 2610	. . . . . . 7 ⊢ (dist‘𝐾) = (dist‘𝐾)
13	11, 12	ressds 15896	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (dist‘𝐾) = (dist‘(𝐾 ↾s 𝐴)))
14	13	adantl 481	. . . . 5 ⊢ ((𝐾 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑉) → (dist‘𝐾) = (dist‘(𝐾 ↾s 𝐴)))
15		incom 3767	. . . . . . 7 ⊢ ((Base‘𝐾) ∩ 𝐴) = (𝐴 ∩ (Base‘𝐾))
16	11, 1	ressbas 15757	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ (Base‘𝐾)) = (Base‘(𝐾 ↾s 𝐴)))
17	16	adantl 481	. . . . . . 7 ⊢ ((𝐾 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑉) → (𝐴 ∩ (Base‘𝐾)) = (Base‘(𝐾 ↾s 𝐴)))
18	15, 17	syl5eq 2656	. . . . . 6 ⊢ ((𝐾 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑉) → ((Base‘𝐾) ∩ 𝐴) = (Base‘(𝐾 ↾s 𝐴)))
19	18	sqxpeqd 5065	. . . . 5 ⊢ ((𝐾 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑉) → (((Base‘𝐾) ∩ 𝐴) × ((Base‘𝐾) ∩ 𝐴)) = ((Base‘(𝐾 ↾s 𝐴)) × (Base‘(𝐾 ↾s 𝐴))))
20	14, 19	reseq12d 5318	. . . 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 2656	. . 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 6107	. . 3 ⊢ ((𝐾 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑉) → (∞Met‘((Base‘𝐾) ∩ 𝐴)) = (∞Met‘(Base‘(𝐾 ↾s 𝐴))))
23	6, 21, 22	3eltr3d 2702	. 2 ⊢ ((𝐾 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑉) → ((dist‘(𝐾 ↾s 𝐴)) ↾ ((Base‘(𝐾 ↾s 𝐴)) × (Base‘(𝐾 ↾s 𝐴)))) ∈ (∞Met‘(Base‘(𝐾 ↾s 𝐴))))
24		eqid 2610	. . . . . . 7 ⊢ (TopOpen‘𝐾) = (TopOpen‘𝐾)
25	24, 1, 2	xmstopn 22066	. . . . . 6 ⊢ (𝐾 ∈ ∞MetSp → (TopOpen‘𝐾) = (MetOpen‘((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))))
26	25	adantr 480	. . . . 5 ⊢ ((𝐾 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑉) → (TopOpen‘𝐾) = (MetOpen‘((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))))
27	26	oveq1d 6564	. . . 4 ⊢ ((𝐾 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑉) → ((TopOpen‘𝐾) ↾t ((Base‘𝐾) ∩ 𝐴)) = ((MetOpen‘((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))) ↾t ((Base‘𝐾) ∩ 𝐴)))
28		inss1 3795	. . . . 5 ⊢ ((Base‘𝐾) ∩ 𝐴) ⊆ (Base‘𝐾)
29		xpss12 5148	. . . . . . . . 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 704	. . . . . . . 8 ⊢ (((Base‘𝐾) ∩ 𝐴) × ((Base‘𝐾) ∩ 𝐴)) ⊆ ((Base‘𝐾) × (Base‘𝐾))
31		resabs1 5347	. . . . . . . 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 2635	. . . . . 6 ⊢ (((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ↾ (𝐴 × 𝐴)) = (((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ↾ (((Base‘𝐾) ∩ 𝐴) × ((Base‘𝐾) ∩ 𝐴)))
34		eqid 2610	. . . . . 6 ⊢ (MetOpen‘((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))) = (MetOpen‘((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))))
35		eqid 2610	. . . . . 6 ⊢ (MetOpen‘(((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ↾ (𝐴 × 𝐴))) = (MetOpen‘(((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ↾ (𝐴 × 𝐴)))
36	33, 34, 35	metrest 22139	. . . . 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 693	. . . 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 2644	. . 3 ⊢ ((𝐾 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑉) → ((TopOpen‘𝐾) ↾t ((Base‘𝐾) ∩ 𝐴)) = (MetOpen‘(((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ↾ (𝐴 × 𝐴))))
39		xmstps 22068	. . . . . . . . 9 ⊢ (𝐾 ∈ ∞MetSp → 𝐾 ∈ TopSp)
40	1, 24	tpsuni 20553	. . . . . . . . 9 ⊢ (𝐾 ∈ TopSp → (Base‘𝐾) = ∪ (TopOpen‘𝐾))
41	39, 40	syl 17	. . . . . . . 8 ⊢ (𝐾 ∈ ∞MetSp → (Base‘𝐾) = ∪ (TopOpen‘𝐾))
42	41	adantr 480	. . . . . . 7 ⊢ ((𝐾 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑉) → (Base‘𝐾) = ∪ (TopOpen‘𝐾))
43	42	ineq2d 3776	. . . . . 6 ⊢ ((𝐾 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑉) → (𝐴 ∩ (Base‘𝐾)) = (𝐴 ∩ ∪ (TopOpen‘𝐾)))
44	15, 43	syl5eq 2656	. . . . 5 ⊢ ((𝐾 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑉) → ((Base‘𝐾) ∩ 𝐴) = (𝐴 ∩ ∪ (TopOpen‘𝐾)))
45	44	oveq2d 6565	. . . 4 ⊢ ((𝐾 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑉) → ((TopOpen‘𝐾) ↾t ((Base‘𝐾) ∩ 𝐴)) = ((TopOpen‘𝐾) ↾t (𝐴 ∩ ∪ (TopOpen‘𝐾))))
46	1, 24	istps 20551	. . . . . 6 ⊢ (𝐾 ∈ TopSp ↔ (TopOpen‘𝐾) ∈ (TopOn‘(Base‘𝐾)))
47	39, 46	sylib 207	. . . . 5 ⊢ (𝐾 ∈ ∞MetSp → (TopOpen‘𝐾) ∈ (TopOn‘(Base‘𝐾)))
48		eqid 2610	. . . . . 6 ⊢ ∪ (TopOpen‘𝐾) = ∪ (TopOpen‘𝐾)
49	48	restin 20780	. . . . 5 ⊢ (((TopOpen‘𝐾) ∈ (TopOn‘(Base‘𝐾)) ∧ 𝐴 ∈ 𝑉) → ((TopOpen‘𝐾) ↾t 𝐴) = ((TopOpen‘𝐾) ↾t (𝐴 ∩ ∪ (TopOpen‘𝐾))))
50	47, 49	sylan 487	. . . 4 ⊢ ((𝐾 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑉) → ((TopOpen‘𝐾) ↾t 𝐴) = ((TopOpen‘𝐾) ↾t (𝐴 ∩ ∪ (TopOpen‘𝐾))))
51	45, 50	eqtr4d 2647	. . 3 ⊢ ((𝐾 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑉) → ((TopOpen‘𝐾) ↾t ((Base‘𝐾) ∩ 𝐴)) = ((TopOpen‘𝐾) ↾t 𝐴))
52	21	fveq2d 6107	. . 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 2652	. 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 20800	. . 3 ⊢ ((TopOpen‘𝐾) ↾t 𝐴) = (TopOpen‘(𝐾 ↾s 𝐴))
55		eqid 2610	. . 3 ⊢ (Base‘(𝐾 ↾s 𝐴)) = (Base‘(𝐾 ↾s 𝐴))
56		eqid 2610	. . 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 22063	. 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 695	1 ⊢ ((𝐾 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑉) → (𝐾 ↾s 𝐴) ∈ ∞MetSp)

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ∩ cin 3539   ⊆ wss 3540  ∪ cuni 4372   × cxp 5036   ↾ cres 5040  ‘cfv 5804  (class class class)co 6549  Basecbs 15695   ↾_s cress 15696  distcds 15777   ↾_t crest 15904  TopOpenctopn 15905  ∞Metcxmt 19552  MetOpencmopn 19557  TopOnctopon 20518  TopSpctps 20519  ∞MetSpcxme 21932

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-sup 8231  df-inf 8232  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-q 11665  df-rp 11709  df-xneg 11822  df-xadd 11823  df-xmul 11824  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-tset 15787  df-ds 15791  df-rest 15906  df-topn 15907  df-topgen 15927  df-psmet 19559  df-xmet 19560  df-bl 19562  df-mopn 19563  df-top 20521  df-bases 20522  df-topon 20523  df-topsp 20524  df-xms 21935

This theorem is referenced by:  ressms  22141  qqhcn  29363  qqhucn  29364