Theorem ressxms 22701

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 2826	. . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2826	. . . . . 6 ⊢ ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))
3	1, 2	xmsxmet 22632	. . . . 5 ⊢ (𝐾 ∈ ∞MetSp → ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ∈ (∞Met‘(Base‘𝐾)))
4	3	adantr 474	. . . 4 ⊢ ((𝐾 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑉) → ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ∈ (∞Met‘(Base‘𝐾)))
5		xmetres 22540	. . . 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 5647	. . . . 5 ⊢ (((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ↾ (𝐴 × 𝐴)) = ((dist‘𝐾) ↾ (((Base‘𝐾) × (Base‘𝐾)) ∩ (𝐴 × 𝐴)))
8		inxp 5488	. . . . . 6 ⊢ (((Base‘𝐾) × (Base‘𝐾)) ∩ (𝐴 × 𝐴)) = (((Base‘𝐾) ∩ 𝐴) × ((Base‘𝐾) ∩ 𝐴))
9	8	reseq2i 5627	. . . . 5 ⊢ ((dist‘𝐾) ↾ (((Base‘𝐾) × (Base‘𝐾)) ∩ (𝐴 × 𝐴))) = ((dist‘𝐾) ↾ (((Base‘𝐾) ∩ 𝐴) × ((Base‘𝐾) ∩ 𝐴)))
10	7, 9	eqtri 2850	. . . 4 ⊢ (((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ↾ (𝐴 × 𝐴)) = ((dist‘𝐾) ↾ (((Base‘𝐾) ∩ 𝐴) × ((Base‘𝐾) ∩ 𝐴)))
11		eqid 2826	. . . . . . 7 ⊢ (𝐾 ↾s 𝐴) = (𝐾 ↾s 𝐴)
12		eqid 2826	. . . . . . 7 ⊢ (dist‘𝐾) = (dist‘𝐾)
13	11, 12	ressds 16427	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (dist‘𝐾) = (dist‘(𝐾 ↾s 𝐴)))
14	13	adantl 475	. . . . 5 ⊢ ((𝐾 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑉) → (dist‘𝐾) = (dist‘(𝐾 ↾s 𝐴)))
15		incom 4033	. . . . . . 7 ⊢ ((Base‘𝐾) ∩ 𝐴) = (𝐴 ∩ (Base‘𝐾))
16	11, 1	ressbas 16294	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ (Base‘𝐾)) = (Base‘(𝐾 ↾s 𝐴)))
17	16	adantl 475	. . . . . . 7 ⊢ ((𝐾 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑉) → (𝐴 ∩ (Base‘𝐾)) = (Base‘(𝐾 ↾s 𝐴)))
18	15, 17	syl5eq 2874	. . . . . 6 ⊢ ((𝐾 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑉) → ((Base‘𝐾) ∩ 𝐴) = (Base‘(𝐾 ↾s 𝐴)))
19	18	sqxpeqd 5375	. . . . 5 ⊢ ((𝐾 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑉) → (((Base‘𝐾) ∩ 𝐴) × ((Base‘𝐾) ∩ 𝐴)) = ((Base‘(𝐾 ↾s 𝐴)) × (Base‘(𝐾 ↾s 𝐴))))
20	14, 19	reseq12d 5631	. . . 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 2874	. . 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 6438	. . 3 ⊢ ((𝐾 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑉) → (∞Met‘((Base‘𝐾) ∩ 𝐴)) = (∞Met‘(Base‘(𝐾 ↾s 𝐴))))
23	6, 21, 22	3eltr3d 2921	. 2 ⊢ ((𝐾 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑉) → ((dist‘(𝐾 ↾s 𝐴)) ↾ ((Base‘(𝐾 ↾s 𝐴)) × (Base‘(𝐾 ↾s 𝐴)))) ∈ (∞Met‘(Base‘(𝐾 ↾s 𝐴))))
24		eqid 2826	. . . . . . 7 ⊢ (TopOpen‘𝐾) = (TopOpen‘𝐾)
25	24, 1, 2	xmstopn 22627	. . . . . 6 ⊢ (𝐾 ∈ ∞MetSp → (TopOpen‘𝐾) = (MetOpen‘((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))))
26	25	adantr 474	. . . . 5 ⊢ ((𝐾 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑉) → (TopOpen‘𝐾) = (MetOpen‘((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))))
27	26	oveq1d 6921	. . . 4 ⊢ ((𝐾 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑉) → ((TopOpen‘𝐾) ↾t ((Base‘𝐾) ∩ 𝐴)) = ((MetOpen‘((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))) ↾t ((Base‘𝐾) ∩ 𝐴)))
28		inss1 4058	. . . . 5 ⊢ ((Base‘𝐾) ∩ 𝐴) ⊆ (Base‘𝐾)
29		xpss12 5358	. . . . . . . . 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 685	. . . . . . . 8 ⊢ (((Base‘𝐾) ∩ 𝐴) × ((Base‘𝐾) ∩ 𝐴)) ⊆ ((Base‘𝐾) × (Base‘𝐾))
31		resabs1 5664	. . . . . . . 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 2853	. . . . . 6 ⊢ (((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ↾ (𝐴 × 𝐴)) = (((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ↾ (((Base‘𝐾) ∩ 𝐴) × ((Base‘𝐾) ∩ 𝐴)))
34		eqid 2826	. . . . . 6 ⊢ (MetOpen‘((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))) = (MetOpen‘((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))))
35		eqid 2826	. . . . . 6 ⊢ (MetOpen‘(((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ↾ (𝐴 × 𝐴))) = (MetOpen‘(((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ↾ (𝐴 × 𝐴)))
36	33, 34, 35	metrest 22700	. . . . 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 582	. . . 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 2862	. . 3 ⊢ ((𝐾 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑉) → ((TopOpen‘𝐾) ↾t ((Base‘𝐾) ∩ 𝐴)) = (MetOpen‘(((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ↾ (𝐴 × 𝐴))))
39		xmstps 22629	. . . . . . . . 9 ⊢ (𝐾 ∈ ∞MetSp → 𝐾 ∈ TopSp)
40	1, 24	tpsuni 21112	. . . . . . . . 9 ⊢ (𝐾 ∈ TopSp → (Base‘𝐾) = ∪ (TopOpen‘𝐾))
41	39, 40	syl 17	. . . . . . . 8 ⊢ (𝐾 ∈ ∞MetSp → (Base‘𝐾) = ∪ (TopOpen‘𝐾))
42	41	adantr 474	. . . . . . 7 ⊢ ((𝐾 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑉) → (Base‘𝐾) = ∪ (TopOpen‘𝐾))
43	42	ineq2d 4042	. . . . . 6 ⊢ ((𝐾 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑉) → (𝐴 ∩ (Base‘𝐾)) = (𝐴 ∩ ∪ (TopOpen‘𝐾)))
44	15, 43	syl5eq 2874	. . . . 5 ⊢ ((𝐾 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑉) → ((Base‘𝐾) ∩ 𝐴) = (𝐴 ∩ ∪ (TopOpen‘𝐾)))
45	44	oveq2d 6922	. . . 4 ⊢ ((𝐾 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑉) → ((TopOpen‘𝐾) ↾t ((Base‘𝐾) ∩ 𝐴)) = ((TopOpen‘𝐾) ↾t (𝐴 ∩ ∪ (TopOpen‘𝐾))))
46	1, 24	istps 21110	. . . . . 6 ⊢ (𝐾 ∈ TopSp ↔ (TopOpen‘𝐾) ∈ (TopOn‘(Base‘𝐾)))
47	39, 46	sylib 210	. . . . 5 ⊢ (𝐾 ∈ ∞MetSp → (TopOpen‘𝐾) ∈ (TopOn‘(Base‘𝐾)))
48		eqid 2826	. . . . . 6 ⊢ ∪ (TopOpen‘𝐾) = ∪ (TopOpen‘𝐾)
49	48	restin 21342	. . . . 5 ⊢ (((TopOpen‘𝐾) ∈ (TopOn‘(Base‘𝐾)) ∧ 𝐴 ∈ 𝑉) → ((TopOpen‘𝐾) ↾t 𝐴) = ((TopOpen‘𝐾) ↾t (𝐴 ∩ ∪ (TopOpen‘𝐾))))
50	47, 49	sylan 577	. . . 4 ⊢ ((𝐾 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑉) → ((TopOpen‘𝐾) ↾t 𝐴) = ((TopOpen‘𝐾) ↾t (𝐴 ∩ ∪ (TopOpen‘𝐾))))
51	45, 50	eqtr4d 2865	. . 3 ⊢ ((𝐾 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑉) → ((TopOpen‘𝐾) ↾t ((Base‘𝐾) ∩ 𝐴)) = ((TopOpen‘𝐾) ↾t 𝐴))
52	21	fveq2d 6438	. . 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 2870	. 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 21362	. . 3 ⊢ ((TopOpen‘𝐾) ↾t 𝐴) = (TopOpen‘(𝐾 ↾s 𝐴))
55		eqid 2826	. . 3 ⊢ (Base‘(𝐾 ↾s 𝐴)) = (Base‘(𝐾 ↾s 𝐴))
56		eqid 2826	. . 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 22624	. 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 580	1 ⊢ ((𝐾 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑉) → (𝐾 ↾s 𝐴) ∈ ∞MetSp)

Syntax hints:   → wi 4   ∧ wa 386   = wceq 1658   ∈ wcel 2166   ∩ cin 3798   ⊆ wss 3799  ∪ cuni 4659   × cxp 5341   ↾ cres 5345  ‘cfv 6124  (class class class)co 6906  Basecbs 16223   ↾_s cress 16224  distcds 16315   ↾_t crest 16435  TopOpenctopn 16436  ∞Metcxmet 20092  MetOpencmopn 20097  TopOnctopon 21086  TopSpctps 21108  ∞MetSpcxms 22493

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-rep 4995  ax-sep 5006  ax-nul 5014  ax-pow 5066  ax-pr 5128  ax-un 7210  ax-cnex 10309  ax-resscn 10310  ax-1cn 10311  ax-icn 10312  ax-addcl 10313  ax-addrcl 10314  ax-mulcl 10315  ax-mulrcl 10316  ax-mulcom 10317  ax-addass 10318  ax-mulass 10319  ax-distr 10320  ax-i2m1 10321  ax-1ne0 10322  ax-1rid 10323  ax-rnegex 10324  ax-rrecex 10325  ax-cnre 10326  ax-pre-lttri 10327  ax-pre-lttrn 10328  ax-pre-ltadd 10329  ax-pre-mulgt0 10330  ax-pre-sup 10331

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ne 3001  df-nel 3104  df-ral 3123  df-rex 3124  df-reu 3125  df-rmo 3126  df-rab 3127  df-v 3417  df-sbc 3664  df-csb 3759  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-pss 3815  df-nul 4146  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4660  df-iun 4743  df-br 4875  df-opab 4937  df-mpt 4954  df-tr 4977  df-id 5251  df-eprel 5256  df-po 5264  df-so 5265  df-fr 5302  df-we 5304  df-xp 5349  df-rel 5350  df-cnv 5351  df-co 5352  df-dm 5353  df-rn 5354  df-res 5355  df-ima 5356  df-pred 5921  df-ord 5967  df-on 5968  df-lim 5969  df-suc 5970  df-iota 6087  df-fun 6126  df-fn 6127  df-f 6128  df-f1 6129  df-fo 6130  df-f1o 6131  df-fv 6132  df-riota 6867  df-ov 6909  df-oprab 6910  df-mpt2 6911  df-om 7328  df-1st 7429  df-2nd 7430  df-wrecs 7673  df-recs 7735  df-rdg 7773  df-er 8010  df-map 8125  df-en 8224  df-dom 8225  df-sdom 8226  df-sup 8618  df-inf 8619  df-pnf 10394  df-mnf 10395  df-xr 10396  df-ltxr 10397  df-le 10398  df-sub 10588  df-neg 10589  df-div 11011  df-nn 11352  df-2 11415  df-3 11416  df-4 11417  df-5 11418  df-6 11419  df-7 11420  df-8 11421  df-9 11422  df-n0 11620  df-z 11706  df-dec 11823  df-uz 11970  df-q 12073  df-rp 12114  df-xneg 12233  df-xadd 12234  df-xmul 12235  df-ndx 16226  df-slot 16227  df-base 16229  df-sets 16230  df-ress 16231  df-tset 16325  df-ds 16328  df-rest 16437  df-topn 16438  df-topgen 16458  df-psmet 20099  df-xmet 20100  df-bl 20102  df-mopn 20103  df-top 21070  df-topon 21087  df-topsp 21109  df-bases 21122  df-xms 22496

This theorem is referenced by:  ressms  22702  qqhcn  30581  qqhucn  30582