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Theorem ressxms 22608
Description: The restriction of a metric space is a metric space. (Contributed by Mario Carneiro, 24-Aug-2015.)
Assertion
Ref Expression
ressxms ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → (𝐾s 𝐴) ∈ ∞MetSp)

Proof of Theorem ressxms
StepHypRef Expression
1 eqid 2764 . . . . . 6 (Base‘𝐾) = (Base‘𝐾)
2 eqid 2764 . . . . . 6 ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))
31, 2xmsxmet 22539 . . . . 5 (𝐾 ∈ ∞MetSp → ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ∈ (∞Met‘(Base‘𝐾)))
43adantr 472 . . . 4 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ∈ (∞Met‘(Base‘𝐾)))
5 xmetres 22447 . . . 4 (((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ∈ (∞Met‘(Base‘𝐾)) → (((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ↾ (𝐴 × 𝐴)) ∈ (∞Met‘((Base‘𝐾) ∩ 𝐴)))
64, 5syl 17 . . 3 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → (((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ↾ (𝐴 × 𝐴)) ∈ (∞Met‘((Base‘𝐾) ∩ 𝐴)))
7 resres 5584 . . . . 5 (((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ↾ (𝐴 × 𝐴)) = ((dist‘𝐾) ↾ (((Base‘𝐾) × (Base‘𝐾)) ∩ (𝐴 × 𝐴)))
8 inxp 5422 . . . . . 6 (((Base‘𝐾) × (Base‘𝐾)) ∩ (𝐴 × 𝐴)) = (((Base‘𝐾) ∩ 𝐴) × ((Base‘𝐾) ∩ 𝐴))
98reseq2i 5561 . . . . 5 ((dist‘𝐾) ↾ (((Base‘𝐾) × (Base‘𝐾)) ∩ (𝐴 × 𝐴))) = ((dist‘𝐾) ↾ (((Base‘𝐾) ∩ 𝐴) × ((Base‘𝐾) ∩ 𝐴)))
107, 9eqtri 2786 . . . 4 (((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ↾ (𝐴 × 𝐴)) = ((dist‘𝐾) ↾ (((Base‘𝐾) ∩ 𝐴) × ((Base‘𝐾) ∩ 𝐴)))
11 eqid 2764 . . . . . . 7 (𝐾s 𝐴) = (𝐾s 𝐴)
12 eqid 2764 . . . . . . 7 (dist‘𝐾) = (dist‘𝐾)
1311, 12ressds 16340 . . . . . 6 (𝐴𝑉 → (dist‘𝐾) = (dist‘(𝐾s 𝐴)))
1413adantl 473 . . . . 5 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → (dist‘𝐾) = (dist‘(𝐾s 𝐴)))
15 incom 3966 . . . . . . 7 ((Base‘𝐾) ∩ 𝐴) = (𝐴 ∩ (Base‘𝐾))
1611, 1ressbas 16203 . . . . . . . 8 (𝐴𝑉 → (𝐴 ∩ (Base‘𝐾)) = (Base‘(𝐾s 𝐴)))
1716adantl 473 . . . . . . 7 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → (𝐴 ∩ (Base‘𝐾)) = (Base‘(𝐾s 𝐴)))
1815, 17syl5eq 2810 . . . . . 6 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → ((Base‘𝐾) ∩ 𝐴) = (Base‘(𝐾s 𝐴)))
1918sqxpeqd 5308 . . . . 5 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → (((Base‘𝐾) ∩ 𝐴) × ((Base‘𝐾) ∩ 𝐴)) = ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴))))
2014, 19reseq12d 5565 . . . 4 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → ((dist‘𝐾) ↾ (((Base‘𝐾) ∩ 𝐴) × ((Base‘𝐾) ∩ 𝐴))) = ((dist‘(𝐾s 𝐴)) ↾ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴)))))
2110, 20syl5eq 2810 . . 3 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → (((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ↾ (𝐴 × 𝐴)) = ((dist‘(𝐾s 𝐴)) ↾ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴)))))
2218fveq2d 6378 . . 3 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → (∞Met‘((Base‘𝐾) ∩ 𝐴)) = (∞Met‘(Base‘(𝐾s 𝐴))))
236, 21, 223eltr3d 2857 . 2 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → ((dist‘(𝐾s 𝐴)) ↾ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴)))) ∈ (∞Met‘(Base‘(𝐾s 𝐴))))
24 eqid 2764 . . . . . . 7 (TopOpen‘𝐾) = (TopOpen‘𝐾)
2524, 1, 2xmstopn 22534 . . . . . 6 (𝐾 ∈ ∞MetSp → (TopOpen‘𝐾) = (MetOpen‘((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))))
2625adantr 472 . . . . 5 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → (TopOpen‘𝐾) = (MetOpen‘((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))))
2726oveq1d 6856 . . . 4 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → ((TopOpen‘𝐾) ↾t ((Base‘𝐾) ∩ 𝐴)) = ((MetOpen‘((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))) ↾t ((Base‘𝐾) ∩ 𝐴)))
28 inss1 3991 . . . . 5 ((Base‘𝐾) ∩ 𝐴) ⊆ (Base‘𝐾)
29 xpss12 5291 . . . . . . . . 9 ((((Base‘𝐾) ∩ 𝐴) ⊆ (Base‘𝐾) ∧ ((Base‘𝐾) ∩ 𝐴) ⊆ (Base‘𝐾)) → (((Base‘𝐾) ∩ 𝐴) × ((Base‘𝐾) ∩ 𝐴)) ⊆ ((Base‘𝐾) × (Base‘𝐾)))
3028, 28, 29mp2an 683 . . . . . . . 8 (((Base‘𝐾) ∩ 𝐴) × ((Base‘𝐾) ∩ 𝐴)) ⊆ ((Base‘𝐾) × (Base‘𝐾))
31 resabs1 5601 . . . . . . . 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))))
3230, 31ax-mp 5 . . . . . . 7 (((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ↾ (((Base‘𝐾) ∩ 𝐴) × ((Base‘𝐾) ∩ 𝐴))) = ((dist‘𝐾) ↾ (((Base‘𝐾) ∩ 𝐴) × ((Base‘𝐾) ∩ 𝐴)))
3310, 32eqtr4i 2789 . . . . . 6 (((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ↾ (𝐴 × 𝐴)) = (((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ↾ (((Base‘𝐾) ∩ 𝐴) × ((Base‘𝐾) ∩ 𝐴)))
34 eqid 2764 . . . . . 6 (MetOpen‘((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))) = (MetOpen‘((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))))
35 eqid 2764 . . . . . 6 (MetOpen‘(((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ↾ (𝐴 × 𝐴))) = (MetOpen‘(((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ↾ (𝐴 × 𝐴)))
3633, 34, 35metrest 22607 . . . . 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))))
374, 28, 36sylancl 580 . . . 4 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → ((MetOpen‘((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))) ↾t ((Base‘𝐾) ∩ 𝐴)) = (MetOpen‘(((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ↾ (𝐴 × 𝐴))))
3827, 37eqtrd 2798 . . 3 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → ((TopOpen‘𝐾) ↾t ((Base‘𝐾) ∩ 𝐴)) = (MetOpen‘(((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ↾ (𝐴 × 𝐴))))
39 xmstps 22536 . . . . . . . . 9 (𝐾 ∈ ∞MetSp → 𝐾 ∈ TopSp)
401, 24tpsuni 21019 . . . . . . . . 9 (𝐾 ∈ TopSp → (Base‘𝐾) = (TopOpen‘𝐾))
4139, 40syl 17 . . . . . . . 8 (𝐾 ∈ ∞MetSp → (Base‘𝐾) = (TopOpen‘𝐾))
4241adantr 472 . . . . . . 7 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → (Base‘𝐾) = (TopOpen‘𝐾))
4342ineq2d 3975 . . . . . 6 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → (𝐴 ∩ (Base‘𝐾)) = (𝐴 (TopOpen‘𝐾)))
4415, 43syl5eq 2810 . . . . 5 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → ((Base‘𝐾) ∩ 𝐴) = (𝐴 (TopOpen‘𝐾)))
4544oveq2d 6857 . . . 4 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → ((TopOpen‘𝐾) ↾t ((Base‘𝐾) ∩ 𝐴)) = ((TopOpen‘𝐾) ↾t (𝐴 (TopOpen‘𝐾))))
461, 24istps 21017 . . . . . 6 (𝐾 ∈ TopSp ↔ (TopOpen‘𝐾) ∈ (TopOn‘(Base‘𝐾)))
4739, 46sylib 209 . . . . 5 (𝐾 ∈ ∞MetSp → (TopOpen‘𝐾) ∈ (TopOn‘(Base‘𝐾)))
48 eqid 2764 . . . . . 6 (TopOpen‘𝐾) = (TopOpen‘𝐾)
4948restin 21249 . . . . 5 (((TopOpen‘𝐾) ∈ (TopOn‘(Base‘𝐾)) ∧ 𝐴𝑉) → ((TopOpen‘𝐾) ↾t 𝐴) = ((TopOpen‘𝐾) ↾t (𝐴 (TopOpen‘𝐾))))
5047, 49sylan 575 . . . 4 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → ((TopOpen‘𝐾) ↾t 𝐴) = ((TopOpen‘𝐾) ↾t (𝐴 (TopOpen‘𝐾))))
5145, 50eqtr4d 2801 . . 3 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → ((TopOpen‘𝐾) ↾t ((Base‘𝐾) ∩ 𝐴)) = ((TopOpen‘𝐾) ↾t 𝐴))
5221fveq2d 6378 . . 3 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → (MetOpen‘(((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ↾ (𝐴 × 𝐴))) = (MetOpen‘((dist‘(𝐾s 𝐴)) ↾ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴))))))
5338, 51, 523eqtr3d 2806 . 2 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → ((TopOpen‘𝐾) ↾t 𝐴) = (MetOpen‘((dist‘(𝐾s 𝐴)) ↾ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴))))))
5411, 24resstopn 21269 . . 3 ((TopOpen‘𝐾) ↾t 𝐴) = (TopOpen‘(𝐾s 𝐴))
55 eqid 2764 . . 3 (Base‘(𝐾s 𝐴)) = (Base‘(𝐾s 𝐴))
56 eqid 2764 . . 3 ((dist‘(𝐾s 𝐴)) ↾ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴)))) = ((dist‘(𝐾s 𝐴)) ↾ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴))))
5754, 55, 56isxms2 22531 . 2 ((𝐾s 𝐴) ∈ ∞MetSp ↔ (((dist‘(𝐾s 𝐴)) ↾ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴)))) ∈ (∞Met‘(Base‘(𝐾s 𝐴))) ∧ ((TopOpen‘𝐾) ↾t 𝐴) = (MetOpen‘((dist‘(𝐾s 𝐴)) ↾ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴)))))))
5823, 53, 57sylanbrc 578 1 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → (𝐾s 𝐴) ∈ ∞MetSp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1652  wcel 2155  cin 3730  wss 3731   cuni 4593   × cxp 5274  cres 5278  cfv 6067  (class class class)co 6841  Basecbs 16131  s cress 16132  distcds 16224  t crest 16348  TopOpenctopn 16349  ∞Metcxmet 20003  MetOpencmopn 20008  TopOnctopon 20993  TopSpctps 21015  ∞MetSpcxms 22400
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2349  ax-ext 2742  ax-rep 4929  ax-sep 4940  ax-nul 4948  ax-pow 5000  ax-pr 5061  ax-un 7146  ax-cnex 10244  ax-resscn 10245  ax-1cn 10246  ax-icn 10247  ax-addcl 10248  ax-addrcl 10249  ax-mulcl 10250  ax-mulrcl 10251  ax-mulcom 10252  ax-addass 10253  ax-mulass 10254  ax-distr 10255  ax-i2m1 10256  ax-1ne0 10257  ax-1rid 10258  ax-rnegex 10259  ax-rrecex 10260  ax-cnre 10261  ax-pre-lttri 10262  ax-pre-lttrn 10263  ax-pre-ltadd 10264  ax-pre-mulgt0 10265  ax-pre-sup 10266
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2564  df-eu 2581  df-clab 2751  df-cleq 2757  df-clel 2760  df-nfc 2895  df-ne 2937  df-nel 3040  df-ral 3059  df-rex 3060  df-reu 3061  df-rmo 3062  df-rab 3063  df-v 3351  df-sbc 3596  df-csb 3691  df-dif 3734  df-un 3736  df-in 3738  df-ss 3745  df-pss 3747  df-nul 4079  df-if 4243  df-pw 4316  df-sn 4334  df-pr 4336  df-tp 4338  df-op 4340  df-uni 4594  df-iun 4677  df-br 4809  df-opab 4871  df-mpt 4888  df-tr 4911  df-id 5184  df-eprel 5189  df-po 5197  df-so 5198  df-fr 5235  df-we 5237  df-xp 5282  df-rel 5283  df-cnv 5284  df-co 5285  df-dm 5286  df-rn 5287  df-res 5288  df-ima 5289  df-pred 5864  df-ord 5910  df-on 5911  df-lim 5912  df-suc 5913  df-iota 6030  df-fun 6069  df-fn 6070  df-f 6071  df-f1 6072  df-fo 6073  df-f1o 6074  df-fv 6075  df-riota 6802  df-ov 6844  df-oprab 6845  df-mpt2 6846  df-om 7263  df-1st 7365  df-2nd 7366  df-wrecs 7609  df-recs 7671  df-rdg 7709  df-er 7946  df-map 8061  df-en 8160  df-dom 8161  df-sdom 8162  df-sup 8554  df-inf 8555  df-pnf 10329  df-mnf 10330  df-xr 10331  df-ltxr 10332  df-le 10333  df-sub 10521  df-neg 10522  df-div 10938  df-nn 11274  df-2 11334  df-3 11335  df-4 11336  df-5 11337  df-6 11338  df-7 11339  df-8 11340  df-9 11341  df-n0 11538  df-z 11624  df-dec 11740  df-uz 11886  df-q 11989  df-rp 12028  df-xneg 12145  df-xadd 12146  df-xmul 12147  df-ndx 16134  df-slot 16135  df-base 16137  df-sets 16138  df-ress 16139  df-tset 16234  df-ds 16237  df-rest 16350  df-topn 16351  df-topgen 16371  df-psmet 20010  df-xmet 20011  df-bl 20013  df-mopn 20014  df-top 20977  df-topon 20994  df-topsp 21016  df-bases 21029  df-xms 22403
This theorem is referenced by:  ressms  22609  qqhcn  30416  qqhucn  30417
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