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Theorem ressxms 23681
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 2738 . . . . . 6 (Base‘𝐾) = (Base‘𝐾)
2 eqid 2738 . . . . . 6 ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))
31, 2xmsxmet 23609 . . . . 5 (𝐾 ∈ ∞MetSp → ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ∈ (∞Met‘(Base‘𝐾)))
43adantr 481 . . . 4 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ∈ (∞Met‘(Base‘𝐾)))
5 xmetres 23517 . . . 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 5904 . . . . 5 (((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ↾ (𝐴 × 𝐴)) = ((dist‘𝐾) ↾ (((Base‘𝐾) × (Base‘𝐾)) ∩ (𝐴 × 𝐴)))
8 inxp 5741 . . . . . 6 (((Base‘𝐾) × (Base‘𝐾)) ∩ (𝐴 × 𝐴)) = (((Base‘𝐾) ∩ 𝐴) × ((Base‘𝐾) ∩ 𝐴))
98reseq2i 5888 . . . . 5 ((dist‘𝐾) ↾ (((Base‘𝐾) × (Base‘𝐾)) ∩ (𝐴 × 𝐴))) = ((dist‘𝐾) ↾ (((Base‘𝐾) ∩ 𝐴) × ((Base‘𝐾) ∩ 𝐴)))
107, 9eqtri 2766 . . . 4 (((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ↾ (𝐴 × 𝐴)) = ((dist‘𝐾) ↾ (((Base‘𝐾) ∩ 𝐴) × ((Base‘𝐾) ∩ 𝐴)))
11 eqid 2738 . . . . . . 7 (𝐾s 𝐴) = (𝐾s 𝐴)
12 eqid 2738 . . . . . . 7 (dist‘𝐾) = (dist‘𝐾)
1311, 12ressds 17120 . . . . . 6 (𝐴𝑉 → (dist‘𝐾) = (dist‘(𝐾s 𝐴)))
1413adantl 482 . . . . 5 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → (dist‘𝐾) = (dist‘(𝐾s 𝐴)))
15 incom 4135 . . . . . . 7 ((Base‘𝐾) ∩ 𝐴) = (𝐴 ∩ (Base‘𝐾))
1611, 1ressbas 16947 . . . . . . . 8 (𝐴𝑉 → (𝐴 ∩ (Base‘𝐾)) = (Base‘(𝐾s 𝐴)))
1716adantl 482 . . . . . . 7 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → (𝐴 ∩ (Base‘𝐾)) = (Base‘(𝐾s 𝐴)))
1815, 17eqtrid 2790 . . . . . 6 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → ((Base‘𝐾) ∩ 𝐴) = (Base‘(𝐾s 𝐴)))
1918sqxpeqd 5621 . . . . 5 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → (((Base‘𝐾) ∩ 𝐴) × ((Base‘𝐾) ∩ 𝐴)) = ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴))))
2014, 19reseq12d 5892 . . . 4 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → ((dist‘𝐾) ↾ (((Base‘𝐾) ∩ 𝐴) × ((Base‘𝐾) ∩ 𝐴))) = ((dist‘(𝐾s 𝐴)) ↾ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴)))))
2110, 20eqtrid 2790 . . 3 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → (((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ↾ (𝐴 × 𝐴)) = ((dist‘(𝐾s 𝐴)) ↾ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴)))))
2218fveq2d 6778 . . 3 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → (∞Met‘((Base‘𝐾) ∩ 𝐴)) = (∞Met‘(Base‘(𝐾s 𝐴))))
236, 21, 223eltr3d 2853 . 2 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → ((dist‘(𝐾s 𝐴)) ↾ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴)))) ∈ (∞Met‘(Base‘(𝐾s 𝐴))))
24 eqid 2738 . . . . . . 7 (TopOpen‘𝐾) = (TopOpen‘𝐾)
2524, 1, 2xmstopn 23604 . . . . . 6 (𝐾 ∈ ∞MetSp → (TopOpen‘𝐾) = (MetOpen‘((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))))
2625adantr 481 . . . . 5 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → (TopOpen‘𝐾) = (MetOpen‘((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))))
2726oveq1d 7290 . . . 4 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → ((TopOpen‘𝐾) ↾t ((Base‘𝐾) ∩ 𝐴)) = ((MetOpen‘((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))) ↾t ((Base‘𝐾) ∩ 𝐴)))
28 inss1 4162 . . . . 5 ((Base‘𝐾) ∩ 𝐴) ⊆ (Base‘𝐾)
29 xpss12 5604 . . . . . . . . 9 ((((Base‘𝐾) ∩ 𝐴) ⊆ (Base‘𝐾) ∧ ((Base‘𝐾) ∩ 𝐴) ⊆ (Base‘𝐾)) → (((Base‘𝐾) ∩ 𝐴) × ((Base‘𝐾) ∩ 𝐴)) ⊆ ((Base‘𝐾) × (Base‘𝐾)))
3028, 28, 29mp2an 689 . . . . . . . 8 (((Base‘𝐾) ∩ 𝐴) × ((Base‘𝐾) ∩ 𝐴)) ⊆ ((Base‘𝐾) × (Base‘𝐾))
31 resabs1 5921 . . . . . . . 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 2769 . . . . . 6 (((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ↾ (𝐴 × 𝐴)) = (((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ↾ (((Base‘𝐾) ∩ 𝐴) × ((Base‘𝐾) ∩ 𝐴)))
34 eqid 2738 . . . . . 6 (MetOpen‘((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))) = (MetOpen‘((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))))
35 eqid 2738 . . . . . 6 (MetOpen‘(((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ↾ (𝐴 × 𝐴))) = (MetOpen‘(((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ↾ (𝐴 × 𝐴)))
3633, 34, 35metrest 23680 . . . . 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 586 . . . 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 2778 . . 3 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → ((TopOpen‘𝐾) ↾t ((Base‘𝐾) ∩ 𝐴)) = (MetOpen‘(((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ↾ (𝐴 × 𝐴))))
39 xmstps 23606 . . . . . . . . 9 (𝐾 ∈ ∞MetSp → 𝐾 ∈ TopSp)
401, 24tpsuni 22085 . . . . . . . . 9 (𝐾 ∈ TopSp → (Base‘𝐾) = (TopOpen‘𝐾))
4139, 40syl 17 . . . . . . . 8 (𝐾 ∈ ∞MetSp → (Base‘𝐾) = (TopOpen‘𝐾))
4241adantr 481 . . . . . . 7 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → (Base‘𝐾) = (TopOpen‘𝐾))
4342ineq2d 4146 . . . . . 6 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → (𝐴 ∩ (Base‘𝐾)) = (𝐴 (TopOpen‘𝐾)))
4415, 43eqtrid 2790 . . . . 5 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → ((Base‘𝐾) ∩ 𝐴) = (𝐴 (TopOpen‘𝐾)))
4544oveq2d 7291 . . . 4 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → ((TopOpen‘𝐾) ↾t ((Base‘𝐾) ∩ 𝐴)) = ((TopOpen‘𝐾) ↾t (𝐴 (TopOpen‘𝐾))))
461, 24istps 22083 . . . . . 6 (𝐾 ∈ TopSp ↔ (TopOpen‘𝐾) ∈ (TopOn‘(Base‘𝐾)))
4739, 46sylib 217 . . . . 5 (𝐾 ∈ ∞MetSp → (TopOpen‘𝐾) ∈ (TopOn‘(Base‘𝐾)))
48 eqid 2738 . . . . . 6 (TopOpen‘𝐾) = (TopOpen‘𝐾)
4948restin 22317 . . . . 5 (((TopOpen‘𝐾) ∈ (TopOn‘(Base‘𝐾)) ∧ 𝐴𝑉) → ((TopOpen‘𝐾) ↾t 𝐴) = ((TopOpen‘𝐾) ↾t (𝐴 (TopOpen‘𝐾))))
5047, 49sylan 580 . . . 4 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → ((TopOpen‘𝐾) ↾t 𝐴) = ((TopOpen‘𝐾) ↾t (𝐴 (TopOpen‘𝐾))))
5145, 50eqtr4d 2781 . . 3 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → ((TopOpen‘𝐾) ↾t ((Base‘𝐾) ∩ 𝐴)) = ((TopOpen‘𝐾) ↾t 𝐴))
5221fveq2d 6778 . . 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 2786 . 2 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → ((TopOpen‘𝐾) ↾t 𝐴) = (MetOpen‘((dist‘(𝐾s 𝐴)) ↾ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴))))))
5411, 24resstopn 22337 . . 3 ((TopOpen‘𝐾) ↾t 𝐴) = (TopOpen‘(𝐾s 𝐴))
55 eqid 2738 . . 3 (Base‘(𝐾s 𝐴)) = (Base‘(𝐾s 𝐴))
56 eqid 2738 . . 3 ((dist‘(𝐾s 𝐴)) ↾ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴)))) = ((dist‘(𝐾s 𝐴)) ↾ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴))))
5754, 55, 56isxms2 23601 . 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 583 1 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → (𝐾s 𝐴) ∈ ∞MetSp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1539  wcel 2106  cin 3886  wss 3887   cuni 4839   × cxp 5587  cres 5591  cfv 6433  (class class class)co 7275  Basecbs 16912  s cress 16941  distcds 16971  t crest 17131  TopOpenctopn 17132  ∞Metcxmet 20582  MetOpencmopn 20587  TopOnctopon 22059  TopSpctps 22081  ∞MetSpcxms 23470
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948  ax-pre-sup 10949
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-er 8498  df-map 8617  df-en 8734  df-dom 8735  df-sdom 8736  df-sup 9201  df-inf 9202  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-div 11633  df-nn 11974  df-2 12036  df-3 12037  df-4 12038  df-5 12039  df-6 12040  df-7 12041  df-8 12042  df-9 12043  df-n0 12234  df-z 12320  df-dec 12438  df-uz 12583  df-q 12689  df-rp 12731  df-xneg 12848  df-xadd 12849  df-xmul 12850  df-sets 16865  df-slot 16883  df-ndx 16895  df-base 16913  df-ress 16942  df-tset 16981  df-ds 16984  df-rest 17133  df-topn 17134  df-topgen 17154  df-psmet 20589  df-xmet 20590  df-bl 20592  df-mopn 20593  df-top 22043  df-topon 22060  df-topsp 22082  df-bases 22096  df-xms 23473
This theorem is referenced by:  ressms  23682  qqhcn  31941  qqhucn  31942
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