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Theorem ressxms 24452
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 2727 . . . . . 6 (Base‘𝐾) = (Base‘𝐾)
2 eqid 2727 . . . . . 6 ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))
31, 2xmsxmet 24380 . . . . 5 (𝐾 ∈ ∞MetSp → ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ∈ (∞Met‘(Base‘𝐾)))
43adantr 479 . . . 4 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ∈ (∞Met‘(Base‘𝐾)))
5 xmetres 24288 . . . 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 6000 . . . . 5 (((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ↾ (𝐴 × 𝐴)) = ((dist‘𝐾) ↾ (((Base‘𝐾) × (Base‘𝐾)) ∩ (𝐴 × 𝐴)))
8 inxp 5836 . . . . . 6 (((Base‘𝐾) × (Base‘𝐾)) ∩ (𝐴 × 𝐴)) = (((Base‘𝐾) ∩ 𝐴) × ((Base‘𝐾) ∩ 𝐴))
98reseq2i 5984 . . . . 5 ((dist‘𝐾) ↾ (((Base‘𝐾) × (Base‘𝐾)) ∩ (𝐴 × 𝐴))) = ((dist‘𝐾) ↾ (((Base‘𝐾) ∩ 𝐴) × ((Base‘𝐾) ∩ 𝐴)))
107, 9eqtri 2755 . . . 4 (((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ↾ (𝐴 × 𝐴)) = ((dist‘𝐾) ↾ (((Base‘𝐾) ∩ 𝐴) × ((Base‘𝐾) ∩ 𝐴)))
11 eqid 2727 . . . . . . 7 (𝐾s 𝐴) = (𝐾s 𝐴)
12 eqid 2727 . . . . . . 7 (dist‘𝐾) = (dist‘𝐾)
1311, 12ressds 17396 . . . . . 6 (𝐴𝑉 → (dist‘𝐾) = (dist‘(𝐾s 𝐴)))
1413adantl 480 . . . . 5 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → (dist‘𝐾) = (dist‘(𝐾s 𝐴)))
15 incom 4201 . . . . . . 7 ((Base‘𝐾) ∩ 𝐴) = (𝐴 ∩ (Base‘𝐾))
1611, 1ressbas 17220 . . . . . . . 8 (𝐴𝑉 → (𝐴 ∩ (Base‘𝐾)) = (Base‘(𝐾s 𝐴)))
1716adantl 480 . . . . . . 7 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → (𝐴 ∩ (Base‘𝐾)) = (Base‘(𝐾s 𝐴)))
1815, 17eqtrid 2779 . . . . . 6 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → ((Base‘𝐾) ∩ 𝐴) = (Base‘(𝐾s 𝐴)))
1918sqxpeqd 5712 . . . . 5 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → (((Base‘𝐾) ∩ 𝐴) × ((Base‘𝐾) ∩ 𝐴)) = ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴))))
2014, 19reseq12d 5988 . . . 4 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → ((dist‘𝐾) ↾ (((Base‘𝐾) ∩ 𝐴) × ((Base‘𝐾) ∩ 𝐴))) = ((dist‘(𝐾s 𝐴)) ↾ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴)))))
2110, 20eqtrid 2779 . . 3 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → (((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ↾ (𝐴 × 𝐴)) = ((dist‘(𝐾s 𝐴)) ↾ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴)))))
2218fveq2d 6904 . . 3 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → (∞Met‘((Base‘𝐾) ∩ 𝐴)) = (∞Met‘(Base‘(𝐾s 𝐴))))
236, 21, 223eltr3d 2842 . 2 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → ((dist‘(𝐾s 𝐴)) ↾ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴)))) ∈ (∞Met‘(Base‘(𝐾s 𝐴))))
24 eqid 2727 . . . . . . 7 (TopOpen‘𝐾) = (TopOpen‘𝐾)
2524, 1, 2xmstopn 24375 . . . . . 6 (𝐾 ∈ ∞MetSp → (TopOpen‘𝐾) = (MetOpen‘((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))))
2625adantr 479 . . . . 5 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → (TopOpen‘𝐾) = (MetOpen‘((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))))
2726oveq1d 7439 . . . 4 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → ((TopOpen‘𝐾) ↾t ((Base‘𝐾) ∩ 𝐴)) = ((MetOpen‘((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))) ↾t ((Base‘𝐾) ∩ 𝐴)))
28 inss1 4229 . . . . 5 ((Base‘𝐾) ∩ 𝐴) ⊆ (Base‘𝐾)
29 xpss12 5695 . . . . . . . . 9 ((((Base‘𝐾) ∩ 𝐴) ⊆ (Base‘𝐾) ∧ ((Base‘𝐾) ∩ 𝐴) ⊆ (Base‘𝐾)) → (((Base‘𝐾) ∩ 𝐴) × ((Base‘𝐾) ∩ 𝐴)) ⊆ ((Base‘𝐾) × (Base‘𝐾)))
3028, 28, 29mp2an 690 . . . . . . . 8 (((Base‘𝐾) ∩ 𝐴) × ((Base‘𝐾) ∩ 𝐴)) ⊆ ((Base‘𝐾) × (Base‘𝐾))
31 resabs1 6014 . . . . . . . 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 2758 . . . . . 6 (((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ↾ (𝐴 × 𝐴)) = (((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ↾ (((Base‘𝐾) ∩ 𝐴) × ((Base‘𝐾) ∩ 𝐴)))
34 eqid 2727 . . . . . 6 (MetOpen‘((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))) = (MetOpen‘((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))))
35 eqid 2727 . . . . . 6 (MetOpen‘(((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ↾ (𝐴 × 𝐴))) = (MetOpen‘(((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ↾ (𝐴 × 𝐴)))
3633, 34, 35metrest 24451 . . . . 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 584 . . . 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 2767 . . 3 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → ((TopOpen‘𝐾) ↾t ((Base‘𝐾) ∩ 𝐴)) = (MetOpen‘(((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ↾ (𝐴 × 𝐴))))
39 xmstps 24377 . . . . . . . . 9 (𝐾 ∈ ∞MetSp → 𝐾 ∈ TopSp)
401, 24tpsuni 22856 . . . . . . . . 9 (𝐾 ∈ TopSp → (Base‘𝐾) = (TopOpen‘𝐾))
4139, 40syl 17 . . . . . . . 8 (𝐾 ∈ ∞MetSp → (Base‘𝐾) = (TopOpen‘𝐾))
4241adantr 479 . . . . . . 7 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → (Base‘𝐾) = (TopOpen‘𝐾))
4342ineq2d 4212 . . . . . 6 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → (𝐴 ∩ (Base‘𝐾)) = (𝐴 (TopOpen‘𝐾)))
4415, 43eqtrid 2779 . . . . 5 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → ((Base‘𝐾) ∩ 𝐴) = (𝐴 (TopOpen‘𝐾)))
4544oveq2d 7440 . . . 4 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → ((TopOpen‘𝐾) ↾t ((Base‘𝐾) ∩ 𝐴)) = ((TopOpen‘𝐾) ↾t (𝐴 (TopOpen‘𝐾))))
461, 24istps 22854 . . . . . 6 (𝐾 ∈ TopSp ↔ (TopOpen‘𝐾) ∈ (TopOn‘(Base‘𝐾)))
4739, 46sylib 217 . . . . 5 (𝐾 ∈ ∞MetSp → (TopOpen‘𝐾) ∈ (TopOn‘(Base‘𝐾)))
48 eqid 2727 . . . . . 6 (TopOpen‘𝐾) = (TopOpen‘𝐾)
4948restin 23088 . . . . 5 (((TopOpen‘𝐾) ∈ (TopOn‘(Base‘𝐾)) ∧ 𝐴𝑉) → ((TopOpen‘𝐾) ↾t 𝐴) = ((TopOpen‘𝐾) ↾t (𝐴 (TopOpen‘𝐾))))
5047, 49sylan 578 . . . 4 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → ((TopOpen‘𝐾) ↾t 𝐴) = ((TopOpen‘𝐾) ↾t (𝐴 (TopOpen‘𝐾))))
5145, 50eqtr4d 2770 . . 3 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → ((TopOpen‘𝐾) ↾t ((Base‘𝐾) ∩ 𝐴)) = ((TopOpen‘𝐾) ↾t 𝐴))
5221fveq2d 6904 . . 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 2775 . 2 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → ((TopOpen‘𝐾) ↾t 𝐴) = (MetOpen‘((dist‘(𝐾s 𝐴)) ↾ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴))))))
5411, 24resstopn 23108 . . 3 ((TopOpen‘𝐾) ↾t 𝐴) = (TopOpen‘(𝐾s 𝐴))
55 eqid 2727 . . 3 (Base‘(𝐾s 𝐴)) = (Base‘(𝐾s 𝐴))
56 eqid 2727 . . 3 ((dist‘(𝐾s 𝐴)) ↾ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴)))) = ((dist‘(𝐾s 𝐴)) ↾ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴))))
5754, 55, 56isxms2 24372 . 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 581 1 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → (𝐾s 𝐴) ∈ ∞MetSp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  cin 3946  wss 3947   cuni 4910   × cxp 5678  cres 5682  cfv 6551  (class class class)co 7424  Basecbs 17185  s cress 17214  distcds 17247  t crest 17407  TopOpenctopn 17408  ∞Metcxmet 21269  MetOpencmopn 21274  TopOnctopon 22830  TopSpctps 22852  ∞MetSpcxms 24241
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-rep 5287  ax-sep 5301  ax-nul 5308  ax-pow 5367  ax-pr 5431  ax-un 7744  ax-cnex 11200  ax-resscn 11201  ax-1cn 11202  ax-icn 11203  ax-addcl 11204  ax-addrcl 11205  ax-mulcl 11206  ax-mulrcl 11207  ax-mulcom 11208  ax-addass 11209  ax-mulass 11210  ax-distr 11211  ax-i2m1 11212  ax-1ne0 11213  ax-1rid 11214  ax-rnegex 11215  ax-rrecex 11216  ax-cnre 11217  ax-pre-lttri 11218  ax-pre-lttrn 11219  ax-pre-ltadd 11220  ax-pre-mulgt0 11221  ax-pre-sup 11222
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-nel 3043  df-ral 3058  df-rex 3067  df-rmo 3372  df-reu 3373  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-iun 5000  df-br 5151  df-opab 5213  df-mpt 5234  df-tr 5268  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5635  df-we 5637  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-pred 6308  df-ord 6375  df-on 6376  df-lim 6377  df-suc 6378  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-f1 6556  df-fo 6557  df-f1o 6558  df-fv 6559  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7875  df-1st 7997  df-2nd 7998  df-frecs 8291  df-wrecs 8322  df-recs 8396  df-rdg 8435  df-er 8729  df-map 8851  df-en 8969  df-dom 8970  df-sdom 8971  df-sup 9471  df-inf 9472  df-pnf 11286  df-mnf 11287  df-xr 11288  df-ltxr 11289  df-le 11290  df-sub 11482  df-neg 11483  df-div 11908  df-nn 12249  df-2 12311  df-3 12312  df-4 12313  df-5 12314  df-6 12315  df-7 12316  df-8 12317  df-9 12318  df-n0 12509  df-z 12595  df-dec 12714  df-uz 12859  df-q 12969  df-rp 13013  df-xneg 13130  df-xadd 13131  df-xmul 13132  df-sets 17138  df-slot 17156  df-ndx 17168  df-base 17186  df-ress 17215  df-tset 17257  df-ds 17260  df-rest 17409  df-topn 17410  df-topgen 17430  df-psmet 21276  df-xmet 21277  df-bl 21279  df-mopn 21280  df-top 22814  df-topon 22831  df-topsp 22853  df-bases 22867  df-xms 24244
This theorem is referenced by:  ressms  24453  qqhcn  33597  qqhucn  33598
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