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Theorem ressxms 22253
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 2621 . . . . . 6 (Base‘𝐾) = (Base‘𝐾)
2 eqid 2621 . . . . . 6 ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))
31, 2xmsxmet 22184 . . . . 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 22092 . . . 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 5373 . . . . 5 (((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ↾ (𝐴 × 𝐴)) = ((dist‘𝐾) ↾ (((Base‘𝐾) × (Base‘𝐾)) ∩ (𝐴 × 𝐴)))
8 inxp 5219 . . . . . 6 (((Base‘𝐾) × (Base‘𝐾)) ∩ (𝐴 × 𝐴)) = (((Base‘𝐾) ∩ 𝐴) × ((Base‘𝐾) ∩ 𝐴))
98reseq2i 5358 . . . . 5 ((dist‘𝐾) ↾ (((Base‘𝐾) × (Base‘𝐾)) ∩ (𝐴 × 𝐴))) = ((dist‘𝐾) ↾ (((Base‘𝐾) ∩ 𝐴) × ((Base‘𝐾) ∩ 𝐴)))
107, 9eqtri 2643 . . . 4 (((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ↾ (𝐴 × 𝐴)) = ((dist‘𝐾) ↾ (((Base‘𝐾) ∩ 𝐴) × ((Base‘𝐾) ∩ 𝐴)))
11 eqid 2621 . . . . . . 7 (𝐾s 𝐴) = (𝐾s 𝐴)
12 eqid 2621 . . . . . . 7 (dist‘𝐾) = (dist‘𝐾)
1311, 12ressds 16005 . . . . . 6 (𝐴𝑉 → (dist‘𝐾) = (dist‘(𝐾s 𝐴)))
1413adantl 482 . . . . 5 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → (dist‘𝐾) = (dist‘(𝐾s 𝐴)))
15 incom 3788 . . . . . . 7 ((Base‘𝐾) ∩ 𝐴) = (𝐴 ∩ (Base‘𝐾))
1611, 1ressbas 15862 . . . . . . . 8 (𝐴𝑉 → (𝐴 ∩ (Base‘𝐾)) = (Base‘(𝐾s 𝐴)))
1716adantl 482 . . . . . . 7 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → (𝐴 ∩ (Base‘𝐾)) = (Base‘(𝐾s 𝐴)))
1815, 17syl5eq 2667 . . . . . 6 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → ((Base‘𝐾) ∩ 𝐴) = (Base‘(𝐾s 𝐴)))
1918sqxpeqd 5106 . . . . 5 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → (((Base‘𝐾) ∩ 𝐴) × ((Base‘𝐾) ∩ 𝐴)) = ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴))))
2014, 19reseq12d 5362 . . . 4 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → ((dist‘𝐾) ↾ (((Base‘𝐾) ∩ 𝐴) × ((Base‘𝐾) ∩ 𝐴))) = ((dist‘(𝐾s 𝐴)) ↾ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴)))))
2110, 20syl5eq 2667 . . 3 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → (((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ↾ (𝐴 × 𝐴)) = ((dist‘(𝐾s 𝐴)) ↾ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴)))))
2218fveq2d 6157 . . 3 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → (∞Met‘((Base‘𝐾) ∩ 𝐴)) = (∞Met‘(Base‘(𝐾s 𝐴))))
236, 21, 223eltr3d 2712 . 2 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → ((dist‘(𝐾s 𝐴)) ↾ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴)))) ∈ (∞Met‘(Base‘(𝐾s 𝐴))))
24 eqid 2621 . . . . . . 7 (TopOpen‘𝐾) = (TopOpen‘𝐾)
2524, 1, 2xmstopn 22179 . . . . . 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 6625 . . . 4 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → ((TopOpen‘𝐾) ↾t ((Base‘𝐾) ∩ 𝐴)) = ((MetOpen‘((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))) ↾t ((Base‘𝐾) ∩ 𝐴)))
28 inss1 3816 . . . . 5 ((Base‘𝐾) ∩ 𝐴) ⊆ (Base‘𝐾)
29 xpss12 5191 . . . . . . . . 9 ((((Base‘𝐾) ∩ 𝐴) ⊆ (Base‘𝐾) ∧ ((Base‘𝐾) ∩ 𝐴) ⊆ (Base‘𝐾)) → (((Base‘𝐾) ∩ 𝐴) × ((Base‘𝐾) ∩ 𝐴)) ⊆ ((Base‘𝐾) × (Base‘𝐾)))
3028, 28, 29mp2an 707 . . . . . . . 8 (((Base‘𝐾) ∩ 𝐴) × ((Base‘𝐾) ∩ 𝐴)) ⊆ ((Base‘𝐾) × (Base‘𝐾))
31 resabs1 5391 . . . . . . . 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 2646 . . . . . 6 (((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ↾ (𝐴 × 𝐴)) = (((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ↾ (((Base‘𝐾) ∩ 𝐴) × ((Base‘𝐾) ∩ 𝐴)))
34 eqid 2621 . . . . . 6 (MetOpen‘((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))) = (MetOpen‘((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))))
35 eqid 2621 . . . . . 6 (MetOpen‘(((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ↾ (𝐴 × 𝐴))) = (MetOpen‘(((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ↾ (𝐴 × 𝐴)))
3633, 34, 35metrest 22252 . . . . 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 693 . . . 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 2655 . . 3 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → ((TopOpen‘𝐾) ↾t ((Base‘𝐾) ∩ 𝐴)) = (MetOpen‘(((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ↾ (𝐴 × 𝐴))))
39 xmstps 22181 . . . . . . . . 9 (𝐾 ∈ ∞MetSp → 𝐾 ∈ TopSp)
401, 24tpsuni 20664 . . . . . . . . 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 3797 . . . . . 6 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → (𝐴 ∩ (Base‘𝐾)) = (𝐴 (TopOpen‘𝐾)))
4415, 43syl5eq 2667 . . . . 5 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → ((Base‘𝐾) ∩ 𝐴) = (𝐴 (TopOpen‘𝐾)))
4544oveq2d 6626 . . . 4 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → ((TopOpen‘𝐾) ↾t ((Base‘𝐾) ∩ 𝐴)) = ((TopOpen‘𝐾) ↾t (𝐴 (TopOpen‘𝐾))))
461, 24istps 20662 . . . . . 6 (𝐾 ∈ TopSp ↔ (TopOpen‘𝐾) ∈ (TopOn‘(Base‘𝐾)))
4739, 46sylib 208 . . . . 5 (𝐾 ∈ ∞MetSp → (TopOpen‘𝐾) ∈ (TopOn‘(Base‘𝐾)))
48 eqid 2621 . . . . . 6 (TopOpen‘𝐾) = (TopOpen‘𝐾)
4948restin 20893 . . . . 5 (((TopOpen‘𝐾) ∈ (TopOn‘(Base‘𝐾)) ∧ 𝐴𝑉) → ((TopOpen‘𝐾) ↾t 𝐴) = ((TopOpen‘𝐾) ↾t (𝐴 (TopOpen‘𝐾))))
5047, 49sylan 488 . . . 4 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → ((TopOpen‘𝐾) ↾t 𝐴) = ((TopOpen‘𝐾) ↾t (𝐴 (TopOpen‘𝐾))))
5145, 50eqtr4d 2658 . . 3 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → ((TopOpen‘𝐾) ↾t ((Base‘𝐾) ∩ 𝐴)) = ((TopOpen‘𝐾) ↾t 𝐴))
5221fveq2d 6157 . . 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 2663 . 2 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → ((TopOpen‘𝐾) ↾t 𝐴) = (MetOpen‘((dist‘(𝐾s 𝐴)) ↾ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴))))))
5411, 24resstopn 20913 . . 3 ((TopOpen‘𝐾) ↾t 𝐴) = (TopOpen‘(𝐾s 𝐴))
55 eqid 2621 . . 3 (Base‘(𝐾s 𝐴)) = (Base‘(𝐾s 𝐴))
56 eqid 2621 . . 3 ((dist‘(𝐾s 𝐴)) ↾ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴)))) = ((dist‘(𝐾s 𝐴)) ↾ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴))))
5754, 55, 56isxms2 22176 . 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 697 1 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → (𝐾s 𝐴) ∈ ∞MetSp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  cin 3558  wss 3559   cuni 4407   × cxp 5077  cres 5081  cfv 5852  (class class class)co 6610  Basecbs 15792  s cress 15793  distcds 15882  t crest 16013  TopOpenctopn 16014  ∞Metcxmt 19663  MetOpencmopn 19668  TopOnctopon 20647  TopSpctps 20660  ∞MetSpcxme 22045
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-cnex 9944  ax-resscn 9945  ax-1cn 9946  ax-icn 9947  ax-addcl 9948  ax-addrcl 9949  ax-mulcl 9950  ax-mulrcl 9951  ax-mulcom 9952  ax-addass 9953  ax-mulass 9954  ax-distr 9955  ax-i2m1 9956  ax-1ne0 9957  ax-1rid 9958  ax-rnegex 9959  ax-rrecex 9960  ax-cnre 9961  ax-pre-lttri 9962  ax-pre-lttrn 9963  ax-pre-ltadd 9964  ax-pre-mulgt0 9965  ax-pre-sup 9966
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-er 7694  df-map 7811  df-en 7908  df-dom 7909  df-sdom 7910  df-sup 8300  df-inf 8301  df-pnf 10028  df-mnf 10029  df-xr 10030  df-ltxr 10031  df-le 10032  df-sub 10220  df-neg 10221  df-div 10637  df-nn 10973  df-2 11031  df-3 11032  df-4 11033  df-5 11034  df-6 11035  df-7 11036  df-8 11037  df-9 11038  df-n0 11245  df-z 11330  df-dec 11446  df-uz 11640  df-q 11741  df-rp 11785  df-xneg 11898  df-xadd 11899  df-xmul 11900  df-ndx 15795  df-slot 15796  df-base 15797  df-sets 15798  df-ress 15799  df-tset 15892  df-ds 15896  df-rest 16015  df-topn 16016  df-topgen 16036  df-psmet 19670  df-xmet 19671  df-bl 19673  df-mopn 19674  df-top 20631  df-topon 20648  df-topsp 20661  df-bases 20674  df-xms 22048
This theorem is referenced by:  ressms  22254  qqhcn  29841  qqhucn  29842
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