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Theorem ressxms 23271
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 23202 . . . . 5 (𝐾 ∈ ∞MetSp → ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ∈ (∞Met‘(Base‘𝐾)))
43adantr 484 . . . 4 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ∈ (∞Met‘(Base‘𝐾)))
5 xmetres 23110 . . . 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 5832 . . . . 5 (((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ↾ (𝐴 × 𝐴)) = ((dist‘𝐾) ↾ (((Base‘𝐾) × (Base‘𝐾)) ∩ (𝐴 × 𝐴)))
8 inxp 5669 . . . . . 6 (((Base‘𝐾) × (Base‘𝐾)) ∩ (𝐴 × 𝐴)) = (((Base‘𝐾) ∩ 𝐴) × ((Base‘𝐾) ∩ 𝐴))
98reseq2i 5816 . . . . 5 ((dist‘𝐾) ↾ (((Base‘𝐾) × (Base‘𝐾)) ∩ (𝐴 × 𝐴))) = ((dist‘𝐾) ↾ (((Base‘𝐾) ∩ 𝐴) × ((Base‘𝐾) ∩ 𝐴)))
107, 9eqtri 2761 . . . 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 16782 . . . . . 6 (𝐴𝑉 → (dist‘𝐾) = (dist‘(𝐾s 𝐴)))
1413adantl 485 . . . . 5 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → (dist‘𝐾) = (dist‘(𝐾s 𝐴)))
15 incom 4089 . . . . . . 7 ((Base‘𝐾) ∩ 𝐴) = (𝐴 ∩ (Base‘𝐾))
1611, 1ressbas 16650 . . . . . . . 8 (𝐴𝑉 → (𝐴 ∩ (Base‘𝐾)) = (Base‘(𝐾s 𝐴)))
1716adantl 485 . . . . . . 7 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → (𝐴 ∩ (Base‘𝐾)) = (Base‘(𝐾s 𝐴)))
1815, 17syl5eq 2785 . . . . . 6 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → ((Base‘𝐾) ∩ 𝐴) = (Base‘(𝐾s 𝐴)))
1918sqxpeqd 5551 . . . . 5 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → (((Base‘𝐾) ∩ 𝐴) × ((Base‘𝐾) ∩ 𝐴)) = ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴))))
2014, 19reseq12d 5820 . . . 4 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → ((dist‘𝐾) ↾ (((Base‘𝐾) ∩ 𝐴) × ((Base‘𝐾) ∩ 𝐴))) = ((dist‘(𝐾s 𝐴)) ↾ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴)))))
2110, 20syl5eq 2785 . . 3 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → (((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ↾ (𝐴 × 𝐴)) = ((dist‘(𝐾s 𝐴)) ↾ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴)))))
2218fveq2d 6672 . . 3 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → (∞Met‘((Base‘𝐾) ∩ 𝐴)) = (∞Met‘(Base‘(𝐾s 𝐴))))
236, 21, 223eltr3d 2847 . 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 23197 . . . . . 6 (𝐾 ∈ ∞MetSp → (TopOpen‘𝐾) = (MetOpen‘((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))))
2625adantr 484 . . . . 5 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → (TopOpen‘𝐾) = (MetOpen‘((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))))
2726oveq1d 7179 . . . 4 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → ((TopOpen‘𝐾) ↾t ((Base‘𝐾) ∩ 𝐴)) = ((MetOpen‘((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))) ↾t ((Base‘𝐾) ∩ 𝐴)))
28 inss1 4117 . . . . 5 ((Base‘𝐾) ∩ 𝐴) ⊆ (Base‘𝐾)
29 xpss12 5534 . . . . . . . . 9 ((((Base‘𝐾) ∩ 𝐴) ⊆ (Base‘𝐾) ∧ ((Base‘𝐾) ∩ 𝐴) ⊆ (Base‘𝐾)) → (((Base‘𝐾) ∩ 𝐴) × ((Base‘𝐾) ∩ 𝐴)) ⊆ ((Base‘𝐾) × (Base‘𝐾)))
3028, 28, 29mp2an 692 . . . . . . . 8 (((Base‘𝐾) ∩ 𝐴) × ((Base‘𝐾) ∩ 𝐴)) ⊆ ((Base‘𝐾) × (Base‘𝐾))
31 resabs1 5849 . . . . . . . 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 2764 . . . . . 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 23270 . . . . 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 589 . . . 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 2773 . . 3 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → ((TopOpen‘𝐾) ↾t ((Base‘𝐾) ∩ 𝐴)) = (MetOpen‘(((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ↾ (𝐴 × 𝐴))))
39 xmstps 23199 . . . . . . . . 9 (𝐾 ∈ ∞MetSp → 𝐾 ∈ TopSp)
401, 24tpsuni 21680 . . . . . . . . 9 (𝐾 ∈ TopSp → (Base‘𝐾) = (TopOpen‘𝐾))
4139, 40syl 17 . . . . . . . 8 (𝐾 ∈ ∞MetSp → (Base‘𝐾) = (TopOpen‘𝐾))
4241adantr 484 . . . . . . 7 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → (Base‘𝐾) = (TopOpen‘𝐾))
4342ineq2d 4101 . . . . . 6 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → (𝐴 ∩ (Base‘𝐾)) = (𝐴 (TopOpen‘𝐾)))
4415, 43syl5eq 2785 . . . . 5 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → ((Base‘𝐾) ∩ 𝐴) = (𝐴 (TopOpen‘𝐾)))
4544oveq2d 7180 . . . 4 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → ((TopOpen‘𝐾) ↾t ((Base‘𝐾) ∩ 𝐴)) = ((TopOpen‘𝐾) ↾t (𝐴 (TopOpen‘𝐾))))
461, 24istps 21678 . . . . . 6 (𝐾 ∈ TopSp ↔ (TopOpen‘𝐾) ∈ (TopOn‘(Base‘𝐾)))
4739, 46sylib 221 . . . . 5 (𝐾 ∈ ∞MetSp → (TopOpen‘𝐾) ∈ (TopOn‘(Base‘𝐾)))
48 eqid 2738 . . . . . 6 (TopOpen‘𝐾) = (TopOpen‘𝐾)
4948restin 21910 . . . . 5 (((TopOpen‘𝐾) ∈ (TopOn‘(Base‘𝐾)) ∧ 𝐴𝑉) → ((TopOpen‘𝐾) ↾t 𝐴) = ((TopOpen‘𝐾) ↾t (𝐴 (TopOpen‘𝐾))))
5047, 49sylan 583 . . . 4 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → ((TopOpen‘𝐾) ↾t 𝐴) = ((TopOpen‘𝐾) ↾t (𝐴 (TopOpen‘𝐾))))
5145, 50eqtr4d 2776 . . 3 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → ((TopOpen‘𝐾) ↾t ((Base‘𝐾) ∩ 𝐴)) = ((TopOpen‘𝐾) ↾t 𝐴))
5221fveq2d 6672 . . 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 2781 . 2 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → ((TopOpen‘𝐾) ↾t 𝐴) = (MetOpen‘((dist‘(𝐾s 𝐴)) ↾ ((Base‘(𝐾s 𝐴)) × (Base‘(𝐾s 𝐴))))))
5411, 24resstopn 21930 . . 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 23194 . 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 586 1 ((𝐾 ∈ ∞MetSp ∧ 𝐴𝑉) → (𝐾s 𝐴) ∈ ∞MetSp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1542  wcel 2113  cin 3840  wss 3841   cuni 4793   × cxp 5517  cres 5521  cfv 6333  (class class class)co 7164  Basecbs 16579  s cress 16580  distcds 16670  t crest 16790  TopOpenctopn 16791  ∞Metcxmet 20195  MetOpencmopn 20200  TopOnctopon 21654  TopSpctps 21676  ∞MetSpcxms 23063
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-rep 5151  ax-sep 5164  ax-nul 5171  ax-pow 5229  ax-pr 5293  ax-un 7473  ax-cnex 10664  ax-resscn 10665  ax-1cn 10666  ax-icn 10667  ax-addcl 10668  ax-addrcl 10669  ax-mulcl 10670  ax-mulrcl 10671  ax-mulcom 10672  ax-addass 10673  ax-mulass 10674  ax-distr 10675  ax-i2m1 10676  ax-1ne0 10677  ax-1rid 10678  ax-rnegex 10679  ax-rrecex 10680  ax-cnre 10681  ax-pre-lttri 10682  ax-pre-lttrn 10683  ax-pre-ltadd 10684  ax-pre-mulgt0 10685  ax-pre-sup 10686
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-nel 3039  df-ral 3058  df-rex 3059  df-reu 3060  df-rmo 3061  df-rab 3062  df-v 3399  df-sbc 3680  df-csb 3789  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-pss 3860  df-nul 4210  df-if 4412  df-pw 4487  df-sn 4514  df-pr 4516  df-tp 4518  df-op 4520  df-uni 4794  df-iun 4880  df-br 5028  df-opab 5090  df-mpt 5108  df-tr 5134  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6123  df-ord 6169  df-on 6170  df-lim 6171  df-suc 6172  df-iota 6291  df-fun 6335  df-fn 6336  df-f 6337  df-f1 6338  df-fo 6339  df-f1o 6340  df-fv 6341  df-riota 7121  df-ov 7167  df-oprab 7168  df-mpo 7169  df-om 7594  df-1st 7707  df-2nd 7708  df-wrecs 7969  df-recs 8030  df-rdg 8068  df-er 8313  df-map 8432  df-en 8549  df-dom 8550  df-sdom 8551  df-sup 8972  df-inf 8973  df-pnf 10748  df-mnf 10749  df-xr 10750  df-ltxr 10751  df-le 10752  df-sub 10943  df-neg 10944  df-div 11369  df-nn 11710  df-2 11772  df-3 11773  df-4 11774  df-5 11775  df-6 11776  df-7 11777  df-8 11778  df-9 11779  df-n0 11970  df-z 12056  df-dec 12173  df-uz 12318  df-q 12424  df-rp 12466  df-xneg 12583  df-xadd 12584  df-xmul 12585  df-ndx 16582  df-slot 16583  df-base 16585  df-sets 16586  df-ress 16587  df-tset 16680  df-ds 16683  df-rest 16792  df-topn 16793  df-topgen 16813  df-psmet 20202  df-xmet 20203  df-bl 20205  df-mopn 20206  df-top 21638  df-topon 21655  df-topsp 21677  df-bases 21690  df-xms 23066
This theorem is referenced by:  ressms  23272  qqhcn  31503  qqhucn  31504
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