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| Mirrors > Home > MPE Home > Th. List > ressms | Structured version Visualization version GIF version | ||
| Description: The restriction of a metric space is a metric space. (Contributed by Mario Carneiro, 24-Aug-2015.) |
| Ref | Expression |
|---|---|
| ressms | ⊢ ((𝐾 ∈ MetSp ∧ 𝐴 ∈ 𝑉) → (𝐾 ↾s 𝐴) ∈ MetSp) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | msxms 24464 | . . 3 ⊢ (𝐾 ∈ MetSp → 𝐾 ∈ ∞MetSp) | |
| 2 | ressxms 24538 | . . 3 ⊢ ((𝐾 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑉) → (𝐾 ↾s 𝐴) ∈ ∞MetSp) | |
| 3 | 1, 2 | sylan 580 | . 2 ⊢ ((𝐾 ∈ MetSp ∧ 𝐴 ∈ 𝑉) → (𝐾 ↾s 𝐴) ∈ ∞MetSp) |
| 4 | eqid 2737 | . . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
| 5 | eqid 2737 | . . . . . 6 ⊢ ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) | |
| 6 | 4, 5 | msmet 24467 | . . . . 5 ⊢ (𝐾 ∈ MetSp → ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ∈ (Met‘(Base‘𝐾))) |
| 7 | 6 | adantr 480 | . . . 4 ⊢ ((𝐾 ∈ MetSp ∧ 𝐴 ∈ 𝑉) → ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ∈ (Met‘(Base‘𝐾))) |
| 8 | metres 24375 | . . . 4 ⊢ (((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ∈ (Met‘(Base‘𝐾)) → (((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ↾ (𝐴 × 𝐴)) ∈ (Met‘((Base‘𝐾) ∩ 𝐴))) | |
| 9 | 7, 8 | syl 17 | . . 3 ⊢ ((𝐾 ∈ MetSp ∧ 𝐴 ∈ 𝑉) → (((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ↾ (𝐴 × 𝐴)) ∈ (Met‘((Base‘𝐾) ∩ 𝐴))) |
| 10 | resres 6010 | . . . . 5 ⊢ (((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ↾ (𝐴 × 𝐴)) = ((dist‘𝐾) ↾ (((Base‘𝐾) × (Base‘𝐾)) ∩ (𝐴 × 𝐴))) | |
| 11 | inxp 5842 | . . . . . 6 ⊢ (((Base‘𝐾) × (Base‘𝐾)) ∩ (𝐴 × 𝐴)) = (((Base‘𝐾) ∩ 𝐴) × ((Base‘𝐾) ∩ 𝐴)) | |
| 12 | 11 | reseq2i 5994 | . . . . 5 ⊢ ((dist‘𝐾) ↾ (((Base‘𝐾) × (Base‘𝐾)) ∩ (𝐴 × 𝐴))) = ((dist‘𝐾) ↾ (((Base‘𝐾) ∩ 𝐴) × ((Base‘𝐾) ∩ 𝐴))) |
| 13 | 10, 12 | eqtri 2765 | . . . 4 ⊢ (((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ↾ (𝐴 × 𝐴)) = ((dist‘𝐾) ↾ (((Base‘𝐾) ∩ 𝐴) × ((Base‘𝐾) ∩ 𝐴))) |
| 14 | eqid 2737 | . . . . . . 7 ⊢ (𝐾 ↾s 𝐴) = (𝐾 ↾s 𝐴) | |
| 15 | eqid 2737 | . . . . . . 7 ⊢ (dist‘𝐾) = (dist‘𝐾) | |
| 16 | 14, 15 | ressds 17454 | . . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (dist‘𝐾) = (dist‘(𝐾 ↾s 𝐴))) |
| 17 | 16 | adantl 481 | . . . . 5 ⊢ ((𝐾 ∈ MetSp ∧ 𝐴 ∈ 𝑉) → (dist‘𝐾) = (dist‘(𝐾 ↾s 𝐴))) |
| 18 | incom 4209 | . . . . . . 7 ⊢ ((Base‘𝐾) ∩ 𝐴) = (𝐴 ∩ (Base‘𝐾)) | |
| 19 | 14, 4 | ressbas 17280 | . . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ (Base‘𝐾)) = (Base‘(𝐾 ↾s 𝐴))) |
| 20 | 19 | adantl 481 | . . . . . . 7 ⊢ ((𝐾 ∈ MetSp ∧ 𝐴 ∈ 𝑉) → (𝐴 ∩ (Base‘𝐾)) = (Base‘(𝐾 ↾s 𝐴))) |
| 21 | 18, 20 | eqtrid 2789 | . . . . . 6 ⊢ ((𝐾 ∈ MetSp ∧ 𝐴 ∈ 𝑉) → ((Base‘𝐾) ∩ 𝐴) = (Base‘(𝐾 ↾s 𝐴))) |
| 22 | 21 | sqxpeqd 5717 | . . . . 5 ⊢ ((𝐾 ∈ MetSp ∧ 𝐴 ∈ 𝑉) → (((Base‘𝐾) ∩ 𝐴) × ((Base‘𝐾) ∩ 𝐴)) = ((Base‘(𝐾 ↾s 𝐴)) × (Base‘(𝐾 ↾s 𝐴)))) |
| 23 | 17, 22 | reseq12d 5998 | . . . 4 ⊢ ((𝐾 ∈ MetSp ∧ 𝐴 ∈ 𝑉) → ((dist‘𝐾) ↾ (((Base‘𝐾) ∩ 𝐴) × ((Base‘𝐾) ∩ 𝐴))) = ((dist‘(𝐾 ↾s 𝐴)) ↾ ((Base‘(𝐾 ↾s 𝐴)) × (Base‘(𝐾 ↾s 𝐴))))) |
| 24 | 13, 23 | eqtrid 2789 | . . 3 ⊢ ((𝐾 ∈ MetSp ∧ 𝐴 ∈ 𝑉) → (((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ↾ (𝐴 × 𝐴)) = ((dist‘(𝐾 ↾s 𝐴)) ↾ ((Base‘(𝐾 ↾s 𝐴)) × (Base‘(𝐾 ↾s 𝐴))))) |
| 25 | 21 | fveq2d 6910 | . . 3 ⊢ ((𝐾 ∈ MetSp ∧ 𝐴 ∈ 𝑉) → (Met‘((Base‘𝐾) ∩ 𝐴)) = (Met‘(Base‘(𝐾 ↾s 𝐴)))) |
| 26 | 9, 24, 25 | 3eltr3d 2855 | . 2 ⊢ ((𝐾 ∈ MetSp ∧ 𝐴 ∈ 𝑉) → ((dist‘(𝐾 ↾s 𝐴)) ↾ ((Base‘(𝐾 ↾s 𝐴)) × (Base‘(𝐾 ↾s 𝐴)))) ∈ (Met‘(Base‘(𝐾 ↾s 𝐴)))) |
| 27 | eqid 2737 | . . . 4 ⊢ (TopOpen‘𝐾) = (TopOpen‘𝐾) | |
| 28 | 14, 27 | resstopn 23194 | . . 3 ⊢ ((TopOpen‘𝐾) ↾t 𝐴) = (TopOpen‘(𝐾 ↾s 𝐴)) |
| 29 | eqid 2737 | . . 3 ⊢ (Base‘(𝐾 ↾s 𝐴)) = (Base‘(𝐾 ↾s 𝐴)) | |
| 30 | eqid 2737 | . . 3 ⊢ ((dist‘(𝐾 ↾s 𝐴)) ↾ ((Base‘(𝐾 ↾s 𝐴)) × (Base‘(𝐾 ↾s 𝐴)))) = ((dist‘(𝐾 ↾s 𝐴)) ↾ ((Base‘(𝐾 ↾s 𝐴)) × (Base‘(𝐾 ↾s 𝐴)))) | |
| 31 | 28, 29, 30 | isms 24459 | . 2 ⊢ ((𝐾 ↾s 𝐴) ∈ MetSp ↔ ((𝐾 ↾s 𝐴) ∈ ∞MetSp ∧ ((dist‘(𝐾 ↾s 𝐴)) ↾ ((Base‘(𝐾 ↾s 𝐴)) × (Base‘(𝐾 ↾s 𝐴)))) ∈ (Met‘(Base‘(𝐾 ↾s 𝐴))))) |
| 32 | 3, 26, 31 | sylanbrc 583 | 1 ⊢ ((𝐾 ∈ MetSp ∧ 𝐴 ∈ 𝑉) → (𝐾 ↾s 𝐴) ∈ MetSp) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∩ cin 3950 × cxp 5683 ↾ cres 5687 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 ↾s cress 17274 distcds 17306 ↾t crest 17465 TopOpenctopn 17466 Metcmet 21350 ∞MetSpcxms 24327 MetSpcms 24328 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-sup 9482 df-inf 9483 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-tset 17316 df-ds 17319 df-rest 17467 df-topn 17468 df-topgen 17488 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-xms 24330 df-ms 24331 |
| This theorem is referenced by: subgngp 24648 cmsss 25385 cmscsscms 25407 cnpwstotbnd 37804 |
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