![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > setsms | Structured version Visualization version GIF version |
Description: The constructed metric space is a metric space iff the provided distance function is a metric. (Contributed by Mario Carneiro, 28-Aug-2015.) |
Ref | Expression |
---|---|
setsms.x | ⊢ (𝜑 → 𝑋 = (Base‘𝑀)) |
setsms.d | ⊢ (𝜑 → 𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋))) |
setsms.k | ⊢ (𝜑 → 𝐾 = (𝑀 sSet 〈(TopSet‘ndx), (MetOpen‘𝐷)〉)) |
setsms.m | ⊢ (𝜑 → 𝑀 ∈ 𝑉) |
Ref | Expression |
---|---|
setsms | ⊢ (𝜑 → (𝐾 ∈ MetSp ↔ 𝐷 ∈ (Met‘𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | setsms.x | . . . 4 ⊢ (𝜑 → 𝑋 = (Base‘𝑀)) | |
2 | setsms.d | . . . 4 ⊢ (𝜑 → 𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋))) | |
3 | setsms.k | . . . 4 ⊢ (𝜑 → 𝐾 = (𝑀 sSet 〈(TopSet‘ndx), (MetOpen‘𝐷)〉)) | |
4 | setsms.m | . . . 4 ⊢ (𝜑 → 𝑀 ∈ 𝑉) | |
5 | 1, 2, 3, 4 | setsxms 22661 | . . 3 ⊢ (𝜑 → (𝐾 ∈ ∞MetSp ↔ 𝐷 ∈ (∞Met‘𝑋))) |
6 | 1, 2, 3 | setsmsds 22658 | . . . . . 6 ⊢ (𝜑 → (dist‘𝑀) = (dist‘𝐾)) |
7 | 1, 2, 3 | setsmsbas 22657 | . . . . . . 7 ⊢ (𝜑 → 𝑋 = (Base‘𝐾)) |
8 | 7 | sqxpeqd 5378 | . . . . . 6 ⊢ (𝜑 → (𝑋 × 𝑋) = ((Base‘𝐾) × (Base‘𝐾))) |
9 | 6, 8 | reseq12d 5634 | . . . . 5 ⊢ (𝜑 → ((dist‘𝑀) ↾ (𝑋 × 𝑋)) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))) |
10 | 2, 9 | eqtr2d 2862 | . . . 4 ⊢ (𝜑 → ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) = 𝐷) |
11 | 7 | fveq2d 6441 | . . . . 5 ⊢ (𝜑 → (Met‘𝑋) = (Met‘(Base‘𝐾))) |
12 | 11 | eqcomd 2831 | . . . 4 ⊢ (𝜑 → (Met‘(Base‘𝐾)) = (Met‘𝑋)) |
13 | 10, 12 | eleq12d 2900 | . . 3 ⊢ (𝜑 → (((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ∈ (Met‘(Base‘𝐾)) ↔ 𝐷 ∈ (Met‘𝑋))) |
14 | 5, 13 | anbi12d 624 | . 2 ⊢ (𝜑 → ((𝐾 ∈ ∞MetSp ∧ ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ∈ (Met‘(Base‘𝐾))) ↔ (𝐷 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (Met‘𝑋)))) |
15 | eqid 2825 | . . 3 ⊢ (TopOpen‘𝐾) = (TopOpen‘𝐾) | |
16 | eqid 2825 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
17 | eqid 2825 | . . 3 ⊢ ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) | |
18 | 15, 16, 17 | isms 22631 | . 2 ⊢ (𝐾 ∈ MetSp ↔ (𝐾 ∈ ∞MetSp ∧ ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ∈ (Met‘(Base‘𝐾)))) |
19 | metxmet 22516 | . . 3 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋)) | |
20 | 19 | pm4.71ri 556 | . 2 ⊢ (𝐷 ∈ (Met‘𝑋) ↔ (𝐷 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (Met‘𝑋))) |
21 | 14, 18, 20 | 3bitr4g 306 | 1 ⊢ (𝜑 → (𝐾 ∈ MetSp ↔ 𝐷 ∈ (Met‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1656 ∈ wcel 2164 〈cop 4405 × cxp 5344 ↾ cres 5348 ‘cfv 6127 (class class class)co 6910 ndxcnx 16226 sSet csts 16227 Basecbs 16229 TopSetcts 16318 distcds 16321 TopOpenctopn 16442 ∞Metcxmet 20098 Metcmet 20099 MetOpencmopn 20103 ∞MetSpcxms 22499 MetSpcms 22500 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 ax-pre-sup 10337 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-om 7332 df-1st 7433 df-2nd 7434 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-er 8014 df-map 8129 df-en 8229 df-dom 8230 df-sdom 8231 df-sup 8623 df-inf 8624 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-div 11017 df-nn 11358 df-2 11421 df-3 11422 df-4 11423 df-5 11424 df-6 11425 df-7 11426 df-8 11427 df-9 11428 df-n0 11626 df-z 11712 df-dec 11829 df-uz 11976 df-q 12079 df-rp 12120 df-xneg 12239 df-xadd 12240 df-xmul 12241 df-ndx 16232 df-slot 16233 df-base 16235 df-sets 16236 df-tset 16331 df-ds 16334 df-rest 16443 df-topn 16444 df-topgen 16464 df-psmet 20105 df-xmet 20106 df-met 20107 df-bl 20108 df-mopn 20109 df-top 21076 df-topon 21093 df-topsp 21115 df-bases 21128 df-xms 22502 df-ms 22503 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |