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Mirrors > Home > MPE Home > Th. List > imasf1oms | Structured version Visualization version GIF version |
Description: The image of a metric space is a metric space. (Contributed by Mario Carneiro, 28-Aug-2015.) |
Ref | Expression |
---|---|
imasf1obl.u | ⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅)) |
imasf1obl.v | ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) |
imasf1obl.f | ⊢ (𝜑 → 𝐹:𝑉–1-1-onto→𝐵) |
imasf1oms.r | ⊢ (𝜑 → 𝑅 ∈ MetSp) |
Ref | Expression |
---|---|
imasf1oms | ⊢ (𝜑 → 𝑈 ∈ MetSp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imasf1obl.u | . . 3 ⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅)) | |
2 | imasf1obl.v | . . 3 ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) | |
3 | imasf1obl.f | . . 3 ⊢ (𝜑 → 𝐹:𝑉–1-1-onto→𝐵) | |
4 | imasf1oms.r | . . . 4 ⊢ (𝜑 → 𝑅 ∈ MetSp) | |
5 | msxms 22667 | . . . 4 ⊢ (𝑅 ∈ MetSp → 𝑅 ∈ ∞MetSp) | |
6 | 4, 5 | syl 17 | . . 3 ⊢ (𝜑 → 𝑅 ∈ ∞MetSp) |
7 | 1, 2, 3, 6 | imasf1oxms 22702 | . 2 ⊢ (𝜑 → 𝑈 ∈ ∞MetSp) |
8 | eqid 2777 | . . . . 5 ⊢ ((dist‘𝑅) ↾ (𝑉 × 𝑉)) = ((dist‘𝑅) ↾ (𝑉 × 𝑉)) | |
9 | eqid 2777 | . . . . 5 ⊢ (dist‘𝑈) = (dist‘𝑈) | |
10 | eqid 2777 | . . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
11 | eqid 2777 | . . . . . . . 8 ⊢ ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))) = ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))) | |
12 | 10, 11 | msmet 22670 | . . . . . . 7 ⊢ (𝑅 ∈ MetSp → ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))) ∈ (Met‘(Base‘𝑅))) |
13 | 4, 12 | syl 17 | . . . . . 6 ⊢ (𝜑 → ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))) ∈ (Met‘(Base‘𝑅))) |
14 | 2 | sqxpeqd 5387 | . . . . . . 7 ⊢ (𝜑 → (𝑉 × 𝑉) = ((Base‘𝑅) × (Base‘𝑅))) |
15 | 14 | reseq2d 5642 | . . . . . 6 ⊢ (𝜑 → ((dist‘𝑅) ↾ (𝑉 × 𝑉)) = ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅)))) |
16 | 2 | fveq2d 6450 | . . . . . 6 ⊢ (𝜑 → (Met‘𝑉) = (Met‘(Base‘𝑅))) |
17 | 13, 15, 16 | 3eltr4d 2873 | . . . . 5 ⊢ (𝜑 → ((dist‘𝑅) ↾ (𝑉 × 𝑉)) ∈ (Met‘𝑉)) |
18 | 1, 2, 3, 4, 8, 9, 17 | imasf1omet 22589 | . . . 4 ⊢ (𝜑 → (dist‘𝑈) ∈ (Met‘𝐵)) |
19 | f1ofo 6398 | . . . . . . 7 ⊢ (𝐹:𝑉–1-1-onto→𝐵 → 𝐹:𝑉–onto→𝐵) | |
20 | 3, 19 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐹:𝑉–onto→𝐵) |
21 | 1, 2, 20, 4 | imasbas 16558 | . . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘𝑈)) |
22 | 21 | fveq2d 6450 | . . . 4 ⊢ (𝜑 → (Met‘𝐵) = (Met‘(Base‘𝑈))) |
23 | 18, 22 | eleqtrd 2860 | . . 3 ⊢ (𝜑 → (dist‘𝑈) ∈ (Met‘(Base‘𝑈))) |
24 | ssid 3841 | . . 3 ⊢ (Base‘𝑈) ⊆ (Base‘𝑈) | |
25 | metres2 22576 | . . 3 ⊢ (((dist‘𝑈) ∈ (Met‘(Base‘𝑈)) ∧ (Base‘𝑈) ⊆ (Base‘𝑈)) → ((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈))) ∈ (Met‘(Base‘𝑈))) | |
26 | 23, 24, 25 | sylancl 580 | . 2 ⊢ (𝜑 → ((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈))) ∈ (Met‘(Base‘𝑈))) |
27 | eqid 2777 | . . 3 ⊢ (TopOpen‘𝑈) = (TopOpen‘𝑈) | |
28 | eqid 2777 | . . 3 ⊢ (Base‘𝑈) = (Base‘𝑈) | |
29 | eqid 2777 | . . 3 ⊢ ((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈))) = ((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈))) | |
30 | 27, 28, 29 | isms 22662 | . 2 ⊢ (𝑈 ∈ MetSp ↔ (𝑈 ∈ ∞MetSp ∧ ((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈))) ∈ (Met‘(Base‘𝑈)))) |
31 | 7, 26, 30 | sylanbrc 578 | 1 ⊢ (𝜑 → 𝑈 ∈ MetSp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2106 ⊆ wss 3791 × cxp 5353 ↾ cres 5357 –onto→wfo 6133 –1-1-onto→wf1o 6134 ‘cfv 6135 (class class class)co 6922 Basecbs 16255 distcds 16347 TopOpenctopn 16468 “s cimas 16550 Metcmet 20128 ∞MetSpcxms 22530 MetSpcms 22531 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-inf2 8835 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-pre-sup 10350 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-int 4711 df-iun 4755 df-iin 4756 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-se 5315 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-isom 6144 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-of 7174 df-om 7344 df-1st 7445 df-2nd 7446 df-supp 7577 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-oadd 7847 df-er 8026 df-map 8142 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-fsupp 8564 df-sup 8636 df-inf 8637 df-oi 8704 df-card 9098 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-div 11033 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-7 11443 df-8 11444 df-9 11445 df-n0 11643 df-z 11729 df-dec 11846 df-uz 11993 df-q 12096 df-rp 12138 df-xneg 12257 df-xadd 12258 df-xmul 12259 df-fz 12644 df-fzo 12785 df-seq 13120 df-hash 13436 df-struct 16257 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-ress 16263 df-plusg 16351 df-mulr 16352 df-sca 16354 df-vsca 16355 df-ip 16356 df-tset 16357 df-ple 16358 df-ds 16360 df-rest 16469 df-topn 16470 df-0g 16488 df-gsum 16489 df-topgen 16490 df-xrs 16548 df-qtop 16553 df-imas 16554 df-mre 16632 df-mrc 16633 df-acs 16635 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-submnd 17722 df-mulg 17928 df-cntz 18133 df-cmn 18581 df-psmet 20134 df-xmet 20135 df-met 20136 df-bl 20137 df-mopn 20138 df-top 21106 df-topon 21123 df-topsp 21145 df-bases 21158 df-xms 22533 df-ms 22534 |
This theorem is referenced by: xpsms 22748 |
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