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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 24380 | . . . 4 ⊢ (𝑅 ∈ MetSp → 𝑅 ∈ ∞MetSp) | |
6 | 4, 5 | syl 17 | . . 3 ⊢ (𝜑 → 𝑅 ∈ ∞MetSp) |
7 | 1, 2, 3, 6 | imasf1oxms 24418 | . 2 ⊢ (𝜑 → 𝑈 ∈ ∞MetSp) |
8 | eqid 2728 | . . . . 5 ⊢ ((dist‘𝑅) ↾ (𝑉 × 𝑉)) = ((dist‘𝑅) ↾ (𝑉 × 𝑉)) | |
9 | eqid 2728 | . . . . 5 ⊢ (dist‘𝑈) = (dist‘𝑈) | |
10 | eqid 2728 | . . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
11 | eqid 2728 | . . . . . . . 8 ⊢ ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))) = ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))) | |
12 | 10, 11 | msmet 24383 | . . . . . . 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 5714 | . . . . . . 7 ⊢ (𝜑 → (𝑉 × 𝑉) = ((Base‘𝑅) × (Base‘𝑅))) |
15 | 14 | reseq2d 5989 | . . . . . 6 ⊢ (𝜑 → ((dist‘𝑅) ↾ (𝑉 × 𝑉)) = ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅)))) |
16 | 2 | fveq2d 6906 | . . . . . 6 ⊢ (𝜑 → (Met‘𝑉) = (Met‘(Base‘𝑅))) |
17 | 13, 15, 16 | 3eltr4d 2844 | . . . . 5 ⊢ (𝜑 → ((dist‘𝑅) ↾ (𝑉 × 𝑉)) ∈ (Met‘𝑉)) |
18 | 1, 2, 3, 4, 8, 9, 17 | imasf1omet 24302 | . . . 4 ⊢ (𝜑 → (dist‘𝑈) ∈ (Met‘𝐵)) |
19 | f1ofo 6851 | . . . . . . 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 17501 | . . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘𝑈)) |
22 | 21 | fveq2d 6906 | . . . 4 ⊢ (𝜑 → (Met‘𝐵) = (Met‘(Base‘𝑈))) |
23 | 18, 22 | eleqtrd 2831 | . . 3 ⊢ (𝜑 → (dist‘𝑈) ∈ (Met‘(Base‘𝑈))) |
24 | ssid 4004 | . . 3 ⊢ (Base‘𝑈) ⊆ (Base‘𝑈) | |
25 | metres2 24289 | . . 3 ⊢ (((dist‘𝑈) ∈ (Met‘(Base‘𝑈)) ∧ (Base‘𝑈) ⊆ (Base‘𝑈)) → ((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈))) ∈ (Met‘(Base‘𝑈))) | |
26 | 23, 24, 25 | sylancl 584 | . 2 ⊢ (𝜑 → ((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈))) ∈ (Met‘(Base‘𝑈))) |
27 | eqid 2728 | . . 3 ⊢ (TopOpen‘𝑈) = (TopOpen‘𝑈) | |
28 | eqid 2728 | . . 3 ⊢ (Base‘𝑈) = (Base‘𝑈) | |
29 | eqid 2728 | . . 3 ⊢ ((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈))) = ((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈))) | |
30 | 27, 28, 29 | isms 24375 | . 2 ⊢ (𝑈 ∈ MetSp ↔ (𝑈 ∈ ∞MetSp ∧ ((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈))) ∈ (Met‘(Base‘𝑈)))) |
31 | 7, 26, 30 | sylanbrc 581 | 1 ⊢ (𝜑 → 𝑈 ∈ MetSp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ⊆ wss 3949 × cxp 5680 ↾ cres 5684 –onto→wfo 6551 –1-1-onto→wf1o 6552 ‘cfv 6553 (class class class)co 7426 Basecbs 17187 distcds 17249 TopOpenctopn 17410 “s cimas 17493 Metcmet 21272 ∞MetSpcxms 24243 MetSpcms 24244 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-iin 5003 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7691 df-om 7877 df-1st 7999 df-2nd 8000 df-supp 8172 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-map 8853 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-fsupp 9394 df-sup 9473 df-inf 9474 df-oi 9541 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-uz 12861 df-q 12971 df-rp 13015 df-xneg 13132 df-xadd 13133 df-xmul 13134 df-fz 13525 df-fzo 13668 df-seq 14007 df-hash 14330 df-struct 17123 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-plusg 17253 df-mulr 17254 df-sca 17256 df-vsca 17257 df-ip 17258 df-tset 17259 df-ple 17260 df-ds 17262 df-rest 17411 df-topn 17412 df-0g 17430 df-gsum 17431 df-topgen 17432 df-xrs 17491 df-qtop 17496 df-imas 17497 df-mre 17573 df-mrc 17574 df-acs 17576 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-submnd 18748 df-mulg 19031 df-cntz 19275 df-cmn 19744 df-psmet 21278 df-xmet 21279 df-met 21280 df-bl 21281 df-mopn 21282 df-top 22816 df-topon 22833 df-topsp 22855 df-bases 22869 df-xms 24246 df-ms 24247 |
This theorem is referenced by: xpsms 24464 |
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