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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 23066 | . . . 4 ⊢ (𝑅 ∈ MetSp → 𝑅 ∈ ∞MetSp) | |
6 | 4, 5 | syl 17 | . . 3 ⊢ (𝜑 → 𝑅 ∈ ∞MetSp) |
7 | 1, 2, 3, 6 | imasf1oxms 23101 | . 2 ⊢ (𝜑 → 𝑈 ∈ ∞MetSp) |
8 | eqid 2823 | . . . . 5 ⊢ ((dist‘𝑅) ↾ (𝑉 × 𝑉)) = ((dist‘𝑅) ↾ (𝑉 × 𝑉)) | |
9 | eqid 2823 | . . . . 5 ⊢ (dist‘𝑈) = (dist‘𝑈) | |
10 | eqid 2823 | . . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
11 | eqid 2823 | . . . . . . . 8 ⊢ ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))) = ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))) | |
12 | 10, 11 | msmet 23069 | . . . . . . 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 5589 | . . . . . . 7 ⊢ (𝜑 → (𝑉 × 𝑉) = ((Base‘𝑅) × (Base‘𝑅))) |
15 | 14 | reseq2d 5855 | . . . . . 6 ⊢ (𝜑 → ((dist‘𝑅) ↾ (𝑉 × 𝑉)) = ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅)))) |
16 | 2 | fveq2d 6676 | . . . . . 6 ⊢ (𝜑 → (Met‘𝑉) = (Met‘(Base‘𝑅))) |
17 | 13, 15, 16 | 3eltr4d 2930 | . . . . 5 ⊢ (𝜑 → ((dist‘𝑅) ↾ (𝑉 × 𝑉)) ∈ (Met‘𝑉)) |
18 | 1, 2, 3, 4, 8, 9, 17 | imasf1omet 22988 | . . . 4 ⊢ (𝜑 → (dist‘𝑈) ∈ (Met‘𝐵)) |
19 | f1ofo 6624 | . . . . . . 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 16787 | . . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘𝑈)) |
22 | 21 | fveq2d 6676 | . . . 4 ⊢ (𝜑 → (Met‘𝐵) = (Met‘(Base‘𝑈))) |
23 | 18, 22 | eleqtrd 2917 | . . 3 ⊢ (𝜑 → (dist‘𝑈) ∈ (Met‘(Base‘𝑈))) |
24 | ssid 3991 | . . 3 ⊢ (Base‘𝑈) ⊆ (Base‘𝑈) | |
25 | metres2 22975 | . . 3 ⊢ (((dist‘𝑈) ∈ (Met‘(Base‘𝑈)) ∧ (Base‘𝑈) ⊆ (Base‘𝑈)) → ((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈))) ∈ (Met‘(Base‘𝑈))) | |
26 | 23, 24, 25 | sylancl 588 | . 2 ⊢ (𝜑 → ((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈))) ∈ (Met‘(Base‘𝑈))) |
27 | eqid 2823 | . . 3 ⊢ (TopOpen‘𝑈) = (TopOpen‘𝑈) | |
28 | eqid 2823 | . . 3 ⊢ (Base‘𝑈) = (Base‘𝑈) | |
29 | eqid 2823 | . . 3 ⊢ ((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈))) = ((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈))) | |
30 | 27, 28, 29 | isms 23061 | . 2 ⊢ (𝑈 ∈ MetSp ↔ (𝑈 ∈ ∞MetSp ∧ ((dist‘𝑈) ↾ ((Base‘𝑈) × (Base‘𝑈))) ∈ (Met‘(Base‘𝑈)))) |
31 | 7, 26, 30 | sylanbrc 585 | 1 ⊢ (𝜑 → 𝑈 ∈ MetSp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ⊆ wss 3938 × cxp 5555 ↾ cres 5559 –onto→wfo 6355 –1-1-onto→wf1o 6356 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 distcds 16576 TopOpenctopn 16697 “s cimas 16779 Metcmet 20533 ∞MetSpcxms 22929 MetSpcms 22930 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-om 7583 df-1st 7691 df-2nd 7692 df-supp 7833 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fsupp 8836 df-sup 8908 df-inf 8909 df-oi 8976 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-fz 12896 df-fzo 13037 df-seq 13373 df-hash 13694 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-sca 16583 df-vsca 16584 df-ip 16585 df-tset 16586 df-ple 16587 df-ds 16589 df-rest 16698 df-topn 16699 df-0g 16717 df-gsum 16718 df-topgen 16719 df-xrs 16777 df-qtop 16782 df-imas 16783 df-mre 16859 df-mrc 16860 df-acs 16862 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-submnd 17959 df-mulg 18227 df-cntz 18449 df-cmn 18910 df-psmet 20539 df-xmet 20540 df-met 20541 df-bl 20542 df-mopn 20543 df-top 21504 df-topon 21521 df-topsp 21543 df-bases 21556 df-xms 22932 df-ms 22933 |
This theorem is referenced by: xpsms 23147 |
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