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Mirrors > Home > MPE Home > Th. List > metnrm | Structured version Visualization version GIF version |
Description: A metric space is normal. (Contributed by Jeff Hankins, 31-Aug-2013.) (Revised by Mario Carneiro, 5-Sep-2015.) (Proof shortened by AV, 30-Sep-2020.) |
Ref | Expression |
---|---|
metnrm.j | ⊢ 𝐽 = (MetOpen‘𝐷) |
Ref | Expression |
---|---|
metnrm | ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Nrm) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | metnrm.j | . . 3 ⊢ 𝐽 = (MetOpen‘𝐷) | |
2 | 1 | mopntop 23047 | . 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top) |
3 | eqid 2798 | . . . . 5 ⊢ (𝑢 ∈ 𝑋 ↦ inf(ran (𝑣 ∈ 𝑥 ↦ (𝑢𝐷𝑣)), ℝ*, < )) = (𝑢 ∈ 𝑋 ↦ inf(ran (𝑣 ∈ 𝑥 ↦ (𝑢𝐷𝑣)), ℝ*, < )) | |
4 | simp1 1133 | . . . . 5 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝑦 ∈ (Clsd‘𝐽)) ∧ (𝑥 ∩ 𝑦) = ∅) → 𝐷 ∈ (∞Met‘𝑋)) | |
5 | simp2l 1196 | . . . . 5 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝑦 ∈ (Clsd‘𝐽)) ∧ (𝑥 ∩ 𝑦) = ∅) → 𝑥 ∈ (Clsd‘𝐽)) | |
6 | simp2r 1197 | . . . . 5 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝑦 ∈ (Clsd‘𝐽)) ∧ (𝑥 ∩ 𝑦) = ∅) → 𝑦 ∈ (Clsd‘𝐽)) | |
7 | simp3 1135 | . . . . 5 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝑦 ∈ (Clsd‘𝐽)) ∧ (𝑥 ∩ 𝑦) = ∅) → (𝑥 ∩ 𝑦) = ∅) | |
8 | eqid 2798 | . . . . 5 ⊢ ∪ 𝑠 ∈ 𝑦 (𝑠(ball‘𝐷)(if(1 ≤ ((𝑢 ∈ 𝑋 ↦ inf(ran (𝑣 ∈ 𝑥 ↦ (𝑢𝐷𝑣)), ℝ*, < ))‘𝑠), 1, ((𝑢 ∈ 𝑋 ↦ inf(ran (𝑣 ∈ 𝑥 ↦ (𝑢𝐷𝑣)), ℝ*, < ))‘𝑠)) / 2)) = ∪ 𝑠 ∈ 𝑦 (𝑠(ball‘𝐷)(if(1 ≤ ((𝑢 ∈ 𝑋 ↦ inf(ran (𝑣 ∈ 𝑥 ↦ (𝑢𝐷𝑣)), ℝ*, < ))‘𝑠), 1, ((𝑢 ∈ 𝑋 ↦ inf(ran (𝑣 ∈ 𝑥 ↦ (𝑢𝐷𝑣)), ℝ*, < ))‘𝑠)) / 2)) | |
9 | eqid 2798 | . . . . 5 ⊢ (𝑢 ∈ 𝑋 ↦ inf(ran (𝑣 ∈ 𝑦 ↦ (𝑢𝐷𝑣)), ℝ*, < )) = (𝑢 ∈ 𝑋 ↦ inf(ran (𝑣 ∈ 𝑦 ↦ (𝑢𝐷𝑣)), ℝ*, < )) | |
10 | eqid 2798 | . . . . 5 ⊢ ∪ 𝑡 ∈ 𝑥 (𝑡(ball‘𝐷)(if(1 ≤ ((𝑢 ∈ 𝑋 ↦ inf(ran (𝑣 ∈ 𝑦 ↦ (𝑢𝐷𝑣)), ℝ*, < ))‘𝑡), 1, ((𝑢 ∈ 𝑋 ↦ inf(ran (𝑣 ∈ 𝑦 ↦ (𝑢𝐷𝑣)), ℝ*, < ))‘𝑡)) / 2)) = ∪ 𝑡 ∈ 𝑥 (𝑡(ball‘𝐷)(if(1 ≤ ((𝑢 ∈ 𝑋 ↦ inf(ran (𝑣 ∈ 𝑦 ↦ (𝑢𝐷𝑣)), ℝ*, < ))‘𝑡), 1, ((𝑢 ∈ 𝑋 ↦ inf(ran (𝑣 ∈ 𝑦 ↦ (𝑢𝐷𝑣)), ℝ*, < ))‘𝑡)) / 2)) | |
11 | 3, 1, 4, 5, 6, 7, 8, 9, 10 | metnrmlem3 23466 | . . . 4 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝑦 ∈ (Clsd‘𝐽)) ∧ (𝑥 ∩ 𝑦) = ∅) → ∃𝑧 ∈ 𝐽 ∃𝑤 ∈ 𝐽 (𝑥 ⊆ 𝑧 ∧ 𝑦 ⊆ 𝑤 ∧ (𝑧 ∩ 𝑤) = ∅)) |
12 | 11 | 3expia 1118 | . . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝑦 ∈ (Clsd‘𝐽))) → ((𝑥 ∩ 𝑦) = ∅ → ∃𝑧 ∈ 𝐽 ∃𝑤 ∈ 𝐽 (𝑥 ⊆ 𝑧 ∧ 𝑦 ⊆ 𝑤 ∧ (𝑧 ∩ 𝑤) = ∅))) |
13 | 12 | ralrimivva 3156 | . 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → ∀𝑥 ∈ (Clsd‘𝐽)∀𝑦 ∈ (Clsd‘𝐽)((𝑥 ∩ 𝑦) = ∅ → ∃𝑧 ∈ 𝐽 ∃𝑤 ∈ 𝐽 (𝑥 ⊆ 𝑧 ∧ 𝑦 ⊆ 𝑤 ∧ (𝑧 ∩ 𝑤) = ∅))) |
14 | isnrm3 21964 | . 2 ⊢ (𝐽 ∈ Nrm ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ (Clsd‘𝐽)∀𝑦 ∈ (Clsd‘𝐽)((𝑥 ∩ 𝑦) = ∅ → ∃𝑧 ∈ 𝐽 ∃𝑤 ∈ 𝐽 (𝑥 ⊆ 𝑧 ∧ 𝑦 ⊆ 𝑤 ∧ (𝑧 ∩ 𝑤) = ∅)))) | |
15 | 2, 13, 14 | sylanbrc 586 | 1 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Nrm) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ∀wral 3106 ∃wrex 3107 ∩ cin 3880 ⊆ wss 3881 ∅c0 4243 ifcif 4425 ∪ ciun 4881 class class class wbr 5030 ↦ cmpt 5110 ran crn 5520 ‘cfv 6324 (class class class)co 7135 infcinf 8889 1c1 10527 ℝ*cxr 10663 < clt 10664 ≤ cle 10665 / cdiv 11286 2c2 11680 ∞Metcxmet 20076 ballcbl 20078 MetOpencmopn 20081 Topctop 21498 Clsdccld 21621 Nrmcnrm 21915 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-ec 8274 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-sup 8890 df-inf 8891 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-n0 11886 df-z 11970 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-icc 12733 df-topgen 16709 df-psmet 20083 df-xmet 20084 df-bl 20086 df-mopn 20087 df-top 21499 df-topon 21516 df-bases 21551 df-cld 21624 df-ntr 21625 df-cls 21626 df-nrm 21922 |
This theorem is referenced by: metreg 23468 |
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