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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 23044 | . 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top) |
3 | eqid 2821 | . . . . 5 ⊢ (𝑢 ∈ 𝑋 ↦ inf(ran (𝑣 ∈ 𝑥 ↦ (𝑢𝐷𝑣)), ℝ*, < )) = (𝑢 ∈ 𝑋 ↦ inf(ran (𝑣 ∈ 𝑥 ↦ (𝑢𝐷𝑣)), ℝ*, < )) | |
4 | simp1 1132 | . . . . 5 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝑦 ∈ (Clsd‘𝐽)) ∧ (𝑥 ∩ 𝑦) = ∅) → 𝐷 ∈ (∞Met‘𝑋)) | |
5 | simp2l 1195 | . . . . 5 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝑦 ∈ (Clsd‘𝐽)) ∧ (𝑥 ∩ 𝑦) = ∅) → 𝑥 ∈ (Clsd‘𝐽)) | |
6 | simp2r 1196 | . . . . 5 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝑦 ∈ (Clsd‘𝐽)) ∧ (𝑥 ∩ 𝑦) = ∅) → 𝑦 ∈ (Clsd‘𝐽)) | |
7 | simp3 1134 | . . . . 5 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝑦 ∈ (Clsd‘𝐽)) ∧ (𝑥 ∩ 𝑦) = ∅) → (𝑥 ∩ 𝑦) = ∅) | |
8 | eqid 2821 | . . . . 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 2821 | . . . . 5 ⊢ (𝑢 ∈ 𝑋 ↦ inf(ran (𝑣 ∈ 𝑦 ↦ (𝑢𝐷𝑣)), ℝ*, < )) = (𝑢 ∈ 𝑋 ↦ inf(ran (𝑣 ∈ 𝑦 ↦ (𝑢𝐷𝑣)), ℝ*, < )) | |
10 | eqid 2821 | . . . . 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 23463 | . . . 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 1117 | . . 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 3191 | . 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → ∀𝑥 ∈ (Clsd‘𝐽)∀𝑦 ∈ (Clsd‘𝐽)((𝑥 ∩ 𝑦) = ∅ → ∃𝑧 ∈ 𝐽 ∃𝑤 ∈ 𝐽 (𝑥 ⊆ 𝑧 ∧ 𝑦 ⊆ 𝑤 ∧ (𝑧 ∩ 𝑤) = ∅))) |
14 | isnrm3 21961 | . 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 585 | 1 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Nrm) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ∀wral 3138 ∃wrex 3139 ∩ cin 3935 ⊆ wss 3936 ∅c0 4291 ifcif 4467 ∪ ciun 4912 class class class wbr 5059 ↦ cmpt 5139 ran crn 5551 ‘cfv 6350 (class class class)co 7150 infcinf 8899 1c1 10532 ℝ*cxr 10668 < clt 10669 ≤ cle 10670 / cdiv 11291 2c2 11686 ∞Metcxmet 20524 ballcbl 20526 MetOpencmopn 20529 Topctop 21495 Clsdccld 21618 Nrmcnrm 21912 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-iin 4915 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-ec 8285 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-sup 8900 df-inf 8901 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-n0 11892 df-z 11976 df-uz 12238 df-q 12343 df-rp 12384 df-xneg 12501 df-xadd 12502 df-xmul 12503 df-icc 12739 df-topgen 16711 df-psmet 20531 df-xmet 20532 df-bl 20534 df-mopn 20535 df-top 21496 df-topon 21513 df-bases 21548 df-cld 21621 df-ntr 21622 df-cls 21623 df-nrm 21919 |
This theorem is referenced by: metreg 23465 |
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