![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 22615 | . 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top) |
3 | eqid 2825 | . . . . 5 ⊢ (𝑢 ∈ 𝑋 ↦ inf(ran (𝑣 ∈ 𝑥 ↦ (𝑢𝐷𝑣)), ℝ*, < )) = (𝑢 ∈ 𝑋 ↦ inf(ran (𝑣 ∈ 𝑥 ↦ (𝑢𝐷𝑣)), ℝ*, < )) | |
4 | simp1 1172 | . . . . 5 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝑦 ∈ (Clsd‘𝐽)) ∧ (𝑥 ∩ 𝑦) = ∅) → 𝐷 ∈ (∞Met‘𝑋)) | |
5 | simp2l 1262 | . . . . 5 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝑦 ∈ (Clsd‘𝐽)) ∧ (𝑥 ∩ 𝑦) = ∅) → 𝑥 ∈ (Clsd‘𝐽)) | |
6 | simp2r 1263 | . . . . 5 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝑦 ∈ (Clsd‘𝐽)) ∧ (𝑥 ∩ 𝑦) = ∅) → 𝑦 ∈ (Clsd‘𝐽)) | |
7 | simp3 1174 | . . . . 5 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝑦 ∈ (Clsd‘𝐽)) ∧ (𝑥 ∩ 𝑦) = ∅) → (𝑥 ∩ 𝑦) = ∅) | |
8 | eqid 2825 | . . . . 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 2825 | . . . . 5 ⊢ (𝑢 ∈ 𝑋 ↦ inf(ran (𝑣 ∈ 𝑦 ↦ (𝑢𝐷𝑣)), ℝ*, < )) = (𝑢 ∈ 𝑋 ↦ inf(ran (𝑣 ∈ 𝑦 ↦ (𝑢𝐷𝑣)), ℝ*, < )) | |
10 | eqid 2825 | . . . . 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 23034 | . . . 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 1156 | . . 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 3180 | . 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → ∀𝑥 ∈ (Clsd‘𝐽)∀𝑦 ∈ (Clsd‘𝐽)((𝑥 ∩ 𝑦) = ∅ → ∃𝑧 ∈ 𝐽 ∃𝑤 ∈ 𝐽 (𝑥 ⊆ 𝑧 ∧ 𝑦 ⊆ 𝑤 ∧ (𝑧 ∩ 𝑤) = ∅))) |
14 | isnrm3 21534 | . 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 580 | 1 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Nrm) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∧ w3a 1113 = wceq 1658 ∈ wcel 2166 ∀wral 3117 ∃wrex 3118 ∩ cin 3797 ⊆ wss 3798 ∅c0 4144 ifcif 4306 ∪ ciun 4740 class class class wbr 4873 ↦ cmpt 4952 ran crn 5343 ‘cfv 6123 (class class class)co 6905 infcinf 8616 1c1 10253 ℝ*cxr 10390 < clt 10391 ≤ cle 10392 / cdiv 11009 2c2 11406 ∞Metcxmet 20091 ballcbl 20093 MetOpencmopn 20096 Topctop 21068 Clsdccld 21191 Nrmcnrm 21485 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 ax-pre-sup 10330 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-int 4698 df-iun 4742 df-iin 4743 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-1st 7428 df-2nd 7429 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-er 8009 df-ec 8011 df-map 8124 df-en 8223 df-dom 8224 df-sdom 8225 df-sup 8617 df-inf 8618 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-div 11010 df-nn 11351 df-2 11414 df-n0 11619 df-z 11705 df-uz 11969 df-q 12072 df-rp 12113 df-xneg 12232 df-xadd 12233 df-xmul 12234 df-icc 12470 df-topgen 16457 df-psmet 20098 df-xmet 20099 df-bl 20101 df-mopn 20102 df-top 21069 df-topon 21086 df-bases 21121 df-cld 21194 df-ntr 21195 df-cls 21196 df-nrm 21492 |
This theorem is referenced by: metreg 23036 |
Copyright terms: Public domain | W3C validator |