|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > metnrmlem2 | Structured version Visualization version GIF version | ||
| Description: Lemma for metnrm 24885. (Contributed by Mario Carneiro, 14-Jan-2014.) (Revised by Mario Carneiro, 5-Sep-2015.) | 
| Ref | Expression | 
|---|---|
| metdscn.f | ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ inf(ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < )) | 
| metdscn.j | ⊢ 𝐽 = (MetOpen‘𝐷) | 
| metnrmlem.1 | ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋)) | 
| metnrmlem.2 | ⊢ (𝜑 → 𝑆 ∈ (Clsd‘𝐽)) | 
| metnrmlem.3 | ⊢ (𝜑 → 𝑇 ∈ (Clsd‘𝐽)) | 
| metnrmlem.4 | ⊢ (𝜑 → (𝑆 ∩ 𝑇) = ∅) | 
| metnrmlem.u | ⊢ 𝑈 = ∪ 𝑡 ∈ 𝑇 (𝑡(ball‘𝐷)(if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2)) | 
| Ref | Expression | 
|---|---|
| metnrmlem2 | ⊢ (𝜑 → (𝑈 ∈ 𝐽 ∧ 𝑇 ⊆ 𝑈)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | metnrmlem.u | . . 3 ⊢ 𝑈 = ∪ 𝑡 ∈ 𝑇 (𝑡(ball‘𝐷)(if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2)) | |
| 2 | metnrmlem.1 | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋)) | |
| 3 | metdscn.j | . . . . . 6 ⊢ 𝐽 = (MetOpen‘𝐷) | |
| 4 | 3 | mopntop 24451 | . . . . 5 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top) | 
| 5 | 2, 4 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐽 ∈ Top) | 
| 6 | 2 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝐷 ∈ (∞Met‘𝑋)) | 
| 7 | metnrmlem.3 | . . . . . . . . 9 ⊢ (𝜑 → 𝑇 ∈ (Clsd‘𝐽)) | |
| 8 | eqid 2736 | . . . . . . . . . 10 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
| 9 | 8 | cldss 23038 | . . . . . . . . 9 ⊢ (𝑇 ∈ (Clsd‘𝐽) → 𝑇 ⊆ ∪ 𝐽) | 
| 10 | 7, 9 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑇 ⊆ ∪ 𝐽) | 
| 11 | 3 | mopnuni 24452 | . . . . . . . . 9 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = ∪ 𝐽) | 
| 12 | 2, 11 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑋 = ∪ 𝐽) | 
| 13 | 10, 12 | sseqtrrd 4020 | . . . . . . 7 ⊢ (𝜑 → 𝑇 ⊆ 𝑋) | 
| 14 | 13 | sselda 3982 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑡 ∈ 𝑋) | 
| 15 | metdscn.f | . . . . . . . . . 10 ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ inf(ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < )) | |
| 16 | metnrmlem.2 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑆 ∈ (Clsd‘𝐽)) | |
| 17 | metnrmlem.4 | . . . . . . . . . 10 ⊢ (𝜑 → (𝑆 ∩ 𝑇) = ∅) | |
| 18 | 15, 3, 2, 16, 7, 17 | metnrmlem1a 24881 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (0 < (𝐹‘𝑡) ∧ if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) ∈ ℝ+)) | 
| 19 | 18 | simprd 495 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) ∈ ℝ+) | 
| 20 | 19 | rphalfcld 13090 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2) ∈ ℝ+) | 
| 21 | 20 | rpxrd 13079 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2) ∈ ℝ*) | 
| 22 | 3 | blopn 24514 | . . . . . 6 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑡 ∈ 𝑋 ∧ (if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2) ∈ ℝ*) → (𝑡(ball‘𝐷)(if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2)) ∈ 𝐽) | 
| 23 | 6, 14, 21, 22 | syl3anc 1372 | . . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝑡(ball‘𝐷)(if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2)) ∈ 𝐽) | 
| 24 | 23 | ralrimiva 3145 | . . . 4 ⊢ (𝜑 → ∀𝑡 ∈ 𝑇 (𝑡(ball‘𝐷)(if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2)) ∈ 𝐽) | 
| 25 | iunopn 22905 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ ∀𝑡 ∈ 𝑇 (𝑡(ball‘𝐷)(if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2)) ∈ 𝐽) → ∪ 𝑡 ∈ 𝑇 (𝑡(ball‘𝐷)(if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2)) ∈ 𝐽) | |
| 26 | 5, 24, 25 | syl2anc 584 | . . 3 ⊢ (𝜑 → ∪ 𝑡 ∈ 𝑇 (𝑡(ball‘𝐷)(if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2)) ∈ 𝐽) | 
| 27 | 1, 26 | eqeltrid 2844 | . 2 ⊢ (𝜑 → 𝑈 ∈ 𝐽) | 
| 28 | blcntr 24424 | . . . . . . 7 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑡 ∈ 𝑋 ∧ (if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2) ∈ ℝ+) → 𝑡 ∈ (𝑡(ball‘𝐷)(if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2))) | |
| 29 | 6, 14, 20, 28 | syl3anc 1372 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑡 ∈ (𝑡(ball‘𝐷)(if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2))) | 
| 30 | 29 | snssd 4808 | . . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → {𝑡} ⊆ (𝑡(ball‘𝐷)(if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2))) | 
| 31 | 30 | ralrimiva 3145 | . . . 4 ⊢ (𝜑 → ∀𝑡 ∈ 𝑇 {𝑡} ⊆ (𝑡(ball‘𝐷)(if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2))) | 
| 32 | ss2iun 5009 | . . . 4 ⊢ (∀𝑡 ∈ 𝑇 {𝑡} ⊆ (𝑡(ball‘𝐷)(if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2)) → ∪ 𝑡 ∈ 𝑇 {𝑡} ⊆ ∪ 𝑡 ∈ 𝑇 (𝑡(ball‘𝐷)(if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2))) | |
| 33 | 31, 32 | syl 17 | . . 3 ⊢ (𝜑 → ∪ 𝑡 ∈ 𝑇 {𝑡} ⊆ ∪ 𝑡 ∈ 𝑇 (𝑡(ball‘𝐷)(if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2))) | 
| 34 | iunid 5059 | . . . 4 ⊢ ∪ 𝑡 ∈ 𝑇 {𝑡} = 𝑇 | |
| 35 | 34 | eqcomi 2745 | . . 3 ⊢ 𝑇 = ∪ 𝑡 ∈ 𝑇 {𝑡} | 
| 36 | 33, 35, 1 | 3sstr4g 4036 | . 2 ⊢ (𝜑 → 𝑇 ⊆ 𝑈) | 
| 37 | 27, 36 | jca 511 | 1 ⊢ (𝜑 → (𝑈 ∈ 𝐽 ∧ 𝑇 ⊆ 𝑈)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ∀wral 3060 ∩ cin 3949 ⊆ wss 3950 ∅c0 4332 ifcif 4524 {csn 4625 ∪ cuni 4906 ∪ ciun 4990 class class class wbr 5142 ↦ cmpt 5224 ran crn 5685 ‘cfv 6560 (class class class)co 7432 infcinf 9482 0cc0 11156 1c1 11157 ℝ*cxr 11295 < clt 11296 ≤ cle 11297 / cdiv 11921 2c2 12322 ℝ+crp 13035 ∞Metcxmet 21350 ballcbl 21352 MetOpencmopn 21355 Topctop 22900 Clsdccld 23025 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 ax-pre-sup 11234 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-iin 4993 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-er 8746 df-map 8869 df-en 8987 df-dom 8988 df-sdom 8989 df-sup 9483 df-inf 9484 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-div 11922 df-nn 12268 df-2 12330 df-n0 12529 df-z 12616 df-uz 12880 df-q 12992 df-rp 13036 df-xneg 13155 df-xadd 13156 df-xmul 13157 df-icc 13395 df-topgen 17489 df-psmet 21357 df-xmet 21358 df-bl 21360 df-mopn 21361 df-top 22901 df-topon 22918 df-bases 22954 df-cld 23028 df-ntr 23029 df-cls 23030 | 
| This theorem is referenced by: metnrmlem3 24884 | 
| Copyright terms: Public domain | W3C validator |