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Mirrors > Home > MPE Home > Th. List > metnrmlem2 | Structured version Visualization version GIF version |
Description: Lemma for metnrm 22866. (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 22446 | . . . . 5 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top) |
5 | 2, 4 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐽 ∈ Top) |
6 | 2 | adantr 472 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝐷 ∈ (∞Met‘𝑋)) |
7 | metnrmlem.3 | . . . . . . . . 9 ⊢ (𝜑 → 𝑇 ∈ (Clsd‘𝐽)) | |
8 | eqid 2760 | . . . . . . . . . 10 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
9 | 8 | cldss 21035 | . . . . . . . . 9 ⊢ (𝑇 ∈ (Clsd‘𝐽) → 𝑇 ⊆ ∪ 𝐽) |
10 | 7, 9 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑇 ⊆ ∪ 𝐽) |
11 | 3 | mopnuni 22447 | . . . . . . . . 9 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = ∪ 𝐽) |
12 | 2, 11 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑋 = ∪ 𝐽) |
13 | 10, 12 | sseqtr4d 3783 | . . . . . . 7 ⊢ (𝜑 → 𝑇 ⊆ 𝑋) |
14 | 13 | sselda 3744 | . . . . . 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 22862 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (0 < (𝐹‘𝑡) ∧ if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) ∈ ℝ+)) |
19 | 18 | simprd 482 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) ∈ ℝ+) |
20 | 19 | rphalfcld 12077 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2) ∈ ℝ+) |
21 | 20 | rpxrd 12066 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2) ∈ ℝ*) |
22 | 3 | blopn 22506 | . . . . . 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 1477 | . . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝑡(ball‘𝐷)(if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2)) ∈ 𝐽) |
24 | 23 | ralrimiva 3104 | . . . 4 ⊢ (𝜑 → ∀𝑡 ∈ 𝑇 (𝑡(ball‘𝐷)(if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2)) ∈ 𝐽) |
25 | iunopn 20905 | . . . 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 696 | . . 3 ⊢ (𝜑 → ∪ 𝑡 ∈ 𝑇 (𝑡(ball‘𝐷)(if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2)) ∈ 𝐽) |
27 | 1, 26 | syl5eqel 2843 | . 2 ⊢ (𝜑 → 𝑈 ∈ 𝐽) |
28 | blcntr 22419 | . . . . . . 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 1477 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑡 ∈ (𝑡(ball‘𝐷)(if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2))) |
30 | 29 | snssd 4485 | . . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → {𝑡} ⊆ (𝑡(ball‘𝐷)(if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2))) |
31 | 30 | ralrimiva 3104 | . . . 4 ⊢ (𝜑 → ∀𝑡 ∈ 𝑇 {𝑡} ⊆ (𝑡(ball‘𝐷)(if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2))) |
32 | ss2iun 4688 | . . . 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 4727 | . . . 4 ⊢ ∪ 𝑡 ∈ 𝑇 {𝑡} = 𝑇 | |
35 | 34 | eqcomi 2769 | . . 3 ⊢ 𝑇 = ∪ 𝑡 ∈ 𝑇 {𝑡} |
36 | 33, 35, 1 | 3sstr4g 3787 | . 2 ⊢ (𝜑 → 𝑇 ⊆ 𝑈) |
37 | 27, 36 | jca 555 | 1 ⊢ (𝜑 → (𝑈 ∈ 𝐽 ∧ 𝑇 ⊆ 𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1632 ∈ wcel 2139 ∀wral 3050 ∩ cin 3714 ⊆ wss 3715 ∅c0 4058 ifcif 4230 {csn 4321 ∪ cuni 4588 ∪ ciun 4672 class class class wbr 4804 ↦ cmpt 4881 ran crn 5267 ‘cfv 6049 (class class class)co 6813 infcinf 8512 0cc0 10128 1c1 10129 ℝ*cxr 10265 < clt 10266 ≤ cle 10267 / cdiv 10876 2c2 11262 ℝ+crp 12025 ∞Metcxmt 19933 ballcbl 19935 MetOpencmopn 19938 Topctop 20900 Clsdccld 21022 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 ax-cnex 10184 ax-resscn 10185 ax-1cn 10186 ax-icn 10187 ax-addcl 10188 ax-addrcl 10189 ax-mulcl 10190 ax-mulrcl 10191 ax-mulcom 10192 ax-addass 10193 ax-mulass 10194 ax-distr 10195 ax-i2m1 10196 ax-1ne0 10197 ax-1rid 10198 ax-rnegex 10199 ax-rrecex 10200 ax-cnre 10201 ax-pre-lttri 10202 ax-pre-lttrn 10203 ax-pre-ltadd 10204 ax-pre-mulgt0 10205 ax-pre-sup 10206 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rmo 3058 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-int 4628 df-iun 4674 df-iin 4675 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6774 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-om 7231 df-1st 7333 df-2nd 7334 df-wrecs 7576 df-recs 7637 df-rdg 7675 df-er 7911 df-map 8025 df-en 8122 df-dom 8123 df-sdom 8124 df-sup 8513 df-inf 8514 df-pnf 10268 df-mnf 10269 df-xr 10270 df-ltxr 10271 df-le 10272 df-sub 10460 df-neg 10461 df-div 10877 df-nn 11213 df-2 11271 df-n0 11485 df-z 11570 df-uz 11880 df-q 11982 df-rp 12026 df-xneg 12139 df-xadd 12140 df-xmul 12141 df-icc 12375 df-topgen 16306 df-psmet 19940 df-xmet 19941 df-bl 19943 df-mopn 19944 df-top 20901 df-topon 20918 df-bases 20952 df-cld 21025 df-ntr 21026 df-cls 21027 |
This theorem is referenced by: metnrmlem3 22865 |
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