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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ovolval4 | Structured version Visualization version GIF version |
Description: The value of the Lebesgue outer measure for subsets of the reals. Similar to ovolval3 41385, but here 𝑓 may represent unordered interval bounds. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
Ref | Expression |
---|---|
ovolval4.a | ⊢ (𝜑 → 𝐴 ⊆ ℝ) |
ovolval4.m | ⊢ 𝑀 = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))} |
Ref | Expression |
---|---|
ovolval4 | ⊢ (𝜑 → (vol*‘𝐴) = inf(𝑀, ℝ*, < )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ovolval4.a | . 2 ⊢ (𝜑 → 𝐴 ⊆ ℝ) | |
2 | ovolval4.m | . 2 ⊢ 𝑀 = {𝑦 ∈ ℝ* ∣ ∃𝑓 ∈ ((ℝ × ℝ) ↑𝑚 ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = (Σ^‘((vol ∘ (,)) ∘ 𝑓)))} | |
3 | fveq2 6353 | . . . . 5 ⊢ (𝑘 = 𝑛 → (𝑓‘𝑘) = (𝑓‘𝑛)) | |
4 | 3 | fveq2d 6357 | . . . 4 ⊢ (𝑘 = 𝑛 → (1st ‘(𝑓‘𝑘)) = (1st ‘(𝑓‘𝑛))) |
5 | 3 | fveq2d 6357 | . . . . . 6 ⊢ (𝑘 = 𝑛 → (2nd ‘(𝑓‘𝑘)) = (2nd ‘(𝑓‘𝑛))) |
6 | 4, 5 | breq12d 4817 | . . . . 5 ⊢ (𝑘 = 𝑛 → ((1st ‘(𝑓‘𝑘)) ≤ (2nd ‘(𝑓‘𝑘)) ↔ (1st ‘(𝑓‘𝑛)) ≤ (2nd ‘(𝑓‘𝑛)))) |
7 | 6, 5, 4 | ifbieq12d 4257 | . . . 4 ⊢ (𝑘 = 𝑛 → if((1st ‘(𝑓‘𝑘)) ≤ (2nd ‘(𝑓‘𝑘)), (2nd ‘(𝑓‘𝑘)), (1st ‘(𝑓‘𝑘))) = if((1st ‘(𝑓‘𝑛)) ≤ (2nd ‘(𝑓‘𝑛)), (2nd ‘(𝑓‘𝑛)), (1st ‘(𝑓‘𝑛)))) |
8 | 4, 7 | opeq12d 4561 | . . 3 ⊢ (𝑘 = 𝑛 → 〈(1st ‘(𝑓‘𝑘)), if((1st ‘(𝑓‘𝑘)) ≤ (2nd ‘(𝑓‘𝑘)), (2nd ‘(𝑓‘𝑘)), (1st ‘(𝑓‘𝑘)))〉 = 〈(1st ‘(𝑓‘𝑛)), if((1st ‘(𝑓‘𝑛)) ≤ (2nd ‘(𝑓‘𝑛)), (2nd ‘(𝑓‘𝑛)), (1st ‘(𝑓‘𝑛)))〉) |
9 | 8 | cbvmptv 4902 | . 2 ⊢ (𝑘 ∈ ℕ ↦ 〈(1st ‘(𝑓‘𝑘)), if((1st ‘(𝑓‘𝑘)) ≤ (2nd ‘(𝑓‘𝑘)), (2nd ‘(𝑓‘𝑘)), (1st ‘(𝑓‘𝑘)))〉) = (𝑛 ∈ ℕ ↦ 〈(1st ‘(𝑓‘𝑛)), if((1st ‘(𝑓‘𝑛)) ≤ (2nd ‘(𝑓‘𝑛)), (2nd ‘(𝑓‘𝑛)), (1st ‘(𝑓‘𝑛)))〉) |
10 | 1, 2, 9 | ovolval4lem2 41388 | 1 ⊢ (𝜑 → (vol*‘𝐴) = inf(𝑀, ℝ*, < )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1632 ∃wrex 3051 {crab 3054 ⊆ wss 3715 ifcif 4230 〈cop 4327 ∪ cuni 4588 class class class wbr 4804 ↦ cmpt 4881 × cxp 5264 ran crn 5267 ∘ ccom 5270 ‘cfv 6049 (class class class)co 6814 1st c1st 7332 2nd c2nd 7333 ↑𝑚 cmap 8025 infcinf 8514 ℝcr 10147 ℝ*cxr 10285 < clt 10286 ≤ cle 10287 ℕcn 11232 (,)cioo 12388 vol*covol 23451 volcvol 23452 Σ^csumge0 41100 |
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 7115 ax-inf2 8713 ax-cnex 10204 ax-resscn 10205 ax-1cn 10206 ax-icn 10207 ax-addcl 10208 ax-addrcl 10209 ax-mulcl 10210 ax-mulrcl 10211 ax-mulcom 10212 ax-addass 10213 ax-mulass 10214 ax-distr 10215 ax-i2m1 10216 ax-1ne0 10217 ax-1rid 10218 ax-rnegex 10219 ax-rrecex 10220 ax-cnre 10221 ax-pre-lttri 10222 ax-pre-lttrn 10223 ax-pre-ltadd 10224 ax-pre-mulgt0 10225 ax-pre-sup 10226 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-fal 1638 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-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-se 5226 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-isom 6058 df-riota 6775 df-ov 6817 df-oprab 6818 df-mpt2 6819 df-of 7063 df-om 7232 df-1st 7334 df-2nd 7335 df-wrecs 7577 df-recs 7638 df-rdg 7676 df-1o 7730 df-2o 7731 df-oadd 7734 df-er 7913 df-map 8027 df-pm 8028 df-en 8124 df-dom 8125 df-sdom 8126 df-fin 8127 df-fi 8484 df-sup 8515 df-inf 8516 df-oi 8582 df-card 8975 df-cda 9202 df-pnf 10288 df-mnf 10289 df-xr 10290 df-ltxr 10291 df-le 10292 df-sub 10480 df-neg 10481 df-div 10897 df-nn 11233 df-2 11291 df-3 11292 df-n0 11505 df-z 11590 df-uz 11900 df-q 12002 df-rp 12046 df-xneg 12159 df-xadd 12160 df-xmul 12161 df-ioo 12392 df-ico 12394 df-icc 12395 df-fz 12540 df-fzo 12680 df-fl 12807 df-seq 13016 df-exp 13075 df-hash 13332 df-cj 14058 df-re 14059 df-im 14060 df-sqrt 14194 df-abs 14195 df-clim 14438 df-rlim 14439 df-sum 14636 df-rest 16305 df-topgen 16326 df-psmet 19960 df-xmet 19961 df-met 19962 df-bl 19963 df-mopn 19964 df-top 20921 df-topon 20938 df-bases 20972 df-cmp 21412 df-ovol 23453 df-vol 23454 df-sumge0 41101 |
This theorem is referenced by: ovolval5 41393 |
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