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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > isvonmbl | Structured version Visualization version GIF version |
Description: The predicate "𝐴 is measurable w.r.t. the n-dimensional Lebesgue measure". A set is measurable if it splits every other set 𝑥 in a "nice" way, that is, if the measure of the pieces 𝑥 ∩ 𝐴 and 𝑥 ∖ 𝐴 sum up to the measure of 𝑥. Definition 114E of [Fremlin1] p. 25. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
Ref | Expression |
---|---|
isvonmbl.1 | ⊢ (𝜑 → 𝑋 ∈ Fin) |
Ref | Expression |
---|---|
isvonmbl | ⊢ (𝜑 → (𝐸 ∈ dom (voln‘𝑋) ↔ (𝐸 ⊆ (ℝ ↑m 𝑋) ∧ ∀𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)(((voln*‘𝑋)‘(𝑎 ∩ 𝐸)) +𝑒 ((voln*‘𝑋)‘(𝑎 ∖ 𝐸))) = ((voln*‘𝑋)‘𝑎)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isvonmbl.1 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ Fin) | |
2 | 1 | dmvon 43245 | . . 3 ⊢ (𝜑 → dom (voln‘𝑋) = (CaraGen‘(voln*‘𝑋))) |
3 | 2 | eleq2d 2875 | . 2 ⊢ (𝜑 → (𝐸 ∈ dom (voln‘𝑋) ↔ 𝐸 ∈ (CaraGen‘(voln*‘𝑋)))) |
4 | 1 | ovnome 43212 | . . 3 ⊢ (𝜑 → (voln*‘𝑋) ∈ OutMeas) |
5 | eqid 2798 | . . 3 ⊢ (CaraGen‘(voln*‘𝑋)) = (CaraGen‘(voln*‘𝑋)) | |
6 | 4, 5 | caragenel 43134 | . 2 ⊢ (𝜑 → (𝐸 ∈ (CaraGen‘(voln*‘𝑋)) ↔ (𝐸 ∈ 𝒫 ∪ dom (voln*‘𝑋) ∧ ∀𝑎 ∈ 𝒫 ∪ dom (voln*‘𝑋)(((voln*‘𝑋)‘(𝑎 ∩ 𝐸)) +𝑒 ((voln*‘𝑋)‘(𝑎 ∖ 𝐸))) = ((voln*‘𝑋)‘𝑎)))) |
7 | elpwi 4506 | . . . . . . 7 ⊢ (𝐸 ∈ 𝒫 ∪ dom (voln*‘𝑋) → 𝐸 ⊆ ∪ dom (voln*‘𝑋)) | |
8 | 7 | adantl 485 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐸 ∈ 𝒫 ∪ dom (voln*‘𝑋)) → 𝐸 ⊆ ∪ dom (voln*‘𝑋)) |
9 | 1 | unidmovn 43252 | . . . . . . 7 ⊢ (𝜑 → ∪ dom (voln*‘𝑋) = (ℝ ↑m 𝑋)) |
10 | 9 | adantr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐸 ∈ 𝒫 ∪ dom (voln*‘𝑋)) → ∪ dom (voln*‘𝑋) = (ℝ ↑m 𝑋)) |
11 | 8, 10 | sseqtrd 3955 | . . . . 5 ⊢ ((𝜑 ∧ 𝐸 ∈ 𝒫 ∪ dom (voln*‘𝑋)) → 𝐸 ⊆ (ℝ ↑m 𝑋)) |
12 | 11 | ex 416 | . . . 4 ⊢ (𝜑 → (𝐸 ∈ 𝒫 ∪ dom (voln*‘𝑋) → 𝐸 ⊆ (ℝ ↑m 𝑋))) |
13 | simpr 488 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐸 ⊆ (ℝ ↑m 𝑋)) → 𝐸 ⊆ (ℝ ↑m 𝑋)) | |
14 | 9 | eqcomd 2804 | . . . . . . . 8 ⊢ (𝜑 → (ℝ ↑m 𝑋) = ∪ dom (voln*‘𝑋)) |
15 | 14 | adantr 484 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐸 ⊆ (ℝ ↑m 𝑋)) → (ℝ ↑m 𝑋) = ∪ dom (voln*‘𝑋)) |
16 | 13, 15 | sseqtrd 3955 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐸 ⊆ (ℝ ↑m 𝑋)) → 𝐸 ⊆ ∪ dom (voln*‘𝑋)) |
17 | ovex 7168 | . . . . . . . . 9 ⊢ (ℝ ↑m 𝑋) ∈ V | |
18 | 17 | ssex 5189 | . . . . . . . 8 ⊢ (𝐸 ⊆ (ℝ ↑m 𝑋) → 𝐸 ∈ V) |
19 | elpwg 4500 | . . . . . . . 8 ⊢ (𝐸 ∈ V → (𝐸 ∈ 𝒫 ∪ dom (voln*‘𝑋) ↔ 𝐸 ⊆ ∪ dom (voln*‘𝑋))) | |
20 | 18, 19 | syl 17 | . . . . . . 7 ⊢ (𝐸 ⊆ (ℝ ↑m 𝑋) → (𝐸 ∈ 𝒫 ∪ dom (voln*‘𝑋) ↔ 𝐸 ⊆ ∪ dom (voln*‘𝑋))) |
21 | 20 | adantl 485 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐸 ⊆ (ℝ ↑m 𝑋)) → (𝐸 ∈ 𝒫 ∪ dom (voln*‘𝑋) ↔ 𝐸 ⊆ ∪ dom (voln*‘𝑋))) |
22 | 16, 21 | mpbird 260 | . . . . 5 ⊢ ((𝜑 ∧ 𝐸 ⊆ (ℝ ↑m 𝑋)) → 𝐸 ∈ 𝒫 ∪ dom (voln*‘𝑋)) |
23 | 22 | ex 416 | . . . 4 ⊢ (𝜑 → (𝐸 ⊆ (ℝ ↑m 𝑋) → 𝐸 ∈ 𝒫 ∪ dom (voln*‘𝑋))) |
24 | 12, 23 | impbid 215 | . . 3 ⊢ (𝜑 → (𝐸 ∈ 𝒫 ∪ dom (voln*‘𝑋) ↔ 𝐸 ⊆ (ℝ ↑m 𝑋))) |
25 | 9 | pweqd 4516 | . . . 4 ⊢ (𝜑 → 𝒫 ∪ dom (voln*‘𝑋) = 𝒫 (ℝ ↑m 𝑋)) |
26 | raleq 3358 | . . . 4 ⊢ (𝒫 ∪ dom (voln*‘𝑋) = 𝒫 (ℝ ↑m 𝑋) → (∀𝑎 ∈ 𝒫 ∪ dom (voln*‘𝑋)(((voln*‘𝑋)‘(𝑎 ∩ 𝐸)) +𝑒 ((voln*‘𝑋)‘(𝑎 ∖ 𝐸))) = ((voln*‘𝑋)‘𝑎) ↔ ∀𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)(((voln*‘𝑋)‘(𝑎 ∩ 𝐸)) +𝑒 ((voln*‘𝑋)‘(𝑎 ∖ 𝐸))) = ((voln*‘𝑋)‘𝑎))) | |
27 | 25, 26 | syl 17 | . . 3 ⊢ (𝜑 → (∀𝑎 ∈ 𝒫 ∪ dom (voln*‘𝑋)(((voln*‘𝑋)‘(𝑎 ∩ 𝐸)) +𝑒 ((voln*‘𝑋)‘(𝑎 ∖ 𝐸))) = ((voln*‘𝑋)‘𝑎) ↔ ∀𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)(((voln*‘𝑋)‘(𝑎 ∩ 𝐸)) +𝑒 ((voln*‘𝑋)‘(𝑎 ∖ 𝐸))) = ((voln*‘𝑋)‘𝑎))) |
28 | 24, 27 | anbi12d 633 | . 2 ⊢ (𝜑 → ((𝐸 ∈ 𝒫 ∪ dom (voln*‘𝑋) ∧ ∀𝑎 ∈ 𝒫 ∪ dom (voln*‘𝑋)(((voln*‘𝑋)‘(𝑎 ∩ 𝐸)) +𝑒 ((voln*‘𝑋)‘(𝑎 ∖ 𝐸))) = ((voln*‘𝑋)‘𝑎)) ↔ (𝐸 ⊆ (ℝ ↑m 𝑋) ∧ ∀𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)(((voln*‘𝑋)‘(𝑎 ∩ 𝐸)) +𝑒 ((voln*‘𝑋)‘(𝑎 ∖ 𝐸))) = ((voln*‘𝑋)‘𝑎)))) |
29 | 3, 6, 28 | 3bitrd 308 | 1 ⊢ (𝜑 → (𝐸 ∈ dom (voln‘𝑋) ↔ (𝐸 ⊆ (ℝ ↑m 𝑋) ∧ ∀𝑎 ∈ 𝒫 (ℝ ↑m 𝑋)(((voln*‘𝑋)‘(𝑎 ∩ 𝐸)) +𝑒 ((voln*‘𝑋)‘(𝑎 ∖ 𝐸))) = ((voln*‘𝑋)‘𝑎)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∀wral 3106 Vcvv 3441 ∖ cdif 3878 ∩ cin 3880 ⊆ wss 3881 𝒫 cpw 4497 ∪ cuni 4800 dom cdm 5519 ‘cfv 6324 (class class class)co 7135 ↑m cmap 8389 Fincfn 8492 ℝcr 10525 +𝑒 cxad 12493 CaraGenccaragen 43130 voln*covoln 43175 volncvoln 43177 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cc 9846 ax-ac2 9874 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 ax-mulf 10606 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-disj 4996 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-tpos 7875 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-map 8391 df-pm 8392 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fi 8859 df-sup 8890 df-inf 8891 df-oi 8958 df-dju 9314 df-card 9352 df-acn 9355 df-ac 9527 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ioo 12730 df-ico 12732 df-icc 12733 df-fz 12886 df-fzo 13029 df-fl 13157 df-seq 13365 df-exp 13426 df-hash 13687 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-clim 14837 df-rlim 14838 df-sum 15035 df-prod 15252 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-rest 16688 df-0g 16707 df-topgen 16709 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-grp 18098 df-minusg 18099 df-subg 18268 df-cmn 18900 df-abl 18901 df-mgp 19233 df-ur 19245 df-ring 19292 df-cring 19293 df-oppr 19369 df-dvdsr 19387 df-unit 19388 df-invr 19418 df-dvr 19429 df-drng 19497 df-psmet 20083 df-xmet 20084 df-met 20085 df-bl 20086 df-mopn 20087 df-cnfld 20092 df-top 21499 df-topon 21516 df-bases 21551 df-cmp 21992 df-ovol 24068 df-vol 24069 df-sumge0 43002 df-ome 43129 df-caragen 43131 df-ovoln 43176 df-voln 43178 |
This theorem is referenced by: vonvolmbl 43300 |
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