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Mirrors > Home > MPE Home > Th. List > ismbl | Structured version Visualization version GIF version |
Description: The predicate "𝐴 is Lebesgue-measurable". 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 (assuming that the measure of 𝑥 is a real number, so that this addition makes sense). (Contributed by Mario Carneiro, 17-Mar-2014.) |
Ref | Expression |
---|---|
ismbl | ⊢ (𝐴 ∈ dom vol ↔ (𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ 𝒫 ℝ((vol*‘𝑥) ∈ ℝ → (vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴)))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ineq2 4182 | . . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑥 ∩ 𝑦) = (𝑥 ∩ 𝐴)) | |
2 | 1 | fveq2d 6668 | . . . . . 6 ⊢ (𝑦 = 𝐴 → (vol*‘(𝑥 ∩ 𝑦)) = (vol*‘(𝑥 ∩ 𝐴))) |
3 | difeq2 4092 | . . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑥 ∖ 𝑦) = (𝑥 ∖ 𝐴)) | |
4 | 3 | fveq2d 6668 | . . . . . 6 ⊢ (𝑦 = 𝐴 → (vol*‘(𝑥 ∖ 𝑦)) = (vol*‘(𝑥 ∖ 𝐴))) |
5 | 2, 4 | oveq12d 7163 | . . . . 5 ⊢ (𝑦 = 𝐴 → ((vol*‘(𝑥 ∩ 𝑦)) + (vol*‘(𝑥 ∖ 𝑦))) = ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴)))) |
6 | 5 | eqeq2d 2832 | . . . 4 ⊢ (𝑦 = 𝐴 → ((vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝑦)) + (vol*‘(𝑥 ∖ 𝑦))) ↔ (vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))))) |
7 | 6 | ralbidv 3197 | . . 3 ⊢ (𝑦 = 𝐴 → (∀𝑥 ∈ (◡vol* “ ℝ)(vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝑦)) + (vol*‘(𝑥 ∖ 𝑦))) ↔ ∀𝑥 ∈ (◡vol* “ ℝ)(vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))))) |
8 | df-vol 23995 | . . . . . 6 ⊢ vol = (vol* ↾ {𝑦 ∣ ∀𝑥 ∈ (◡vol* “ ℝ)(vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝑦)) + (vol*‘(𝑥 ∖ 𝑦)))}) | |
9 | 8 | dmeqi 5767 | . . . . 5 ⊢ dom vol = dom (vol* ↾ {𝑦 ∣ ∀𝑥 ∈ (◡vol* “ ℝ)(vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝑦)) + (vol*‘(𝑥 ∖ 𝑦)))}) |
10 | dmres 5869 | . . . . 5 ⊢ dom (vol* ↾ {𝑦 ∣ ∀𝑥 ∈ (◡vol* “ ℝ)(vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝑦)) + (vol*‘(𝑥 ∖ 𝑦)))}) = ({𝑦 ∣ ∀𝑥 ∈ (◡vol* “ ℝ)(vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝑦)) + (vol*‘(𝑥 ∖ 𝑦)))} ∩ dom vol*) | |
11 | ovolf 24012 | . . . . . . 7 ⊢ vol*:𝒫 ℝ⟶(0[,]+∞) | |
12 | 11 | fdmi 6518 | . . . . . 6 ⊢ dom vol* = 𝒫 ℝ |
13 | 12 | ineq2i 4185 | . . . . 5 ⊢ ({𝑦 ∣ ∀𝑥 ∈ (◡vol* “ ℝ)(vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝑦)) + (vol*‘(𝑥 ∖ 𝑦)))} ∩ dom vol*) = ({𝑦 ∣ ∀𝑥 ∈ (◡vol* “ ℝ)(vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝑦)) + (vol*‘(𝑥 ∖ 𝑦)))} ∩ 𝒫 ℝ) |
14 | 9, 10, 13 | 3eqtri 2848 | . . . 4 ⊢ dom vol = ({𝑦 ∣ ∀𝑥 ∈ (◡vol* “ ℝ)(vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝑦)) + (vol*‘(𝑥 ∖ 𝑦)))} ∩ 𝒫 ℝ) |
15 | dfrab2 4278 | . . . 4 ⊢ {𝑦 ∈ 𝒫 ℝ ∣ ∀𝑥 ∈ (◡vol* “ ℝ)(vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝑦)) + (vol*‘(𝑥 ∖ 𝑦)))} = ({𝑦 ∣ ∀𝑥 ∈ (◡vol* “ ℝ)(vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝑦)) + (vol*‘(𝑥 ∖ 𝑦)))} ∩ 𝒫 ℝ) | |
16 | 14, 15 | eqtr4i 2847 | . . 3 ⊢ dom vol = {𝑦 ∈ 𝒫 ℝ ∣ ∀𝑥 ∈ (◡vol* “ ℝ)(vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝑦)) + (vol*‘(𝑥 ∖ 𝑦)))} |
17 | 7, 16 | elrab2 3682 | . 2 ⊢ (𝐴 ∈ dom vol ↔ (𝐴 ∈ 𝒫 ℝ ∧ ∀𝑥 ∈ (◡vol* “ ℝ)(vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))))) |
18 | reex 10617 | . . . 4 ⊢ ℝ ∈ V | |
19 | 18 | elpw2 5240 | . . 3 ⊢ (𝐴 ∈ 𝒫 ℝ ↔ 𝐴 ⊆ ℝ) |
20 | ffn 6508 | . . . . . . 7 ⊢ (vol*:𝒫 ℝ⟶(0[,]+∞) → vol* Fn 𝒫 ℝ) | |
21 | elpreima 6821 | . . . . . . 7 ⊢ (vol* Fn 𝒫 ℝ → (𝑥 ∈ (◡vol* “ ℝ) ↔ (𝑥 ∈ 𝒫 ℝ ∧ (vol*‘𝑥) ∈ ℝ))) | |
22 | 11, 20, 21 | mp2b 10 | . . . . . 6 ⊢ (𝑥 ∈ (◡vol* “ ℝ) ↔ (𝑥 ∈ 𝒫 ℝ ∧ (vol*‘𝑥) ∈ ℝ)) |
23 | 22 | imbi1i 351 | . . . . 5 ⊢ ((𝑥 ∈ (◡vol* “ ℝ) → (vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴)))) ↔ ((𝑥 ∈ 𝒫 ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))))) |
24 | impexp 451 | . . . . 5 ⊢ (((𝑥 ∈ 𝒫 ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴)))) ↔ (𝑥 ∈ 𝒫 ℝ → ((vol*‘𝑥) ∈ ℝ → (vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴)))))) | |
25 | 23, 24 | bitri 276 | . . . 4 ⊢ ((𝑥 ∈ (◡vol* “ ℝ) → (vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴)))) ↔ (𝑥 ∈ 𝒫 ℝ → ((vol*‘𝑥) ∈ ℝ → (vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴)))))) |
26 | 25 | ralbii2 3163 | . . 3 ⊢ (∀𝑥 ∈ (◡vol* “ ℝ)(vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))) ↔ ∀𝑥 ∈ 𝒫 ℝ((vol*‘𝑥) ∈ ℝ → (vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))))) |
27 | 19, 26 | anbi12i 626 | . 2 ⊢ ((𝐴 ∈ 𝒫 ℝ ∧ ∀𝑥 ∈ (◡vol* “ ℝ)(vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴)))) ↔ (𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ 𝒫 ℝ((vol*‘𝑥) ∈ ℝ → (vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴)))))) |
28 | 17, 27 | bitri 276 | 1 ⊢ (𝐴 ∈ dom vol ↔ (𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ 𝒫 ℝ((vol*‘𝑥) ∈ ℝ → (vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴)))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 {cab 2799 ∀wral 3138 {crab 3142 ∖ cdif 3932 ∩ cin 3934 ⊆ wss 3935 𝒫 cpw 4537 ◡ccnv 5548 dom cdm 5549 ↾ cres 5551 “ cima 5552 Fn wfn 6344 ⟶wf 6345 ‘cfv 6349 (class class class)co 7145 ℝcr 10525 0cc0 10526 + caddc 10529 +∞cpnf 10661 [,]cicc 12731 vol*covol 23992 volcvol 23993 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 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 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4833 df-iun 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7569 df-1st 7680 df-2nd 7681 df-wrecs 7938 df-recs 7999 df-rdg 8037 df-er 8279 df-map 8398 df-en 8499 df-dom 8500 df-sdom 8501 df-sup 8895 df-inf 8896 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 11628 df-2 11689 df-3 11690 df-n0 11887 df-z 11971 df-uz 12233 df-rp 12380 df-ico 12734 df-icc 12735 df-fz 12883 df-seq 13360 df-exp 13420 df-cj 14448 df-re 14449 df-im 14450 df-sqrt 14584 df-abs 14585 df-ovol 23994 df-vol 23995 |
This theorem is referenced by: ismbl2 24057 mblss 24061 mblsplit 24062 cmmbl 24064 shftmbl 24068 voliunlem2 24081 |
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