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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 4171 | . . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑥 ∩ 𝑦) = (𝑥 ∩ 𝐴)) | |
2 | 1 | fveq2d 6851 | . . . . . 6 ⊢ (𝑦 = 𝐴 → (vol*‘(𝑥 ∩ 𝑦)) = (vol*‘(𝑥 ∩ 𝐴))) |
3 | difeq2 4081 | . . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑥 ∖ 𝑦) = (𝑥 ∖ 𝐴)) | |
4 | 3 | fveq2d 6851 | . . . . . 6 ⊢ (𝑦 = 𝐴 → (vol*‘(𝑥 ∖ 𝑦)) = (vol*‘(𝑥 ∖ 𝐴))) |
5 | 2, 4 | oveq12d 7380 | . . . . 5 ⊢ (𝑦 = 𝐴 → ((vol*‘(𝑥 ∩ 𝑦)) + (vol*‘(𝑥 ∖ 𝑦))) = ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴)))) |
6 | 5 | eqeq2d 2748 | . . . 4 ⊢ (𝑦 = 𝐴 → ((vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝑦)) + (vol*‘(𝑥 ∖ 𝑦))) ↔ (vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))))) |
7 | 6 | ralbidv 3175 | . . 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 24845 | . . . . . 6 ⊢ vol = (vol* ↾ {𝑦 ∣ ∀𝑥 ∈ (◡vol* “ ℝ)(vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝑦)) + (vol*‘(𝑥 ∖ 𝑦)))}) | |
9 | 8 | dmeqi 5865 | . . . . 5 ⊢ dom vol = dom (vol* ↾ {𝑦 ∣ ∀𝑥 ∈ (◡vol* “ ℝ)(vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝑦)) + (vol*‘(𝑥 ∖ 𝑦)))}) |
10 | dmres 5964 | . . . . 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 24862 | . . . . . . 7 ⊢ vol*:𝒫 ℝ⟶(0[,]+∞) | |
12 | 11 | fdmi 6685 | . . . . . 6 ⊢ dom vol* = 𝒫 ℝ |
13 | 12 | ineq2i 4174 | . . . . 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 2769 | . . . 4 ⊢ dom vol = ({𝑦 ∣ ∀𝑥 ∈ (◡vol* “ ℝ)(vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝑦)) + (vol*‘(𝑥 ∖ 𝑦)))} ∩ 𝒫 ℝ) |
15 | dfrab2 4275 | . . . 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 2768 | . . 3 ⊢ dom vol = {𝑦 ∈ 𝒫 ℝ ∣ ∀𝑥 ∈ (◡vol* “ ℝ)(vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝑦)) + (vol*‘(𝑥 ∖ 𝑦)))} |
17 | 7, 16 | elrab2 3653 | . 2 ⊢ (𝐴 ∈ dom vol ↔ (𝐴 ∈ 𝒫 ℝ ∧ ∀𝑥 ∈ (◡vol* “ ℝ)(vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))))) |
18 | reex 11149 | . . . 4 ⊢ ℝ ∈ V | |
19 | 18 | elpw2 5307 | . . 3 ⊢ (𝐴 ∈ 𝒫 ℝ ↔ 𝐴 ⊆ ℝ) |
20 | ffn 6673 | . . . . . . 7 ⊢ (vol*:𝒫 ℝ⟶(0[,]+∞) → vol* Fn 𝒫 ℝ) | |
21 | elpreima 7013 | . . . . . . 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 350 | . . . . 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 452 | . . . . 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 275 | . . . 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 3093 | . . 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 628 | . 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 275 | 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 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 {cab 2714 ∀wral 3065 {crab 3410 ∖ cdif 3912 ∩ cin 3914 ⊆ wss 3915 𝒫 cpw 4565 ◡ccnv 5637 dom cdm 5638 ↾ cres 5640 “ cima 5641 Fn wfn 6496 ⟶wf 6497 ‘cfv 6501 (class class class)co 7362 ℝcr 11057 0cc0 11058 + caddc 11061 +∞cpnf 11193 [,]cicc 13274 vol*covol 24842 volcvol 24843 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 ax-pre-sup 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-er 8655 df-map 8774 df-en 8891 df-dom 8892 df-sdom 8893 df-sup 9385 df-inf 9386 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-div 11820 df-nn 12161 df-2 12223 df-3 12224 df-n0 12421 df-z 12507 df-uz 12771 df-rp 12923 df-ico 13277 df-icc 13278 df-fz 13432 df-seq 13914 df-exp 13975 df-cj 14991 df-re 14992 df-im 14993 df-sqrt 15127 df-abs 15128 df-ovol 24844 df-vol 24845 |
This theorem is referenced by: ismbl2 24907 mblss 24911 mblsplit 24912 cmmbl 24914 shftmbl 24918 voliunlem2 24931 |
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