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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 3808 | . . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑥 ∩ 𝑦) = (𝑥 ∩ 𝐴)) | |
2 | 1 | fveq2d 6195 | . . . . . 6 ⊢ (𝑦 = 𝐴 → (vol*‘(𝑥 ∩ 𝑦)) = (vol*‘(𝑥 ∩ 𝐴))) |
3 | difeq2 3722 | . . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑥 ∖ 𝑦) = (𝑥 ∖ 𝐴)) | |
4 | 3 | fveq2d 6195 | . . . . . 6 ⊢ (𝑦 = 𝐴 → (vol*‘(𝑥 ∖ 𝑦)) = (vol*‘(𝑥 ∖ 𝐴))) |
5 | 2, 4 | oveq12d 6668 | . . . . 5 ⊢ (𝑦 = 𝐴 → ((vol*‘(𝑥 ∩ 𝑦)) + (vol*‘(𝑥 ∖ 𝑦))) = ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴)))) |
6 | 5 | eqeq2d 2632 | . . . 4 ⊢ (𝑦 = 𝐴 → ((vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝑦)) + (vol*‘(𝑥 ∖ 𝑦))) ↔ (vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))))) |
7 | 6 | ralbidv 2986 | . . 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 23234 | . . . . . 6 ⊢ vol = (vol* ↾ {𝑦 ∣ ∀𝑥 ∈ (◡vol* “ ℝ)(vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝑦)) + (vol*‘(𝑥 ∖ 𝑦)))}) | |
9 | 8 | dmeqi 5325 | . . . . 5 ⊢ dom vol = dom (vol* ↾ {𝑦 ∣ ∀𝑥 ∈ (◡vol* “ ℝ)(vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝑦)) + (vol*‘(𝑥 ∖ 𝑦)))}) |
10 | dmres 5419 | . . . . 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 23250 | . . . . . . 7 ⊢ vol*:𝒫 ℝ⟶(0[,]+∞) | |
12 | 11 | fdmi 6052 | . . . . . 6 ⊢ dom vol* = 𝒫 ℝ |
13 | 12 | ineq2i 3811 | . . . . 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 2648 | . . . 4 ⊢ dom vol = ({𝑦 ∣ ∀𝑥 ∈ (◡vol* “ ℝ)(vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝑦)) + (vol*‘(𝑥 ∖ 𝑦)))} ∩ 𝒫 ℝ) |
15 | dfrab2 3903 | . . . 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 2647 | . . 3 ⊢ dom vol = {𝑦 ∈ 𝒫 ℝ ∣ ∀𝑥 ∈ (◡vol* “ ℝ)(vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝑦)) + (vol*‘(𝑥 ∖ 𝑦)))} |
17 | 7, 16 | elrab2 3366 | . 2 ⊢ (𝐴 ∈ dom vol ↔ (𝐴 ∈ 𝒫 ℝ ∧ ∀𝑥 ∈ (◡vol* “ ℝ)(vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))))) |
18 | reex 10027 | . . . 4 ⊢ ℝ ∈ V | |
19 | 18 | elpw2 4828 | . . 3 ⊢ (𝐴 ∈ 𝒫 ℝ ↔ 𝐴 ⊆ ℝ) |
20 | ffn 6045 | . . . . . . 7 ⊢ (vol*:𝒫 ℝ⟶(0[,]+∞) → vol* Fn 𝒫 ℝ) | |
21 | elpreima 6337 | . . . . . . 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 339 | . . . . 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 462 | . . . . 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 264 | . . . 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 2978 | . . 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 733 | . 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 264 | 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 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 {cab 2608 ∀wral 2912 {crab 2916 ∖ cdif 3571 ∩ cin 3573 ⊆ wss 3574 𝒫 cpw 4158 ◡ccnv 5113 dom cdm 5114 ↾ cres 5116 “ cima 5117 Fn wfn 5883 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ℝcr 9935 0cc0 9936 + caddc 9939 +∞cpnf 10071 [,]cicc 12178 vol*covol 23231 volcvol 23232 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-inf 8349 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-ico 12181 df-icc 12182 df-fz 12327 df-seq 12802 df-exp 12861 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-ovol 23233 df-vol 23234 |
This theorem is referenced by: ismbl2 23295 mblss 23299 mblsplit 23300 cmmbl 23302 shftmbl 23306 voliunlem2 23319 |
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