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Mirrors > Home > MPE Home > Th. List > Mathboxes > vonvolmbllem | Structured version Visualization version GIF version |
Description: If a subset 𝐵 of real numbers is Lebesgue measurable, then its corresponding 1-dimensional set is measurable w.r.t. the n-dimensional Lebesgue measure, (with 𝑛 equal to 1). (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
Ref | Expression |
---|---|
vonvolmbllem.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
vonvolmbllem.b | ⊢ (𝜑 → 𝐵 ⊆ ℝ) |
vonvolmbllem.e | ⊢ (𝜑 → ∀𝑦 ∈ 𝒫 ℝ(vol*‘𝑦) = ((vol*‘(𝑦 ∩ 𝐵)) +𝑒 (vol*‘(𝑦 ∖ 𝐵)))) |
vonvolmbllem.x | ⊢ (𝜑 → 𝑋 ⊆ (ℝ ↑𝑚 {𝐴})) |
vonvolmbllem.y | ⊢ 𝑌 = ∪ 𝑓 ∈ 𝑋 ran 𝑓 |
Ref | Expression |
---|---|
vonvolmbllem | ⊢ (𝜑 → (((voln*‘{𝐴})‘(𝑋 ∩ (𝐵 ↑𝑚 {𝐴}))) +𝑒 ((voln*‘{𝐴})‘(𝑋 ∖ (𝐵 ↑𝑚 {𝐴})))) = ((voln*‘{𝐴})‘𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2793 | . . . . . . . 8 ⊢ Ⅎ𝑓𝑌 | |
2 | vonvolmbllem.a | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
3 | vonvolmbllem.x | . . . . . . . 8 ⊢ (𝜑 → 𝑋 ⊆ (ℝ ↑𝑚 {𝐴})) | |
4 | vonvolmbllem.y | . . . . . . . 8 ⊢ 𝑌 = ∪ 𝑓 ∈ 𝑋 ran 𝑓 | |
5 | 1, 2, 3, 4 | ssmapsn 39722 | . . . . . . 7 ⊢ (𝜑 → 𝑋 = (𝑌 ↑𝑚 {𝐴})) |
6 | 5 | ineq1d 3846 | . . . . . 6 ⊢ (𝜑 → (𝑋 ∩ (𝐵 ↑𝑚 {𝐴})) = ((𝑌 ↑𝑚 {𝐴}) ∩ (𝐵 ↑𝑚 {𝐴}))) |
7 | reex 10065 | . . . . . . . . 9 ⊢ ℝ ∈ V | |
8 | 7 | a1i 11 | . . . . . . . 8 ⊢ (𝜑 → ℝ ∈ V) |
9 | 3 | sselda 3636 | . . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝑓 ∈ (ℝ ↑𝑚 {𝐴})) |
10 | elmapi 7921 | . . . . . . . . . . . . 13 ⊢ (𝑓 ∈ (ℝ ↑𝑚 {𝐴}) → 𝑓:{𝐴}⟶ℝ) | |
11 | 9, 10 | syl 17 | . . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝑓:{𝐴}⟶ℝ) |
12 | frn 6091 | . . . . . . . . . . . 12 ⊢ (𝑓:{𝐴}⟶ℝ → ran 𝑓 ⊆ ℝ) | |
13 | 11, 12 | syl 17 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → ran 𝑓 ⊆ ℝ) |
14 | 13 | ralrimiva 2995 | . . . . . . . . . 10 ⊢ (𝜑 → ∀𝑓 ∈ 𝑋 ran 𝑓 ⊆ ℝ) |
15 | iunss 4593 | . . . . . . . . . 10 ⊢ (∪ 𝑓 ∈ 𝑋 ran 𝑓 ⊆ ℝ ↔ ∀𝑓 ∈ 𝑋 ran 𝑓 ⊆ ℝ) | |
16 | 14, 15 | sylibr 224 | . . . . . . . . 9 ⊢ (𝜑 → ∪ 𝑓 ∈ 𝑋 ran 𝑓 ⊆ ℝ) |
17 | 4, 16 | syl5eqss 3682 | . . . . . . . 8 ⊢ (𝜑 → 𝑌 ⊆ ℝ) |
18 | 8, 17 | ssexd 4838 | . . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ V) |
19 | vonvolmbllem.b | . . . . . . . 8 ⊢ (𝜑 → 𝐵 ⊆ ℝ) | |
20 | 8, 19 | ssexd 4838 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ V) |
21 | snex 4938 | . . . . . . . 8 ⊢ {𝐴} ∈ V | |
22 | 21 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → {𝐴} ∈ V) |
23 | 18, 20, 22 | inmap 39715 | . . . . . 6 ⊢ (𝜑 → ((𝑌 ↑𝑚 {𝐴}) ∩ (𝐵 ↑𝑚 {𝐴})) = ((𝑌 ∩ 𝐵) ↑𝑚 {𝐴})) |
24 | 6, 23 | eqtrd 2685 | . . . . 5 ⊢ (𝜑 → (𝑋 ∩ (𝐵 ↑𝑚 {𝐴})) = ((𝑌 ∩ 𝐵) ↑𝑚 {𝐴})) |
25 | 24 | fveq2d 6233 | . . . 4 ⊢ (𝜑 → ((voln*‘{𝐴})‘(𝑋 ∩ (𝐵 ↑𝑚 {𝐴}))) = ((voln*‘{𝐴})‘((𝑌 ∩ 𝐵) ↑𝑚 {𝐴}))) |
26 | 17 | ssinss1d 39528 | . . . . 5 ⊢ (𝜑 → (𝑌 ∩ 𝐵) ⊆ ℝ) |
27 | 2, 26 | ovnovol 41194 | . . . 4 ⊢ (𝜑 → ((voln*‘{𝐴})‘((𝑌 ∩ 𝐵) ↑𝑚 {𝐴})) = (vol*‘(𝑌 ∩ 𝐵))) |
28 | 25, 27 | eqtrd 2685 | . . 3 ⊢ (𝜑 → ((voln*‘{𝐴})‘(𝑋 ∩ (𝐵 ↑𝑚 {𝐴}))) = (vol*‘(𝑌 ∩ 𝐵))) |
29 | 5 | difeq1d 3760 | . . . . . 6 ⊢ (𝜑 → (𝑋 ∖ (𝐵 ↑𝑚 {𝐴})) = ((𝑌 ↑𝑚 {𝐴}) ∖ (𝐵 ↑𝑚 {𝐴}))) |
30 | 18, 20, 2 | difmapsn 39718 | . . . . . 6 ⊢ (𝜑 → ((𝑌 ↑𝑚 {𝐴}) ∖ (𝐵 ↑𝑚 {𝐴})) = ((𝑌 ∖ 𝐵) ↑𝑚 {𝐴})) |
31 | 29, 30 | eqtrd 2685 | . . . . 5 ⊢ (𝜑 → (𝑋 ∖ (𝐵 ↑𝑚 {𝐴})) = ((𝑌 ∖ 𝐵) ↑𝑚 {𝐴})) |
32 | 31 | fveq2d 6233 | . . . 4 ⊢ (𝜑 → ((voln*‘{𝐴})‘(𝑋 ∖ (𝐵 ↑𝑚 {𝐴}))) = ((voln*‘{𝐴})‘((𝑌 ∖ 𝐵) ↑𝑚 {𝐴}))) |
33 | 17 | ssdifssd 3781 | . . . . 5 ⊢ (𝜑 → (𝑌 ∖ 𝐵) ⊆ ℝ) |
34 | 2, 33 | ovnovol 41194 | . . . 4 ⊢ (𝜑 → ((voln*‘{𝐴})‘((𝑌 ∖ 𝐵) ↑𝑚 {𝐴})) = (vol*‘(𝑌 ∖ 𝐵))) |
35 | 32, 34 | eqtrd 2685 | . . 3 ⊢ (𝜑 → ((voln*‘{𝐴})‘(𝑋 ∖ (𝐵 ↑𝑚 {𝐴}))) = (vol*‘(𝑌 ∖ 𝐵))) |
36 | 28, 35 | oveq12d 6708 | . 2 ⊢ (𝜑 → (((voln*‘{𝐴})‘(𝑋 ∩ (𝐵 ↑𝑚 {𝐴}))) +𝑒 ((voln*‘{𝐴})‘(𝑋 ∖ (𝐵 ↑𝑚 {𝐴})))) = ((vol*‘(𝑌 ∩ 𝐵)) +𝑒 (vol*‘(𝑌 ∖ 𝐵)))) |
37 | 5 | fveq2d 6233 | . . 3 ⊢ (𝜑 → ((voln*‘{𝐴})‘𝑋) = ((voln*‘{𝐴})‘(𝑌 ↑𝑚 {𝐴}))) |
38 | 2, 17 | ovnovol 41194 | . . 3 ⊢ (𝜑 → ((voln*‘{𝐴})‘(𝑌 ↑𝑚 {𝐴})) = (vol*‘𝑌)) |
39 | 18, 17 | elpwd 4200 | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝒫 ℝ) |
40 | vonvolmbllem.e | . . . 4 ⊢ (𝜑 → ∀𝑦 ∈ 𝒫 ℝ(vol*‘𝑦) = ((vol*‘(𝑦 ∩ 𝐵)) +𝑒 (vol*‘(𝑦 ∖ 𝐵)))) | |
41 | fveq2 6229 | . . . . . 6 ⊢ (𝑦 = 𝑌 → (vol*‘𝑦) = (vol*‘𝑌)) | |
42 | ineq1 3840 | . . . . . . . 8 ⊢ (𝑦 = 𝑌 → (𝑦 ∩ 𝐵) = (𝑌 ∩ 𝐵)) | |
43 | 42 | fveq2d 6233 | . . . . . . 7 ⊢ (𝑦 = 𝑌 → (vol*‘(𝑦 ∩ 𝐵)) = (vol*‘(𝑌 ∩ 𝐵))) |
44 | difeq1 3754 | . . . . . . . 8 ⊢ (𝑦 = 𝑌 → (𝑦 ∖ 𝐵) = (𝑌 ∖ 𝐵)) | |
45 | 44 | fveq2d 6233 | . . . . . . 7 ⊢ (𝑦 = 𝑌 → (vol*‘(𝑦 ∖ 𝐵)) = (vol*‘(𝑌 ∖ 𝐵))) |
46 | 43, 45 | oveq12d 6708 | . . . . . 6 ⊢ (𝑦 = 𝑌 → ((vol*‘(𝑦 ∩ 𝐵)) +𝑒 (vol*‘(𝑦 ∖ 𝐵))) = ((vol*‘(𝑌 ∩ 𝐵)) +𝑒 (vol*‘(𝑌 ∖ 𝐵)))) |
47 | 41, 46 | eqeq12d 2666 | . . . . 5 ⊢ (𝑦 = 𝑌 → ((vol*‘𝑦) = ((vol*‘(𝑦 ∩ 𝐵)) +𝑒 (vol*‘(𝑦 ∖ 𝐵))) ↔ (vol*‘𝑌) = ((vol*‘(𝑌 ∩ 𝐵)) +𝑒 (vol*‘(𝑌 ∖ 𝐵))))) |
48 | 47 | rspcva 3338 | . . . 4 ⊢ ((𝑌 ∈ 𝒫 ℝ ∧ ∀𝑦 ∈ 𝒫 ℝ(vol*‘𝑦) = ((vol*‘(𝑦 ∩ 𝐵)) +𝑒 (vol*‘(𝑦 ∖ 𝐵)))) → (vol*‘𝑌) = ((vol*‘(𝑌 ∩ 𝐵)) +𝑒 (vol*‘(𝑌 ∖ 𝐵)))) |
49 | 39, 40, 48 | syl2anc 694 | . . 3 ⊢ (𝜑 → (vol*‘𝑌) = ((vol*‘(𝑌 ∩ 𝐵)) +𝑒 (vol*‘(𝑌 ∖ 𝐵)))) |
50 | 37, 38, 49 | 3eqtrd 2689 | . 2 ⊢ (𝜑 → ((voln*‘{𝐴})‘𝑋) = ((vol*‘(𝑌 ∩ 𝐵)) +𝑒 (vol*‘(𝑌 ∖ 𝐵)))) |
51 | 36, 50 | eqtr4d 2688 | 1 ⊢ (𝜑 → (((voln*‘{𝐴})‘(𝑋 ∩ (𝐵 ↑𝑚 {𝐴}))) +𝑒 ((voln*‘{𝐴})‘(𝑋 ∖ (𝐵 ↑𝑚 {𝐴})))) = ((voln*‘{𝐴})‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1523 ∈ wcel 2030 ∀wral 2941 Vcvv 3231 ∖ cdif 3604 ∩ cin 3606 ⊆ wss 3607 𝒫 cpw 4191 {csn 4210 ∪ ciun 4552 ran crn 5144 ⟶wf 5922 ‘cfv 5926 (class class class)co 6690 ↑𝑚 cmap 7899 ℝcr 9973 +𝑒 cxad 11982 vol*covol 23277 voln*covoln 41071 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-inf2 8576 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 ax-pre-sup 10052 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-fal 1529 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-se 5103 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-isom 5935 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-of 6939 df-om 7108 df-1st 7210 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-2o 7606 df-oadd 7609 df-er 7787 df-map 7901 df-pm 7902 df-ixp 7951 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-fi 8358 df-sup 8389 df-inf 8390 df-oi 8456 df-card 8803 df-cda 9028 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-div 10723 df-nn 11059 df-2 11117 df-3 11118 df-n0 11331 df-z 11416 df-uz 11726 df-q 11827 df-rp 11871 df-xneg 11984 df-xadd 11985 df-xmul 11986 df-ioo 12217 df-ico 12219 df-icc 12220 df-fz 12365 df-fzo 12505 df-fl 12633 df-seq 12842 df-exp 12901 df-hash 13158 df-cj 13883 df-re 13884 df-im 13885 df-sqrt 14019 df-abs 14020 df-clim 14263 df-rlim 14264 df-sum 14461 df-prod 14680 df-rest 16130 df-topgen 16151 df-psmet 19786 df-xmet 19787 df-met 19788 df-bl 19789 df-mopn 19790 df-top 20747 df-topon 20764 df-bases 20798 df-cmp 21238 df-ovol 23279 df-vol 23280 df-sumge0 40898 df-ovoln 41072 |
This theorem is referenced by: vonvolmbl 41196 |
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