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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 2974 | . . . . . . . 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 41355 | . . . . . . 7 ⊢ (𝜑 → 𝑋 = (𝑌 ↑m {𝐴})) |
6 | 5 | ineq1d 4185 | . . . . . 6 ⊢ (𝜑 → (𝑋 ∩ (𝐵 ↑m {𝐴})) = ((𝑌 ↑m {𝐴}) ∩ (𝐵 ↑m {𝐴}))) |
7 | reex 10616 | . . . . . . . . 9 ⊢ ℝ ∈ V | |
8 | 7 | a1i 11 | . . . . . . . 8 ⊢ (𝜑 → ℝ ∈ V) |
9 | 3 | sselda 3964 | . . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝑓 ∈ (ℝ ↑m {𝐴})) |
10 | elmapi 8417 | . . . . . . . . . . . . 13 ⊢ (𝑓 ∈ (ℝ ↑m {𝐴}) → 𝑓:{𝐴}⟶ℝ) | |
11 | 9, 10 | syl 17 | . . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝑓:{𝐴}⟶ℝ) |
12 | 11 | frnd 6514 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → ran 𝑓 ⊆ ℝ) |
13 | 12 | ralrimiva 3179 | . . . . . . . . . 10 ⊢ (𝜑 → ∀𝑓 ∈ 𝑋 ran 𝑓 ⊆ ℝ) |
14 | iunss 4960 | . . . . . . . . . 10 ⊢ (∪ 𝑓 ∈ 𝑋 ran 𝑓 ⊆ ℝ ↔ ∀𝑓 ∈ 𝑋 ran 𝑓 ⊆ ℝ) | |
15 | 13, 14 | sylibr 235 | . . . . . . . . 9 ⊢ (𝜑 → ∪ 𝑓 ∈ 𝑋 ran 𝑓 ⊆ ℝ) |
16 | 4, 15 | eqsstrid 4012 | . . . . . . . 8 ⊢ (𝜑 → 𝑌 ⊆ ℝ) |
17 | 8, 16 | ssexd 5219 | . . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ V) |
18 | vonvolmbllem.b | . . . . . . . 8 ⊢ (𝜑 → 𝐵 ⊆ ℝ) | |
19 | 8, 18 | ssexd 5219 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ V) |
20 | snex 5322 | . . . . . . . 8 ⊢ {𝐴} ∈ V | |
21 | 20 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → {𝐴} ∈ V) |
22 | 17, 19, 21 | inmap 41348 | . . . . . 6 ⊢ (𝜑 → ((𝑌 ↑m {𝐴}) ∩ (𝐵 ↑m {𝐴})) = ((𝑌 ∩ 𝐵) ↑m {𝐴})) |
23 | 6, 22 | eqtrd 2853 | . . . . 5 ⊢ (𝜑 → (𝑋 ∩ (𝐵 ↑m {𝐴})) = ((𝑌 ∩ 𝐵) ↑m {𝐴})) |
24 | 23 | fveq2d 6667 | . . . 4 ⊢ (𝜑 → ((voln*‘{𝐴})‘(𝑋 ∩ (𝐵 ↑m {𝐴}))) = ((voln*‘{𝐴})‘((𝑌 ∩ 𝐵) ↑m {𝐴}))) |
25 | 16 | ssinss1d 41187 | . . . . 5 ⊢ (𝜑 → (𝑌 ∩ 𝐵) ⊆ ℝ) |
26 | 2, 25 | ovnovol 42818 | . . . 4 ⊢ (𝜑 → ((voln*‘{𝐴})‘((𝑌 ∩ 𝐵) ↑m {𝐴})) = (vol*‘(𝑌 ∩ 𝐵))) |
27 | 24, 26 | eqtrd 2853 | . . 3 ⊢ (𝜑 → ((voln*‘{𝐴})‘(𝑋 ∩ (𝐵 ↑m {𝐴}))) = (vol*‘(𝑌 ∩ 𝐵))) |
28 | 5 | difeq1d 4095 | . . . . . 6 ⊢ (𝜑 → (𝑋 ∖ (𝐵 ↑m {𝐴})) = ((𝑌 ↑m {𝐴}) ∖ (𝐵 ↑m {𝐴}))) |
29 | 17, 19, 2 | difmapsn 41351 | . . . . . 6 ⊢ (𝜑 → ((𝑌 ↑m {𝐴}) ∖ (𝐵 ↑m {𝐴})) = ((𝑌 ∖ 𝐵) ↑m {𝐴})) |
30 | 28, 29 | eqtrd 2853 | . . . . 5 ⊢ (𝜑 → (𝑋 ∖ (𝐵 ↑m {𝐴})) = ((𝑌 ∖ 𝐵) ↑m {𝐴})) |
31 | 30 | fveq2d 6667 | . . . 4 ⊢ (𝜑 → ((voln*‘{𝐴})‘(𝑋 ∖ (𝐵 ↑m {𝐴}))) = ((voln*‘{𝐴})‘((𝑌 ∖ 𝐵) ↑m {𝐴}))) |
32 | 16 | ssdifssd 4116 | . . . . 5 ⊢ (𝜑 → (𝑌 ∖ 𝐵) ⊆ ℝ) |
33 | 2, 32 | ovnovol 42818 | . . . 4 ⊢ (𝜑 → ((voln*‘{𝐴})‘((𝑌 ∖ 𝐵) ↑m {𝐴})) = (vol*‘(𝑌 ∖ 𝐵))) |
34 | 31, 33 | eqtrd 2853 | . . 3 ⊢ (𝜑 → ((voln*‘{𝐴})‘(𝑋 ∖ (𝐵 ↑m {𝐴}))) = (vol*‘(𝑌 ∖ 𝐵))) |
35 | 27, 34 | oveq12d 7163 | . 2 ⊢ (𝜑 → (((voln*‘{𝐴})‘(𝑋 ∩ (𝐵 ↑m {𝐴}))) +𝑒 ((voln*‘{𝐴})‘(𝑋 ∖ (𝐵 ↑m {𝐴})))) = ((vol*‘(𝑌 ∩ 𝐵)) +𝑒 (vol*‘(𝑌 ∖ 𝐵)))) |
36 | 5 | fveq2d 6667 | . . 3 ⊢ (𝜑 → ((voln*‘{𝐴})‘𝑋) = ((voln*‘{𝐴})‘(𝑌 ↑m {𝐴}))) |
37 | 2, 16 | ovnovol 42818 | . . 3 ⊢ (𝜑 → ((voln*‘{𝐴})‘(𝑌 ↑m {𝐴})) = (vol*‘𝑌)) |
38 | 17, 16 | elpwd 4546 | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝒫 ℝ) |
39 | vonvolmbllem.e | . . . 4 ⊢ (𝜑 → ∀𝑦 ∈ 𝒫 ℝ(vol*‘𝑦) = ((vol*‘(𝑦 ∩ 𝐵)) +𝑒 (vol*‘(𝑦 ∖ 𝐵)))) | |
40 | fveq2 6663 | . . . . . 6 ⊢ (𝑦 = 𝑌 → (vol*‘𝑦) = (vol*‘𝑌)) | |
41 | ineq1 4178 | . . . . . . . 8 ⊢ (𝑦 = 𝑌 → (𝑦 ∩ 𝐵) = (𝑌 ∩ 𝐵)) | |
42 | 41 | fveq2d 6667 | . . . . . . 7 ⊢ (𝑦 = 𝑌 → (vol*‘(𝑦 ∩ 𝐵)) = (vol*‘(𝑌 ∩ 𝐵))) |
43 | difeq1 4089 | . . . . . . . 8 ⊢ (𝑦 = 𝑌 → (𝑦 ∖ 𝐵) = (𝑌 ∖ 𝐵)) | |
44 | 43 | fveq2d 6667 | . . . . . . 7 ⊢ (𝑦 = 𝑌 → (vol*‘(𝑦 ∖ 𝐵)) = (vol*‘(𝑌 ∖ 𝐵))) |
45 | 42, 44 | oveq12d 7163 | . . . . . 6 ⊢ (𝑦 = 𝑌 → ((vol*‘(𝑦 ∩ 𝐵)) +𝑒 (vol*‘(𝑦 ∖ 𝐵))) = ((vol*‘(𝑌 ∩ 𝐵)) +𝑒 (vol*‘(𝑌 ∖ 𝐵)))) |
46 | 40, 45 | eqeq12d 2834 | . . . . 5 ⊢ (𝑦 = 𝑌 → ((vol*‘𝑦) = ((vol*‘(𝑦 ∩ 𝐵)) +𝑒 (vol*‘(𝑦 ∖ 𝐵))) ↔ (vol*‘𝑌) = ((vol*‘(𝑌 ∩ 𝐵)) +𝑒 (vol*‘(𝑌 ∖ 𝐵))))) |
47 | 46 | rspcva 3618 | . . . 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)))) |
48 | 38, 39, 47 | syl2anc 584 | . . 3 ⊢ (𝜑 → (vol*‘𝑌) = ((vol*‘(𝑌 ∩ 𝐵)) +𝑒 (vol*‘(𝑌 ∖ 𝐵)))) |
49 | 36, 37, 48 | 3eqtrd 2857 | . 2 ⊢ (𝜑 → ((voln*‘{𝐴})‘𝑋) = ((vol*‘(𝑌 ∩ 𝐵)) +𝑒 (vol*‘(𝑌 ∖ 𝐵)))) |
50 | 35, 49 | eqtr4d 2856 | 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 396 = wceq 1528 ∈ wcel 2105 ∀wral 3135 Vcvv 3492 ∖ cdif 3930 ∩ cin 3932 ⊆ wss 3933 𝒫 cpw 4535 {csn 4557 ∪ ciun 4910 ran crn 5549 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 ↑m cmap 8395 ℝcr 10524 +𝑒 cxad 12493 vol*covol 23990 voln*covoln 42695 |
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 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-inf2 9092 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-2o 8092 df-oadd 8095 df-er 8278 df-map 8397 df-pm 8398 df-ixp 8450 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-fi 8863 df-sup 8894 df-inf 8895 df-oi 8962 df-dju 9318 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 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 12881 df-fzo 13022 df-fl 13150 df-seq 13358 df-exp 13418 df-hash 13679 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-clim 14833 df-rlim 14834 df-sum 15031 df-prod 15248 df-rest 16684 df-topgen 16705 df-psmet 20465 df-xmet 20466 df-met 20467 df-bl 20468 df-mopn 20469 df-top 21430 df-topon 21447 df-bases 21482 df-cmp 21923 df-ovol 23992 df-vol 23993 df-sumge0 42522 df-ovoln 42696 |
This theorem is referenced by: vonvolmbl 42820 |
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