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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 2977 | . . . . . . . 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 41472 | . . . . . . 7 ⊢ (𝜑 → 𝑋 = (𝑌 ↑m {𝐴})) |
6 | 5 | ineq1d 4187 | . . . . . 6 ⊢ (𝜑 → (𝑋 ∩ (𝐵 ↑m {𝐴})) = ((𝑌 ↑m {𝐴}) ∩ (𝐵 ↑m {𝐴}))) |
7 | reex 10622 | . . . . . . . . 9 ⊢ ℝ ∈ V | |
8 | 7 | a1i 11 | . . . . . . . 8 ⊢ (𝜑 → ℝ ∈ V) |
9 | 3 | sselda 3966 | . . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝑓 ∈ (ℝ ↑m {𝐴})) |
10 | elmapi 8422 | . . . . . . . . . . . . 13 ⊢ (𝑓 ∈ (ℝ ↑m {𝐴}) → 𝑓:{𝐴}⟶ℝ) | |
11 | 9, 10 | syl 17 | . . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝑓:{𝐴}⟶ℝ) |
12 | 11 | frnd 6515 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → ran 𝑓 ⊆ ℝ) |
13 | 12 | ralrimiva 3182 | . . . . . . . . . 10 ⊢ (𝜑 → ∀𝑓 ∈ 𝑋 ran 𝑓 ⊆ ℝ) |
14 | iunss 4961 | . . . . . . . . . 10 ⊢ (∪ 𝑓 ∈ 𝑋 ran 𝑓 ⊆ ℝ ↔ ∀𝑓 ∈ 𝑋 ran 𝑓 ⊆ ℝ) | |
15 | 13, 14 | sylibr 236 | . . . . . . . . 9 ⊢ (𝜑 → ∪ 𝑓 ∈ 𝑋 ran 𝑓 ⊆ ℝ) |
16 | 4, 15 | eqsstrid 4014 | . . . . . . . 8 ⊢ (𝜑 → 𝑌 ⊆ ℝ) |
17 | 8, 16 | ssexd 5220 | . . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ V) |
18 | vonvolmbllem.b | . . . . . . . 8 ⊢ (𝜑 → 𝐵 ⊆ ℝ) | |
19 | 8, 18 | ssexd 5220 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ V) |
20 | snex 5323 | . . . . . . . 8 ⊢ {𝐴} ∈ V | |
21 | 20 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → {𝐴} ∈ V) |
22 | 17, 19, 21 | inmap 41465 | . . . . . 6 ⊢ (𝜑 → ((𝑌 ↑m {𝐴}) ∩ (𝐵 ↑m {𝐴})) = ((𝑌 ∩ 𝐵) ↑m {𝐴})) |
23 | 6, 22 | eqtrd 2856 | . . . . 5 ⊢ (𝜑 → (𝑋 ∩ (𝐵 ↑m {𝐴})) = ((𝑌 ∩ 𝐵) ↑m {𝐴})) |
24 | 23 | fveq2d 6668 | . . . 4 ⊢ (𝜑 → ((voln*‘{𝐴})‘(𝑋 ∩ (𝐵 ↑m {𝐴}))) = ((voln*‘{𝐴})‘((𝑌 ∩ 𝐵) ↑m {𝐴}))) |
25 | 16 | ssinss1d 41303 | . . . . 5 ⊢ (𝜑 → (𝑌 ∩ 𝐵) ⊆ ℝ) |
26 | 2, 25 | ovnovol 42935 | . . . 4 ⊢ (𝜑 → ((voln*‘{𝐴})‘((𝑌 ∩ 𝐵) ↑m {𝐴})) = (vol*‘(𝑌 ∩ 𝐵))) |
27 | 24, 26 | eqtrd 2856 | . . 3 ⊢ (𝜑 → ((voln*‘{𝐴})‘(𝑋 ∩ (𝐵 ↑m {𝐴}))) = (vol*‘(𝑌 ∩ 𝐵))) |
28 | 5 | difeq1d 4097 | . . . . . 6 ⊢ (𝜑 → (𝑋 ∖ (𝐵 ↑m {𝐴})) = ((𝑌 ↑m {𝐴}) ∖ (𝐵 ↑m {𝐴}))) |
29 | 17, 19, 2 | difmapsn 41468 | . . . . . 6 ⊢ (𝜑 → ((𝑌 ↑m {𝐴}) ∖ (𝐵 ↑m {𝐴})) = ((𝑌 ∖ 𝐵) ↑m {𝐴})) |
30 | 28, 29 | eqtrd 2856 | . . . . 5 ⊢ (𝜑 → (𝑋 ∖ (𝐵 ↑m {𝐴})) = ((𝑌 ∖ 𝐵) ↑m {𝐴})) |
31 | 30 | fveq2d 6668 | . . . 4 ⊢ (𝜑 → ((voln*‘{𝐴})‘(𝑋 ∖ (𝐵 ↑m {𝐴}))) = ((voln*‘{𝐴})‘((𝑌 ∖ 𝐵) ↑m {𝐴}))) |
32 | 16 | ssdifssd 4118 | . . . . 5 ⊢ (𝜑 → (𝑌 ∖ 𝐵) ⊆ ℝ) |
33 | 2, 32 | ovnovol 42935 | . . . 4 ⊢ (𝜑 → ((voln*‘{𝐴})‘((𝑌 ∖ 𝐵) ↑m {𝐴})) = (vol*‘(𝑌 ∖ 𝐵))) |
34 | 31, 33 | eqtrd 2856 | . . 3 ⊢ (𝜑 → ((voln*‘{𝐴})‘(𝑋 ∖ (𝐵 ↑m {𝐴}))) = (vol*‘(𝑌 ∖ 𝐵))) |
35 | 27, 34 | oveq12d 7168 | . 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 6668 | . . 3 ⊢ (𝜑 → ((voln*‘{𝐴})‘𝑋) = ((voln*‘{𝐴})‘(𝑌 ↑m {𝐴}))) |
37 | 2, 16 | ovnovol 42935 | . . 3 ⊢ (𝜑 → ((voln*‘{𝐴})‘(𝑌 ↑m {𝐴})) = (vol*‘𝑌)) |
38 | 17, 16 | elpwd 4549 | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝒫 ℝ) |
39 | vonvolmbllem.e | . . . 4 ⊢ (𝜑 → ∀𝑦 ∈ 𝒫 ℝ(vol*‘𝑦) = ((vol*‘(𝑦 ∩ 𝐵)) +𝑒 (vol*‘(𝑦 ∖ 𝐵)))) | |
40 | fveq2 6664 | . . . . . 6 ⊢ (𝑦 = 𝑌 → (vol*‘𝑦) = (vol*‘𝑌)) | |
41 | ineq1 4180 | . . . . . . . 8 ⊢ (𝑦 = 𝑌 → (𝑦 ∩ 𝐵) = (𝑌 ∩ 𝐵)) | |
42 | 41 | fveq2d 6668 | . . . . . . 7 ⊢ (𝑦 = 𝑌 → (vol*‘(𝑦 ∩ 𝐵)) = (vol*‘(𝑌 ∩ 𝐵))) |
43 | difeq1 4091 | . . . . . . . 8 ⊢ (𝑦 = 𝑌 → (𝑦 ∖ 𝐵) = (𝑌 ∖ 𝐵)) | |
44 | 43 | fveq2d 6668 | . . . . . . 7 ⊢ (𝑦 = 𝑌 → (vol*‘(𝑦 ∖ 𝐵)) = (vol*‘(𝑌 ∖ 𝐵))) |
45 | 42, 44 | oveq12d 7168 | . . . . . 6 ⊢ (𝑦 = 𝑌 → ((vol*‘(𝑦 ∩ 𝐵)) +𝑒 (vol*‘(𝑦 ∖ 𝐵))) = ((vol*‘(𝑌 ∩ 𝐵)) +𝑒 (vol*‘(𝑌 ∖ 𝐵)))) |
46 | 40, 45 | eqeq12d 2837 | . . . . 5 ⊢ (𝑦 = 𝑌 → ((vol*‘𝑦) = ((vol*‘(𝑦 ∩ 𝐵)) +𝑒 (vol*‘(𝑦 ∖ 𝐵))) ↔ (vol*‘𝑌) = ((vol*‘(𝑌 ∩ 𝐵)) +𝑒 (vol*‘(𝑌 ∖ 𝐵))))) |
47 | 46 | rspcva 3620 | . . . 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 586 | . . 3 ⊢ (𝜑 → (vol*‘𝑌) = ((vol*‘(𝑌 ∩ 𝐵)) +𝑒 (vol*‘(𝑌 ∖ 𝐵)))) |
49 | 36, 37, 48 | 3eqtrd 2860 | . 2 ⊢ (𝜑 → ((voln*‘{𝐴})‘𝑋) = ((vol*‘(𝑌 ∩ 𝐵)) +𝑒 (vol*‘(𝑌 ∖ 𝐵)))) |
50 | 35, 49 | eqtr4d 2859 | 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 398 = wceq 1533 ∈ wcel 2110 ∀wral 3138 Vcvv 3494 ∖ cdif 3932 ∩ cin 3934 ⊆ wss 3935 𝒫 cpw 4538 {csn 4560 ∪ ciun 4911 ran crn 5550 ⟶wf 6345 ‘cfv 6349 (class class class)co 7150 ↑m cmap 8400 ℝcr 10530 +𝑒 cxad 12499 vol*covol 24057 voln*covoln 42812 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-inf2 9098 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 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 3496 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 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 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-se 5509 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-isom 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-oadd 8100 df-er 8283 df-map 8402 df-pm 8403 df-ixp 8456 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-fi 8869 df-sup 8900 df-inf 8901 df-oi 8968 df-dju 9324 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-n0 11892 df-z 11976 df-uz 12238 df-q 12343 df-rp 12384 df-xneg 12501 df-xadd 12502 df-xmul 12503 df-ioo 12736 df-ico 12738 df-icc 12739 df-fz 12887 df-fzo 13028 df-fl 13156 df-seq 13364 df-exp 13424 df-hash 13685 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-clim 14839 df-rlim 14840 df-sum 15037 df-prod 15254 df-rest 16690 df-topgen 16711 df-psmet 20531 df-xmet 20532 df-met 20533 df-bl 20534 df-mopn 20535 df-top 21496 df-topon 21513 df-bases 21548 df-cmp 21989 df-ovol 24059 df-vol 24060 df-sumge0 42639 df-ovoln 42813 |
This theorem is referenced by: vonvolmbl 42937 |
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