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Mirrors > Home > MPE Home > Th. List > Mathboxes > vonvolmbl2 | Structured version Visualization version GIF version |
Description: A subset 𝑋 of the space of 1-dimensional Real numbers is Lebesgue measurable if and only if its projection 𝑌 on the Real numbers is Lebesgue measure. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
Ref | Expression |
---|---|
vonvolmbl2.f | ⊢ Ⅎ𝑓𝑌 |
vonvolmbl2.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
vonvolmbl2.x | ⊢ (𝜑 → 𝑋 ⊆ (ℝ ↑m {𝐴})) |
vonvolmbl2.y | ⊢ 𝑌 = ∪ 𝑓 ∈ 𝑋 ran 𝑓 |
Ref | Expression |
---|---|
vonvolmbl2 | ⊢ (𝜑 → (𝑋 ∈ dom (voln‘{𝐴}) ↔ 𝑌 ∈ dom vol)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vonvolmbl2.f | . . . 4 ⊢ Ⅎ𝑓𝑌 | |
2 | vonvolmbl2.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
3 | vonvolmbl2.x | . . . 4 ⊢ (𝜑 → 𝑋 ⊆ (ℝ ↑m {𝐴})) | |
4 | vonvolmbl2.y | . . . 4 ⊢ 𝑌 = ∪ 𝑓 ∈ 𝑋 ran 𝑓 | |
5 | 1, 2, 3, 4 | ssmapsn 42278 | . . 3 ⊢ (𝜑 → 𝑋 = (𝑌 ↑m {𝐴})) |
6 | 5 | eleq1d 2817 | . 2 ⊢ (𝜑 → (𝑋 ∈ dom (voln‘{𝐴}) ↔ (𝑌 ↑m {𝐴}) ∈ dom (voln‘{𝐴}))) |
7 | 3 | adantr 484 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝑋 ⊆ (ℝ ↑m {𝐴})) |
8 | simpr 488 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝑓 ∈ 𝑋) | |
9 | 7, 8 | sseldd 3876 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝑓 ∈ (ℝ ↑m {𝐴})) |
10 | elmapi 8452 | . . . . . . 7 ⊢ (𝑓 ∈ (ℝ ↑m {𝐴}) → 𝑓:{𝐴}⟶ℝ) | |
11 | frn 6505 | . . . . . . 7 ⊢ (𝑓:{𝐴}⟶ℝ → ran 𝑓 ⊆ ℝ) | |
12 | 9, 10, 11 | 3syl 18 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → ran 𝑓 ⊆ ℝ) |
13 | 12 | ralrimiva 3096 | . . . . 5 ⊢ (𝜑 → ∀𝑓 ∈ 𝑋 ran 𝑓 ⊆ ℝ) |
14 | iunss 4928 | . . . . 5 ⊢ (∪ 𝑓 ∈ 𝑋 ran 𝑓 ⊆ ℝ ↔ ∀𝑓 ∈ 𝑋 ran 𝑓 ⊆ ℝ) | |
15 | 13, 14 | sylibr 237 | . . . 4 ⊢ (𝜑 → ∪ 𝑓 ∈ 𝑋 ran 𝑓 ⊆ ℝ) |
16 | 4, 15 | eqsstrid 3923 | . . 3 ⊢ (𝜑 → 𝑌 ⊆ ℝ) |
17 | 2, 16 | vonvolmbl 43725 | . 2 ⊢ (𝜑 → ((𝑌 ↑m {𝐴}) ∈ dom (voln‘{𝐴}) ↔ 𝑌 ∈ dom vol)) |
18 | 6, 17 | bitrd 282 | 1 ⊢ (𝜑 → (𝑋 ∈ dom (voln‘{𝐴}) ↔ 𝑌 ∈ dom vol)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1542 ∈ wcel 2113 Ⅎwnfc 2879 ∀wral 3053 ⊆ wss 3841 {csn 4513 ∪ ciun 4878 dom cdm 5519 ran crn 5520 ⟶wf 6329 ‘cfv 6333 (class class class)co 7164 ↑m cmap 8430 ℝcr 10607 volcvol 24208 volncvoln 43602 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-rep 5151 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 ax-inf2 9170 ax-cc 9928 ax-ac2 9956 ax-cnex 10664 ax-resscn 10665 ax-1cn 10666 ax-icn 10667 ax-addcl 10668 ax-addrcl 10669 ax-mulcl 10670 ax-mulrcl 10671 ax-mulcom 10672 ax-addass 10673 ax-mulass 10674 ax-distr 10675 ax-i2m1 10676 ax-1ne0 10677 ax-1rid 10678 ax-rnegex 10679 ax-rrecex 10680 ax-cnre 10681 ax-pre-lttri 10682 ax-pre-lttrn 10683 ax-pre-ltadd 10684 ax-pre-mulgt0 10685 ax-pre-sup 10686 ax-addf 10687 ax-mulf 10688 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-pss 3860 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-tp 4518 df-op 4520 df-uni 4794 df-int 4834 df-iun 4880 df-disj 4993 df-br 5028 df-opab 5090 df-mpt 5108 df-tr 5134 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6123 df-ord 6169 df-on 6170 df-lim 6171 df-suc 6172 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-isom 6342 df-riota 7121 df-ov 7167 df-oprab 7168 df-mpo 7169 df-of 7419 df-om 7594 df-1st 7707 df-2nd 7708 df-tpos 7914 df-wrecs 7969 df-recs 8030 df-rdg 8068 df-1o 8124 df-2o 8125 df-er 8313 df-map 8432 df-pm 8433 df-ixp 8501 df-en 8549 df-dom 8550 df-sdom 8551 df-fin 8552 df-fi 8941 df-sup 8972 df-inf 8973 df-oi 9040 df-dju 9396 df-card 9434 df-acn 9437 df-ac 9609 df-pnf 10748 df-mnf 10749 df-xr 10750 df-ltxr 10751 df-le 10752 df-sub 10943 df-neg 10944 df-div 11369 df-nn 11710 df-2 11772 df-3 11773 df-4 11774 df-5 11775 df-6 11776 df-7 11777 df-8 11778 df-9 11779 df-n0 11970 df-z 12056 df-dec 12173 df-uz 12318 df-q 12424 df-rp 12466 df-xneg 12583 df-xadd 12584 df-xmul 12585 df-ioo 12818 df-ico 12820 df-icc 12821 df-fz 12975 df-fzo 13118 df-fl 13246 df-seq 13454 df-exp 13515 df-hash 13776 df-cj 14541 df-re 14542 df-im 14543 df-sqrt 14677 df-abs 14678 df-clim 14928 df-rlim 14929 df-sum 15129 df-prod 15345 df-struct 16581 df-ndx 16582 df-slot 16583 df-base 16585 df-sets 16586 df-ress 16587 df-plusg 16674 df-mulr 16675 df-starv 16676 df-tset 16680 df-ple 16681 df-ds 16683 df-unif 16684 df-rest 16792 df-0g 16811 df-topgen 16813 df-mgm 17961 df-sgrp 18010 df-mnd 18021 df-grp 18215 df-minusg 18216 df-subg 18387 df-cmn 19019 df-abl 19020 df-mgp 19352 df-ur 19364 df-ring 19411 df-cring 19412 df-oppr 19488 df-dvdsr 19506 df-unit 19507 df-invr 19537 df-dvr 19548 df-drng 19616 df-psmet 20202 df-xmet 20203 df-met 20204 df-bl 20205 df-mopn 20206 df-cnfld 20211 df-top 21638 df-topon 21655 df-bases 21690 df-cmp 22131 df-ovol 24209 df-vol 24210 df-sumge0 43427 df-ome 43554 df-caragen 43556 df-ovoln 43601 df-voln 43603 |
This theorem is referenced by: (None) |
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