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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > mbfresmf | Structured version Visualization version GIF version |
Description: A real-valued measurable function is a sigma-measurable function (w.r.t. the Lebesgue measure on the Reals). (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
mbfresmf.1 | ⊢ (𝜑 → 𝐹 ∈ MblFn) |
mbfresmf.2 | ⊢ (𝜑 → ran 𝐹 ⊆ ℝ) |
mbfresmf.3 | ⊢ 𝑆 = dom vol |
Ref | Expression |
---|---|
mbfresmf | ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1957 | . 2 ⊢ Ⅎ𝑎𝜑 | |
2 | mbfresmf.3 | . . . 4 ⊢ 𝑆 = dom vol | |
3 | 2 | a1i 11 | . . 3 ⊢ (𝜑 → 𝑆 = dom vol) |
4 | dmvolsal 41488 | . . . 4 ⊢ dom vol ∈ SAlg | |
5 | 4 | a1i 11 | . . 3 ⊢ (𝜑 → dom vol ∈ SAlg) |
6 | 3, 5 | eqeltrd 2859 | . 2 ⊢ (𝜑 → 𝑆 ∈ SAlg) |
7 | mbfresmf.1 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ MblFn) | |
8 | mbfdmssre 41144 | . . . 4 ⊢ (𝐹 ∈ MblFn → dom 𝐹 ⊆ ℝ) | |
9 | 7, 8 | syl 17 | . . 3 ⊢ (𝜑 → dom 𝐹 ⊆ ℝ) |
10 | 2 | unieqi 4680 | . . . 4 ⊢ ∪ 𝑆 = ∪ dom vol |
11 | unidmvol 23745 | . . . 4 ⊢ ∪ dom vol = ℝ | |
12 | 10, 11 | eqtri 2802 | . . 3 ⊢ ∪ 𝑆 = ℝ |
13 | 9, 12 | syl6sseqr 3871 | . 2 ⊢ (𝜑 → dom 𝐹 ⊆ ∪ 𝑆) |
14 | mbff 23829 | . . . . 5 ⊢ (𝐹 ∈ MblFn → 𝐹:dom 𝐹⟶ℂ) | |
15 | ffn 6291 | . . . . 5 ⊢ (𝐹:dom 𝐹⟶ℂ → 𝐹 Fn dom 𝐹) | |
16 | 7, 14, 15 | 3syl 18 | . . . 4 ⊢ (𝜑 → 𝐹 Fn dom 𝐹) |
17 | mbfresmf.2 | . . . 4 ⊢ (𝜑 → ran 𝐹 ⊆ ℝ) | |
18 | 16, 17 | jca 507 | . . 3 ⊢ (𝜑 → (𝐹 Fn dom 𝐹 ∧ ran 𝐹 ⊆ ℝ)) |
19 | df-f 6139 | . . 3 ⊢ (𝐹:dom 𝐹⟶ℝ ↔ (𝐹 Fn dom 𝐹 ∧ ran 𝐹 ⊆ ℝ)) | |
20 | 18, 19 | sylibr 226 | . 2 ⊢ (𝜑 → 𝐹:dom 𝐹⟶ℝ) |
21 | 20 | adantr 474 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝐹:dom 𝐹⟶ℝ) |
22 | rexr 10422 | . . . . . 6 ⊢ (𝑎 ∈ ℝ → 𝑎 ∈ ℝ*) | |
23 | 22 | adantl 475 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝑎 ∈ ℝ*) |
24 | 21, 23 | preimaioomnf 41856 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → (◡𝐹 “ (-∞(,)𝑎)) = {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎}) |
25 | 24 | eqcomd 2784 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (◡𝐹 “ (-∞(,)𝑎))) |
26 | 4 | elexi 3415 | . . . . . 6 ⊢ dom vol ∈ V |
27 | 2, 26 | eqeltri 2855 | . . . . 5 ⊢ 𝑆 ∈ V |
28 | 27 | a1i 11 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝑆 ∈ V) |
29 | 7 | dmexd 7377 | . . . . 5 ⊢ (𝜑 → dom 𝐹 ∈ V) |
30 | 29 | adantr 474 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → dom 𝐹 ∈ V) |
31 | mbfima 23834 | . . . . . . 7 ⊢ ((𝐹 ∈ MblFn ∧ 𝐹:dom 𝐹⟶ℝ) → (◡𝐹 “ (-∞(,)𝑎)) ∈ dom vol) | |
32 | 7, 20, 31 | syl2anc 579 | . . . . . 6 ⊢ (𝜑 → (◡𝐹 “ (-∞(,)𝑎)) ∈ dom vol) |
33 | 32, 3 | eleqtrrd 2862 | . . . . 5 ⊢ (𝜑 → (◡𝐹 “ (-∞(,)𝑎)) ∈ 𝑆) |
34 | 33 | adantr 474 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → (◡𝐹 “ (-∞(,)𝑎)) ∈ 𝑆) |
35 | cnvimass 5739 | . . . . 5 ⊢ (◡𝐹 “ (-∞(,)𝑎)) ⊆ dom 𝐹 | |
36 | dfss 3807 | . . . . . 6 ⊢ ((◡𝐹 “ (-∞(,)𝑎)) ⊆ dom 𝐹 ↔ (◡𝐹 “ (-∞(,)𝑎)) = ((◡𝐹 “ (-∞(,)𝑎)) ∩ dom 𝐹)) | |
37 | 36 | biimpi 208 | . . . . 5 ⊢ ((◡𝐹 “ (-∞(,)𝑎)) ⊆ dom 𝐹 → (◡𝐹 “ (-∞(,)𝑎)) = ((◡𝐹 “ (-∞(,)𝑎)) ∩ dom 𝐹)) |
38 | 35, 37 | ax-mp 5 | . . . 4 ⊢ (◡𝐹 “ (-∞(,)𝑎)) = ((◡𝐹 “ (-∞(,)𝑎)) ∩ dom 𝐹) |
39 | 28, 30, 34, 38 | elrestd 40220 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → (◡𝐹 “ (-∞(,)𝑎)) ∈ (𝑆 ↾t dom 𝐹)) |
40 | 25, 39 | eqeltrd 2859 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t dom 𝐹)) |
41 | 1, 6, 13, 20, 40 | issmfd 41871 | 1 ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2107 {crab 3094 Vcvv 3398 ∩ cin 3791 ⊆ wss 3792 ∪ cuni 4671 class class class wbr 4886 ◡ccnv 5354 dom cdm 5355 ran crn 5356 “ cima 5358 Fn wfn 6130 ⟶wf 6131 ‘cfv 6135 (class class class)co 6922 ℂcc 10270 ℝcr 10271 -∞cmnf 10409 ℝ*cxr 10410 < clt 10411 (,)cioo 12487 ↾t crest 16467 volcvol 23667 MblFncmbf 23818 SAlgcsalg 41452 SMblFncsmblfn 41836 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-inf2 8835 ax-cc 9592 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-pre-sup 10350 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-int 4711 df-iun 4755 df-disj 4855 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-se 5315 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-isom 6144 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-of 7174 df-om 7344 df-1st 7445 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-2o 7844 df-oadd 7847 df-er 8026 df-map 8142 df-pm 8143 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-sup 8636 df-inf 8637 df-oi 8704 df-card 9098 df-cda 9325 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-div 11033 df-nn 11375 df-2 11438 df-3 11439 df-n0 11643 df-z 11729 df-uz 11993 df-q 12096 df-rp 12138 df-xadd 12258 df-ioo 12491 df-ico 12493 df-icc 12494 df-fz 12644 df-fzo 12785 df-fl 12912 df-seq 13120 df-exp 13179 df-hash 13436 df-cj 14246 df-re 14247 df-im 14248 df-sqrt 14382 df-abs 14383 df-clim 14627 df-rlim 14628 df-sum 14825 df-rest 16469 df-xmet 20135 df-met 20136 df-ovol 23668 df-vol 23669 df-mbf 23823 df-salg 41453 df-smblfn 41837 |
This theorem is referenced by: mbfpsssmf 41918 |
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