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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 1915 | . 2 ⊢ Ⅎ𝑎𝜑 | |
2 | mbfresmf.3 | . . . 4 ⊢ 𝑆 = dom vol | |
3 | 2 | a1i 11 | . . 3 ⊢ (𝜑 → 𝑆 = dom vol) |
4 | dmvolsal 42986 | . . . 4 ⊢ dom vol ∈ SAlg | |
5 | 4 | a1i 11 | . . 3 ⊢ (𝜑 → dom vol ∈ SAlg) |
6 | 3, 5 | eqeltrd 2890 | . 2 ⊢ (𝜑 → 𝑆 ∈ SAlg) |
7 | mbfresmf.1 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ MblFn) | |
8 | mbfdmssre 42642 | . . . 4 ⊢ (𝐹 ∈ MblFn → dom 𝐹 ⊆ ℝ) | |
9 | 7, 8 | syl 17 | . . 3 ⊢ (𝜑 → dom 𝐹 ⊆ ℝ) |
10 | 2 | unieqi 4813 | . . . 4 ⊢ ∪ 𝑆 = ∪ dom vol |
11 | unidmvol 24145 | . . . 4 ⊢ ∪ dom vol = ℝ | |
12 | 10, 11 | eqtri 2821 | . . 3 ⊢ ∪ 𝑆 = ℝ |
13 | 9, 12 | sseqtrrdi 3966 | . 2 ⊢ (𝜑 → dom 𝐹 ⊆ ∪ 𝑆) |
14 | mbff 24229 | . . . . 5 ⊢ (𝐹 ∈ MblFn → 𝐹:dom 𝐹⟶ℂ) | |
15 | ffn 6487 | . . . . 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 515 | . . 3 ⊢ (𝜑 → (𝐹 Fn dom 𝐹 ∧ ran 𝐹 ⊆ ℝ)) |
19 | df-f 6328 | . . 3 ⊢ (𝐹:dom 𝐹⟶ℝ ↔ (𝐹 Fn dom 𝐹 ∧ ran 𝐹 ⊆ ℝ)) | |
20 | 18, 19 | sylibr 237 | . 2 ⊢ (𝜑 → 𝐹:dom 𝐹⟶ℝ) |
21 | 20 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝐹:dom 𝐹⟶ℝ) |
22 | rexr 10676 | . . . . . 6 ⊢ (𝑎 ∈ ℝ → 𝑎 ∈ ℝ*) | |
23 | 22 | adantl 485 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝑎 ∈ ℝ*) |
24 | 21, 23 | preimaioomnf 43354 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → (◡𝐹 “ (-∞(,)𝑎)) = {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎}) |
25 | 24 | eqcomd 2804 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (◡𝐹 “ (-∞(,)𝑎))) |
26 | 4 | elexi 3460 | . . . . . 6 ⊢ dom vol ∈ V |
27 | 2, 26 | eqeltri 2886 | . . . . 5 ⊢ 𝑆 ∈ V |
28 | 27 | a1i 11 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝑆 ∈ V) |
29 | 7 | dmexd 7596 | . . . . 5 ⊢ (𝜑 → dom 𝐹 ∈ V) |
30 | 29 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → dom 𝐹 ∈ V) |
31 | mbfima 24234 | . . . . . . 7 ⊢ ((𝐹 ∈ MblFn ∧ 𝐹:dom 𝐹⟶ℝ) → (◡𝐹 “ (-∞(,)𝑎)) ∈ dom vol) | |
32 | 7, 20, 31 | syl2anc 587 | . . . . . 6 ⊢ (𝜑 → (◡𝐹 “ (-∞(,)𝑎)) ∈ dom vol) |
33 | 32, 3 | eleqtrrd 2893 | . . . . 5 ⊢ (𝜑 → (◡𝐹 “ (-∞(,)𝑎)) ∈ 𝑆) |
34 | 33 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → (◡𝐹 “ (-∞(,)𝑎)) ∈ 𝑆) |
35 | cnvimass 5916 | . . . . 5 ⊢ (◡𝐹 “ (-∞(,)𝑎)) ⊆ dom 𝐹 | |
36 | dfss 3899 | . . . . . 6 ⊢ ((◡𝐹 “ (-∞(,)𝑎)) ⊆ dom 𝐹 ↔ (◡𝐹 “ (-∞(,)𝑎)) = ((◡𝐹 “ (-∞(,)𝑎)) ∩ dom 𝐹)) | |
37 | 36 | biimpi 219 | . . . . 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 41744 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → (◡𝐹 “ (-∞(,)𝑎)) ∈ (𝑆 ↾t dom 𝐹)) |
40 | 25, 39 | eqeltrd 2890 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t dom 𝐹)) |
41 | 1, 6, 13, 20, 40 | issmfd 43369 | 1 ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 {crab 3110 Vcvv 3441 ∩ cin 3880 ⊆ wss 3881 ∪ cuni 4800 class class class wbr 5030 ◡ccnv 5518 dom cdm 5519 ran crn 5520 “ cima 5522 Fn wfn 6319 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 ℂcc 10524 ℝcr 10525 -∞cmnf 10662 ℝ*cxr 10663 < clt 10664 (,)cioo 12726 ↾t crest 16686 volcvol 24067 MblFncmbf 24218 SAlgcsalg 42950 SMblFncsmblfn 43334 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cc 9846 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-disj 4996 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 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 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-map 8391 df-pm 8392 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-inf 8891 df-oi 8958 df-dju 9314 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-q 12337 df-rp 12378 df-xadd 12496 df-ioo 12730 df-ico 12732 df-icc 12733 df-fz 12886 df-fzo 13029 df-fl 13157 df-seq 13365 df-exp 13426 df-hash 13687 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-clim 14837 df-rlim 14838 df-sum 15035 df-rest 16688 df-xmet 20084 df-met 20085 df-ovol 24068 df-vol 24069 df-mbf 24223 df-salg 42951 df-smblfn 43335 |
This theorem is referenced by: mbfpsssmf 43416 |
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