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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 1909 | . 2 ⊢ Ⅎ𝑎𝜑 | |
2 | mbfresmf.3 | . . . 4 ⊢ 𝑆 = dom vol | |
3 | 2 | a1i 11 | . . 3 ⊢ (𝜑 → 𝑆 = dom vol) |
4 | dmvolsal 45877 | . . . 4 ⊢ dom vol ∈ SAlg | |
5 | 4 | a1i 11 | . . 3 ⊢ (𝜑 → dom vol ∈ SAlg) |
6 | 3, 5 | eqeltrd 2825 | . 2 ⊢ (𝜑 → 𝑆 ∈ SAlg) |
7 | mbfresmf.1 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ MblFn) | |
8 | mbfdmssre 45531 | . . . 4 ⊢ (𝐹 ∈ MblFn → dom 𝐹 ⊆ ℝ) | |
9 | 7, 8 | syl 17 | . . 3 ⊢ (𝜑 → dom 𝐹 ⊆ ℝ) |
10 | 2 | unieqi 4921 | . . . 4 ⊢ ∪ 𝑆 = ∪ dom vol |
11 | unidmvol 25531 | . . . 4 ⊢ ∪ dom vol = ℝ | |
12 | 10, 11 | eqtri 2753 | . . 3 ⊢ ∪ 𝑆 = ℝ |
13 | 9, 12 | sseqtrrdi 4028 | . 2 ⊢ (𝜑 → dom 𝐹 ⊆ ∪ 𝑆) |
14 | mbff 25615 | . . . . 5 ⊢ (𝐹 ∈ MblFn → 𝐹:dom 𝐹⟶ℂ) | |
15 | ffn 6723 | . . . . 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 510 | . . 3 ⊢ (𝜑 → (𝐹 Fn dom 𝐹 ∧ ran 𝐹 ⊆ ℝ)) |
19 | df-f 6553 | . . 3 ⊢ (𝐹:dom 𝐹⟶ℝ ↔ (𝐹 Fn dom 𝐹 ∧ ran 𝐹 ⊆ ℝ)) | |
20 | 18, 19 | sylibr 233 | . 2 ⊢ (𝜑 → 𝐹:dom 𝐹⟶ℝ) |
21 | 20 | adantr 479 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝐹:dom 𝐹⟶ℝ) |
22 | rexr 11297 | . . . . . 6 ⊢ (𝑎 ∈ ℝ → 𝑎 ∈ ℝ*) | |
23 | 22 | adantl 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝑎 ∈ ℝ*) |
24 | 21, 23 | preimaioomnf 46250 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → (◡𝐹 “ (-∞(,)𝑎)) = {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎}) |
25 | 24 | eqcomd 2731 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (◡𝐹 “ (-∞(,)𝑎))) |
26 | 4 | elexi 3482 | . . . . . 6 ⊢ dom vol ∈ V |
27 | 2, 26 | eqeltri 2821 | . . . . 5 ⊢ 𝑆 ∈ V |
28 | 27 | a1i 11 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝑆 ∈ V) |
29 | 7 | dmexd 7911 | . . . . 5 ⊢ (𝜑 → dom 𝐹 ∈ V) |
30 | 29 | adantr 479 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → dom 𝐹 ∈ V) |
31 | mbfima 25620 | . . . . . . 7 ⊢ ((𝐹 ∈ MblFn ∧ 𝐹:dom 𝐹⟶ℝ) → (◡𝐹 “ (-∞(,)𝑎)) ∈ dom vol) | |
32 | 7, 20, 31 | syl2anc 582 | . . . . . 6 ⊢ (𝜑 → (◡𝐹 “ (-∞(,)𝑎)) ∈ dom vol) |
33 | 32, 3 | eleqtrrd 2828 | . . . . 5 ⊢ (𝜑 → (◡𝐹 “ (-∞(,)𝑎)) ∈ 𝑆) |
34 | 33 | adantr 479 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → (◡𝐹 “ (-∞(,)𝑎)) ∈ 𝑆) |
35 | cnvimass 6086 | . . . . 5 ⊢ (◡𝐹 “ (-∞(,)𝑎)) ⊆ dom 𝐹 | |
36 | dfss 3963 | . . . . . 6 ⊢ ((◡𝐹 “ (-∞(,)𝑎)) ⊆ dom 𝐹 ↔ (◡𝐹 “ (-∞(,)𝑎)) = ((◡𝐹 “ (-∞(,)𝑎)) ∩ dom 𝐹)) | |
37 | 36 | biimpi 215 | . . . . 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 44619 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → (◡𝐹 “ (-∞(,)𝑎)) ∈ (𝑆 ↾t dom 𝐹)) |
40 | 25, 39 | eqeltrd 2825 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t dom 𝐹)) |
41 | 1, 6, 13, 20, 40 | issmfd 46266 | 1 ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 {crab 3418 Vcvv 3461 ∩ cin 3943 ⊆ wss 3944 ∪ cuni 4909 class class class wbr 5149 ◡ccnv 5677 dom cdm 5678 ran crn 5679 “ cima 5681 Fn wfn 6544 ⟶wf 6545 ‘cfv 6549 (class class class)co 7419 ℂcc 11143 ℝcr 11144 -∞cmnf 11283 ℝ*cxr 11284 < clt 11285 (,)cioo 13364 ↾t crest 17421 volcvol 25453 MblFncmbf 25604 SAlgcsalg 45839 SMblFncsmblfn 46226 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-inf2 9671 ax-cc 10465 ax-cnex 11201 ax-resscn 11202 ax-1cn 11203 ax-icn 11204 ax-addcl 11205 ax-addrcl 11206 ax-mulcl 11207 ax-mulrcl 11208 ax-mulcom 11209 ax-addass 11210 ax-mulass 11211 ax-distr 11212 ax-i2m1 11213 ax-1ne0 11214 ax-1rid 11215 ax-rnegex 11216 ax-rrecex 11217 ax-cnre 11218 ax-pre-lttri 11219 ax-pre-lttrn 11220 ax-pre-ltadd 11221 ax-pre-mulgt0 11222 ax-pre-sup 11223 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-disj 5115 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-isom 6558 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7685 df-om 7872 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-2o 8488 df-er 8725 df-map 8847 df-pm 8848 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-sup 9472 df-inf 9473 df-oi 9540 df-dju 9931 df-card 9969 df-pnf 11287 df-mnf 11288 df-xr 11289 df-ltxr 11290 df-le 11291 df-sub 11483 df-neg 11484 df-div 11909 df-nn 12251 df-2 12313 df-3 12314 df-n0 12511 df-z 12597 df-uz 12861 df-q 12971 df-rp 13015 df-xadd 13133 df-ioo 13368 df-ico 13370 df-icc 13371 df-fz 13525 df-fzo 13668 df-fl 13798 df-seq 14008 df-exp 14068 df-hash 14334 df-cj 15090 df-re 15091 df-im 15092 df-sqrt 15226 df-abs 15227 df-clim 15476 df-rlim 15477 df-sum 15677 df-rest 17423 df-xmet 21306 df-met 21307 df-ovol 25454 df-vol 25455 df-mbf 25609 df-salg 45840 df-smblfn 46227 |
This theorem is referenced by: mbfpsssmf 46314 |
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