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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 1914 | . 2 ⊢ Ⅎ𝑎𝜑 | |
| 2 | mbfresmf.3 | . . . 4 ⊢ 𝑆 = dom vol | |
| 3 | 2 | a1i 11 | . . 3 ⊢ (𝜑 → 𝑆 = dom vol) |
| 4 | dmvolsal 46361 | . . . 4 ⊢ dom vol ∈ SAlg | |
| 5 | 4 | a1i 11 | . . 3 ⊢ (𝜑 → dom vol ∈ SAlg) |
| 6 | 3, 5 | eqeltrd 2841 | . 2 ⊢ (𝜑 → 𝑆 ∈ SAlg) |
| 7 | mbfresmf.1 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ MblFn) | |
| 8 | mbfdmssre 46015 | . . . 4 ⊢ (𝐹 ∈ MblFn → dom 𝐹 ⊆ ℝ) | |
| 9 | 7, 8 | syl 17 | . . 3 ⊢ (𝜑 → dom 𝐹 ⊆ ℝ) |
| 10 | 2 | unieqi 4919 | . . . 4 ⊢ ∪ 𝑆 = ∪ dom vol |
| 11 | unidmvol 25576 | . . . 4 ⊢ ∪ dom vol = ℝ | |
| 12 | 10, 11 | eqtri 2765 | . . 3 ⊢ ∪ 𝑆 = ℝ |
| 13 | 9, 12 | sseqtrrdi 4025 | . 2 ⊢ (𝜑 → dom 𝐹 ⊆ ∪ 𝑆) |
| 14 | mbff 25660 | . . . . 5 ⊢ (𝐹 ∈ MblFn → 𝐹:dom 𝐹⟶ℂ) | |
| 15 | ffn 6736 | . . . . 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 511 | . . 3 ⊢ (𝜑 → (𝐹 Fn dom 𝐹 ∧ ran 𝐹 ⊆ ℝ)) |
| 19 | df-f 6565 | . . 3 ⊢ (𝐹:dom 𝐹⟶ℝ ↔ (𝐹 Fn dom 𝐹 ∧ ran 𝐹 ⊆ ℝ)) | |
| 20 | 18, 19 | sylibr 234 | . 2 ⊢ (𝜑 → 𝐹:dom 𝐹⟶ℝ) |
| 21 | 20 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝐹:dom 𝐹⟶ℝ) |
| 22 | rexr 11307 | . . . . . 6 ⊢ (𝑎 ∈ ℝ → 𝑎 ∈ ℝ*) | |
| 23 | 22 | adantl 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝑎 ∈ ℝ*) |
| 24 | 21, 23 | preimaioomnf 46734 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → (◡𝐹 “ (-∞(,)𝑎)) = {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎}) |
| 25 | 24 | eqcomd 2743 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = (◡𝐹 “ (-∞(,)𝑎))) |
| 26 | 4 | elexi 3503 | . . . . . 6 ⊢ dom vol ∈ V |
| 27 | 2, 26 | eqeltri 2837 | . . . . 5 ⊢ 𝑆 ∈ V |
| 28 | 27 | a1i 11 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝑆 ∈ V) |
| 29 | 7 | dmexd 7925 | . . . . 5 ⊢ (𝜑 → dom 𝐹 ∈ V) |
| 30 | 29 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → dom 𝐹 ∈ V) |
| 31 | mbfima 25665 | . . . . . . 7 ⊢ ((𝐹 ∈ MblFn ∧ 𝐹:dom 𝐹⟶ℝ) → (◡𝐹 “ (-∞(,)𝑎)) ∈ dom vol) | |
| 32 | 7, 20, 31 | syl2anc 584 | . . . . . 6 ⊢ (𝜑 → (◡𝐹 “ (-∞(,)𝑎)) ∈ dom vol) |
| 33 | 32, 3 | eleqtrrd 2844 | . . . . 5 ⊢ (𝜑 → (◡𝐹 “ (-∞(,)𝑎)) ∈ 𝑆) |
| 34 | 33 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → (◡𝐹 “ (-∞(,)𝑎)) ∈ 𝑆) |
| 35 | cnvimass 6100 | . . . . 5 ⊢ (◡𝐹 “ (-∞(,)𝑎)) ⊆ dom 𝐹 | |
| 36 | dfss 3970 | . . . . . 6 ⊢ ((◡𝐹 “ (-∞(,)𝑎)) ⊆ dom 𝐹 ↔ (◡𝐹 “ (-∞(,)𝑎)) = ((◡𝐹 “ (-∞(,)𝑎)) ∩ dom 𝐹)) | |
| 37 | 36 | biimpi 216 | . . . . 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 45113 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → (◡𝐹 “ (-∞(,)𝑎)) ∈ (𝑆 ↾t dom 𝐹)) |
| 40 | 25, 39 | eqeltrd 2841 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t dom 𝐹)) |
| 41 | 1, 6, 13, 20, 40 | issmfd 46750 | 1 ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 {crab 3436 Vcvv 3480 ∩ cin 3950 ⊆ wss 3951 ∪ cuni 4907 class class class wbr 5143 ◡ccnv 5684 dom cdm 5685 ran crn 5686 “ cima 5688 Fn wfn 6556 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 ℝcr 11154 -∞cmnf 11293 ℝ*cxr 11294 < clt 11295 (,)cioo 13387 ↾t crest 17465 volcvol 25498 MblFncmbf 25649 SAlgcsalg 46323 SMblFncsmblfn 46710 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cc 10475 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-disj 5111 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-pm 8869 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-inf 9483 df-oi 9550 df-dju 9941 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12614 df-uz 12879 df-q 12991 df-rp 13035 df-xadd 13155 df-ioo 13391 df-ico 13393 df-icc 13394 df-fz 13548 df-fzo 13695 df-fl 13832 df-seq 14043 df-exp 14103 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-clim 15524 df-rlim 15525 df-sum 15723 df-rest 17467 df-xmet 21357 df-met 21358 df-ovol 25499 df-vol 25500 df-mbf 25654 df-salg 46324 df-smblfn 46711 |
| This theorem is referenced by: mbfpsssmf 46798 |
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