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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > issmfle2d | Structured version Visualization version GIF version |
Description: A sufficient condition for "𝐹 being a measurable function w.r.t. to the sigma-algebra 𝑆". (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
Ref | Expression |
---|---|
issmfle2d.a | ⊢ Ⅎ𝑎𝜑 |
issmfle2d.s | ⊢ (𝜑 → 𝑆 ∈ SAlg) |
issmfle2d.d | ⊢ (𝜑 → 𝐷 ⊆ ∪ 𝑆) |
issmfle2d.f | ⊢ (𝜑 → 𝐹:𝐷⟶ℝ) |
issmfle2d.l | ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → (◡𝐹 “ (-∞(,]𝑎)) ∈ (𝑆 ↾t 𝐷)) |
Ref | Expression |
---|---|
issmfle2d | ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | issmfle2d.a | . 2 ⊢ Ⅎ𝑎𝜑 | |
2 | issmfle2d.s | . 2 ⊢ (𝜑 → 𝑆 ∈ SAlg) | |
3 | issmfle2d.d | . 2 ⊢ (𝜑 → 𝐷 ⊆ ∪ 𝑆) | |
4 | issmfle2d.f | . 2 ⊢ (𝜑 → 𝐹:𝐷⟶ℝ) | |
5 | 4 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝐹:𝐷⟶ℝ) |
6 | rexr 10533 | . . . . 5 ⊢ (𝑎 ∈ ℝ → 𝑎 ∈ ℝ*) | |
7 | 6 | adantl 482 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝑎 ∈ ℝ*) |
8 | 5, 7 | preimaiocmnf 41398 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → (◡𝐹 “ (-∞(,]𝑎)) = {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) ≤ 𝑎}) |
9 | issmfle2d.l | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → (◡𝐹 “ (-∞(,]𝑎)) ∈ (𝑆 ↾t 𝐷)) | |
10 | 8, 9 | eqeltrrd 2884 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) ≤ 𝑎} ∈ (𝑆 ↾t 𝐷)) |
11 | 1, 2, 3, 4, 10 | issmfled 42596 | 1 ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 Ⅎwnf 1765 ∈ wcel 2081 {crab 3109 ⊆ wss 3859 ∪ cuni 4745 class class class wbr 4962 ◡ccnv 5442 “ cima 5446 ⟶wf 6221 ‘cfv 6225 (class class class)co 7016 ℝcr 10382 -∞cmnf 10519 ℝ*cxr 10520 ≤ cle 10522 (,]cioc 12589 ↾t crest 16523 SAlgcsalg 42155 SMblFncsmblfn 42539 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5081 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-inf2 8950 ax-cc 9703 ax-ac2 9731 ax-cnex 10439 ax-resscn 10440 ax-1cn 10441 ax-icn 10442 ax-addcl 10443 ax-addrcl 10444 ax-mulcl 10445 ax-mulrcl 10446 ax-mulcom 10447 ax-addass 10448 ax-mulass 10449 ax-distr 10450 ax-i2m1 10451 ax-1ne0 10452 ax-1rid 10453 ax-rnegex 10454 ax-rrecex 10455 ax-cnre 10456 ax-pre-lttri 10457 ax-pre-lttrn 10458 ax-pre-ltadd 10459 ax-pre-mulgt0 10460 ax-pre-sup 10461 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-pss 3876 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-tp 4477 df-op 4479 df-uni 4746 df-int 4783 df-iun 4827 df-iin 4828 df-br 4963 df-opab 5025 df-mpt 5042 df-tr 5064 df-id 5348 df-eprel 5353 df-po 5362 df-so 5363 df-fr 5402 df-se 5403 df-we 5404 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-pred 6023 df-ord 6069 df-on 6070 df-lim 6071 df-suc 6072 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-isom 6234 df-riota 6977 df-ov 7019 df-oprab 7020 df-mpo 7021 df-om 7437 df-1st 7545 df-2nd 7546 df-wrecs 7798 df-recs 7860 df-rdg 7898 df-1o 7953 df-oadd 7957 df-er 8139 df-map 8258 df-pm 8259 df-en 8358 df-dom 8359 df-sdom 8360 df-fin 8361 df-sup 8752 df-inf 8753 df-card 9214 df-acn 9217 df-ac 9388 df-pnf 10523 df-mnf 10524 df-xr 10525 df-ltxr 10526 df-le 10527 df-sub 10719 df-neg 10720 df-div 11146 df-nn 11487 df-n0 11746 df-z 11830 df-uz 12094 df-q 12198 df-rp 12240 df-ioo 12592 df-ioc 12593 df-ico 12594 df-fl 13012 df-rest 16525 df-salg 42156 df-smblfn 42540 |
This theorem is referenced by: smfsuplem3 42649 |
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