Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > orvclteel | Structured version Visualization version GIF version |
Description: Preimage maps produced by the "less than or equal to" relation are measurable sets. (Contributed by Thierry Arnoux, 4-Feb-2017.) |
Ref | Expression |
---|---|
dstfrv.1 | ⊢ (𝜑 → 𝑃 ∈ Prob) |
dstfrv.2 | ⊢ (𝜑 → 𝑋 ∈ (rRndVar‘𝑃)) |
orvclteel.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
Ref | Expression |
---|---|
orvclteel | ⊢ (𝜑 → (𝑋∘RV/𝑐 ≤ 𝐴) ∈ dom 𝑃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dstfrv.1 | . 2 ⊢ (𝜑 → 𝑃 ∈ Prob) | |
2 | dstfrv.2 | . 2 ⊢ (𝜑 → 𝑋 ∈ (rRndVar‘𝑃)) | |
3 | orvclteel.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
4 | rexr 10862 | . . . . . . . 8 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ*) | |
5 | 4 | ad2antrl 728 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝑥 ≤ 𝐴)) → 𝑥 ∈ ℝ*) |
6 | mnflt 12698 | . . . . . . . . 9 ⊢ (𝑥 ∈ ℝ → -∞ < 𝑥) | |
7 | 6 | ad2antrl 728 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝑥 ≤ 𝐴)) → -∞ < 𝑥) |
8 | simprr 773 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝑥 ≤ 𝐴)) → 𝑥 ≤ 𝐴) | |
9 | 7, 8 | jca 515 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝑥 ≤ 𝐴)) → (-∞ < 𝑥 ∧ 𝑥 ≤ 𝐴)) |
10 | 5, 9 | jca 515 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝑥 ≤ 𝐴)) → (𝑥 ∈ ℝ* ∧ (-∞ < 𝑥 ∧ 𝑥 ≤ 𝐴))) |
11 | simprl 771 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ* ∧ (-∞ < 𝑥 ∧ 𝑥 ≤ 𝐴))) → 𝑥 ∈ ℝ*) | |
12 | 3 | adantr 484 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ* ∧ (-∞ < 𝑥 ∧ 𝑥 ≤ 𝐴))) → 𝐴 ∈ ℝ) |
13 | simprrl 781 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ* ∧ (-∞ < 𝑥 ∧ 𝑥 ≤ 𝐴))) → -∞ < 𝑥) | |
14 | simprrr 782 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ* ∧ (-∞ < 𝑥 ∧ 𝑥 ≤ 𝐴))) → 𝑥 ≤ 𝐴) | |
15 | xrre 12742 | . . . . . . . 8 ⊢ (((𝑥 ∈ ℝ* ∧ 𝐴 ∈ ℝ) ∧ (-∞ < 𝑥 ∧ 𝑥 ≤ 𝐴)) → 𝑥 ∈ ℝ) | |
16 | 11, 12, 13, 14, 15 | syl22anc 839 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ* ∧ (-∞ < 𝑥 ∧ 𝑥 ≤ 𝐴))) → 𝑥 ∈ ℝ) |
17 | 16, 14 | jca 515 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ* ∧ (-∞ < 𝑥 ∧ 𝑥 ≤ 𝐴))) → (𝑥 ∈ ℝ ∧ 𝑥 ≤ 𝐴)) |
18 | 10, 17 | impbida 801 | . . . . 5 ⊢ (𝜑 → ((𝑥 ∈ ℝ ∧ 𝑥 ≤ 𝐴) ↔ (𝑥 ∈ ℝ* ∧ (-∞ < 𝑥 ∧ 𝑥 ≤ 𝐴)))) |
19 | 18 | rabbidva2 3379 | . . . 4 ⊢ (𝜑 → {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴} = {𝑥 ∈ ℝ* ∣ (-∞ < 𝑥 ∧ 𝑥 ≤ 𝐴)}) |
20 | mnfxr 10873 | . . . . 5 ⊢ -∞ ∈ ℝ* | |
21 | 3 | rexrd 10866 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
22 | iocval 12955 | . . . . 5 ⊢ ((-∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ*) → (-∞(,]𝐴) = {𝑥 ∈ ℝ* ∣ (-∞ < 𝑥 ∧ 𝑥 ≤ 𝐴)}) | |
23 | 20, 21, 22 | sylancr 590 | . . . 4 ⊢ (𝜑 → (-∞(,]𝐴) = {𝑥 ∈ ℝ* ∣ (-∞ < 𝑥 ∧ 𝑥 ≤ 𝐴)}) |
24 | 19, 23 | eqtr4d 2777 | . . 3 ⊢ (𝜑 → {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴} = (-∞(,]𝐴)) |
25 | iocmnfcld 23638 | . . . 4 ⊢ (𝐴 ∈ ℝ → (-∞(,]𝐴) ∈ (Clsd‘(topGen‘ran (,)))) | |
26 | 3, 25 | syl 17 | . . 3 ⊢ (𝜑 → (-∞(,]𝐴) ∈ (Clsd‘(topGen‘ran (,)))) |
27 | 24, 26 | eqeltrd 2834 | . 2 ⊢ (𝜑 → {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴} ∈ (Clsd‘(topGen‘ran (,)))) |
28 | 1, 2, 3, 27 | orrvccel 32117 | 1 ⊢ (𝜑 → (𝑋∘RV/𝑐 ≤ 𝐴) ∈ dom 𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 {crab 3058 class class class wbr 5043 dom cdm 5540 ran crn 5541 ‘cfv 6369 (class class class)co 7202 ℝcr 10711 -∞cmnf 10848 ℝ*cxr 10849 < clt 10850 ≤ cle 10851 (,)cioo 12918 (,]cioc 12919 topGenctg 16914 Clsdccld 21885 Probcprb 32058 rRndVarcrrv 32091 ∘RV/𝑐corvc 32106 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-inf2 9245 ax-ac2 10060 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 ax-pre-sup 10790 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-int 4850 df-iun 4896 df-iin 4897 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-se 5499 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-isom 6378 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-om 7634 df-1st 7750 df-2nd 7751 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-1o 8191 df-2o 8192 df-er 8380 df-map 8499 df-en 8616 df-dom 8617 df-sdom 8618 df-fin 8619 df-sup 9047 df-inf 9048 df-oi 9115 df-dju 9500 df-card 9538 df-acn 9541 df-ac 9713 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-div 11473 df-nn 11814 df-n0 12074 df-z 12160 df-uz 12422 df-q 12528 df-ioo 12922 df-ioc 12923 df-topgen 16920 df-top 21763 df-bases 21815 df-cld 21888 df-esum 31680 df-siga 31761 df-sigagen 31791 df-brsiga 31834 df-meas 31848 df-mbfm 31902 df-prob 32059 df-rrv 32092 df-orvc 32107 |
This theorem is referenced by: dstfrvunirn 32125 dstfrvinc 32127 dstfrvclim1 32128 |
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