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 10681 | . . . . . . . 8 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ*) | |
5 | 4 | ad2antrl 726 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝑥 ≤ 𝐴)) → 𝑥 ∈ ℝ*) |
6 | mnflt 12512 | . . . . . . . . 9 ⊢ (𝑥 ∈ ℝ → -∞ < 𝑥) | |
7 | 6 | ad2antrl 726 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝑥 ≤ 𝐴)) → -∞ < 𝑥) |
8 | simprr 771 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝑥 ≤ 𝐴)) → 𝑥 ≤ 𝐴) | |
9 | 7, 8 | jca 514 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝑥 ≤ 𝐴)) → (-∞ < 𝑥 ∧ 𝑥 ≤ 𝐴)) |
10 | 5, 9 | jca 514 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝑥 ≤ 𝐴)) → (𝑥 ∈ ℝ* ∧ (-∞ < 𝑥 ∧ 𝑥 ≤ 𝐴))) |
11 | simprl 769 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ* ∧ (-∞ < 𝑥 ∧ 𝑥 ≤ 𝐴))) → 𝑥 ∈ ℝ*) | |
12 | 3 | adantr 483 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ* ∧ (-∞ < 𝑥 ∧ 𝑥 ≤ 𝐴))) → 𝐴 ∈ ℝ) |
13 | simprrl 779 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ* ∧ (-∞ < 𝑥 ∧ 𝑥 ≤ 𝐴))) → -∞ < 𝑥) | |
14 | simprrr 780 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ* ∧ (-∞ < 𝑥 ∧ 𝑥 ≤ 𝐴))) → 𝑥 ≤ 𝐴) | |
15 | xrre 12556 | . . . . . . . 8 ⊢ (((𝑥 ∈ ℝ* ∧ 𝐴 ∈ ℝ) ∧ (-∞ < 𝑥 ∧ 𝑥 ≤ 𝐴)) → 𝑥 ∈ ℝ) | |
16 | 11, 12, 13, 14, 15 | syl22anc 836 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ* ∧ (-∞ < 𝑥 ∧ 𝑥 ≤ 𝐴))) → 𝑥 ∈ ℝ) |
17 | 16, 14 | jca 514 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ* ∧ (-∞ < 𝑥 ∧ 𝑥 ≤ 𝐴))) → (𝑥 ∈ ℝ ∧ 𝑥 ≤ 𝐴)) |
18 | 10, 17 | impbida 799 | . . . . 5 ⊢ (𝜑 → ((𝑥 ∈ ℝ ∧ 𝑥 ≤ 𝐴) ↔ (𝑥 ∈ ℝ* ∧ (-∞ < 𝑥 ∧ 𝑥 ≤ 𝐴)))) |
19 | 18 | rabbidva2 3477 | . . . 4 ⊢ (𝜑 → {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴} = {𝑥 ∈ ℝ* ∣ (-∞ < 𝑥 ∧ 𝑥 ≤ 𝐴)}) |
20 | mnfxr 10692 | . . . . 5 ⊢ -∞ ∈ ℝ* | |
21 | 3 | rexrd 10685 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
22 | iocval 12769 | . . . . 5 ⊢ ((-∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ*) → (-∞(,]𝐴) = {𝑥 ∈ ℝ* ∣ (-∞ < 𝑥 ∧ 𝑥 ≤ 𝐴)}) | |
23 | 20, 21, 22 | sylancr 589 | . . . 4 ⊢ (𝜑 → (-∞(,]𝐴) = {𝑥 ∈ ℝ* ∣ (-∞ < 𝑥 ∧ 𝑥 ≤ 𝐴)}) |
24 | 19, 23 | eqtr4d 2859 | . . 3 ⊢ (𝜑 → {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴} = (-∞(,]𝐴)) |
25 | iocmnfcld 23371 | . . . 4 ⊢ (𝐴 ∈ ℝ → (-∞(,]𝐴) ∈ (Clsd‘(topGen‘ran (,)))) | |
26 | 3, 25 | syl 17 | . . 3 ⊢ (𝜑 → (-∞(,]𝐴) ∈ (Clsd‘(topGen‘ran (,)))) |
27 | 24, 26 | eqeltrd 2913 | . 2 ⊢ (𝜑 → {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴} ∈ (Clsd‘(topGen‘ran (,)))) |
28 | 1, 2, 3, 27 | orrvccel 31719 | 1 ⊢ (𝜑 → (𝑋∘RV/𝑐 ≤ 𝐴) ∈ dom 𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 {crab 3142 class class class wbr 5059 dom cdm 5550 ran crn 5551 ‘cfv 6350 (class class class)co 7150 ℝcr 10530 -∞cmnf 10667 ℝ*cxr 10668 < clt 10669 ≤ cle 10670 (,)cioo 12732 (,]cioc 12733 topGenctg 16705 Clsdccld 21618 Probcprb 31660 rRndVarcrrv 31693 ∘RV/𝑐corvc 31708 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-inf2 9098 ax-ac2 9879 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-iin 4915 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-se 5510 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-isom 6359 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-oadd 8100 df-er 8283 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-sup 8900 df-inf 8901 df-oi 8968 df-dju 9324 df-card 9362 df-acn 9365 df-ac 9536 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-n0 11892 df-z 11976 df-uz 12238 df-q 12343 df-ioo 12736 df-ioc 12737 df-topgen 16711 df-top 21496 df-bases 21548 df-cld 21621 df-esum 31282 df-siga 31363 df-sigagen 31393 df-brsiga 31436 df-meas 31450 df-mbfm 31504 df-prob 31661 df-rrv 31694 df-orvc 31709 |
This theorem is referenced by: dstfrvunirn 31727 dstfrvinc 31729 dstfrvclim1 31730 |
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