Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > orvcgteel | Structured version Visualization version GIF version |
Description: Preimage maps produced by the "greater than or equal to" relation are measurable sets. (Contributed by Thierry Arnoux, 5-Feb-2017.) |
Ref | Expression |
---|---|
orvcgteel.1 | ⊢ (𝜑 → 𝑃 ∈ Prob) |
orvcgteel.2 | ⊢ (𝜑 → 𝑋 ∈ (rRndVar‘𝑃)) |
orvcgteel.3 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
Ref | Expression |
---|---|
orvcgteel | ⊢ (𝜑 → (𝑋∘RV/𝑐◡ ≤ 𝐴) ∈ dom 𝑃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | orvcgteel.1 | . 2 ⊢ (𝜑 → 𝑃 ∈ Prob) | |
2 | orvcgteel.2 | . 2 ⊢ (𝜑 → 𝑋 ∈ (rRndVar‘𝑃)) | |
3 | orvcgteel.3 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
4 | simpr 485 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℝ) | |
5 | 3 | adantr 481 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 𝐴 ∈ ℝ) |
6 | brcnvg 5743 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝑥◡ ≤ 𝐴 ↔ 𝐴 ≤ 𝑥)) | |
7 | 4, 5, 6 | syl2anc 584 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑥◡ ≤ 𝐴 ↔ 𝐴 ≤ 𝑥)) |
8 | 7 | pm5.32da 579 | . . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ ℝ ∧ 𝑥◡ ≤ 𝐴) ↔ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥))) |
9 | rexr 10675 | . . . . . . . . 9 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ*) | |
10 | 9 | ad2antrl 724 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥)) → 𝑥 ∈ ℝ*) |
11 | simprr 769 | . . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥)) → 𝐴 ≤ 𝑥) | |
12 | ltpnf 12503 | . . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ → 𝑥 < +∞) | |
13 | 12 | ad2antrl 724 | . . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥)) → 𝑥 < +∞) |
14 | 11, 13 | jca 512 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥)) → (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞)) |
15 | 10, 14 | jca 512 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥)) → (𝑥 ∈ ℝ* ∧ (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞))) |
16 | simprl 767 | . . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ* ∧ (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞))) → 𝑥 ∈ ℝ*) | |
17 | 3 | adantr 481 | . . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ* ∧ (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞))) → 𝐴 ∈ ℝ) |
18 | simprrl 777 | . . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ* ∧ (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞))) → 𝐴 ≤ 𝑥) | |
19 | simprrr 778 | . . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ* ∧ (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞))) → 𝑥 < +∞) | |
20 | xrre3 12552 | . . . . . . . . 9 ⊢ (((𝑥 ∈ ℝ* ∧ 𝐴 ∈ ℝ) ∧ (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞)) → 𝑥 ∈ ℝ) | |
21 | 16, 17, 18, 19, 20 | syl22anc 834 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ* ∧ (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞))) → 𝑥 ∈ ℝ) |
22 | 21, 18 | jca 512 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ* ∧ (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞))) → (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥)) |
23 | 15, 22 | impbida 797 | . . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥) ↔ (𝑥 ∈ ℝ* ∧ (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞)))) |
24 | 8, 23 | bitrd 280 | . . . . 5 ⊢ (𝜑 → ((𝑥 ∈ ℝ ∧ 𝑥◡ ≤ 𝐴) ↔ (𝑥 ∈ ℝ* ∧ (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞)))) |
25 | 24 | rabbidva2 3474 | . . . 4 ⊢ (𝜑 → {𝑥 ∈ ℝ ∣ 𝑥◡ ≤ 𝐴} = {𝑥 ∈ ℝ* ∣ (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞)}) |
26 | 3 | rexrd 10679 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
27 | pnfxr 10683 | . . . . 5 ⊢ +∞ ∈ ℝ* | |
28 | icoval 12764 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (𝐴[,)+∞) = {𝑥 ∈ ℝ* ∣ (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞)}) | |
29 | 26, 27, 28 | sylancl 586 | . . . 4 ⊢ (𝜑 → (𝐴[,)+∞) = {𝑥 ∈ ℝ* ∣ (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞)}) |
30 | 25, 29 | eqtr4d 2856 | . . 3 ⊢ (𝜑 → {𝑥 ∈ ℝ ∣ 𝑥◡ ≤ 𝐴} = (𝐴[,)+∞)) |
31 | icopnfcld 23303 | . . . 4 ⊢ (𝐴 ∈ ℝ → (𝐴[,)+∞) ∈ (Clsd‘(topGen‘ran (,)))) | |
32 | 3, 31 | syl 17 | . . 3 ⊢ (𝜑 → (𝐴[,)+∞) ∈ (Clsd‘(topGen‘ran (,)))) |
33 | 30, 32 | eqeltrd 2910 | . 2 ⊢ (𝜑 → {𝑥 ∈ ℝ ∣ 𝑥◡ ≤ 𝐴} ∈ (Clsd‘(topGen‘ran (,)))) |
34 | 1, 2, 3, 33 | orrvccel 31623 | 1 ⊢ (𝜑 → (𝑋∘RV/𝑐◡ ≤ 𝐴) ∈ dom 𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 {crab 3139 class class class wbr 5057 ◡ccnv 5547 dom cdm 5548 ran crn 5549 ‘cfv 6348 (class class class)co 7145 ℝcr 10524 +∞cpnf 10660 ℝ*cxr 10662 < clt 10663 ≤ cle 10664 (,)cioo 12726 [,)cico 12728 topGenctg 16699 Clsdccld 21552 Probcprb 31564 rRndVarcrrv 31597 ∘RV/𝑐corvc 31612 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-inf2 9092 ax-ac2 9873 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-2o 8092 df-oadd 8095 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-sup 8894 df-inf 8895 df-oi 8962 df-dju 9318 df-card 9356 df-acn 9359 df-ac 9530 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-n0 11886 df-z 11970 df-uz 12232 df-q 12337 df-ioo 12730 df-ico 12732 df-topgen 16705 df-top 21430 df-bases 21482 df-cld 21555 df-esum 31186 df-siga 31267 df-sigagen 31297 df-brsiga 31340 df-meas 31354 df-mbfm 31408 df-prob 31565 df-rrv 31598 df-orvc 31613 |
This theorem is referenced by: (None) |
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