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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 477 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℝ) | |
5 | 3 | adantr 473 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 𝐴 ∈ ℝ) |
6 | brcnvg 5604 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝑥◡ ≤ 𝐴 ↔ 𝐴 ≤ 𝑥)) | |
7 | 4, 5, 6 | syl2anc 576 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑥◡ ≤ 𝐴 ↔ 𝐴 ≤ 𝑥)) |
8 | 7 | pm5.32da 571 | . . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ ℝ ∧ 𝑥◡ ≤ 𝐴) ↔ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥))) |
9 | rexr 10492 | . . . . . . . . 9 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ*) | |
10 | 9 | ad2antrl 716 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥)) → 𝑥 ∈ ℝ*) |
11 | simprr 761 | . . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥)) → 𝐴 ≤ 𝑥) | |
12 | ltpnf 12338 | . . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ → 𝑥 < +∞) | |
13 | 12 | ad2antrl 716 | . . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥)) → 𝑥 < +∞) |
14 | 11, 13 | jca 504 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥)) → (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞)) |
15 | 10, 14 | jca 504 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥)) → (𝑥 ∈ ℝ* ∧ (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞))) |
16 | simprl 759 | . . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ* ∧ (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞))) → 𝑥 ∈ ℝ*) | |
17 | 3 | adantr 473 | . . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ* ∧ (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞))) → 𝐴 ∈ ℝ) |
18 | simprrl 769 | . . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ* ∧ (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞))) → 𝐴 ≤ 𝑥) | |
19 | simprrr 770 | . . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ* ∧ (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞))) → 𝑥 < +∞) | |
20 | xrre3 12387 | . . . . . . . . 9 ⊢ (((𝑥 ∈ ℝ* ∧ 𝐴 ∈ ℝ) ∧ (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞)) → 𝑥 ∈ ℝ) | |
21 | 16, 17, 18, 19, 20 | syl22anc 827 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ* ∧ (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞))) → 𝑥 ∈ ℝ) |
22 | 21, 18 | jca 504 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ* ∧ (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞))) → (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥)) |
23 | 15, 22 | impbida 789 | . . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥) ↔ (𝑥 ∈ ℝ* ∧ (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞)))) |
24 | 8, 23 | bitrd 271 | . . . . 5 ⊢ (𝜑 → ((𝑥 ∈ ℝ ∧ 𝑥◡ ≤ 𝐴) ↔ (𝑥 ∈ ℝ* ∧ (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞)))) |
25 | 24 | rabbidva2 3402 | . . . 4 ⊢ (𝜑 → {𝑥 ∈ ℝ ∣ 𝑥◡ ≤ 𝐴} = {𝑥 ∈ ℝ* ∣ (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞)}) |
26 | 3 | rexrd 10496 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
27 | pnfxr 10500 | . . . . 5 ⊢ +∞ ∈ ℝ* | |
28 | icoval 12598 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (𝐴[,)+∞) = {𝑥 ∈ ℝ* ∣ (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞)}) | |
29 | 26, 27, 28 | sylancl 578 | . . . 4 ⊢ (𝜑 → (𝐴[,)+∞) = {𝑥 ∈ ℝ* ∣ (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞)}) |
30 | 25, 29 | eqtr4d 2819 | . . 3 ⊢ (𝜑 → {𝑥 ∈ ℝ ∣ 𝑥◡ ≤ 𝐴} = (𝐴[,)+∞)) |
31 | icopnfcld 23094 | . . . 4 ⊢ (𝐴 ∈ ℝ → (𝐴[,)+∞) ∈ (Clsd‘(topGen‘ran (,)))) | |
32 | 3, 31 | syl 17 | . . 3 ⊢ (𝜑 → (𝐴[,)+∞) ∈ (Clsd‘(topGen‘ran (,)))) |
33 | 30, 32 | eqeltrd 2868 | . 2 ⊢ (𝜑 → {𝑥 ∈ ℝ ∣ 𝑥◡ ≤ 𝐴} ∈ (Clsd‘(topGen‘ran (,)))) |
34 | 1, 2, 3, 33 | orrvccel 31402 | 1 ⊢ (𝜑 → (𝑋∘RV/𝑐◡ ≤ 𝐴) ∈ dom 𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 = wceq 1508 ∈ wcel 2051 {crab 3094 class class class wbr 4934 ◡ccnv 5410 dom cdm 5411 ran crn 5412 ‘cfv 6193 (class class class)co 6982 ℝcr 10340 +∞cpnf 10477 ℝ*cxr 10479 < clt 10480 ≤ cle 10481 (,)cioo 12560 [,)cico 12562 topGenctg 16573 Clsdccld 21343 Probcprb 31343 rRndVarcrrv 31376 ∘RV/𝑐corvc 31391 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2752 ax-rep 5053 ax-sep 5064 ax-nul 5071 ax-pow 5123 ax-pr 5190 ax-un 7285 ax-inf2 8904 ax-ac2 9689 ax-cnex 10397 ax-resscn 10398 ax-1cn 10399 ax-icn 10400 ax-addcl 10401 ax-addrcl 10402 ax-mulcl 10403 ax-mulrcl 10404 ax-mulcom 10405 ax-addass 10406 ax-mulass 10407 ax-distr 10408 ax-i2m1 10409 ax-1ne0 10410 ax-1rid 10411 ax-rnegex 10412 ax-rrecex 10413 ax-cnre 10414 ax-pre-lttri 10415 ax-pre-lttrn 10416 ax-pre-ltadd 10417 ax-pre-mulgt0 10418 ax-pre-sup 10419 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-fal 1521 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2551 df-eu 2589 df-clab 2761 df-cleq 2773 df-clel 2848 df-nfc 2920 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3419 df-sbc 3684 df-csb 3789 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-pss 3847 df-nul 4182 df-if 4354 df-pw 4427 df-sn 4445 df-pr 4447 df-tp 4449 df-op 4451 df-uni 4718 df-int 4755 df-iun 4799 df-iin 4800 df-br 4935 df-opab 4997 df-mpt 5014 df-tr 5036 df-id 5316 df-eprel 5321 df-po 5330 df-so 5331 df-fr 5370 df-se 5371 df-we 5372 df-xp 5417 df-rel 5418 df-cnv 5419 df-co 5420 df-dm 5421 df-rn 5422 df-res 5423 df-ima 5424 df-pred 5991 df-ord 6037 df-on 6038 df-lim 6039 df-suc 6040 df-iota 6157 df-fun 6195 df-fn 6196 df-f 6197 df-f1 6198 df-fo 6199 df-f1o 6200 df-fv 6201 df-isom 6202 df-riota 6943 df-ov 6985 df-oprab 6986 df-mpo 6987 df-om 7403 df-1st 7507 df-2nd 7508 df-wrecs 7756 df-recs 7818 df-rdg 7856 df-1o 7911 df-2o 7912 df-oadd 7915 df-er 8095 df-map 8214 df-en 8313 df-dom 8314 df-sdom 8315 df-fin 8316 df-sup 8707 df-inf 8708 df-oi 8775 df-dju 9130 df-card 9168 df-acn 9171 df-ac 9342 df-pnf 10482 df-mnf 10483 df-xr 10484 df-ltxr 10485 df-le 10486 df-sub 10678 df-neg 10679 df-div 11105 df-nn 11446 df-n0 11714 df-z 11800 df-uz 12065 df-q 12169 df-ioo 12564 df-ico 12566 df-topgen 16579 df-top 21221 df-bases 21273 df-cld 21346 df-esum 30963 df-siga 31044 df-sigagen 31075 df-brsiga 31118 df-meas 31132 df-mbfm 31186 df-prob 31344 df-rrv 31377 df-orvc 31392 |
This theorem is referenced by: (None) |
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