| Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
||
| 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 484 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℝ) | |
| 5 | 3 | adantr 480 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 𝐴 ∈ ℝ) |
| 6 | brcnvg 5890 | . . . . . . . 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 11307 | . . . . . . . . 9 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ*) | |
| 10 | 9 | ad2antrl 728 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥)) → 𝑥 ∈ ℝ*) |
| 11 | simprr 773 | . . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥)) → 𝐴 ≤ 𝑥) | |
| 12 | ltpnf 13162 | . . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ → 𝑥 < +∞) | |
| 13 | 12 | ad2antrl 728 | . . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥)) → 𝑥 < +∞) |
| 14 | 11, 13 | jca 511 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥)) → (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞)) |
| 15 | 10, 14 | jca 511 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥)) → (𝑥 ∈ ℝ* ∧ (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞))) |
| 16 | simprl 771 | . . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ* ∧ (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞))) → 𝑥 ∈ ℝ*) | |
| 17 | 3 | adantr 480 | . . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ* ∧ (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞))) → 𝐴 ∈ ℝ) |
| 18 | simprrl 781 | . . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ* ∧ (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞))) → 𝐴 ≤ 𝑥) | |
| 19 | simprrr 782 | . . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ* ∧ (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞))) → 𝑥 < +∞) | |
| 20 | xrre3 13213 | . . . . . . . . 9 ⊢ (((𝑥 ∈ ℝ* ∧ 𝐴 ∈ ℝ) ∧ (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞)) → 𝑥 ∈ ℝ) | |
| 21 | 16, 17, 18, 19, 20 | syl22anc 839 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ* ∧ (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞))) → 𝑥 ∈ ℝ) |
| 22 | 21, 18 | jca 511 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ* ∧ (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞))) → (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥)) |
| 23 | 15, 22 | impbida 801 | . . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥) ↔ (𝑥 ∈ ℝ* ∧ (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞)))) |
| 24 | 8, 23 | bitrd 279 | . . . . 5 ⊢ (𝜑 → ((𝑥 ∈ ℝ ∧ 𝑥◡ ≤ 𝐴) ↔ (𝑥 ∈ ℝ* ∧ (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞)))) |
| 25 | 24 | rabbidva2 3438 | . . . 4 ⊢ (𝜑 → {𝑥 ∈ ℝ ∣ 𝑥◡ ≤ 𝐴} = {𝑥 ∈ ℝ* ∣ (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞)}) |
| 26 | 3 | rexrd 11311 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
| 27 | pnfxr 11315 | . . . . 5 ⊢ +∞ ∈ ℝ* | |
| 28 | icoval 13425 | . . . . 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 2780 | . . 3 ⊢ (𝜑 → {𝑥 ∈ ℝ ∣ 𝑥◡ ≤ 𝐴} = (𝐴[,)+∞)) |
| 31 | icopnfcld 24788 | . . . 4 ⊢ (𝐴 ∈ ℝ → (𝐴[,)+∞) ∈ (Clsd‘(topGen‘ran (,)))) | |
| 32 | 3, 31 | syl 17 | . . 3 ⊢ (𝜑 → (𝐴[,)+∞) ∈ (Clsd‘(topGen‘ran (,)))) |
| 33 | 30, 32 | eqeltrd 2841 | . 2 ⊢ (𝜑 → {𝑥 ∈ ℝ ∣ 𝑥◡ ≤ 𝐴} ∈ (Clsd‘(topGen‘ran (,)))) |
| 34 | 1, 2, 3, 33 | orrvccel 34469 | 1 ⊢ (𝜑 → (𝑋∘RV/𝑐◡ ≤ 𝐴) ∈ dom 𝑃) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 {crab 3436 class class class wbr 5143 ◡ccnv 5684 dom cdm 5685 ran crn 5686 ‘cfv 6561 (class class class)co 7431 ℝcr 11154 +∞cpnf 11292 ℝ*cxr 11294 < clt 11295 ≤ cle 11296 (,)cioo 13387 [,)cico 13389 topGenctg 17482 Clsdccld 23024 Probcprb 34409 rRndVarcrrv 34442 ∘RV/𝑐corvc 34458 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-ac2 10503 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-inf 9483 df-oi 9550 df-dju 9941 df-card 9979 df-acn 9982 df-ac 10156 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-n0 12527 df-z 12614 df-uz 12879 df-q 12991 df-ioo 13391 df-ico 13393 df-topgen 17488 df-top 22900 df-bases 22953 df-cld 23027 df-esum 34029 df-siga 34110 df-sigagen 34140 df-brsiga 34183 df-meas 34197 df-mbfm 34251 df-prob 34410 df-rrv 34443 df-orvc 34459 |
| This theorem is referenced by: (None) |
| Copyright terms: Public domain | W3C validator |