orvcgteel - Mathbox for Thierry Arnoux

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Theorem orvcgteel 34459

Description: Preimage maps produced by the "greater than or equal to" relation are measurable sets. (Contributed by Thierry Arnoux, 5-Feb-2017.)

**Hypotheses**
Ref	Expression
orvcgteel.1	⊢ (𝜑 → 𝑃 ∈ Prob)
orvcgteel.2	⊢ (𝜑 → 𝑋 ∈ (rRndVar‘𝑃))
orvcgteel.3	⊢ (𝜑 → 𝐴 ∈ ℝ)

**Assertion**
Ref	Expression
orvcgteel	⊢ (𝜑 → (𝑋∘_RV/𝑐^◡ ≤ 𝐴) ∈ dom 𝑃)

Proof of Theorem orvcgteel Dummy variable 𝑥 is distinct from all other variables.

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 5843	. . . . . . . 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 11220	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ^*)
10	9	ad2antrl 728	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥)) → 𝑥 ∈ ℝ^*)
11		simprr 772	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥)) → 𝐴 ≤ 𝑥)
12		ltpnf 13080	. . . . . . . . . 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 770	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞))) → 𝑥 ∈ ℝ^*)
17	3	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞))) → 𝐴 ∈ ℝ)
18		simprrl 780	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞))) → 𝐴 ≤ 𝑥)
19		simprrr 781	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞))) → 𝑥 < +∞)
20		xrre3 13131	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℝ^* ∧ 𝐴 ∈ ℝ) ∧ (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞)) → 𝑥 ∈ ℝ)
21	16, 17, 18, 19, 20	syl22anc 838	. . . . . . . 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 800	. . . . . 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 3407	. . . 4 ⊢ (𝜑 → {𝑥 ∈ ℝ ∣ 𝑥^◡ ≤ 𝐴} = {𝑥 ∈ ℝ^* ∣ (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞)})
26	3	rexrd 11224	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
27		pnfxr 11228	. . . . 5 ⊢ +∞ ∈ ℝ^*
28		icoval 13344	. . . . 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 2767	. . 3 ⊢ (𝜑 → {𝑥 ∈ ℝ ∣ 𝑥^◡ ≤ 𝐴} = (𝐴[,)+∞))
31		icopnfcld 24655	. . . 4 ⊢ (𝐴 ∈ ℝ → (𝐴[,)+∞) ∈ (Clsd‘(topGen‘ran (,))))
32	3, 31	syl 17	. . 3 ⊢ (𝜑 → (𝐴[,)+∞) ∈ (Clsd‘(topGen‘ran (,))))
33	30, 32	eqeltrd 2828	. 2 ⊢ (𝜑 → {𝑥 ∈ ℝ ∣ 𝑥^◡ ≤ 𝐴} ∈ (Clsd‘(topGen‘ran (,))))
34	1, 2, 3, 33	orrvccel 34458	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 2109 {crab 3405 class class class wbr 5107 ^◡ccnv 5637 dom cdm 5638 ran crn 5639 ‘cfv 6511 (class class class)co 7387 ℝcr 11067 +∞cpnf 11205 ℝ^*cxr 11207 < clt 11208 ≤ cle 11209 (,)cioo 13306 [,)cico 13308 topGenctg 17400 Clsdccld 22903 Probcprb 34398 rRndVarcrrv 34431 ∘_RV/𝑐corvc 34447

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-inf2 9594 ax-ac2 10416 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-iin 4958 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-se 5592 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-isom 6520 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-om 7843 df-1st 7968 df-2nd 7969 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-2o 8435 df-er 8671 df-map 8801 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-sup 9393 df-inf 9394 df-oi 9463 df-dju 9854 df-card 9892 df-acn 9895 df-ac 10069 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-n0 12443 df-z 12530 df-uz 12794 df-q 12908 df-ioo 13310 df-ico 13312 df-topgen 17406 df-top 22781 df-bases 22833 df-cld 22906 df-esum 34018 df-siga 34099 df-sigagen 34129 df-brsiga 34172 df-meas 34186 df-mbfm 34240 df-prob 34399 df-rrv 34432 df-orvc 34448

This theorem is referenced by: (None)

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