orvcgteel - Mathbox for Thierry Arnoux

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Theorem orvcgteel 32334

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 5777	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝑥^◡ ≤ 𝐴 ↔ 𝐴 ≤ 𝑥))
7	4, 5, 6	syl2anc 583	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑥^◡ ≤ 𝐴 ↔ 𝐴 ≤ 𝑥))
8	7	pm5.32da 578	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ ℝ ∧ 𝑥^◡ ≤ 𝐴) ↔ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥)))
9		rexr 10952	. . . . . . . . 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 12785	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ → 𝑥 < +∞)
13	12	ad2antrl 724	. . . . . . . . 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 767	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞))) → 𝑥 ∈ ℝ^*)
17	3	adantr 480	. . . . . . . . 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 12834	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℝ^* ∧ 𝐴 ∈ ℝ) ∧ (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞)) → 𝑥 ∈ ℝ)
21	16, 17, 18, 19, 20	syl22anc 835	. . . . . . . 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 797	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥) ↔ (𝑥 ∈ ℝ^* ∧ (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞))))
24	8, 23	bitrd 278	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ ℝ ∧ 𝑥^◡ ≤ 𝐴) ↔ (𝑥 ∈ ℝ^* ∧ (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞))))
25	24	rabbidva2 3400	. . . 4 ⊢ (𝜑 → {𝑥 ∈ ℝ ∣ 𝑥^◡ ≤ 𝐴} = {𝑥 ∈ ℝ^* ∣ (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞)})
26	3	rexrd 10956	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
27		pnfxr 10960	. . . . 5 ⊢ +∞ ∈ ℝ^*
28		icoval 13046	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ +∞ ∈ ℝ^) → (𝐴[,)+∞) = {𝑥 ∈ ℝ^ ∣ (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞)})
29	26, 27, 28	sylancl 585	. . . 4 ⊢ (𝜑 → (𝐴[,)+∞) = {𝑥 ∈ ℝ^* ∣ (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞)})
30	25, 29	eqtr4d 2781	. . 3 ⊢ (𝜑 → {𝑥 ∈ ℝ ∣ 𝑥^◡ ≤ 𝐴} = (𝐴[,)+∞))
31		icopnfcld 23837	. . . 4 ⊢ (𝐴 ∈ ℝ → (𝐴[,)+∞) ∈ (Clsd‘(topGen‘ran (,))))
32	3, 31	syl 17	. . 3 ⊢ (𝜑 → (𝐴[,)+∞) ∈ (Clsd‘(topGen‘ran (,))))
33	30, 32	eqeltrd 2839	. 2 ⊢ (𝜑 → {𝑥 ∈ ℝ ∣ 𝑥^◡ ≤ 𝐴} ∈ (Clsd‘(topGen‘ran (,))))
34	1, 2, 3, 33	orrvccel 32333	1 ⊢ (𝜑 → (𝑋∘_RV/𝑐^◡ ≤ 𝐴) ∈ dom 𝑃)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1539 ∈ wcel 2108 {crab 3067 class class class wbr 5070 ^◡ccnv 5579 dom cdm 5580 ran crn 5581 ‘cfv 6418 (class class class)co 7255 ℝcr 10801 +∞cpnf 10937 ℝ^*cxr 10939 < clt 10940 ≤ cle 10941 (,)cioo 13008 [,)cico 13010 topGenctg 17065 Clsdccld 22075 Probcprb 32274 rRndVarcrrv 32307 ∘_RV/𝑐corvc 32322

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-inf2 9329 ax-ac2 10150 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-2o 8268 df-er 8456 df-map 8575 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-sup 9131 df-inf 9132 df-oi 9199 df-dju 9590 df-card 9628 df-acn 9631 df-ac 9803 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-n0 12164 df-z 12250 df-uz 12512 df-q 12618 df-ioo 13012 df-ico 13014 df-topgen 17071 df-top 21951 df-bases 22004 df-cld 22078 df-esum 31896 df-siga 31977 df-sigagen 32007 df-brsiga 32050 df-meas 32064 df-mbfm 32118 df-prob 32275 df-rrv 32308 df-orvc 32323

This theorem is referenced by: (None)

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