orvclteel - Mathbox for Thierry Arnoux

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Theorem orvclteel 34475

Description: Preimage maps produced by the "less than or equal to" relation are measurable sets. (Contributed by Thierry Arnoux, 4-Feb-2017.)

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

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

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

Step	Hyp	Ref	Expression
1		dstfrv.1	. 2 ⊢ (𝜑 → 𝑃 ∈ Prob)
2		dstfrv.2	. 2 ⊢ (𝜑 → 𝑋 ∈ (rRndVar‘𝑃))
3		orvclteel.1	. 2 ⊢ (𝜑 → 𝐴 ∈ ℝ)
4		rexr 11307	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ^*)
5	4	ad2antrl 728	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝑥 ≤ 𝐴)) → 𝑥 ∈ ℝ^*)
6		mnflt 13165	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ → -∞ < 𝑥)
7	6	ad2antrl 728	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝑥 ≤ 𝐴)) → -∞ < 𝑥)
8		simprr 773	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝑥 ≤ 𝐴)) → 𝑥 ≤ 𝐴)
9	7, 8	jca 511	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝑥 ≤ 𝐴)) → (-∞ < 𝑥 ∧ 𝑥 ≤ 𝐴))
10	5, 9	jca 511	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝑥 ≤ 𝐴)) → (𝑥 ∈ ℝ^* ∧ (-∞ < 𝑥 ∧ 𝑥 ≤ 𝐴)))
11		simprl 771	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ (-∞ < 𝑥 ∧ 𝑥 ≤ 𝐴))) → 𝑥 ∈ ℝ^*)
12	3	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ (-∞ < 𝑥 ∧ 𝑥 ≤ 𝐴))) → 𝐴 ∈ ℝ)
13		simprrl 781	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ (-∞ < 𝑥 ∧ 𝑥 ≤ 𝐴))) → -∞ < 𝑥)
14		simprrr 782	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ (-∞ < 𝑥 ∧ 𝑥 ≤ 𝐴))) → 𝑥 ≤ 𝐴)
15		xrre 13211	. . . . . . . 8 ⊢ (((𝑥 ∈ ℝ^* ∧ 𝐴 ∈ ℝ) ∧ (-∞ < 𝑥 ∧ 𝑥 ≤ 𝐴)) → 𝑥 ∈ ℝ)
16	11, 12, 13, 14, 15	syl22anc 839	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ (-∞ < 𝑥 ∧ 𝑥 ≤ 𝐴))) → 𝑥 ∈ ℝ)
17	16, 14	jca 511	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ (-∞ < 𝑥 ∧ 𝑥 ≤ 𝐴))) → (𝑥 ∈ ℝ ∧ 𝑥 ≤ 𝐴))
18	10, 17	impbida 801	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ ℝ ∧ 𝑥 ≤ 𝐴) ↔ (𝑥 ∈ ℝ^* ∧ (-∞ < 𝑥 ∧ 𝑥 ≤ 𝐴))))
19	18	rabbidva2 3438	. . . 4 ⊢ (𝜑 → {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴} = {𝑥 ∈ ℝ^* ∣ (-∞ < 𝑥 ∧ 𝑥 ≤ 𝐴)})
20		mnfxr 11318	. . . . 5 ⊢ -∞ ∈ ℝ^*
21	3	rexrd 11311	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
22		iocval 13424	. . . . 5 ⊢ ((-∞ ∈ ℝ^* ∧ 𝐴 ∈ ℝ^) → (-∞(,]𝐴) = {𝑥 ∈ ℝ^ ∣ (-∞ < 𝑥 ∧ 𝑥 ≤ 𝐴)})
23	20, 21, 22	sylancr 587	. . . 4 ⊢ (𝜑 → (-∞(,]𝐴) = {𝑥 ∈ ℝ^* ∣ (-∞ < 𝑥 ∧ 𝑥 ≤ 𝐴)})
24	19, 23	eqtr4d 2780	. . 3 ⊢ (𝜑 → {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴} = (-∞(,]𝐴))
25		iocmnfcld 24789	. . . 4 ⊢ (𝐴 ∈ ℝ → (-∞(,]𝐴) ∈ (Clsd‘(topGen‘ran (,))))
26	3, 25	syl 17	. . 3 ⊢ (𝜑 → (-∞(,]𝐴) ∈ (Clsd‘(topGen‘ran (,))))
27	24, 26	eqeltrd 2841	. 2 ⊢ (𝜑 → {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴} ∈ (Clsd‘(topGen‘ran (,))))
28	1, 2, 3, 27	orrvccel 34469	1 ⊢ (𝜑 → (𝑋∘_RV/𝑐 ≤ 𝐴) ∈ dom 𝑃)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 {crab 3436 class class class wbr 5143 dom cdm 5685 ran crn 5686 ‘cfv 6561 (class class class)co 7431 ℝcr 11154 -∞cmnf 11293 ℝ^*cxr 11294 < clt 11295 ≤ cle 11296 (,)cioo 13387 (,]cioc 13388 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-ioc 13392 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: dstfrvunirn 34477 dstfrvinc 34479 dstfrvclim1 34480

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