orvclteel - Mathbox for Thierry Arnoux

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Theorem orvclteel 29861

Description: Preimage maps produced by the "lower than or equal" 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 9964	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ^*)
5	4	ad2antrl 760	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝑥 ≤ 𝐴)) → 𝑥 ∈ ℝ^*)
6		mnflt 11833	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ → -∞ < 𝑥)
7	6	ad2antrl 760	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝑥 ≤ 𝐴)) → -∞ < 𝑥)
8		simprr 792	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝑥 ≤ 𝐴)) → 𝑥 ≤ 𝐴)
9	7, 8	jca 553	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝑥 ≤ 𝐴)) → (-∞ < 𝑥 ∧ 𝑥 ≤ 𝐴))
10	5, 9	jca 553	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝑥 ≤ 𝐴)) → (𝑥 ∈ ℝ^* ∧ (-∞ < 𝑥 ∧ 𝑥 ≤ 𝐴)))
11		simprl 790	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ (-∞ < 𝑥 ∧ 𝑥 ≤ 𝐴))) → 𝑥 ∈ ℝ^*)
12	3	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ (-∞ < 𝑥 ∧ 𝑥 ≤ 𝐴))) → 𝐴 ∈ ℝ)
13		simprrl 800	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ (-∞ < 𝑥 ∧ 𝑥 ≤ 𝐴))) → -∞ < 𝑥)
14		simprrr 801	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ (-∞ < 𝑥 ∧ 𝑥 ≤ 𝐴))) → 𝑥 ≤ 𝐴)
15		xrre 11874	. . . . . . . 8 ⊢ (((𝑥 ∈ ℝ^* ∧ 𝐴 ∈ ℝ) ∧ (-∞ < 𝑥 ∧ 𝑥 ≤ 𝐴)) → 𝑥 ∈ ℝ)
16	11, 12, 13, 14, 15	syl22anc 1319	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ (-∞ < 𝑥 ∧ 𝑥 ≤ 𝐴))) → 𝑥 ∈ ℝ)
17	16, 14	jca 553	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ (-∞ < 𝑥 ∧ 𝑥 ≤ 𝐴))) → (𝑥 ∈ ℝ ∧ 𝑥 ≤ 𝐴))
18	10, 17	impbida 873	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ ℝ ∧ 𝑥 ≤ 𝐴) ↔ (𝑥 ∈ ℝ^* ∧ (-∞ < 𝑥 ∧ 𝑥 ≤ 𝐴))))
19	18	rabbidva2 3162	. . . 4 ⊢ (𝜑 → {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴} = {𝑥 ∈ ℝ^* ∣ (-∞ < 𝑥 ∧ 𝑥 ≤ 𝐴)})
20		mnfxr 9975	. . . . 5 ⊢ -∞ ∈ ℝ^*
21	3	rexrd 9968	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
22		iocval 12083	. . . . 5 ⊢ ((-∞ ∈ ℝ^* ∧ 𝐴 ∈ ℝ^) → (-∞(,]𝐴) = {𝑥 ∈ ℝ^ ∣ (-∞ < 𝑥 ∧ 𝑥 ≤ 𝐴)})
23	20, 21, 22	sylancr 694	. . . 4 ⊢ (𝜑 → (-∞(,]𝐴) = {𝑥 ∈ ℝ^* ∣ (-∞ < 𝑥 ∧ 𝑥 ≤ 𝐴)})
24	19, 23	eqtr4d 2647	. . 3 ⊢ (𝜑 → {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴} = (-∞(,]𝐴))
25		iocmnfcld 22382	. . . 4 ⊢ (𝐴 ∈ ℝ → (-∞(,]𝐴) ∈ (Clsd‘(topGen‘ran (,))))
26	3, 25	syl 17	. . 3 ⊢ (𝜑 → (-∞(,]𝐴) ∈ (Clsd‘(topGen‘ran (,))))
27	24, 26	eqeltrd 2688	. 2 ⊢ (𝜑 → {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴} ∈ (Clsd‘(topGen‘ran (,))))
28	1, 2, 3, 27	orrvccel 29855	1 ⊢ (𝜑 → (𝑋∘_RV/𝑐 ≤ 𝐴) ∈ dom 𝑃)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 {crab 2900 class class class wbr 4583 dom cdm 5038 ran crn 5039 ‘cfv 5804 (class class class)co 6549 ℝcr 9814 -∞cmnf 9951 ℝ^*cxr 9952 < clt 9953 ≤ cle 9954 (,)cioo 12046 (,]cioc 12047 topGenctg 15921 Clsdccld 20630 Probcprb 29796 rRndVarcrrv 29829 ∘_RV/𝑐corvc 29844

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-ac2 9168 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-acn 8651 df-ac 8822 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-q 11665 df-ioo 12050 df-ioc 12051 df-topgen 15927 df-top 20521 df-bases 20522 df-cld 20633 df-esum 29417 df-siga 29498 df-sigagen 29529 df-brsiga 29572 df-meas 29586 df-mbfm 29640 df-prob 29797 df-rrv 29830 df-orvc 29845

This theorem is referenced by: dstfrvunirn 29863 dstfrvinc 29865 dstfrvclim1 29866

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