orvclteel - Mathbox for Thierry Arnoux

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Theorem orvclteel 35151

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 11358	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ^*)
5	4	ad2antrl 741	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝑥 ≤ 𝐴)) → 𝑥 ∈ ℝ^*)
6		mnflt 13255	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ → -∞ < 𝑥)
7	6	ad2antrl 741	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝑥 ≤ 𝐴)) → -∞ < 𝑥)
8		simprr 785	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝑥 ≤ 𝐴)) → 𝑥 ≤ 𝐴)
9	7, 8	jca 521	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝑥 ≤ 𝐴)) → (-∞ < 𝑥 ∧ 𝑥 ≤ 𝐴))
10	5, 9	jca 521	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝑥 ≤ 𝐴)) → (𝑥 ∈ ℝ^* ∧ (-∞ < 𝑥 ∧ 𝑥 ≤ 𝐴)))
11		simprl 783	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ (-∞ < 𝑥 ∧ 𝑥 ≤ 𝐴))) → 𝑥 ∈ ℝ^*)
12	3	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ (-∞ < 𝑥 ∧ 𝑥 ≤ 𝐴))) → 𝐴 ∈ ℝ)
13		simprrl 793	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ (-∞ < 𝑥 ∧ 𝑥 ≤ 𝐴))) → -∞ < 𝑥)
14		simprrr 794	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ (-∞ < 𝑥 ∧ 𝑥 ≤ 𝐴))) → 𝑥 ≤ 𝐴)
15		xrre 13302	. . . . . . . 8 ⊢ (((𝑥 ∈ ℝ^* ∧ 𝐴 ∈ ℝ) ∧ (-∞ < 𝑥 ∧ 𝑥 ≤ 𝐴)) → 𝑥 ∈ ℝ)
16	11, 12, 13, 14, 15	syl22anc 852	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ (-∞ < 𝑥 ∧ 𝑥 ≤ 𝐴))) → 𝑥 ∈ ℝ)
17	16, 14	jca 521	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ (-∞ < 𝑥 ∧ 𝑥 ≤ 𝐴))) → (𝑥 ∈ ℝ ∧ 𝑥 ≤ 𝐴))
18	10, 17	impbida 813	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ ℝ ∧ 𝑥 ≤ 𝐴) ↔ (𝑥 ∈ ℝ^* ∧ (-∞ < 𝑥 ∧ 𝑥 ≤ 𝐴))))
19	18	rabbidva2 3414	. . . 4 ⊢ (𝜑 → {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴} = {𝑥 ∈ ℝ^* ∣ (-∞ < 𝑥 ∧ 𝑥 ≤ 𝐴)})
20		mnfxr 11369	. . . . 5 ⊢ -∞ ∈ ℝ^*
21	3	rexrd 11362	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
22		iocval 13516	. . . . 5 ⊢ ((-∞ ∈ ℝ^* ∧ 𝐴 ∈ ℝ^) → (-∞(,]𝐴) = {𝑥 ∈ ℝ^ ∣ (-∞ < 𝑥 ∧ 𝑥 ≤ 𝐴)})
23	20, 21, 22	sylancr 599	. . . 4 ⊢ (𝜑 → (-∞(,]𝐴) = {𝑥 ∈ ℝ^* ∣ (-∞ < 𝑥 ∧ 𝑥 ≤ 𝐴)})
24	19, 23	eqtr4d 2798	. . 3 ⊢ (𝜑 → {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴} = (-∞(,]𝐴))
25		iocmnfcld 25092	. . . 4 ⊢ (𝐴 ∈ ℝ → (-∞(,]𝐴) ∈ (Clsd‘(topGen‘ran (,))))
26	3, 25	syl 18	. . 3 ⊢ (𝜑 → (-∞(,]𝐴) ∈ (Clsd‘(topGen‘ran (,))))
27	24, 26	eqeltrd 2860	. 2 ⊢ (𝜑 → {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴} ∈ (Clsd‘(topGen‘ran (,))))
28	1, 2, 3, 27	orrvccel 35145	1 ⊢ (𝜑 → (𝑋∘_RV/𝑐 ≤ 𝐴) ∈ dom 𝑃)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1569 ∈ wcel 2144 {crab 3412 class class class wbr 5102 dom cdm 5650 ran crn 5651 ‘cfv 6538 (class class class)co 7420 ℝcr 11202 -∞cmnf 11344 ℝ^*cxr 11345 < clt 11346 ≤ cle 11347 (,)cioo 13479 (,]cioc 13480 topGenctg 17613 Clsdccld 23339 Probcprb 35085 rRndVarcrrv 35118 ∘_RV/𝑐corvc 35134

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7751 ax-inf2 9645 ax-cnex 11259 ax-resscn 11260 ax-1cn 11261 ax-icn 11262 ax-addcl 11263 ax-addrcl 11264 ax-mulcl 11265 ax-mulrcl 11266 ax-mulcom 11267 ax-addass 11268 ax-mulass 11269 ax-distr 11270 ax-i2m1 11271 ax-1ne0 11272 ax-1rid 11273 ax-rnegex 11274 ax-rrecex 11275 ax-cnre 11276 ax-pre-lttri 11277 ax-pre-lttrn 11278 ax-pre-ltadd 11279 ax-pre-mulgt0 11280 ax-pre-sup 11281

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-op 4835 df-uni 4867 df-int 4907 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-se 5604 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-isom 6547 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7878 df-1st 8001 df-2nd 8002 df-frecs 8302 df-wrecs 8333 df-recs 8382 df-rdg 8421 df-1o 8479 df-2o 8480 df-er 8720 df-map 8852 df-en 8977 df-dom 8978 df-sdom 8979 df-fin 8980 df-sup 9437 df-inf 9438 df-oi 9507 df-dju 9985 df-card 10023 df-pnf 11348 df-mnf 11349 df-xr 11350 df-ltxr 11351 df-le 11352 df-sub 11546 df-neg 11547 df-div 11977 df-nn 12339 df-n0 12610 df-z 12697 df-uz 12969 df-q 13079 df-ioo 13483 df-ioc 13484 df-topgen 17619 df-top 23217 df-bases 23269 df-cld 23342 df-esum 34706 df-siga 34787 df-sigagen 34818 df-brsiga 34861 df-meas 34875 df-mbfm 34929 df-prob 35086 df-rrv 35119 df-orvc 35135

This theorem is used by: dstfrvunirn 35153 dstfrvinc 35155 dstfrvclim1 35156

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