orvclteel - Mathbox for Thierry Arnoux

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Theorem orvclteel 35088

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 11336	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ^*)
5	4	ad2antrl 741	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝑥 ≤ 𝐴)) → 𝑥 ∈ ℝ^*)
6		mnflt 13233	. . . . . . . . 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 13280	. . . . . . . 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 3415	. . . 4 ⊢ (𝜑 → {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴} = {𝑥 ∈ ℝ^* ∣ (-∞ < 𝑥 ∧ 𝑥 ≤ 𝐴)})
20		mnfxr 11347	. . . . 5 ⊢ -∞ ∈ ℝ^*
21	3	rexrd 11340	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
22		iocval 13494	. . . . 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 2799	. . 3 ⊢ (𝜑 → {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴} = (-∞(,]𝐴))
25		iocmnfcld 25067	. . . 4 ⊢ (𝐴 ∈ ℝ → (-∞(,]𝐴) ∈ (Clsd‘(topGen‘ran (,))))
26	3, 25	syl 18	. . 3 ⊢ (𝜑 → (-∞(,]𝐴) ∈ (Clsd‘(topGen‘ran (,))))
27	24, 26	eqeltrd 2861	. 2 ⊢ (𝜑 → {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴} ∈ (Clsd‘(topGen‘ran (,))))
28	1, 2, 3, 27	orrvccel 35082	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 1570 ∈ wcel 2145 {crab 3413 class class class wbr 5103 dom cdm 5651 ran crn 5652 ‘cfv 6531 (class class class)co 7412 ℝcr 11180 -∞cmnf 11322 ℝ^*cxr 11323 < clt 11324 ≤ cle 11325 (,)cioo 13457 (,]cioc 13458 topGenctg 17588 Clsdccld 23314 Probcprb 35022 rRndVarcrrv 35055 ∘_RV/𝑐corvc 35071

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7740 ax-inf2 9626 ax-cnex 11237 ax-resscn 11238 ax-1cn 11239 ax-icn 11240 ax-addcl 11241 ax-addrcl 11242 ax-mulcl 11243 ax-mulrcl 11244 ax-mulcom 11245 ax-addass 11246 ax-mulass 11247 ax-distr 11248 ax-i2m1 11249 ax-1ne0 11250 ax-1rid 11251 ax-rnegex 11252 ax-rrecex 11253 ax-cnre 11254 ax-pre-lttri 11255 ax-pre-lttrn 11256 ax-pre-ltadd 11257 ax-pre-mulgt0 11258 ax-pre-sup 11259

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6297 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6487 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-isom 6540 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7867 df-1st 7990 df-2nd 7991 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-1o 8460 df-2o 8461 df-er 8701 df-map 8833 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-sup 9418 df-inf 9419 df-oi 9488 df-dju 9963 df-card 10001 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11524 df-neg 11525 df-div 11955 df-nn 12317 df-n0 12588 df-z 12675 df-uz 12947 df-q 13057 df-ioo 13461 df-ioc 13462 df-topgen 17594 df-top 23192 df-bases 23244 df-cld 23317 df-esum 34642 df-siga 34723 df-sigagen 34754 df-brsiga 34797 df-meas 34811 df-mbfm 34865 df-prob 35023 df-rrv 35056 df-orvc 35072

This theorem is used by: dstfrvunirn 35090 dstfrvinc 35092 dstfrvclim1 35093

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