orvclteel - Mathbox for Thierry Arnoux

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Theorem orvclteel 31367

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 10482	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ^*)
5	4	ad2antrl 715	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝑥 ≤ 𝐴)) → 𝑥 ∈ ℝ^*)
6		mnflt 12332	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ → -∞ < 𝑥)
7	6	ad2antrl 715	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝑥 ≤ 𝐴)) → -∞ < 𝑥)
8		simprr 760	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝑥 ≤ 𝐴)) → 𝑥 ≤ 𝐴)
9	7, 8	jca 504	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝑥 ≤ 𝐴)) → (-∞ < 𝑥 ∧ 𝑥 ≤ 𝐴))
10	5, 9	jca 504	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝑥 ≤ 𝐴)) → (𝑥 ∈ ℝ^* ∧ (-∞ < 𝑥 ∧ 𝑥 ≤ 𝐴)))
11		simprl 758	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ (-∞ < 𝑥 ∧ 𝑥 ≤ 𝐴))) → 𝑥 ∈ ℝ^*)
12	3	adantr 473	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ (-∞ < 𝑥 ∧ 𝑥 ≤ 𝐴))) → 𝐴 ∈ ℝ)
13		simprrl 768	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ (-∞ < 𝑥 ∧ 𝑥 ≤ 𝐴))) → -∞ < 𝑥)
14		simprrr 769	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ (-∞ < 𝑥 ∧ 𝑥 ≤ 𝐴))) → 𝑥 ≤ 𝐴)
15		xrre 12376	. . . . . . . 8 ⊢ (((𝑥 ∈ ℝ^* ∧ 𝐴 ∈ ℝ) ∧ (-∞ < 𝑥 ∧ 𝑥 ≤ 𝐴)) → 𝑥 ∈ ℝ)
16	11, 12, 13, 14, 15	syl22anc 826	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ (-∞ < 𝑥 ∧ 𝑥 ≤ 𝐴))) → 𝑥 ∈ ℝ)
17	16, 14	jca 504	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ (-∞ < 𝑥 ∧ 𝑥 ≤ 𝐴))) → (𝑥 ∈ ℝ ∧ 𝑥 ≤ 𝐴))
18	10, 17	impbida 788	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ ℝ ∧ 𝑥 ≤ 𝐴) ↔ (𝑥 ∈ ℝ^* ∧ (-∞ < 𝑥 ∧ 𝑥 ≤ 𝐴))))
19	18	rabbidva2 3397	. . . 4 ⊢ (𝜑 → {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴} = {𝑥 ∈ ℝ^* ∣ (-∞ < 𝑥 ∧ 𝑥 ≤ 𝐴)})
20		mnfxr 10494	. . . . 5 ⊢ -∞ ∈ ℝ^*
21	3	rexrd 10486	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
22		iocval 12588	. . . . 5 ⊢ ((-∞ ∈ ℝ^* ∧ 𝐴 ∈ ℝ^) → (-∞(,]𝐴) = {𝑥 ∈ ℝ^ ∣ (-∞ < 𝑥 ∧ 𝑥 ≤ 𝐴)})
23	20, 21, 22	sylancr 578	. . . 4 ⊢ (𝜑 → (-∞(,]𝐴) = {𝑥 ∈ ℝ^* ∣ (-∞ < 𝑥 ∧ 𝑥 ≤ 𝐴)})
24	19, 23	eqtr4d 2814	. . 3 ⊢ (𝜑 → {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴} = (-∞(,]𝐴))
25		iocmnfcld 23074	. . . 4 ⊢ (𝐴 ∈ ℝ → (-∞(,]𝐴) ∈ (Clsd‘(topGen‘ran (,))))
26	3, 25	syl 17	. . 3 ⊢ (𝜑 → (-∞(,]𝐴) ∈ (Clsd‘(topGen‘ran (,))))
27	24, 26	eqeltrd 2863	. 2 ⊢ (𝜑 → {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴} ∈ (Clsd‘(topGen‘ran (,))))
28	1, 2, 3, 27	orrvccel 31361	1 ⊢ (𝜑 → (𝑋∘_RV/𝑐 ≤ 𝐴) ∈ dom 𝑃)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 387 = wceq 1507 ∈ wcel 2048 {crab 3089 class class class wbr 4927 dom cdm 5404 ran crn 5405 ‘cfv 6186 (class class class)co 6974 ℝcr 10330 -∞cmnf 10468 ℝ^*cxr 10469 < clt 10470 ≤ cle 10471 (,)cioo 12551 (,]cioc 12552 topGenctg 16561 Clsdccld 21322 Probcprb 31302 rRndVarcrrv 31335 ∘_RV/𝑐corvc 31350

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2747 ax-rep 5047 ax-sep 5058 ax-nul 5065 ax-pow 5117 ax-pr 5184 ax-un 7277 ax-inf2 8894 ax-ac2 9679 ax-cnex 10387 ax-resscn 10388 ax-1cn 10389 ax-icn 10390 ax-addcl 10391 ax-addrcl 10392 ax-mulcl 10393 ax-mulrcl 10394 ax-mulcom 10395 ax-addass 10396 ax-mulass 10397 ax-distr 10398 ax-i2m1 10399 ax-1ne0 10400 ax-1rid 10401 ax-rnegex 10402 ax-rrecex 10403 ax-cnre 10404 ax-pre-lttri 10405 ax-pre-lttrn 10406 ax-pre-ltadd 10407 ax-pre-mulgt0 10408 ax-pre-sup 10409

This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-fal 1520 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2756 df-cleq 2768 df-clel 2843 df-nfc 2915 df-ne 2965 df-nel 3071 df-ral 3090 df-rex 3091 df-reu 3092 df-rmo 3093 df-rab 3094 df-v 3414 df-sbc 3681 df-csb 3786 df-dif 3831 df-un 3833 df-in 3835 df-ss 3842 df-pss 3844 df-nul 4178 df-if 4349 df-pw 4422 df-sn 4440 df-pr 4442 df-tp 4444 df-op 4446 df-uni 4711 df-int 4748 df-iun 4792 df-iin 4793 df-br 4928 df-opab 4990 df-mpt 5007 df-tr 5029 df-id 5309 df-eprel 5314 df-po 5323 df-so 5324 df-fr 5363 df-se 5364 df-we 5365 df-xp 5410 df-rel 5411 df-cnv 5412 df-co 5413 df-dm 5414 df-rn 5415 df-res 5416 df-ima 5417 df-pred 5984 df-ord 6030 df-on 6031 df-lim 6032 df-suc 6033 df-iota 6150 df-fun 6188 df-fn 6189 df-f 6190 df-f1 6191 df-fo 6192 df-f1o 6193 df-fv 6194 df-isom 6195 df-riota 6935 df-ov 6977 df-oprab 6978 df-mpo 6979 df-om 7395 df-1st 7498 df-2nd 7499 df-wrecs 7747 df-recs 7809 df-rdg 7847 df-1o 7901 df-2o 7902 df-oadd 7905 df-er 8085 df-map 8204 df-en 8303 df-dom 8304 df-sdom 8305 df-fin 8306 df-sup 8697 df-inf 8698 df-oi 8765 df-dju 9120 df-card 9158 df-acn 9161 df-ac 9332 df-pnf 10472 df-mnf 10473 df-xr 10474 df-ltxr 10475 df-le 10476 df-sub 10668 df-neg 10669 df-div 11095 df-nn 11436 df-n0 11705 df-z 11791 df-uz 12056 df-q 12160 df-ioo 12555 df-ioc 12556 df-topgen 16567 df-top 21200 df-bases 21252 df-cld 21325 df-esum 30922 df-siga 31003 df-sigagen 31034 df-brsiga 31077 df-meas 31091 df-mbfm 31145 df-prob 31303 df-rrv 31336 df-orvc 31351

This theorem is referenced by: dstfrvunirn 31369 dstfrvinc 31371 dstfrvclim1 31372

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