orvcgteel - Mathbox for Thierry Arnoux

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Theorem orvcgteel 31403

Description: Preimage maps produced by the "greater than or equal to" relation are measurable sets. (Contributed by Thierry Arnoux, 5-Feb-2017.)

**Hypotheses**
Ref	Expression
orvcgteel.1	⊢ (𝜑 → 𝑃 ∈ Prob)
orvcgteel.2	⊢ (𝜑 → 𝑋 ∈ (rRndVar‘𝑃))
orvcgteel.3	⊢ (𝜑 → 𝐴 ∈ ℝ)

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

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

Step	Hyp	Ref	Expression
1		orvcgteel.1	. 2 ⊢ (𝜑 → 𝑃 ∈ Prob)
2		orvcgteel.2	. 2 ⊢ (𝜑 → 𝑋 ∈ (rRndVar‘𝑃))
3		orvcgteel.3	. 2 ⊢ (𝜑 → 𝐴 ∈ ℝ)
4		simpr 477	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℝ)
5	3	adantr 473	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 𝐴 ∈ ℝ)
6		brcnvg 5604	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝑥^◡ ≤ 𝐴 ↔ 𝐴 ≤ 𝑥))
7	4, 5, 6	syl2anc 576	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑥^◡ ≤ 𝐴 ↔ 𝐴 ≤ 𝑥))
8	7	pm5.32da 571	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ ℝ ∧ 𝑥^◡ ≤ 𝐴) ↔ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥)))
9		rexr 10492	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ^*)
10	9	ad2antrl 716	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥)) → 𝑥 ∈ ℝ^*)
11		simprr 761	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥)) → 𝐴 ≤ 𝑥)
12		ltpnf 12338	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ → 𝑥 < +∞)
13	12	ad2antrl 716	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥)) → 𝑥 < +∞)
14	11, 13	jca 504	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥)) → (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞))
15	10, 14	jca 504	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥)) → (𝑥 ∈ ℝ^* ∧ (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞)))
16		simprl 759	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞))) → 𝑥 ∈ ℝ^*)
17	3	adantr 473	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞))) → 𝐴 ∈ ℝ)
18		simprrl 769	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞))) → 𝐴 ≤ 𝑥)
19		simprrr 770	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞))) → 𝑥 < +∞)
20		xrre3 12387	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℝ^* ∧ 𝐴 ∈ ℝ) ∧ (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞)) → 𝑥 ∈ ℝ)
21	16, 17, 18, 19, 20	syl22anc 827	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞))) → 𝑥 ∈ ℝ)
22	21, 18	jca 504	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞))) → (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥))
23	15, 22	impbida 789	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥) ↔ (𝑥 ∈ ℝ^* ∧ (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞))))
24	8, 23	bitrd 271	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ ℝ ∧ 𝑥^◡ ≤ 𝐴) ↔ (𝑥 ∈ ℝ^* ∧ (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞))))
25	24	rabbidva2 3402	. . . 4 ⊢ (𝜑 → {𝑥 ∈ ℝ ∣ 𝑥^◡ ≤ 𝐴} = {𝑥 ∈ ℝ^* ∣ (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞)})
26	3	rexrd 10496	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
27		pnfxr 10500	. . . . 5 ⊢ +∞ ∈ ℝ^*
28		icoval 12598	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ +∞ ∈ ℝ^) → (𝐴[,)+∞) = {𝑥 ∈ ℝ^ ∣ (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞)})
29	26, 27, 28	sylancl 578	. . . 4 ⊢ (𝜑 → (𝐴[,)+∞) = {𝑥 ∈ ℝ^* ∣ (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞)})
30	25, 29	eqtr4d 2819	. . 3 ⊢ (𝜑 → {𝑥 ∈ ℝ ∣ 𝑥^◡ ≤ 𝐴} = (𝐴[,)+∞))
31		icopnfcld 23094	. . . 4 ⊢ (𝐴 ∈ ℝ → (𝐴[,)+∞) ∈ (Clsd‘(topGen‘ran (,))))
32	3, 31	syl 17	. . 3 ⊢ (𝜑 → (𝐴[,)+∞) ∈ (Clsd‘(topGen‘ran (,))))
33	30, 32	eqeltrd 2868	. 2 ⊢ (𝜑 → {𝑥 ∈ ℝ ∣ 𝑥^◡ ≤ 𝐴} ∈ (Clsd‘(topGen‘ran (,))))
34	1, 2, 3, 33	orrvccel 31402	1 ⊢ (𝜑 → (𝑋∘_RV/𝑐^◡ ≤ 𝐴) ∈ dom 𝑃)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 = wceq 1508 ∈ wcel 2051 {crab 3094 class class class wbr 4934 ^◡ccnv 5410 dom cdm 5411 ran crn 5412 ‘cfv 6193 (class class class)co 6982 ℝcr 10340 +∞cpnf 10477 ℝ^*cxr 10479 < clt 10480 ≤ cle 10481 (,)cioo 12560 [,)cico 12562 topGenctg 16573 Clsdccld 21343 Probcprb 31343 rRndVarcrrv 31376 ∘_RV/𝑐corvc 31391

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2752 ax-rep 5053 ax-sep 5064 ax-nul 5071 ax-pow 5123 ax-pr 5190 ax-un 7285 ax-inf2 8904 ax-ac2 9689 ax-cnex 10397 ax-resscn 10398 ax-1cn 10399 ax-icn 10400 ax-addcl 10401 ax-addrcl 10402 ax-mulcl 10403 ax-mulrcl 10404 ax-mulcom 10405 ax-addass 10406 ax-mulass 10407 ax-distr 10408 ax-i2m1 10409 ax-1ne0 10410 ax-1rid 10411 ax-rnegex 10412 ax-rrecex 10413 ax-cnre 10414 ax-pre-lttri 10415 ax-pre-lttrn 10416 ax-pre-ltadd 10417 ax-pre-mulgt0 10418 ax-pre-sup 10419

This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-fal 1521 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2551 df-eu 2589 df-clab 2761 df-cleq 2773 df-clel 2848 df-nfc 2920 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3419 df-sbc 3684 df-csb 3789 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-pss 3847 df-nul 4182 df-if 4354 df-pw 4427 df-sn 4445 df-pr 4447 df-tp 4449 df-op 4451 df-uni 4718 df-int 4755 df-iun 4799 df-iin 4800 df-br 4935 df-opab 4997 df-mpt 5014 df-tr 5036 df-id 5316 df-eprel 5321 df-po 5330 df-so 5331 df-fr 5370 df-se 5371 df-we 5372 df-xp 5417 df-rel 5418 df-cnv 5419 df-co 5420 df-dm 5421 df-rn 5422 df-res 5423 df-ima 5424 df-pred 5991 df-ord 6037 df-on 6038 df-lim 6039 df-suc 6040 df-iota 6157 df-fun 6195 df-fn 6196 df-f 6197 df-f1 6198 df-fo 6199 df-f1o 6200 df-fv 6201 df-isom 6202 df-riota 6943 df-ov 6985 df-oprab 6986 df-mpo 6987 df-om 7403 df-1st 7507 df-2nd 7508 df-wrecs 7756 df-recs 7818 df-rdg 7856 df-1o 7911 df-2o 7912 df-oadd 7915 df-er 8095 df-map 8214 df-en 8313 df-dom 8314 df-sdom 8315 df-fin 8316 df-sup 8707 df-inf 8708 df-oi 8775 df-dju 9130 df-card 9168 df-acn 9171 df-ac 9342 df-pnf 10482 df-mnf 10483 df-xr 10484 df-ltxr 10485 df-le 10486 df-sub 10678 df-neg 10679 df-div 11105 df-nn 11446 df-n0 11714 df-z 11800 df-uz 12065 df-q 12169 df-ioo 12564 df-ico 12566 df-topgen 16579 df-top 21221 df-bases 21273 df-cld 21346 df-esum 30963 df-siga 31044 df-sigagen 31075 df-brsiga 31118 df-meas 31132 df-mbfm 31186 df-prob 31344 df-rrv 31377 df-orvc 31392

This theorem is referenced by: (None)

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