orvcgteel - Mathbox for Thierry Arnoux

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Theorem orvcgteel 34432

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 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℝ)
5	3	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 𝐴 ∈ ℝ)
6		brcnvg 5904	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝑥^◡ ≤ 𝐴 ↔ 𝐴 ≤ 𝑥))
7	4, 5, 6	syl2anc 583	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑥^◡ ≤ 𝐴 ↔ 𝐴 ≤ 𝑥))
8	7	pm5.32da 578	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ ℝ ∧ 𝑥^◡ ≤ 𝐴) ↔ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥)))
9		rexr 11336	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ^*)
10	9	ad2antrl 727	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥)) → 𝑥 ∈ ℝ^*)
11		simprr 772	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥)) → 𝐴 ≤ 𝑥)
12		ltpnf 13183	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ → 𝑥 < +∞)
13	12	ad2antrl 727	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥)) → 𝑥 < +∞)
14	11, 13	jca 511	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥)) → (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞))
15	10, 14	jca 511	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥)) → (𝑥 ∈ ℝ^* ∧ (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞)))
16		simprl 770	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞))) → 𝑥 ∈ ℝ^*)
17	3	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞))) → 𝐴 ∈ ℝ)
18		simprrl 780	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞))) → 𝐴 ≤ 𝑥)
19		simprrr 781	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞))) → 𝑥 < +∞)
20		xrre3 13233	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℝ^* ∧ 𝐴 ∈ ℝ) ∧ (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞)) → 𝑥 ∈ ℝ)
21	16, 17, 18, 19, 20	syl22anc 838	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞))) → 𝑥 ∈ ℝ)
22	21, 18	jca 511	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞))) → (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥))
23	15, 22	impbida 800	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥) ↔ (𝑥 ∈ ℝ^* ∧ (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞))))
24	8, 23	bitrd 279	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ ℝ ∧ 𝑥^◡ ≤ 𝐴) ↔ (𝑥 ∈ ℝ^* ∧ (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞))))
25	24	rabbidva2 3445	. . . 4 ⊢ (𝜑 → {𝑥 ∈ ℝ ∣ 𝑥^◡ ≤ 𝐴} = {𝑥 ∈ ℝ^* ∣ (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞)})
26	3	rexrd 11340	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
27		pnfxr 11344	. . . . 5 ⊢ +∞ ∈ ℝ^*
28		icoval 13445	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ +∞ ∈ ℝ^) → (𝐴[,)+∞) = {𝑥 ∈ ℝ^ ∣ (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞)})
29	26, 27, 28	sylancl 585	. . . 4 ⊢ (𝜑 → (𝐴[,)+∞) = {𝑥 ∈ ℝ^* ∣ (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞)})
30	25, 29	eqtr4d 2783	. . 3 ⊢ (𝜑 → {𝑥 ∈ ℝ ∣ 𝑥^◡ ≤ 𝐴} = (𝐴[,)+∞))
31		icopnfcld 24809	. . . 4 ⊢ (𝐴 ∈ ℝ → (𝐴[,)+∞) ∈ (Clsd‘(topGen‘ran (,))))
32	3, 31	syl 17	. . 3 ⊢ (𝜑 → (𝐴[,)+∞) ∈ (Clsd‘(topGen‘ran (,))))
33	30, 32	eqeltrd 2844	. 2 ⊢ (𝜑 → {𝑥 ∈ ℝ ∣ 𝑥^◡ ≤ 𝐴} ∈ (Clsd‘(topGen‘ran (,))))
34	1, 2, 3, 33	orrvccel 34431	1 ⊢ (𝜑 → (𝑋∘_RV/𝑐^◡ ≤ 𝐴) ∈ dom 𝑃)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2108 {crab 3443 class class class wbr 5166 ^◡ccnv 5699 dom cdm 5700 ran crn 5701 ‘cfv 6573 (class class class)co 7448 ℝcr 11183 +∞cpnf 11321 ℝ^*cxr 11323 < clt 11324 ≤ cle 11325 (,)cioo 13407 [,)cico 13409 topGenctg 17497 Clsdccld 23045 Probcprb 34372 rRndVarcrrv 34405 ∘_RV/𝑐corvc 34420

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-inf2 9710 ax-ac2 10532 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-1st 8030 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-er 8763 df-map 8886 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-sup 9511 df-inf 9512 df-oi 9579 df-dju 9970 df-card 10008 df-acn 10011 df-ac 10185 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-n0 12554 df-z 12640 df-uz 12904 df-q 13014 df-ioo 13411 df-ico 13413 df-topgen 17503 df-top 22921 df-bases 22974 df-cld 23048 df-esum 33992 df-siga 34073 df-sigagen 34103 df-brsiga 34146 df-meas 34160 df-mbfm 34214 df-prob 34373 df-rrv 34406 df-orvc 34421

This theorem is referenced by: (None)

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