orvcgteel - Mathbox for Thierry Arnoux

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Theorem orvcgteel 34725

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 488	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℝ)
5	3	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 𝐴 ∈ ℝ)
6		brcnvg 5847	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝑥^◡ ≤ 𝐴 ↔ 𝐴 ≤ 𝑥))
7	4, 5, 6	syl2anc 593	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑥^◡ ≤ 𝐴 ↔ 𝐴 ≤ 𝑥))
8	7	pm5.32da 587	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ ℝ ∧ 𝑥^◡ ≤ 𝐴) ↔ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥)))
9		rexr 11221	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ^*)
10	9	ad2antrl 738	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥)) → 𝑥 ∈ ℝ^*)
11		simprr 782	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥)) → 𝐴 ≤ 𝑥)
12		ltpnf 13115	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ → 𝑥 < +∞)
13	12	ad2antrl 738	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥)) → 𝑥 < +∞)
14	11, 13	jca 519	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥)) → (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞))
15	10, 14	jca 519	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥)) → (𝑥 ∈ ℝ^* ∧ (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞)))
16		simprl 780	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞))) → 𝑥 ∈ ℝ^*)
17	3	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞))) → 𝐴 ∈ ℝ)
18		simprrl 790	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞))) → 𝐴 ≤ 𝑥)
19		simprrr 791	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞))) → 𝑥 < +∞)
20		xrre3 13167	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℝ^* ∧ 𝐴 ∈ ℝ) ∧ (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞)) → 𝑥 ∈ ℝ)
21	16, 17, 18, 19, 20	syl22anc 849	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞))) → 𝑥 ∈ ℝ)
22	21, 18	jca 519	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ^* ∧ (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞))) → (𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥))
23	15, 22	impbida 810	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ ℝ ∧ 𝐴 ≤ 𝑥) ↔ (𝑥 ∈ ℝ^* ∧ (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞))))
24	8, 23	bitrd 281	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ ℝ ∧ 𝑥^◡ ≤ 𝐴) ↔ (𝑥 ∈ ℝ^* ∧ (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞))))
25	24	rabbidva2 3415	. . . 4 ⊢ (𝜑 → {𝑥 ∈ ℝ ∣ 𝑥^◡ ≤ 𝐴} = {𝑥 ∈ ℝ^* ∣ (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞)})
26	3	rexrd 11225	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
27		pnfxr 11229	. . . . 5 ⊢ +∞ ∈ ℝ^*
28		icoval 13380	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ +∞ ∈ ℝ^) → (𝐴[,)+∞) = {𝑥 ∈ ℝ^ ∣ (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞)})
29	26, 27, 28	sylancl 595	. . . 4 ⊢ (𝜑 → (𝐴[,)+∞) = {𝑥 ∈ ℝ^* ∣ (𝐴 ≤ 𝑥 ∧ 𝑥 < +∞)})
30	25, 29	eqtr4d 2799	. . 3 ⊢ (𝜑 → {𝑥 ∈ ℝ ∣ 𝑥^◡ ≤ 𝐴} = (𝐴[,)+∞))
31		icopnfcld 24814	. . . 4 ⊢ (𝐴 ∈ ℝ → (𝐴[,)+∞) ∈ (Clsd‘(topGen‘ran (,))))
32	3, 31	syl 17	. . 3 ⊢ (𝜑 → (𝐴[,)+∞) ∈ (Clsd‘(topGen‘ran (,))))
33	30, 32	eqeltrd 2861	. 2 ⊢ (𝜑 → {𝑥 ∈ ℝ ∣ 𝑥^◡ ≤ 𝐴} ∈ (Clsd‘(topGen‘ran (,))))
34	1, 2, 3, 33	orrvccel 34724	1 ⊢ (𝜑 → (𝑋∘_RV/𝑐^◡ ≤ 𝐴) ∈ dom 𝑃)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 399 = wceq 1559 ∈ wcel 2141 {crab 3413 class class class wbr 5097 ^◡ccnv 5642 dom cdm 5643 ran crn 5644 ‘cfv 6515 (class class class)co 7390 ℝcr 11065 +∞cpnf 11206 ℝ^*cxr 11208 < clt 11209 ≤ cle 11210 (,)cioo 13342 [,)cico 13344 topGenctg 17456 Clsdccld 23063 Probcprb 34664 rRndVarcrrv 34697 ∘_RV/𝑐corvc 34713

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5224 ax-sep 5243 ax-nul 5253 ax-pow 5319 ax-pr 5387 ax-un 7712 ax-inf2 9589 ax-ac2 10413 ax-cnex 11122 ax-resscn 11123 ax-1cn 11124 ax-icn 11125 ax-addcl 11126 ax-addrcl 11127 ax-mulcl 11128 ax-mulrcl 11129 ax-mulcom 11130 ax-addass 11131 ax-mulass 11132 ax-distr 11133 ax-i2m1 11134 ax-1ne0 11135 ax-1rid 11136 ax-rnegex 11137 ax-rrecex 11138 ax-cnre 11139 ax-pre-lttri 11140 ax-pre-lttrn 11141 ax-pre-ltadd 11142 ax-pre-mulgt0 11143 ax-pre-sup 11144

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4863 df-int 4903 df-iun 4948 df-iin 4949 df-br 5098 df-opab 5160 df-mpt 5179 df-tr 5205 df-id 5538 df-eprel 5543 df-po 5551 df-so 5552 df-fr 5596 df-se 5597 df-we 5598 df-xp 5649 df-rel 5650 df-cnv 5651 df-co 5652 df-dm 5653 df-rn 5654 df-res 5655 df-ima 5656 df-pred 6282 df-ord 6343 df-on 6344 df-lim 6345 df-suc 6346 df-iota 6471 df-fun 6517 df-fn 6518 df-f 6519 df-f1 6520 df-fo 6521 df-f1o 6522 df-fv 6523 df-isom 6524 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7841 df-1st 7964 df-2nd 7965 df-frecs 8255 df-wrecs 8286 df-recs 8335 df-rdg 8374 df-1o 8430 df-2o 8431 df-er 8671 df-map 8803 df-en 8921 df-dom 8922 df-sdom 8923 df-fin 8924 df-sup 9381 df-inf 9382 df-oi 9451 df-dju 9852 df-card 9890 df-acn 9893 df-ac 10065 df-pnf 11211 df-mnf 11212 df-xr 11213 df-ltxr 11214 df-le 11215 df-sub 11409 df-neg 11410 df-div 11838 df-nn 12204 df-n0 12475 df-z 12562 df-uz 12833 df-q 12943 df-ioo 13346 df-ico 13348 df-topgen 17462 df-top 22941 df-bases 22993 df-cld 23066 df-esum 34285 df-siga 34366 df-sigagen 34396 df-brsiga 34439 df-meas 34453 df-mbfm 34507 df-prob 34665 df-rrv 34698 df-orvc 34714

This theorem is referenced by: (None)

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