Theorem 0rrv 34442

Description: The constant function equal to zero is a random variable. (Contributed by Thierry Arnoux, 16-Jan-2017.) (Revised by Thierry Arnoux, 30-Jan-2017.)

Hypothesis

Ref	Expression
0rrv.1	⊢ (𝜑 → 𝑃 ∈ Prob)

Assertion

Ref	Expression
0rrv	⊢ (𝜑 → (𝑥 ∈ ∪ dom 𝑃 ↦ 0) ∈ (rRndVar‘𝑃))

Distinct variable group:   𝑥,𝑃

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem 0rrv

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		0re 11176	. . . . 5 ⊢ 0 ∈ ℝ
2	1	rgenw 3048	. . . 4 ⊢ ∀𝑥 ∈ ∪ dom 𝑃0 ∈ ℝ
3		eqid 2729	. . . . 5 ⊢ (𝑥 ∈ ∪ dom 𝑃 ↦ 0) = (𝑥 ∈ ∪ dom 𝑃 ↦ 0)
4	3	fmpt 7082	. . . 4 ⊢ (∀𝑥 ∈ ∪ dom 𝑃0 ∈ ℝ ↔ (𝑥 ∈ ∪ dom 𝑃 ↦ 0):∪ dom 𝑃⟶ℝ)
5	2, 4	mpbi 230	. . 3 ⊢ (𝑥 ∈ ∪ dom 𝑃 ↦ 0):∪ dom 𝑃⟶ℝ
6	5	a1i 11	. 2 ⊢ (𝜑 → (𝑥 ∈ ∪ dom 𝑃 ↦ 0):∪ dom 𝑃⟶ℝ)
7		fconstmpt 5700	. . . . . . . . . 10 ⊢ (∪ dom 𝑃 × {0}) = (𝑥 ∈ ∪ dom 𝑃 ↦ 0)
8	7	cnveqi 5838	. . . . . . . . 9 ⊢ ◡(∪ dom 𝑃 × {0}) = ◡(𝑥 ∈ ∪ dom 𝑃 ↦ 0)
9		cnvxp 6130	. . . . . . . . 9 ⊢ ◡(∪ dom 𝑃 × {0}) = ({0} × ∪ dom 𝑃)
10	8, 9	eqtr3i 2754	. . . . . . . 8 ⊢ ◡(𝑥 ∈ ∪ dom 𝑃 ↦ 0) = ({0} × ∪ dom 𝑃)
11	10	imaeq1i 6028	. . . . . . 7 ⊢ (◡(𝑥 ∈ ∪ dom 𝑃 ↦ 0) “ 𝑦) = (({0} × ∪ dom 𝑃) “ 𝑦)
12		df-ima 5651	. . . . . . 7 ⊢ (({0} × ∪ dom 𝑃) “ 𝑦) = ran (({0} × ∪ dom 𝑃) ↾ 𝑦)
13		df-rn 5649	. . . . . . 7 ⊢ ran (({0} × ∪ dom 𝑃) ↾ 𝑦) = dom ◡(({0} × ∪ dom 𝑃) ↾ 𝑦)
14	11, 12, 13	3eqtri 2756	. . . . . 6 ⊢ (◡(𝑥 ∈ ∪ dom 𝑃 ↦ 0) “ 𝑦) = dom ◡(({0} × ∪ dom 𝑃) ↾ 𝑦)
15		df-res 5650	. . . . . . . . 9 ⊢ (({0} × ∪ dom 𝑃) ↾ 𝑦) = (({0} × ∪ dom 𝑃) ∩ (𝑦 × V))
16		inxp 5795	. . . . . . . . 9 ⊢ (({0} × ∪ dom 𝑃) ∩ (𝑦 × V)) = (({0} ∩ 𝑦) × (∪ dom 𝑃 ∩ V))
17		inv1 4361	. . . . . . . . . 10 ⊢ (∪ dom 𝑃 ∩ V) = ∪ dom 𝑃
18	17	xpeq2i 5665	. . . . . . . . 9 ⊢ (({0} ∩ 𝑦) × (∪ dom 𝑃 ∩ V)) = (({0} ∩ 𝑦) × ∪ dom 𝑃)
19	15, 16, 18	3eqtri 2756	. . . . . . . 8 ⊢ (({0} × ∪ dom 𝑃) ↾ 𝑦) = (({0} ∩ 𝑦) × ∪ dom 𝑃)
20	19	cnveqi 5838	. . . . . . 7 ⊢ ◡(({0} × ∪ dom 𝑃) ↾ 𝑦) = ◡(({0} ∩ 𝑦) × ∪ dom 𝑃)
21	20	dmeqi 5868	. . . . . 6 ⊢ dom ◡(({0} × ∪ dom 𝑃) ↾ 𝑦) = dom ◡(({0} ∩ 𝑦) × ∪ dom 𝑃)
22		cnvxp 6130	. . . . . . 7 ⊢ ◡(({0} ∩ 𝑦) × ∪ dom 𝑃) = (∪ dom 𝑃 × ({0} ∩ 𝑦))
23	22	dmeqi 5868	. . . . . 6 ⊢ dom ◡(({0} ∩ 𝑦) × ∪ dom 𝑃) = dom (∪ dom 𝑃 × ({0} ∩ 𝑦))
24	14, 21, 23	3eqtri 2756	. . . . 5 ⊢ (◡(𝑥 ∈ ∪ dom 𝑃 ↦ 0) “ 𝑦) = dom (∪ dom 𝑃 × ({0} ∩ 𝑦))
25		xpeq2 5659	. . . . . . . . . 10 ⊢ (({0} ∩ 𝑦) = ∅ → (∪ dom 𝑃 × ({0} ∩ 𝑦)) = (∪ dom 𝑃 × ∅))
26		xp0 6131	. . . . . . . . . 10 ⊢ (∪ dom 𝑃 × ∅) = ∅
27	25, 26	eqtrdi 2780	. . . . . . . . 9 ⊢ (({0} ∩ 𝑦) = ∅ → (∪ dom 𝑃 × ({0} ∩ 𝑦)) = ∅)
28	27	dmeqd 5869	. . . . . . . 8 ⊢ (({0} ∩ 𝑦) = ∅ → dom (∪ dom 𝑃 × ({0} ∩ 𝑦)) = dom ∅)
29		dm0 5884	. . . . . . . 8 ⊢ dom ∅ = ∅
30	28, 29	eqtrdi 2780	. . . . . . 7 ⊢ (({0} ∩ 𝑦) = ∅ → dom (∪ dom 𝑃 × ({0} ∩ 𝑦)) = ∅)
31	30	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ ({0} ∩ 𝑦) = ∅) → dom (∪ dom 𝑃 × ({0} ∩ 𝑦)) = ∅)
32		0rrv.1	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ Prob)
33		domprobsiga 34402	. . . . . . . 8 ⊢ (𝑃 ∈ Prob → dom 𝑃 ∈ ∪ ran sigAlgebra)
34		0elsiga 34104	. . . . . . . 8 ⊢ (dom 𝑃 ∈ ∪ ran sigAlgebra → ∅ ∈ dom 𝑃)
35	32, 33, 34	3syl 18	. . . . . . 7 ⊢ (𝜑 → ∅ ∈ dom 𝑃)
36	35	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ({0} ∩ 𝑦) = ∅) → ∅ ∈ dom 𝑃)
37	31, 36	eqeltrd 2828	. . . . 5 ⊢ ((𝜑 ∧ ({0} ∩ 𝑦) = ∅) → dom (∪ dom 𝑃 × ({0} ∩ 𝑦)) ∈ dom 𝑃)
38	24, 37	eqeltrid 2832	. . . 4 ⊢ ((𝜑 ∧ ({0} ∩ 𝑦) = ∅) → (◡(𝑥 ∈ ∪ dom 𝑃 ↦ 0) “ 𝑦) ∈ dom 𝑃)
39		dmxp 5892	. . . . . . 7 ⊢ (({0} ∩ 𝑦) ≠ ∅ → dom (∪ dom 𝑃 × ({0} ∩ 𝑦)) = ∪ dom 𝑃)
40	39	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ ({0} ∩ 𝑦) ≠ ∅) → dom (∪ dom 𝑃 × ({0} ∩ 𝑦)) = ∪ dom 𝑃)
41	32	unveldomd 34406	. . . . . . 7 ⊢ (𝜑 → ∪ dom 𝑃 ∈ dom 𝑃)
42	41	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ({0} ∩ 𝑦) ≠ ∅) → ∪ dom 𝑃 ∈ dom 𝑃)
43	40, 42	eqeltrd 2828	. . . . 5 ⊢ ((𝜑 ∧ ({0} ∩ 𝑦) ≠ ∅) → dom (∪ dom 𝑃 × ({0} ∩ 𝑦)) ∈ dom 𝑃)
44	24, 43	eqeltrid 2832	. . . 4 ⊢ ((𝜑 ∧ ({0} ∩ 𝑦) ≠ ∅) → (◡(𝑥 ∈ ∪ dom 𝑃 ↦ 0) “ 𝑦) ∈ dom 𝑃)
45	38, 44	pm2.61dane 3012	. . 3 ⊢ (𝜑 → (◡(𝑥 ∈ ∪ dom 𝑃 ↦ 0) “ 𝑦) ∈ dom 𝑃)
46	45	ralrimivw 3129	. 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝔅ℝ (◡(𝑥 ∈ ∪ dom 𝑃 ↦ 0) “ 𝑦) ∈ dom 𝑃)
47	32	isrrvv 34434	. 2 ⊢ (𝜑 → ((𝑥 ∈ ∪ dom 𝑃 ↦ 0) ∈ (rRndVar‘𝑃) ↔ ((𝑥 ∈ ∪ dom 𝑃 ↦ 0):∪ dom 𝑃⟶ℝ ∧ ∀𝑦 ∈ 𝔅ℝ (◡(𝑥 ∈ ∪ dom 𝑃 ↦ 0) “ 𝑦) ∈ dom 𝑃)))
48	6, 46, 47	mpbir2and 713	1 ⊢ (𝜑 → (𝑥 ∈ ∪ dom 𝑃 ↦ 0) ∈ (rRndVar‘𝑃))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  Vcvv 3447   ∩ cin 3913  ∅c0 4296  {csn 4589  ∪ cuni 4871   ↦ cmpt 5188   × cxp 5636  ^◡ccnv 5637  dom cdm 5638  ran crn 5639   ↾ cres 5640   “ cima 5641  ⟶wf 6507  ‘cfv 6511  ℝcr 11067  0cc0 11068  sigAlgebracsiga 34098  𝔅_ℝcbrsiga 34171  Probcprb 34398  rRndVarcrrv 34431

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-addrcl 11129  ax-rnegex 11139  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-po 5546  df-so 5547  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-1st 7968  df-2nd 7969  df-er 8671  df-map 8801  df-en 8919  df-dom 8920  df-sdom 8921  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-ioo 13310  df-topgen 17406  df-top 22781  df-bases 22833  df-esum 34018  df-siga 34099  df-sigagen 34129  df-brsiga 34172  df-meas 34186  df-mbfm 34240  df-prob 34399  df-rrv 34432

This theorem is referenced by: (None)