Theorem 0rrv 34415

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 11152	. . . . 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 7064	. . . 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 5693	. . . . . . . . . 10 ⊢ (∪ dom 𝑃 × {0}) = (𝑥 ∈ ∪ dom 𝑃 ↦ 0)
8	7	cnveqi 5828	. . . . . . . . 9 ⊢ ◡(∪ dom 𝑃 × {0}) = ◡(𝑥 ∈ ∪ dom 𝑃 ↦ 0)
9		cnvxp 6118	. . . . . . . . 9 ⊢ ◡(∪ dom 𝑃 × {0}) = ({0} × ∪ dom 𝑃)
10	8, 9	eqtr3i 2754	. . . . . . . 8 ⊢ ◡(𝑥 ∈ ∪ dom 𝑃 ↦ 0) = ({0} × ∪ dom 𝑃)
11	10	imaeq1i 6017	. . . . . . 7 ⊢ (◡(𝑥 ∈ ∪ dom 𝑃 ↦ 0) “ 𝑦) = (({0} × ∪ dom 𝑃) “ 𝑦)
12		df-ima 5644	. . . . . . 7 ⊢ (({0} × ∪ dom 𝑃) “ 𝑦) = ran (({0} × ∪ dom 𝑃) ↾ 𝑦)
13		df-rn 5642	. . . . . . 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 5643	. . . . . . . . 9 ⊢ (({0} × ∪ dom 𝑃) ↾ 𝑦) = (({0} × ∪ dom 𝑃) ∩ (𝑦 × V))
16		inxp 5785	. . . . . . . . 9 ⊢ (({0} × ∪ dom 𝑃) ∩ (𝑦 × V)) = (({0} ∩ 𝑦) × (∪ dom 𝑃 ∩ V))
17		inv1 4357	. . . . . . . . . 10 ⊢ (∪ dom 𝑃 ∩ V) = ∪ dom 𝑃
18	17	xpeq2i 5658	. . . . . . . . 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 5828	. . . . . . 7 ⊢ ◡(({0} × ∪ dom 𝑃) ↾ 𝑦) = ◡(({0} ∩ 𝑦) × ∪ dom 𝑃)
21	20	dmeqi 5858	. . . . . 6 ⊢ dom ◡(({0} × ∪ dom 𝑃) ↾ 𝑦) = dom ◡(({0} ∩ 𝑦) × ∪ dom 𝑃)
22		cnvxp 6118	. . . . . . 7 ⊢ ◡(({0} ∩ 𝑦) × ∪ dom 𝑃) = (∪ dom 𝑃 × ({0} ∩ 𝑦))
23	22	dmeqi 5858	. . . . . 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 5652	. . . . . . . . . 10 ⊢ (({0} ∩ 𝑦) = ∅ → (∪ dom 𝑃 × ({0} ∩ 𝑦)) = (∪ dom 𝑃 × ∅))
26		xp0 6119	. . . . . . . . . 10 ⊢ (∪ dom 𝑃 × ∅) = ∅
27	25, 26	eqtrdi 2780	. . . . . . . . 9 ⊢ (({0} ∩ 𝑦) = ∅ → (∪ dom 𝑃 × ({0} ∩ 𝑦)) = ∅)
28	27	dmeqd 5859	. . . . . . . 8 ⊢ (({0} ∩ 𝑦) = ∅ → dom (∪ dom 𝑃 × ({0} ∩ 𝑦)) = dom ∅)
29		dm0 5874	. . . . . . . 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 34375	. . . . . . . 8 ⊢ (𝑃 ∈ Prob → dom 𝑃 ∈ ∪ ran sigAlgebra)
34		0elsiga 34077	. . . . . . . 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 5882	. . . . . . 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 34379	. . . . . . 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 34407	. 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 3444   ∩ cin 3910  ∅c0 4292  {csn 4585  ∪ cuni 4867   ↦ cmpt 5183   × cxp 5629  ^◡ccnv 5630  dom cdm 5631  ran crn 5632   ↾ cres 5633   “ cima 5634  ⟶wf 6495  ‘cfv 6499  ℝcr 11043  0cc0 11044  sigAlgebracsiga 34071  𝔅_ℝcbrsiga 34144  Probcprb 34371  rRndVarcrrv 34404

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 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-cnex 11100  ax-resscn 11101  ax-1cn 11102  ax-addrcl 11105  ax-rnegex 11115  ax-cnre 11117  ax-pre-lttri 11118  ax-pre-lttrn 11119

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 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-int 4907  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-po 5539  df-so 5540  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-ov 7372  df-oprab 7373  df-mpo 7374  df-1st 7947  df-2nd 7948  df-er 8648  df-map 8778  df-en 8896  df-dom 8897  df-sdom 8898  df-pnf 11186  df-mnf 11187  df-xr 11188  df-ltxr 11189  df-le 11190  df-ioo 13286  df-topgen 17382  df-top 22757  df-bases 22809  df-esum 33991  df-siga 34072  df-sigagen 34102  df-brsiga 34145  df-meas 34159  df-mbfm 34213  df-prob 34372  df-rrv 34405

This theorem is referenced by: (None)