Theorem 0rrv 34614

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 11140	. . . . 5 ⊢ 0 ∈ ℝ
2	1	rgenw 3056	. . . 4 ⊢ ∀𝑥 ∈ ∪ dom 𝑃0 ∈ ℝ
3		eqid 2737	. . . . 5 ⊢ (𝑥 ∈ ∪ dom 𝑃 ↦ 0) = (𝑥 ∈ ∪ dom 𝑃 ↦ 0)
4	3	fmpt 7057	. . . 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 5687	. . . . . . . . . 10 ⊢ (∪ dom 𝑃 × {0}) = (𝑥 ∈ ∪ dom 𝑃 ↦ 0)
8	7	cnveqi 5824	. . . . . . . . 9 ⊢ ◡(∪ dom 𝑃 × {0}) = ◡(𝑥 ∈ ∪ dom 𝑃 ↦ 0)
9		cnvxp 6116	. . . . . . . . 9 ⊢ ◡(∪ dom 𝑃 × {0}) = ({0} × ∪ dom 𝑃)
10	8, 9	eqtr3i 2762	. . . . . . . 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 5638	. . . . . . 7 ⊢ (({0} × ∪ dom 𝑃) “ 𝑦) = ran (({0} × ∪ dom 𝑃) ↾ 𝑦)
13		df-rn 5636	. . . . . . 7 ⊢ ran (({0} × ∪ dom 𝑃) ↾ 𝑦) = dom ◡(({0} × ∪ dom 𝑃) ↾ 𝑦)
14	11, 12, 13	3eqtri 2764	. . . . . 6 ⊢ (◡(𝑥 ∈ ∪ dom 𝑃 ↦ 0) “ 𝑦) = dom ◡(({0} × ∪ dom 𝑃) ↾ 𝑦)
15		df-res 5637	. . . . . . . . 9 ⊢ (({0} × ∪ dom 𝑃) ↾ 𝑦) = (({0} × ∪ dom 𝑃) ∩ (𝑦 × V))
16		inxp 5781	. . . . . . . . 9 ⊢ (({0} × ∪ dom 𝑃) ∩ (𝑦 × V)) = (({0} ∩ 𝑦) × (∪ dom 𝑃 ∩ V))
17		inv1 4339	. . . . . . . . . 10 ⊢ (∪ dom 𝑃 ∩ V) = ∪ dom 𝑃
18	17	xpeq2i 5652	. . . . . . . . 9 ⊢ (({0} ∩ 𝑦) × (∪ dom 𝑃 ∩ V)) = (({0} ∩ 𝑦) × ∪ dom 𝑃)
19	15, 16, 18	3eqtri 2764	. . . . . . . 8 ⊢ (({0} × ∪ dom 𝑃) ↾ 𝑦) = (({0} ∩ 𝑦) × ∪ dom 𝑃)
20	19	cnveqi 5824	. . . . . . 7 ⊢ ◡(({0} × ∪ dom 𝑃) ↾ 𝑦) = ◡(({0} ∩ 𝑦) × ∪ dom 𝑃)
21	20	dmeqi 5854	. . . . . 6 ⊢ dom ◡(({0} × ∪ dom 𝑃) ↾ 𝑦) = dom ◡(({0} ∩ 𝑦) × ∪ dom 𝑃)
22		cnvxp 6116	. . . . . . 7 ⊢ ◡(({0} ∩ 𝑦) × ∪ dom 𝑃) = (∪ dom 𝑃 × ({0} ∩ 𝑦))
23	22	dmeqi 5854	. . . . . 6 ⊢ dom ◡(({0} ∩ 𝑦) × ∪ dom 𝑃) = dom (∪ dom 𝑃 × ({0} ∩ 𝑦))
24	14, 21, 23	3eqtri 2764	. . . . 5 ⊢ (◡(𝑥 ∈ ∪ dom 𝑃 ↦ 0) “ 𝑦) = dom (∪ dom 𝑃 × ({0} ∩ 𝑦))
25		xpeq2 5646	. . . . . . . . . 10 ⊢ (({0} ∩ 𝑦) = ∅ → (∪ dom 𝑃 × ({0} ∩ 𝑦)) = (∪ dom 𝑃 × ∅))
26		xp0 5725	. . . . . . . . . 10 ⊢ (∪ dom 𝑃 × ∅) = ∅
27	25, 26	eqtrdi 2788	. . . . . . . . 9 ⊢ (({0} ∩ 𝑦) = ∅ → (∪ dom 𝑃 × ({0} ∩ 𝑦)) = ∅)
28	27	dmeqd 5855	. . . . . . . 8 ⊢ (({0} ∩ 𝑦) = ∅ → dom (∪ dom 𝑃 × ({0} ∩ 𝑦)) = dom ∅)
29		dm0 5870	. . . . . . . 8 ⊢ dom ∅ = ∅
30	28, 29	eqtrdi 2788	. . . . . . 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 34574	. . . . . . . 8 ⊢ (𝑃 ∈ Prob → dom 𝑃 ∈ ∪ ran sigAlgebra)
34		0elsiga 34277	. . . . . . . 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 2837	. . . . 5 ⊢ ((𝜑 ∧ ({0} ∩ 𝑦) = ∅) → dom (∪ dom 𝑃 × ({0} ∩ 𝑦)) ∈ dom 𝑃)
38	24, 37	eqeltrid 2841	. . . 4 ⊢ ((𝜑 ∧ ({0} ∩ 𝑦) = ∅) → (◡(𝑥 ∈ ∪ dom 𝑃 ↦ 0) “ 𝑦) ∈ dom 𝑃)
39		dmxp 5879	. . . . . . 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 34578	. . . . . . 7 ⊢ (𝜑 → ∪ dom 𝑃 ∈ dom 𝑃)
42	41	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ({0} ∩ 𝑦) ≠ ∅) → ∪ dom 𝑃 ∈ dom 𝑃)
43	40, 42	eqeltrd 2837	. . . . 5 ⊢ ((𝜑 ∧ ({0} ∩ 𝑦) ≠ ∅) → dom (∪ dom 𝑃 × ({0} ∩ 𝑦)) ∈ dom 𝑃)
44	24, 43	eqeltrid 2841	. . . 4 ⊢ ((𝜑 ∧ ({0} ∩ 𝑦) ≠ ∅) → (◡(𝑥 ∈ ∪ dom 𝑃 ↦ 0) “ 𝑦) ∈ dom 𝑃)
45	38, 44	pm2.61dane 3020	. . 3 ⊢ (𝜑 → (◡(𝑥 ∈ ∪ dom 𝑃 ↦ 0) “ 𝑦) ∈ dom 𝑃)
46	45	ralrimivw 3134	. 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝔅ℝ (◡(𝑥 ∈ ∪ dom 𝑃 ↦ 0) “ 𝑦) ∈ dom 𝑃)
47	32	isrrvv 34606	. 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 714	1 ⊢ (𝜑 → (𝑥 ∈ ∪ dom 𝑃 ↦ 0) ∈ (rRndVar‘𝑃))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  Vcvv 3430   ∩ cin 3889  ∅c0 4274  {csn 4568  ∪ cuni 4851   ↦ cmpt 5167   × cxp 5623  ^◡ccnv 5624  dom cdm 5625  ran crn 5626   ↾ cres 5627   “ cima 5628  ⟶wf 6489  ‘cfv 6493  ℝcr 11031  0cc0 11032  sigAlgebracsiga 34271  𝔅_ℝcbrsiga 34344  Probcprb 34570  rRndVarcrrv 34603

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683  ax-cnex 11088  ax-resscn 11089  ax-1cn 11090  ax-addrcl 11093  ax-rnegex 11103  ax-cnre 11105  ax-pre-lttri 11106  ax-pre-lttrn 11107

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5520  df-po 5533  df-so 5534  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7364  df-oprab 7365  df-mpo 7366  df-1st 7936  df-2nd 7937  df-er 8637  df-map 8769  df-en 8888  df-dom 8889  df-sdom 8890  df-pnf 11175  df-mnf 11176  df-xr 11177  df-ltxr 11178  df-le 11179  df-ioo 13296  df-topgen 17400  df-top 22872  df-bases 22924  df-esum 34191  df-siga 34272  df-sigagen 34302  df-brsiga 34345  df-meas 34359  df-mbfm 34413  df-prob 34571  df-rrv 34604

This theorem is referenced by: (None)