Theorem 0rrv 34635

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 11137	. . . . 5 ⊢ 0 ∈ ℝ
2	1	rgenw 3057	. . . 4 ⊢ ∀𝑥 ∈ ∪ dom 𝑃0 ∈ ℝ
3		eqid 2739	. . . . 5 ⊢ (𝑥 ∈ ∪ dom 𝑃 ↦ 0) = (𝑥 ∈ ∪ dom 𝑃 ↦ 0)
4	3	fmpt 7051	. . . 4 ⊢ (∀𝑥 ∈ ∪ dom 𝑃0 ∈ ℝ ↔ (𝑥 ∈ ∪ dom 𝑃 ↦ 0):∪ dom 𝑃⟶ℝ)
5	2, 4	mpbi 231	. . 3 ⊢ (𝑥 ∈ ∪ dom 𝑃 ↦ 0):∪ dom 𝑃⟶ℝ
6	5	a1i 11	. 2 ⊢ (𝜑 → (𝑥 ∈ ∪ dom 𝑃 ↦ 0):∪ dom 𝑃⟶ℝ)
7		fconstmpt 5680	. . . . . . . . . 10 ⊢ (∪ dom 𝑃 × {0}) = (𝑥 ∈ ∪ dom 𝑃 ↦ 0)
8	7	cnveqi 5816	. . . . . . . . 9 ⊢ ◡(∪ dom 𝑃 × {0}) = ◡(𝑥 ∈ ∪ dom 𝑃 ↦ 0)
9		cnvxp 6108	. . . . . . . . 9 ⊢ ◡(∪ dom 𝑃 × {0}) = ({0} × ∪ dom 𝑃)
10	8, 9	eqtr3i 2764	. . . . . . . 8 ⊢ ◡(𝑥 ∈ ∪ dom 𝑃 ↦ 0) = ({0} × ∪ dom 𝑃)
11	10	imaeq1i 6009	. . . . . . 7 ⊢ (◡(𝑥 ∈ ∪ dom 𝑃 ↦ 0) “ 𝑦) = (({0} × ∪ dom 𝑃) “ 𝑦)
12		df-ima 5631	. . . . . . 7 ⊢ (({0} × ∪ dom 𝑃) “ 𝑦) = ran (({0} × ∪ dom 𝑃) ↾ 𝑦)
13		df-rn 5629	. . . . . . 7 ⊢ ran (({0} × ∪ dom 𝑃) ↾ 𝑦) = dom ◡(({0} × ∪ dom 𝑃) ↾ 𝑦)
14	11, 12, 13	3eqtri 2766	. . . . . 6 ⊢ (◡(𝑥 ∈ ∪ dom 𝑃 ↦ 0) “ 𝑦) = dom ◡(({0} × ∪ dom 𝑃) ↾ 𝑦)
15		df-res 5630	. . . . . . . . 9 ⊢ (({0} × ∪ dom 𝑃) ↾ 𝑦) = (({0} × ∪ dom 𝑃) ∩ (𝑦 × V))
16		inxp 5774	. . . . . . . . 9 ⊢ (({0} × ∪ dom 𝑃) ∩ (𝑦 × V)) = (({0} ∩ 𝑦) × (∪ dom 𝑃 ∩ V))
17		inv1 4326	. . . . . . . . . 10 ⊢ (∪ dom 𝑃 ∩ V) = ∪ dom 𝑃
18	17	xpeq2i 5645	. . . . . . . . 9 ⊢ (({0} ∩ 𝑦) × (∪ dom 𝑃 ∩ V)) = (({0} ∩ 𝑦) × ∪ dom 𝑃)
19	15, 16, 18	3eqtri 2766	. . . . . . . 8 ⊢ (({0} × ∪ dom 𝑃) ↾ 𝑦) = (({0} ∩ 𝑦) × ∪ dom 𝑃)
20	19	cnveqi 5816	. . . . . . 7 ⊢ ◡(({0} × ∪ dom 𝑃) ↾ 𝑦) = ◡(({0} ∩ 𝑦) × ∪ dom 𝑃)
21	20	dmeqi 5846	. . . . . 6 ⊢ dom ◡(({0} × ∪ dom 𝑃) ↾ 𝑦) = dom ◡(({0} ∩ 𝑦) × ∪ dom 𝑃)
22		cnvxp 6108	. . . . . . 7 ⊢ ◡(({0} ∩ 𝑦) × ∪ dom 𝑃) = (∪ dom 𝑃 × ({0} ∩ 𝑦))
23	22	dmeqi 5846	. . . . . 6 ⊢ dom ◡(({0} ∩ 𝑦) × ∪ dom 𝑃) = dom (∪ dom 𝑃 × ({0} ∩ 𝑦))
24	14, 21, 23	3eqtri 2766	. . . . 5 ⊢ (◡(𝑥 ∈ ∪ dom 𝑃 ↦ 0) “ 𝑦) = dom (∪ dom 𝑃 × ({0} ∩ 𝑦))
25		xpeq2 5639	. . . . . . . . . 10 ⊢ (({0} ∩ 𝑦) = ∅ → (∪ dom 𝑃 × ({0} ∩ 𝑦)) = (∪ dom 𝑃 × ∅))
26		xp0 5718	. . . . . . . . . 10 ⊢ (∪ dom 𝑃 × ∅) = ∅
27	25, 26	eqtrdi 2790	. . . . . . . . 9 ⊢ (({0} ∩ 𝑦) = ∅ → (∪ dom 𝑃 × ({0} ∩ 𝑦)) = ∅)
28	27	dmeqd 5847	. . . . . . . 8 ⊢ (({0} ∩ 𝑦) = ∅ → dom (∪ dom 𝑃 × ({0} ∩ 𝑦)) = dom ∅)
29		dm0 5862	. . . . . . . 8 ⊢ dom ∅ = ∅
30	28, 29	eqtrdi 2790	. . . . . . 7 ⊢ (({0} ∩ 𝑦) = ∅ → dom (∪ dom 𝑃 × ({0} ∩ 𝑦)) = ∅)
31	30	adantl 482	. . . . . 6 ⊢ ((𝜑 ∧ ({0} ∩ 𝑦) = ∅) → dom (∪ dom 𝑃 × ({0} ∩ 𝑦)) = ∅)
32		0rrv.1	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ Prob)
33		domprobsiga 34595	. . . . . . . 8 ⊢ (𝑃 ∈ Prob → dom 𝑃 ∈ ∪ ran sigAlgebra)
34		0elsiga 34298	. . . . . . . 8 ⊢ (dom 𝑃 ∈ ∪ ran sigAlgebra → ∅ ∈ dom 𝑃)
35	32, 33, 34	3syl 18	. . . . . . 7 ⊢ (𝜑 → ∅ ∈ dom 𝑃)
36	35	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ ({0} ∩ 𝑦) = ∅) → ∅ ∈ dom 𝑃)
37	31, 36	eqeltrd 2839	. . . . 5 ⊢ ((𝜑 ∧ ({0} ∩ 𝑦) = ∅) → dom (∪ dom 𝑃 × ({0} ∩ 𝑦)) ∈ dom 𝑃)
38	24, 37	eqeltrid 2843	. . . 4 ⊢ ((𝜑 ∧ ({0} ∩ 𝑦) = ∅) → (◡(𝑥 ∈ ∪ dom 𝑃 ↦ 0) “ 𝑦) ∈ dom 𝑃)
39		dmxp 5871	. . . . . . 7 ⊢ (({0} ∩ 𝑦) ≠ ∅ → dom (∪ dom 𝑃 × ({0} ∩ 𝑦)) = ∪ dom 𝑃)
40	39	adantl 482	. . . . . 6 ⊢ ((𝜑 ∧ ({0} ∩ 𝑦) ≠ ∅) → dom (∪ dom 𝑃 × ({0} ∩ 𝑦)) = ∪ dom 𝑃)
41	32	unveldomd 34599	. . . . . . 7 ⊢ (𝜑 → ∪ dom 𝑃 ∈ dom 𝑃)
42	41	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ ({0} ∩ 𝑦) ≠ ∅) → ∪ dom 𝑃 ∈ dom 𝑃)
43	40, 42	eqeltrd 2839	. . . . 5 ⊢ ((𝜑 ∧ ({0} ∩ 𝑦) ≠ ∅) → dom (∪ dom 𝑃 × ({0} ∩ 𝑦)) ∈ dom 𝑃)
44	24, 43	eqeltrid 2843	. . . 4 ⊢ ((𝜑 ∧ ({0} ∩ 𝑦) ≠ ∅) → (◡(𝑥 ∈ ∪ dom 𝑃 ↦ 0) “ 𝑦) ∈ dom 𝑃)
45	38, 44	pm2.61dane 3021	. . 3 ⊢ (𝜑 → (◡(𝑥 ∈ ∪ dom 𝑃 ↦ 0) “ 𝑦) ∈ dom 𝑃)
46	45	ralrimivw 3135	. 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝔅ℝ (◡(𝑥 ∈ ∪ dom 𝑃 ↦ 0) “ 𝑦) ∈ dom 𝑃)
47	32	isrrvv 34627	. 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 719	1 ⊢ (𝜑 → (𝑥 ∈ ∪ dom 𝑃 ↦ 0) ∈ (rRndVar‘𝑃))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119   ≠ wne 2934  ∀wral 3053  Vcvv 3431   ∩ cin 3882  ∅c0 4261  {csn 4555  ∪ cuni 4838   ↦ cmpt 5153   × cxp 5616  ^◡ccnv 5617  dom cdm 5618  ran crn 5619   ↾ cres 5620   “ cima 5621  ⟶wf 6481  ‘cfv 6485  ℝcr 11028  0cc0 11029  sigAlgebracsiga 34292  𝔅_ℝcbrsiga 34365  Probcprb 34591  rRndVarcrrv 34624

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-addrcl 11090  ax-rnegex 11100  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-int 4878  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-po 5526  df-so 5527  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-er 8633  df-map 8765  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-ioo 13293  df-topgen 17397  df-top 22877  df-bases 22929  df-esum 34212  df-siga 34293  df-sigagen 34323  df-brsiga 34366  df-meas 34380  df-mbfm 34434  df-prob 34592  df-rrv 34625

This theorem is referenced by: (None)