Theorem 0rrv 32397

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 10961	. . . . 5 ⊢ 0 ∈ ℝ
2	1	rgenw 3077	. . . 4 ⊢ ∀𝑥 ∈ ∪ dom 𝑃0 ∈ ℝ
3		eqid 2739	. . . . 5 ⊢ (𝑥 ∈ ∪ dom 𝑃 ↦ 0) = (𝑥 ∈ ∪ dom 𝑃 ↦ 0)
4	3	fmpt 6978	. . . 4 ⊢ (∀𝑥 ∈ ∪ dom 𝑃0 ∈ ℝ ↔ (𝑥 ∈ ∪ dom 𝑃 ↦ 0):∪ dom 𝑃⟶ℝ)
5	2, 4	mpbi 229	. . 3 ⊢ (𝑥 ∈ ∪ dom 𝑃 ↦ 0):∪ dom 𝑃⟶ℝ
6	5	a1i 11	. 2 ⊢ (𝜑 → (𝑥 ∈ ∪ dom 𝑃 ↦ 0):∪ dom 𝑃⟶ℝ)
7		fconstmpt 5648	. . . . . . . . . 10 ⊢ (∪ dom 𝑃 × {0}) = (𝑥 ∈ ∪ dom 𝑃 ↦ 0)
8	7	cnveqi 5780	. . . . . . . . 9 ⊢ ◡(∪ dom 𝑃 × {0}) = ◡(𝑥 ∈ ∪ dom 𝑃 ↦ 0)
9		cnvxp 6057	. . . . . . . . 9 ⊢ ◡(∪ dom 𝑃 × {0}) = ({0} × ∪ dom 𝑃)
10	8, 9	eqtr3i 2769	. . . . . . . 8 ⊢ ◡(𝑥 ∈ ∪ dom 𝑃 ↦ 0) = ({0} × ∪ dom 𝑃)
11	10	imaeq1i 5963	. . . . . . 7 ⊢ (◡(𝑥 ∈ ∪ dom 𝑃 ↦ 0) “ 𝑦) = (({0} × ∪ dom 𝑃) “ 𝑦)
12		df-ima 5601	. . . . . . 7 ⊢ (({0} × ∪ dom 𝑃) “ 𝑦) = ran (({0} × ∪ dom 𝑃) ↾ 𝑦)
13		df-rn 5599	. . . . . . 7 ⊢ ran (({0} × ∪ dom 𝑃) ↾ 𝑦) = dom ◡(({0} × ∪ dom 𝑃) ↾ 𝑦)
14	11, 12, 13	3eqtri 2771	. . . . . 6 ⊢ (◡(𝑥 ∈ ∪ dom 𝑃 ↦ 0) “ 𝑦) = dom ◡(({0} × ∪ dom 𝑃) ↾ 𝑦)
15		df-res 5600	. . . . . . . . 9 ⊢ (({0} × ∪ dom 𝑃) ↾ 𝑦) = (({0} × ∪ dom 𝑃) ∩ (𝑦 × V))
16		inxp 5738	. . . . . . . . 9 ⊢ (({0} × ∪ dom 𝑃) ∩ (𝑦 × V)) = (({0} ∩ 𝑦) × (∪ dom 𝑃 ∩ V))
17		inv1 4333	. . . . . . . . . 10 ⊢ (∪ dom 𝑃 ∩ V) = ∪ dom 𝑃
18	17	xpeq2i 5615	. . . . . . . . 9 ⊢ (({0} ∩ 𝑦) × (∪ dom 𝑃 ∩ V)) = (({0} ∩ 𝑦) × ∪ dom 𝑃)
19	15, 16, 18	3eqtri 2771	. . . . . . . 8 ⊢ (({0} × ∪ dom 𝑃) ↾ 𝑦) = (({0} ∩ 𝑦) × ∪ dom 𝑃)
20	19	cnveqi 5780	. . . . . . 7 ⊢ ◡(({0} × ∪ dom 𝑃) ↾ 𝑦) = ◡(({0} ∩ 𝑦) × ∪ dom 𝑃)
21	20	dmeqi 5810	. . . . . 6 ⊢ dom ◡(({0} × ∪ dom 𝑃) ↾ 𝑦) = dom ◡(({0} ∩ 𝑦) × ∪ dom 𝑃)
22		cnvxp 6057	. . . . . . 7 ⊢ ◡(({0} ∩ 𝑦) × ∪ dom 𝑃) = (∪ dom 𝑃 × ({0} ∩ 𝑦))
23	22	dmeqi 5810	. . . . . 6 ⊢ dom ◡(({0} ∩ 𝑦) × ∪ dom 𝑃) = dom (∪ dom 𝑃 × ({0} ∩ 𝑦))
24	14, 21, 23	3eqtri 2771	. . . . 5 ⊢ (◡(𝑥 ∈ ∪ dom 𝑃 ↦ 0) “ 𝑦) = dom (∪ dom 𝑃 × ({0} ∩ 𝑦))
25		xpeq2 5609	. . . . . . . . . 10 ⊢ (({0} ∩ 𝑦) = ∅ → (∪ dom 𝑃 × ({0} ∩ 𝑦)) = (∪ dom 𝑃 × ∅))
26		xp0 6058	. . . . . . . . . 10 ⊢ (∪ dom 𝑃 × ∅) = ∅
27	25, 26	eqtrdi 2795	. . . . . . . . 9 ⊢ (({0} ∩ 𝑦) = ∅ → (∪ dom 𝑃 × ({0} ∩ 𝑦)) = ∅)
28	27	dmeqd 5811	. . . . . . . 8 ⊢ (({0} ∩ 𝑦) = ∅ → dom (∪ dom 𝑃 × ({0} ∩ 𝑦)) = dom ∅)
29		dm0 5826	. . . . . . . 8 ⊢ dom ∅ = ∅
30	28, 29	eqtrdi 2795	. . . . . . 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 32357	. . . . . . . 8 ⊢ (𝑃 ∈ Prob → dom 𝑃 ∈ ∪ ran sigAlgebra)
34		0elsiga 32061	. . . . . . . 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 2840	. . . . 5 ⊢ ((𝜑 ∧ ({0} ∩ 𝑦) = ∅) → dom (∪ dom 𝑃 × ({0} ∩ 𝑦)) ∈ dom 𝑃)
38	24, 37	eqeltrid 2844	. . . 4 ⊢ ((𝜑 ∧ ({0} ∩ 𝑦) = ∅) → (◡(𝑥 ∈ ∪ dom 𝑃 ↦ 0) “ 𝑦) ∈ dom 𝑃)
39		dmxp 5835	. . . . . . 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 32361	. . . . . . 7 ⊢ (𝜑 → ∪ dom 𝑃 ∈ dom 𝑃)
42	41	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ({0} ∩ 𝑦) ≠ ∅) → ∪ dom 𝑃 ∈ dom 𝑃)
43	40, 42	eqeltrd 2840	. . . . 5 ⊢ ((𝜑 ∧ ({0} ∩ 𝑦) ≠ ∅) → dom (∪ dom 𝑃 × ({0} ∩ 𝑦)) ∈ dom 𝑃)
44	24, 43	eqeltrid 2844	. . . 4 ⊢ ((𝜑 ∧ ({0} ∩ 𝑦) ≠ ∅) → (◡(𝑥 ∈ ∪ dom 𝑃 ↦ 0) “ 𝑦) ∈ dom 𝑃)
45	38, 44	pm2.61dane 3033	. . 3 ⊢ (𝜑 → (◡(𝑥 ∈ ∪ dom 𝑃 ↦ 0) “ 𝑦) ∈ dom 𝑃)
46	45	ralrimivw 3110	. 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝔅ℝ (◡(𝑥 ∈ ∪ dom 𝑃 ↦ 0) “ 𝑦) ∈ dom 𝑃)
47	32	isrrvv 32389	. 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 709	1 ⊢ (𝜑 → (𝑥 ∈ ∪ dom 𝑃 ↦ 0) ∈ (rRndVar‘𝑃))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2109   ≠ wne 2944  ∀wral 3065  Vcvv 3430   ∩ cin 3890  ∅c0 4261  {csn 4566  ∪ cuni 4844   ↦ cmpt 5161   × cxp 5586  ^◡ccnv 5587  dom cdm 5588  ran crn 5589   ↾ cres 5590   “ cima 5591  ⟶wf 6426  ‘cfv 6430  ℝcr 10854  0cc0 10855  sigAlgebracsiga 32055  𝔅_ℝcbrsiga 32128  Probcprb 32353  rRndVarcrrv 32386

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-sep 5226  ax-nul 5233  ax-pow 5291  ax-pr 5355  ax-un 7579  ax-cnex 10911  ax-resscn 10912  ax-1cn 10913  ax-addrcl 10916  ax-rnegex 10926  ax-cnre 10928  ax-pre-lttri 10929  ax-pre-lttrn 10930

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3070  df-rex 3071  df-rab 3074  df-v 3432  df-sbc 3720  df-csb 3837  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4845  df-int 4885  df-iun 4931  df-br 5079  df-opab 5141  df-mpt 5162  df-id 5488  df-po 5502  df-so 5503  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-res 5600  df-ima 5601  df-iota 6388  df-fun 6432  df-fn 6433  df-f 6434  df-f1 6435  df-fo 6436  df-f1o 6437  df-fv 6438  df-ov 7271  df-oprab 7272  df-mpo 7273  df-1st 7817  df-2nd 7818  df-er 8472  df-map 8591  df-en 8708  df-dom 8709  df-sdom 8710  df-pnf 10995  df-mnf 10996  df-xr 10997  df-ltxr 10998  df-le 10999  df-ioo 13065  df-topgen 17135  df-top 22024  df-bases 22077  df-esum 31975  df-siga 32056  df-sigagen 32086  df-brsiga 32129  df-meas 32143  df-mbfm 32197  df-prob 32354  df-rrv 32387

This theorem is referenced by: (None)