Mathbox for Thierry Arnoux < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  0rrv Structured version   Visualization version   GIF version

Theorem 0rrv 31055
 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 (𝜑𝑃 ∈ 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.
StepHypRef Expression
1 0re 10365 . . . . 5 0 ∈ ℝ
21rgenw 3133 . . . 4 𝑥 dom 𝑃0 ∈ ℝ
3 eqid 2825 . . . . 5 (𝑥 dom 𝑃 ↦ 0) = (𝑥 dom 𝑃 ↦ 0)
43fmpt 6634 . . . 4 (∀𝑥 dom 𝑃0 ∈ ℝ ↔ (𝑥 dom 𝑃 ↦ 0): dom 𝑃⟶ℝ)
52, 4mpbi 222 . . 3 (𝑥 dom 𝑃 ↦ 0): dom 𝑃⟶ℝ
65a1i 11 . 2 (𝜑 → (𝑥 dom 𝑃 ↦ 0): dom 𝑃⟶ℝ)
7 fconstmpt 5402 . . . . . . . . . 10 ( dom 𝑃 × {0}) = (𝑥 dom 𝑃 ↦ 0)
87cnveqi 5533 . . . . . . . . 9 ( dom 𝑃 × {0}) = (𝑥 dom 𝑃 ↦ 0)
9 cnvxp 5796 . . . . . . . . 9 ( dom 𝑃 × {0}) = ({0} × dom 𝑃)
108, 9eqtr3i 2851 . . . . . . . 8 (𝑥 dom 𝑃 ↦ 0) = ({0} × dom 𝑃)
1110imaeq1i 5708 . . . . . . 7 ((𝑥 dom 𝑃 ↦ 0) “ 𝑦) = (({0} × dom 𝑃) “ 𝑦)
12 df-ima 5359 . . . . . . 7 (({0} × dom 𝑃) “ 𝑦) = ran (({0} × dom 𝑃) ↾ 𝑦)
13 df-rn 5357 . . . . . . 7 ran (({0} × dom 𝑃) ↾ 𝑦) = dom (({0} × dom 𝑃) ↾ 𝑦)
1411, 12, 133eqtri 2853 . . . . . 6 ((𝑥 dom 𝑃 ↦ 0) “ 𝑦) = dom (({0} × dom 𝑃) ↾ 𝑦)
15 df-res 5358 . . . . . . . . 9 (({0} × dom 𝑃) ↾ 𝑦) = (({0} × dom 𝑃) ∩ (𝑦 × V))
16 inxp 5491 . . . . . . . . 9 (({0} × dom 𝑃) ∩ (𝑦 × V)) = (({0} ∩ 𝑦) × ( dom 𝑃 ∩ V))
17 inv1 4197 . . . . . . . . . 10 ( dom 𝑃 ∩ V) = dom 𝑃
1817xpeq2i 5373 . . . . . . . . 9 (({0} ∩ 𝑦) × ( dom 𝑃 ∩ V)) = (({0} ∩ 𝑦) × dom 𝑃)
1915, 16, 183eqtri 2853 . . . . . . . 8 (({0} × dom 𝑃) ↾ 𝑦) = (({0} ∩ 𝑦) × dom 𝑃)
2019cnveqi 5533 . . . . . . 7 (({0} × dom 𝑃) ↾ 𝑦) = (({0} ∩ 𝑦) × dom 𝑃)
2120dmeqi 5561 . . . . . 6 dom (({0} × dom 𝑃) ↾ 𝑦) = dom (({0} ∩ 𝑦) × dom 𝑃)
22 cnvxp 5796 . . . . . . 7 (({0} ∩ 𝑦) × dom 𝑃) = ( dom 𝑃 × ({0} ∩ 𝑦))
2322dmeqi 5561 . . . . . 6 dom (({0} ∩ 𝑦) × dom 𝑃) = dom ( dom 𝑃 × ({0} ∩ 𝑦))
2414, 21, 233eqtri 2853 . . . . 5 ((𝑥 dom 𝑃 ↦ 0) “ 𝑦) = dom ( dom 𝑃 × ({0} ∩ 𝑦))
25 xpeq2 5367 . . . . . . . . . 10 (({0} ∩ 𝑦) = ∅ → ( dom 𝑃 × ({0} ∩ 𝑦)) = ( dom 𝑃 × ∅))
26 xp0 5797 . . . . . . . . . 10 ( dom 𝑃 × ∅) = ∅
2725, 26syl6eq 2877 . . . . . . . . 9 (({0} ∩ 𝑦) = ∅ → ( dom 𝑃 × ({0} ∩ 𝑦)) = ∅)
2827dmeqd 5562 . . . . . . . 8 (({0} ∩ 𝑦) = ∅ → dom ( dom 𝑃 × ({0} ∩ 𝑦)) = dom ∅)
29 dm0 5575 . . . . . . . 8 dom ∅ = ∅
3028, 29syl6eq 2877 . . . . . . 7 (({0} ∩ 𝑦) = ∅ → dom ( dom 𝑃 × ({0} ∩ 𝑦)) = ∅)
3130adantl 475 . . . . . 6 ((𝜑 ∧ ({0} ∩ 𝑦) = ∅) → dom ( dom 𝑃 × ({0} ∩ 𝑦)) = ∅)
32 0rrv.1 . . . . . . . 8 (𝜑𝑃 ∈ Prob)
33 domprobsiga 31015 . . . . . . . 8 (𝑃 ∈ Prob → dom 𝑃 ran sigAlgebra)
34 0elsiga 30718 . . . . . . . 8 (dom 𝑃 ran sigAlgebra → ∅ ∈ dom 𝑃)
3532, 33, 343syl 18 . . . . . . 7 (𝜑 → ∅ ∈ dom 𝑃)
3635adantr 474 . . . . . 6 ((𝜑 ∧ ({0} ∩ 𝑦) = ∅) → ∅ ∈ dom 𝑃)
3731, 36eqeltrd 2906 . . . . 5 ((𝜑 ∧ ({0} ∩ 𝑦) = ∅) → dom ( dom 𝑃 × ({0} ∩ 𝑦)) ∈ dom 𝑃)
3824, 37syl5eqel 2910 . . . 4 ((𝜑 ∧ ({0} ∩ 𝑦) = ∅) → ((𝑥 dom 𝑃 ↦ 0) “ 𝑦) ∈ dom 𝑃)
39 dmxp 5580 . . . . . . 7 (({0} ∩ 𝑦) ≠ ∅ → dom ( dom 𝑃 × ({0} ∩ 𝑦)) = dom 𝑃)
4039adantl 475 . . . . . 6 ((𝜑 ∧ ({0} ∩ 𝑦) ≠ ∅) → dom ( dom 𝑃 × ({0} ∩ 𝑦)) = dom 𝑃)
4132unveldomd 31019 . . . . . . 7 (𝜑 dom 𝑃 ∈ dom 𝑃)
4241adantr 474 . . . . . 6 ((𝜑 ∧ ({0} ∩ 𝑦) ≠ ∅) → dom 𝑃 ∈ dom 𝑃)
4340, 42eqeltrd 2906 . . . . 5 ((𝜑 ∧ ({0} ∩ 𝑦) ≠ ∅) → dom ( dom 𝑃 × ({0} ∩ 𝑦)) ∈ dom 𝑃)
4424, 43syl5eqel 2910 . . . 4 ((𝜑 ∧ ({0} ∩ 𝑦) ≠ ∅) → ((𝑥 dom 𝑃 ↦ 0) “ 𝑦) ∈ dom 𝑃)
4538, 44pm2.61dane 3086 . . 3 (𝜑 → ((𝑥 dom 𝑃 ↦ 0) “ 𝑦) ∈ dom 𝑃)
4645ralrimivw 3176 . 2 (𝜑 → ∀𝑦 ∈ 𝔅 ((𝑥 dom 𝑃 ↦ 0) “ 𝑦) ∈ dom 𝑃)
4732isrrvv 31047 . 2 (𝜑 → ((𝑥 dom 𝑃 ↦ 0) ∈ (rRndVar‘𝑃) ↔ ((𝑥 dom 𝑃 ↦ 0): dom 𝑃⟶ℝ ∧ ∀𝑦 ∈ 𝔅 ((𝑥 dom 𝑃 ↦ 0) “ 𝑦) ∈ dom 𝑃)))
486, 46, 47mpbir2and 704 1 (𝜑 → (𝑥 dom 𝑃 ↦ 0) ∈ (rRndVar‘𝑃))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 386   = wceq 1656   ∈ wcel 2164   ≠ wne 2999  ∀wral 3117  Vcvv 3414   ∩ cin 3797  ∅c0 4146  {csn 4399  ∪ cuni 4660   ↦ cmpt 4954   × cxp 5344  ◡ccnv 5345  dom cdm 5346  ran crn 5347   ↾ cres 5348   “ cima 5349  ⟶wf 6123  ‘cfv 6127  ℝcr 10258  0cc0 10259  sigAlgebracsiga 30711  𝔅ℝcbrsiga 30785  Probcprb 31011  rRndVarcrrv 31044 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214  ax-cnex 10315  ax-resscn 10316  ax-1cn 10317  ax-addrcl 10320  ax-rnegex 10330  ax-cnre 10332  ax-pre-lttri 10333  ax-pre-lttrn 10334 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-fal 1670  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-int 4700  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-po 5265  df-so 5266  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-1st 7433  df-2nd 7434  df-er 8014  df-map 8129  df-en 8229  df-dom 8230  df-sdom 8231  df-pnf 10400  df-mnf 10401  df-xr 10402  df-ltxr 10403  df-le 10404  df-ioo 12474  df-topgen 16464  df-top 21076  df-bases 21128  df-esum 30631  df-siga 30712  df-sigagen 30743  df-brsiga 30786  df-meas 30800  df-mbfm 30854  df-prob 31012  df-rrv 31045 This theorem is referenced by: (None)
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