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Theorem 0rrv 34432
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 11260 . . . . 5 0 ∈ ℝ
21rgenw 3062 . . . 4 𝑥 dom 𝑃0 ∈ ℝ
3 eqid 2734 . . . . 5 (𝑥 dom 𝑃 ↦ 0) = (𝑥 dom 𝑃 ↦ 0)
43fmpt 7129 . . . 4 (∀𝑥 dom 𝑃0 ∈ ℝ ↔ (𝑥 dom 𝑃 ↦ 0): dom 𝑃⟶ℝ)
52, 4mpbi 230 . . 3 (𝑥 dom 𝑃 ↦ 0): dom 𝑃⟶ℝ
65a1i 11 . 2 (𝜑 → (𝑥 dom 𝑃 ↦ 0): dom 𝑃⟶ℝ)
7 fconstmpt 5750 . . . . . . . . . 10 ( dom 𝑃 × {0}) = (𝑥 dom 𝑃 ↦ 0)
87cnveqi 5887 . . . . . . . . 9 ( dom 𝑃 × {0}) = (𝑥 dom 𝑃 ↦ 0)
9 cnvxp 6178 . . . . . . . . 9 ( dom 𝑃 × {0}) = ({0} × dom 𝑃)
108, 9eqtr3i 2764 . . . . . . . 8 (𝑥 dom 𝑃 ↦ 0) = ({0} × dom 𝑃)
1110imaeq1i 6076 . . . . . . 7 ((𝑥 dom 𝑃 ↦ 0) “ 𝑦) = (({0} × dom 𝑃) “ 𝑦)
12 df-ima 5701 . . . . . . 7 (({0} × dom 𝑃) “ 𝑦) = ran (({0} × dom 𝑃) ↾ 𝑦)
13 df-rn 5699 . . . . . . 7 ran (({0} × dom 𝑃) ↾ 𝑦) = dom (({0} × dom 𝑃) ↾ 𝑦)
1411, 12, 133eqtri 2766 . . . . . 6 ((𝑥 dom 𝑃 ↦ 0) “ 𝑦) = dom (({0} × dom 𝑃) ↾ 𝑦)
15 df-res 5700 . . . . . . . . 9 (({0} × dom 𝑃) ↾ 𝑦) = (({0} × dom 𝑃) ∩ (𝑦 × V))
16 inxp 5844 . . . . . . . . 9 (({0} × dom 𝑃) ∩ (𝑦 × V)) = (({0} ∩ 𝑦) × ( dom 𝑃 ∩ V))
17 inv1 4403 . . . . . . . . . 10 ( dom 𝑃 ∩ V) = dom 𝑃
1817xpeq2i 5715 . . . . . . . . 9 (({0} ∩ 𝑦) × ( dom 𝑃 ∩ V)) = (({0} ∩ 𝑦) × dom 𝑃)
1915, 16, 183eqtri 2766 . . . . . . . 8 (({0} × dom 𝑃) ↾ 𝑦) = (({0} ∩ 𝑦) × dom 𝑃)
2019cnveqi 5887 . . . . . . 7 (({0} × dom 𝑃) ↾ 𝑦) = (({0} ∩ 𝑦) × dom 𝑃)
2120dmeqi 5917 . . . . . 6 dom (({0} × dom 𝑃) ↾ 𝑦) = dom (({0} ∩ 𝑦) × dom 𝑃)
22 cnvxp 6178 . . . . . . 7 (({0} ∩ 𝑦) × dom 𝑃) = ( dom 𝑃 × ({0} ∩ 𝑦))
2322dmeqi 5917 . . . . . 6 dom (({0} ∩ 𝑦) × dom 𝑃) = dom ( dom 𝑃 × ({0} ∩ 𝑦))
2414, 21, 233eqtri 2766 . . . . 5 ((𝑥 dom 𝑃 ↦ 0) “ 𝑦) = dom ( dom 𝑃 × ({0} ∩ 𝑦))
25 xpeq2 5709 . . . . . . . . . 10 (({0} ∩ 𝑦) = ∅ → ( dom 𝑃 × ({0} ∩ 𝑦)) = ( dom 𝑃 × ∅))
26 xp0 6179 . . . . . . . . . 10 ( dom 𝑃 × ∅) = ∅
2725, 26eqtrdi 2790 . . . . . . . . 9 (({0} ∩ 𝑦) = ∅ → ( dom 𝑃 × ({0} ∩ 𝑦)) = ∅)
2827dmeqd 5918 . . . . . . . 8 (({0} ∩ 𝑦) = ∅ → dom ( dom 𝑃 × ({0} ∩ 𝑦)) = dom ∅)
29 dm0 5933 . . . . . . . 8 dom ∅ = ∅
3028, 29eqtrdi 2790 . . . . . . 7 (({0} ∩ 𝑦) = ∅ → dom ( dom 𝑃 × ({0} ∩ 𝑦)) = ∅)
3130adantl 481 . . . . . 6 ((𝜑 ∧ ({0} ∩ 𝑦) = ∅) → dom ( dom 𝑃 × ({0} ∩ 𝑦)) = ∅)
32 0rrv.1 . . . . . . . 8 (𝜑𝑃 ∈ Prob)
33 domprobsiga 34392 . . . . . . . 8 (𝑃 ∈ Prob → dom 𝑃 ran sigAlgebra)
34 0elsiga 34094 . . . . . . . 8 (dom 𝑃 ran sigAlgebra → ∅ ∈ dom 𝑃)
3532, 33, 343syl 18 . . . . . . 7 (𝜑 → ∅ ∈ dom 𝑃)
3635adantr 480 . . . . . 6 ((𝜑 ∧ ({0} ∩ 𝑦) = ∅) → ∅ ∈ dom 𝑃)
3731, 36eqeltrd 2838 . . . . 5 ((𝜑 ∧ ({0} ∩ 𝑦) = ∅) → dom ( dom 𝑃 × ({0} ∩ 𝑦)) ∈ dom 𝑃)
3824, 37eqeltrid 2842 . . . 4 ((𝜑 ∧ ({0} ∩ 𝑦) = ∅) → ((𝑥 dom 𝑃 ↦ 0) “ 𝑦) ∈ dom 𝑃)
39 dmxp 5941 . . . . . . 7 (({0} ∩ 𝑦) ≠ ∅ → dom ( dom 𝑃 × ({0} ∩ 𝑦)) = dom 𝑃)
4039adantl 481 . . . . . 6 ((𝜑 ∧ ({0} ∩ 𝑦) ≠ ∅) → dom ( dom 𝑃 × ({0} ∩ 𝑦)) = dom 𝑃)
4132unveldomd 34396 . . . . . . 7 (𝜑 dom 𝑃 ∈ dom 𝑃)
4241adantr 480 . . . . . 6 ((𝜑 ∧ ({0} ∩ 𝑦) ≠ ∅) → dom 𝑃 ∈ dom 𝑃)
4340, 42eqeltrd 2838 . . . . 5 ((𝜑 ∧ ({0} ∩ 𝑦) ≠ ∅) → dom ( dom 𝑃 × ({0} ∩ 𝑦)) ∈ dom 𝑃)
4424, 43eqeltrid 2842 . . . 4 ((𝜑 ∧ ({0} ∩ 𝑦) ≠ ∅) → ((𝑥 dom 𝑃 ↦ 0) “ 𝑦) ∈ dom 𝑃)
4538, 44pm2.61dane 3026 . . 3 (𝜑 → ((𝑥 dom 𝑃 ↦ 0) “ 𝑦) ∈ dom 𝑃)
4645ralrimivw 3147 . 2 (𝜑 → ∀𝑦 ∈ 𝔅 ((𝑥 dom 𝑃 ↦ 0) “ 𝑦) ∈ dom 𝑃)
4732isrrvv 34424 . 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 713 1 (𝜑 → (𝑥 dom 𝑃 ↦ 0) ∈ (rRndVar‘𝑃))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1536  wcel 2105  wne 2937  wral 3058  Vcvv 3477  cin 3961  c0 4338  {csn 4630   cuni 4911  cmpt 5230   × cxp 5686  ccnv 5687  dom cdm 5688  ran crn 5689  cres 5690  cima 5691  wf 6558  cfv 6562  cr 11151  0cc0 11152  sigAlgebracsiga 34088  𝔅cbrsiga 34161  Probcprb 34388  rRndVarcrrv 34421
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-addrcl 11213  ax-rnegex 11223  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-po 5596  df-so 5597  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-1st 8012  df-2nd 8013  df-er 8743  df-map 8866  df-en 8984  df-dom 8985  df-sdom 8986  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-ioo 13387  df-topgen 17489  df-top 22915  df-bases 22968  df-esum 34008  df-siga 34089  df-sigagen 34119  df-brsiga 34162  df-meas 34176  df-mbfm 34230  df-prob 34389  df-rrv 34422
This theorem is referenced by: (None)
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