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Theorem 0rrv 33916
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 11223 . . . . 5 0 ∈ ℝ
21rgenw 3064 . . . 4 𝑥 dom 𝑃0 ∈ ℝ
3 eqid 2731 . . . . 5 (𝑥 dom 𝑃 ↦ 0) = (𝑥 dom 𝑃 ↦ 0)
43fmpt 7111 . . . 4 (∀𝑥 dom 𝑃0 ∈ ℝ ↔ (𝑥 dom 𝑃 ↦ 0): dom 𝑃⟶ℝ)
52, 4mpbi 229 . . 3 (𝑥 dom 𝑃 ↦ 0): dom 𝑃⟶ℝ
65a1i 11 . 2 (𝜑 → (𝑥 dom 𝑃 ↦ 0): dom 𝑃⟶ℝ)
7 fconstmpt 5738 . . . . . . . . . 10 ( dom 𝑃 × {0}) = (𝑥 dom 𝑃 ↦ 0)
87cnveqi 5874 . . . . . . . . 9 ( dom 𝑃 × {0}) = (𝑥 dom 𝑃 ↦ 0)
9 cnvxp 6156 . . . . . . . . 9 ( dom 𝑃 × {0}) = ({0} × dom 𝑃)
108, 9eqtr3i 2761 . . . . . . . 8 (𝑥 dom 𝑃 ↦ 0) = ({0} × dom 𝑃)
1110imaeq1i 6056 . . . . . . 7 ((𝑥 dom 𝑃 ↦ 0) “ 𝑦) = (({0} × dom 𝑃) “ 𝑦)
12 df-ima 5689 . . . . . . 7 (({0} × dom 𝑃) “ 𝑦) = ran (({0} × dom 𝑃) ↾ 𝑦)
13 df-rn 5687 . . . . . . 7 ran (({0} × dom 𝑃) ↾ 𝑦) = dom (({0} × dom 𝑃) ↾ 𝑦)
1411, 12, 133eqtri 2763 . . . . . 6 ((𝑥 dom 𝑃 ↦ 0) “ 𝑦) = dom (({0} × dom 𝑃) ↾ 𝑦)
15 df-res 5688 . . . . . . . . 9 (({0} × dom 𝑃) ↾ 𝑦) = (({0} × dom 𝑃) ∩ (𝑦 × V))
16 inxp 5832 . . . . . . . . 9 (({0} × dom 𝑃) ∩ (𝑦 × V)) = (({0} ∩ 𝑦) × ( dom 𝑃 ∩ V))
17 inv1 4394 . . . . . . . . . 10 ( dom 𝑃 ∩ V) = dom 𝑃
1817xpeq2i 5703 . . . . . . . . 9 (({0} ∩ 𝑦) × ( dom 𝑃 ∩ V)) = (({0} ∩ 𝑦) × dom 𝑃)
1915, 16, 183eqtri 2763 . . . . . . . 8 (({0} × dom 𝑃) ↾ 𝑦) = (({0} ∩ 𝑦) × dom 𝑃)
2019cnveqi 5874 . . . . . . 7 (({0} × dom 𝑃) ↾ 𝑦) = (({0} ∩ 𝑦) × dom 𝑃)
2120dmeqi 5904 . . . . . 6 dom (({0} × dom 𝑃) ↾ 𝑦) = dom (({0} ∩ 𝑦) × dom 𝑃)
22 cnvxp 6156 . . . . . . 7 (({0} ∩ 𝑦) × dom 𝑃) = ( dom 𝑃 × ({0} ∩ 𝑦))
2322dmeqi 5904 . . . . . 6 dom (({0} ∩ 𝑦) × dom 𝑃) = dom ( dom 𝑃 × ({0} ∩ 𝑦))
2414, 21, 233eqtri 2763 . . . . 5 ((𝑥 dom 𝑃 ↦ 0) “ 𝑦) = dom ( dom 𝑃 × ({0} ∩ 𝑦))
25 xpeq2 5697 . . . . . . . . . 10 (({0} ∩ 𝑦) = ∅ → ( dom 𝑃 × ({0} ∩ 𝑦)) = ( dom 𝑃 × ∅))
26 xp0 6157 . . . . . . . . . 10 ( dom 𝑃 × ∅) = ∅
2725, 26eqtrdi 2787 . . . . . . . . 9 (({0} ∩ 𝑦) = ∅ → ( dom 𝑃 × ({0} ∩ 𝑦)) = ∅)
2827dmeqd 5905 . . . . . . . 8 (({0} ∩ 𝑦) = ∅ → dom ( dom 𝑃 × ({0} ∩ 𝑦)) = dom ∅)
29 dm0 5920 . . . . . . . 8 dom ∅ = ∅
3028, 29eqtrdi 2787 . . . . . . 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 33876 . . . . . . . 8 (𝑃 ∈ Prob → dom 𝑃 ran sigAlgebra)
34 0elsiga 33578 . . . . . . . 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 2832 . . . . 5 ((𝜑 ∧ ({0} ∩ 𝑦) = ∅) → dom ( dom 𝑃 × ({0} ∩ 𝑦)) ∈ dom 𝑃)
3824, 37eqeltrid 2836 . . . 4 ((𝜑 ∧ ({0} ∩ 𝑦) = ∅) → ((𝑥 dom 𝑃 ↦ 0) “ 𝑦) ∈ dom 𝑃)
39 dmxp 5928 . . . . . . 7 (({0} ∩ 𝑦) ≠ ∅ → dom ( dom 𝑃 × ({0} ∩ 𝑦)) = dom 𝑃)
4039adantl 481 . . . . . 6 ((𝜑 ∧ ({0} ∩ 𝑦) ≠ ∅) → dom ( dom 𝑃 × ({0} ∩ 𝑦)) = dom 𝑃)
4132unveldomd 33880 . . . . . . 7 (𝜑 dom 𝑃 ∈ dom 𝑃)
4241adantr 480 . . . . . 6 ((𝜑 ∧ ({0} ∩ 𝑦) ≠ ∅) → dom 𝑃 ∈ dom 𝑃)
4340, 42eqeltrd 2832 . . . . 5 ((𝜑 ∧ ({0} ∩ 𝑦) ≠ ∅) → dom ( dom 𝑃 × ({0} ∩ 𝑦)) ∈ dom 𝑃)
4424, 43eqeltrid 2836 . . . 4 ((𝜑 ∧ ({0} ∩ 𝑦) ≠ ∅) → ((𝑥 dom 𝑃 ↦ 0) “ 𝑦) ∈ dom 𝑃)
4538, 44pm2.61dane 3028 . . 3 (𝜑 → ((𝑥 dom 𝑃 ↦ 0) “ 𝑦) ∈ dom 𝑃)
4645ralrimivw 3149 . 2 (𝜑 → ∀𝑦 ∈ 𝔅 ((𝑥 dom 𝑃 ↦ 0) “ 𝑦) ∈ dom 𝑃)
4732isrrvv 33908 . 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 710 1 (𝜑 → (𝑥 dom 𝑃 ↦ 0) ∈ (rRndVar‘𝑃))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2105  wne 2939  wral 3060  Vcvv 3473  cin 3947  c0 4322  {csn 4628   cuni 4908  cmpt 5231   × cxp 5674  ccnv 5675  dom cdm 5676  ran crn 5677  cres 5678  cima 5679  wf 6539  cfv 6543  cr 11115  0cc0 11116  sigAlgebracsiga 33572  𝔅cbrsiga 33645  Probcprb 33872  rRndVarcrrv 33905
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-cnex 11172  ax-resscn 11173  ax-1cn 11174  ax-addrcl 11177  ax-rnegex 11187  ax-cnre 11189  ax-pre-lttri 11190  ax-pre-lttrn 11191
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-po 5588  df-so 5589  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7979  df-2nd 7980  df-er 8709  df-map 8828  df-en 8946  df-dom 8947  df-sdom 8948  df-pnf 11257  df-mnf 11258  df-xr 11259  df-ltxr 11260  df-le 11261  df-ioo 13335  df-topgen 17396  df-top 22717  df-bases 22770  df-esum 33492  df-siga 33573  df-sigagen 33603  df-brsiga 33646  df-meas 33660  df-mbfm 33714  df-prob 33873  df-rrv 33906
This theorem is referenced by: (None)
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