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Theorem 0rrv 31021
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 10328 . . . . 5 0 ∈ ℝ
21rgenw 3103 . . . 4 𝑥 dom 𝑃0 ∈ ℝ
3 eqid 2797 . . . . 5 (𝑥 dom 𝑃 ↦ 0) = (𝑥 dom 𝑃 ↦ 0)
43fmpt 6604 . . . 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 5366 . . . . . . . . . 10 ( dom 𝑃 × {0}) = (𝑥 dom 𝑃 ↦ 0)
87cnveqi 5498 . . . . . . . . 9 ( dom 𝑃 × {0}) = (𝑥 dom 𝑃 ↦ 0)
9 cnvxp 5766 . . . . . . . . 9 ( dom 𝑃 × {0}) = ({0} × dom 𝑃)
108, 9eqtr3i 2821 . . . . . . . 8 (𝑥 dom 𝑃 ↦ 0) = ({0} × dom 𝑃)
1110imaeq1i 5678 . . . . . . 7 ((𝑥 dom 𝑃 ↦ 0) “ 𝑦) = (({0} × dom 𝑃) “ 𝑦)
12 df-ima 5323 . . . . . . 7 (({0} × dom 𝑃) “ 𝑦) = ran (({0} × dom 𝑃) ↾ 𝑦)
13 df-rn 5321 . . . . . . 7 ran (({0} × dom 𝑃) ↾ 𝑦) = dom (({0} × dom 𝑃) ↾ 𝑦)
1411, 12, 133eqtri 2823 . . . . . 6 ((𝑥 dom 𝑃 ↦ 0) “ 𝑦) = dom (({0} × dom 𝑃) ↾ 𝑦)
15 df-res 5322 . . . . . . . . 9 (({0} × dom 𝑃) ↾ 𝑦) = (({0} × dom 𝑃) ∩ (𝑦 × V))
16 inxp 5456 . . . . . . . . 9 (({0} × dom 𝑃) ∩ (𝑦 × V)) = (({0} ∩ 𝑦) × ( dom 𝑃 ∩ V))
17 inv1 4164 . . . . . . . . . 10 ( dom 𝑃 ∩ V) = dom 𝑃
1817xpeq2i 5337 . . . . . . . . 9 (({0} ∩ 𝑦) × ( dom 𝑃 ∩ V)) = (({0} ∩ 𝑦) × dom 𝑃)
1915, 16, 183eqtri 2823 . . . . . . . 8 (({0} × dom 𝑃) ↾ 𝑦) = (({0} ∩ 𝑦) × dom 𝑃)
2019cnveqi 5498 . . . . . . 7 (({0} × dom 𝑃) ↾ 𝑦) = (({0} ∩ 𝑦) × dom 𝑃)
2120dmeqi 5526 . . . . . 6 dom (({0} × dom 𝑃) ↾ 𝑦) = dom (({0} ∩ 𝑦) × dom 𝑃)
22 cnvxp 5766 . . . . . . 7 (({0} ∩ 𝑦) × dom 𝑃) = ( dom 𝑃 × ({0} ∩ 𝑦))
2322dmeqi 5526 . . . . . 6 dom (({0} ∩ 𝑦) × dom 𝑃) = dom ( dom 𝑃 × ({0} ∩ 𝑦))
2414, 21, 233eqtri 2823 . . . . 5 ((𝑥 dom 𝑃 ↦ 0) “ 𝑦) = dom ( dom 𝑃 × ({0} ∩ 𝑦))
25 xpeq2 5331 . . . . . . . . . 10 (({0} ∩ 𝑦) = ∅ → ( dom 𝑃 × ({0} ∩ 𝑦)) = ( dom 𝑃 × ∅))
26 xp0 5767 . . . . . . . . . 10 ( dom 𝑃 × ∅) = ∅
2725, 26syl6eq 2847 . . . . . . . . 9 (({0} ∩ 𝑦) = ∅ → ( dom 𝑃 × ({0} ∩ 𝑦)) = ∅)
2827dmeqd 5527 . . . . . . . 8 (({0} ∩ 𝑦) = ∅ → dom ( dom 𝑃 × ({0} ∩ 𝑦)) = dom ∅)
29 dm0 5540 . . . . . . . 8 dom ∅ = ∅
3028, 29syl6eq 2847 . . . . . . 7 (({0} ∩ 𝑦) = ∅ → dom ( dom 𝑃 × ({0} ∩ 𝑦)) = ∅)
3130adantl 474 . . . . . 6 ((𝜑 ∧ ({0} ∩ 𝑦) = ∅) → dom ( dom 𝑃 × ({0} ∩ 𝑦)) = ∅)
32 0rrv.1 . . . . . . . 8 (𝜑𝑃 ∈ Prob)
33 domprobsiga 30981 . . . . . . . 8 (𝑃 ∈ Prob → dom 𝑃 ran sigAlgebra)
34 0elsiga 30684 . . . . . . . 8 (dom 𝑃 ran sigAlgebra → ∅ ∈ dom 𝑃)
3532, 33, 343syl 18 . . . . . . 7 (𝜑 → ∅ ∈ dom 𝑃)
3635adantr 473 . . . . . 6 ((𝜑 ∧ ({0} ∩ 𝑦) = ∅) → ∅ ∈ dom 𝑃)
3731, 36eqeltrd 2876 . . . . 5 ((𝜑 ∧ ({0} ∩ 𝑦) = ∅) → dom ( dom 𝑃 × ({0} ∩ 𝑦)) ∈ dom 𝑃)
3824, 37syl5eqel 2880 . . . 4 ((𝜑 ∧ ({0} ∩ 𝑦) = ∅) → ((𝑥 dom 𝑃 ↦ 0) “ 𝑦) ∈ dom 𝑃)
39 dmxp 5545 . . . . . . 7 (({0} ∩ 𝑦) ≠ ∅ → dom ( dom 𝑃 × ({0} ∩ 𝑦)) = dom 𝑃)
4039adantl 474 . . . . . 6 ((𝜑 ∧ ({0} ∩ 𝑦) ≠ ∅) → dom ( dom 𝑃 × ({0} ∩ 𝑦)) = dom 𝑃)
4132unveldomd 30985 . . . . . . 7 (𝜑 dom 𝑃 ∈ dom 𝑃)
4241adantr 473 . . . . . 6 ((𝜑 ∧ ({0} ∩ 𝑦) ≠ ∅) → dom 𝑃 ∈ dom 𝑃)
4340, 42eqeltrd 2876 . . . . 5 ((𝜑 ∧ ({0} ∩ 𝑦) ≠ ∅) → dom ( dom 𝑃 × ({0} ∩ 𝑦)) ∈ dom 𝑃)
4424, 43syl5eqel 2880 . . . 4 ((𝜑 ∧ ({0} ∩ 𝑦) ≠ ∅) → ((𝑥 dom 𝑃 ↦ 0) “ 𝑦) ∈ dom 𝑃)
4538, 44pm2.61dane 3056 . . 3 (𝜑 → ((𝑥 dom 𝑃 ↦ 0) “ 𝑦) ∈ dom 𝑃)
4645ralrimivw 3146 . 2 (𝜑 → ∀𝑦 ∈ 𝔅 ((𝑥 dom 𝑃 ↦ 0) “ 𝑦) ∈ dom 𝑃)
4732isrrvv 31013 . 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 705 1 (𝜑 → (𝑥 dom 𝑃 ↦ 0) ∈ (rRndVar‘𝑃))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385   = wceq 1653  wcel 2157  wne 2969  wral 3087  Vcvv 3383  cin 3766  c0 4113  {csn 4366   cuni 4626  cmpt 4920   × cxp 5308  ccnv 5309  dom cdm 5310  ran crn 5311  cres 5312  cima 5313  wf 6095  cfv 6099  cr 10221  0cc0 10222  sigAlgebracsiga 30677  𝔅cbrsiga 30751  Probcprb 30977  rRndVarcrrv 31010
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2375  ax-ext 2775  ax-sep 4973  ax-nul 4981  ax-pow 5033  ax-pr 5095  ax-un 7181  ax-cnex 10278  ax-resscn 10279  ax-1cn 10280  ax-addrcl 10283  ax-rnegex 10293  ax-cnre 10295  ax-pre-lttri 10296  ax-pre-lttrn 10297
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-fal 1667  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-nel 3073  df-ral 3092  df-rex 3093  df-rab 3096  df-v 3385  df-sbc 3632  df-csb 3727  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-pw 4349  df-sn 4367  df-pr 4369  df-op 4373  df-uni 4627  df-int 4666  df-iun 4710  df-br 4842  df-opab 4904  df-mpt 4921  df-id 5218  df-po 5231  df-so 5232  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-rn 5321  df-res 5322  df-ima 5323  df-iota 6062  df-fun 6101  df-fn 6102  df-f 6103  df-f1 6104  df-fo 6105  df-f1o 6106  df-fv 6107  df-ov 6879  df-oprab 6880  df-mpt2 6881  df-1st 7399  df-2nd 7400  df-er 7980  df-map 8095  df-en 8194  df-dom 8195  df-sdom 8196  df-pnf 10363  df-mnf 10364  df-xr 10365  df-ltxr 10366  df-le 10367  df-ioo 12424  df-topgen 16415  df-top 21023  df-bases 21075  df-esum 30597  df-siga 30678  df-sigagen 30709  df-brsiga 30752  df-meas 30766  df-mbfm 30820  df-prob 30978  df-rrv 31011
This theorem is referenced by: (None)
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