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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rrvmbfm | Structured version Visualization version GIF version |
Description: A real-valued random variable is a measurable function from its sample space to the Borel sigma-algebra. (Contributed by Thierry Arnoux, 25-Jan-2017.) |
Ref | Expression |
---|---|
isrrvv.1 | ⊢ (𝜑 → 𝑃 ∈ Prob) |
Ref | Expression |
---|---|
rrvmbfm | ⊢ (𝜑 → (𝑋 ∈ (rRndVar‘𝑃) ↔ 𝑋 ∈ (dom 𝑃MblFnM𝔅ℝ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isrrvv.1 | . . 3 ⊢ (𝜑 → 𝑃 ∈ Prob) | |
2 | dmeq 5569 | . . . . 5 ⊢ (𝑝 = 𝑃 → dom 𝑝 = dom 𝑃) | |
3 | 2 | oveq1d 6937 | . . . 4 ⊢ (𝑝 = 𝑃 → (dom 𝑝MblFnM𝔅ℝ) = (dom 𝑃MblFnM𝔅ℝ)) |
4 | df-rrv 31102 | . . . 4 ⊢ rRndVar = (𝑝 ∈ Prob ↦ (dom 𝑝MblFnM𝔅ℝ)) | |
5 | ovex 6954 | . . . 4 ⊢ (dom 𝑃MblFnM𝔅ℝ) ∈ V | |
6 | 3, 4, 5 | fvmpt 6542 | . . 3 ⊢ (𝑃 ∈ Prob → (rRndVar‘𝑃) = (dom 𝑃MblFnM𝔅ℝ)) |
7 | 1, 6 | syl 17 | . 2 ⊢ (𝜑 → (rRndVar‘𝑃) = (dom 𝑃MblFnM𝔅ℝ)) |
8 | 7 | eleq2d 2844 | 1 ⊢ (𝜑 → (𝑋 ∈ (rRndVar‘𝑃) ↔ 𝑋 ∈ (dom 𝑃MblFnM𝔅ℝ))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 = wceq 1601 ∈ wcel 2106 dom cdm 5355 ‘cfv 6135 (class class class)co 6922 𝔅ℝcbrsiga 30842 MblFnMcmbfm 30910 Probcprb 31068 rRndVarcrrv 31101 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-sep 5017 ax-nul 5025 ax-pr 5138 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3399 df-sbc 3652 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-nul 4141 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4672 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-iota 6099 df-fun 6137 df-fv 6143 df-ov 6925 df-rrv 31102 |
This theorem is referenced by: isrrvv 31104 rrvadd 31113 rrvmulc 31114 orrvcval4 31125 orrvcoel 31126 orrvccel 31127 |
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