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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 5736 | . . . . 5 ⊢ (𝑝 = 𝑃 → dom 𝑝 = dom 𝑃) | |
3 | 2 | oveq1d 7150 | . . . 4 ⊢ (𝑝 = 𝑃 → (dom 𝑝MblFnM𝔅ℝ) = (dom 𝑃MblFnM𝔅ℝ)) |
4 | df-rrv 31809 | . . . 4 ⊢ rRndVar = (𝑝 ∈ Prob ↦ (dom 𝑝MblFnM𝔅ℝ)) | |
5 | ovex 7168 | . . . 4 ⊢ (dom 𝑃MblFnM𝔅ℝ) ∈ V | |
6 | 3, 4, 5 | fvmpt 6745 | . . 3 ⊢ (𝑃 ∈ Prob → (rRndVar‘𝑃) = (dom 𝑃MblFnM𝔅ℝ)) |
7 | 1, 6 | syl 17 | . 2 ⊢ (𝜑 → (rRndVar‘𝑃) = (dom 𝑃MblFnM𝔅ℝ)) |
8 | 7 | eleq2d 2875 | 1 ⊢ (𝜑 → (𝑋 ∈ (rRndVar‘𝑃) ↔ 𝑋 ∈ (dom 𝑃MblFnM𝔅ℝ))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1538 ∈ wcel 2111 dom cdm 5519 ‘cfv 6324 (class class class)co 7135 𝔅ℝcbrsiga 31550 MblFnMcmbfm 31618 Probcprb 31775 rRndVarcrrv 31808 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6283 df-fun 6326 df-fv 6332 df-ov 7138 df-rrv 31809 |
This theorem is referenced by: isrrvv 31811 rrvadd 31820 rrvmulc 31821 orrvcval4 31832 orrvcoel 31833 orrvccel 31834 |
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