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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 5902 | . . . . 5 ⊢ (𝑝 = 𝑃 → dom 𝑝 = dom 𝑃) | |
3 | 2 | oveq1d 7431 | . . . 4 ⊢ (𝑝 = 𝑃 → (dom 𝑝MblFnM𝔅ℝ) = (dom 𝑃MblFnM𝔅ℝ)) |
4 | df-rrv 34288 | . . . 4 ⊢ rRndVar = (𝑝 ∈ Prob ↦ (dom 𝑝MblFnM𝔅ℝ)) | |
5 | ovex 7449 | . . . 4 ⊢ (dom 𝑃MblFnM𝔅ℝ) ∈ V | |
6 | 3, 4, 5 | fvmpt 7001 | . . 3 ⊢ (𝑃 ∈ Prob → (rRndVar‘𝑃) = (dom 𝑃MblFnM𝔅ℝ)) |
7 | 1, 6 | syl 17 | . 2 ⊢ (𝜑 → (rRndVar‘𝑃) = (dom 𝑃MblFnM𝔅ℝ)) |
8 | 7 | eleq2d 2812 | 1 ⊢ (𝜑 → (𝑋 ∈ (rRndVar‘𝑃) ↔ 𝑋 ∈ (dom 𝑃MblFnM𝔅ℝ))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1534 ∈ wcel 2099 dom cdm 5674 ‘cfv 6546 (class class class)co 7416 𝔅ℝcbrsiga 34027 MblFnMcmbfm 34095 Probcprb 34254 rRndVarcrrv 34287 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5296 ax-nul 5303 ax-pr 5425 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-dif 3949 df-un 3951 df-ss 3963 df-nul 4323 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-br 5146 df-opab 5208 df-mpt 5229 df-id 5572 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-iota 6498 df-fun 6548 df-fv 6554 df-ov 7419 df-rrv 34288 |
This theorem is referenced by: isrrvv 34290 rrvadd 34299 rrvmulc 34300 orrvcval4 34311 orrvcoel 34312 orrvccel 34313 |
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