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Mirrors > Home > MPE Home > Th. List > Mathboxes > isrrvv | Structured version Visualization version GIF version |
Description: Elementhood to the set of real-valued random variables with respect to the probability 𝑃. (Contributed by Thierry Arnoux, 25-Jan-2017.) |
Ref | Expression |
---|---|
isrrvv.1 | ⊢ (𝜑 → 𝑃 ∈ Prob) |
Ref | Expression |
---|---|
isrrvv | ⊢ (𝜑 → (𝑋 ∈ (rRndVar‘𝑃) ↔ (𝑋:∪ dom 𝑃⟶ℝ ∧ ∀𝑦 ∈ 𝔅ℝ (◡𝑋 “ 𝑦) ∈ dom 𝑃))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isrrvv.1 | . . 3 ⊢ (𝜑 → 𝑃 ∈ Prob) | |
2 | 1 | rrvmbfm 29831 | . 2 ⊢ (𝜑 → (𝑋 ∈ (rRndVar‘𝑃) ↔ 𝑋 ∈ (dom 𝑃MblFnM𝔅ℝ))) |
3 | domprobsiga 29800 | . . . 4 ⊢ (𝑃 ∈ Prob → dom 𝑃 ∈ ∪ ran sigAlgebra) | |
4 | 1, 3 | syl 17 | . . 3 ⊢ (𝜑 → dom 𝑃 ∈ ∪ ran sigAlgebra) |
5 | brsigarn 29574 | . . . 4 ⊢ 𝔅ℝ ∈ (sigAlgebra‘ℝ) | |
6 | elrnsiga 29516 | . . . 4 ⊢ (𝔅ℝ ∈ (sigAlgebra‘ℝ) → 𝔅ℝ ∈ ∪ ran sigAlgebra) | |
7 | 5, 6 | mp1i 13 | . . 3 ⊢ (𝜑 → 𝔅ℝ ∈ ∪ ran sigAlgebra) |
8 | 4, 7 | ismbfm 29641 | . 2 ⊢ (𝜑 → (𝑋 ∈ (dom 𝑃MblFnM𝔅ℝ) ↔ (𝑋 ∈ (∪ 𝔅ℝ ↑𝑚 ∪ dom 𝑃) ∧ ∀𝑦 ∈ 𝔅ℝ (◡𝑋 “ 𝑦) ∈ dom 𝑃))) |
9 | unibrsiga 29576 | . . . . . 6 ⊢ ∪ 𝔅ℝ = ℝ | |
10 | 9 | oveq1i 6559 | . . . . 5 ⊢ (∪ 𝔅ℝ ↑𝑚 ∪ dom 𝑃) = (ℝ ↑𝑚 ∪ dom 𝑃) |
11 | 10 | eleq2i 2680 | . . . 4 ⊢ (𝑋 ∈ (∪ 𝔅ℝ ↑𝑚 ∪ dom 𝑃) ↔ 𝑋 ∈ (ℝ ↑𝑚 ∪ dom 𝑃)) |
12 | reex 9906 | . . . . 5 ⊢ ℝ ∈ V | |
13 | uniexg 6853 | . . . . . 6 ⊢ (dom 𝑃 ∈ ∪ ran sigAlgebra → ∪ dom 𝑃 ∈ V) | |
14 | 4, 13 | syl 17 | . . . . 5 ⊢ (𝜑 → ∪ dom 𝑃 ∈ V) |
15 | elmapg 7757 | . . . . 5 ⊢ ((ℝ ∈ V ∧ ∪ dom 𝑃 ∈ V) → (𝑋 ∈ (ℝ ↑𝑚 ∪ dom 𝑃) ↔ 𝑋:∪ dom 𝑃⟶ℝ)) | |
16 | 12, 14, 15 | sylancr 694 | . . . 4 ⊢ (𝜑 → (𝑋 ∈ (ℝ ↑𝑚 ∪ dom 𝑃) ↔ 𝑋:∪ dom 𝑃⟶ℝ)) |
17 | 11, 16 | syl5bb 271 | . . 3 ⊢ (𝜑 → (𝑋 ∈ (∪ 𝔅ℝ ↑𝑚 ∪ dom 𝑃) ↔ 𝑋:∪ dom 𝑃⟶ℝ)) |
18 | 17 | anbi1d 737 | . 2 ⊢ (𝜑 → ((𝑋 ∈ (∪ 𝔅ℝ ↑𝑚 ∪ dom 𝑃) ∧ ∀𝑦 ∈ 𝔅ℝ (◡𝑋 “ 𝑦) ∈ dom 𝑃) ↔ (𝑋:∪ dom 𝑃⟶ℝ ∧ ∀𝑦 ∈ 𝔅ℝ (◡𝑋 “ 𝑦) ∈ dom 𝑃))) |
19 | 2, 8, 18 | 3bitrd 293 | 1 ⊢ (𝜑 → (𝑋 ∈ (rRndVar‘𝑃) ↔ (𝑋:∪ dom 𝑃⟶ℝ ∧ ∀𝑦 ∈ 𝔅ℝ (◡𝑋 “ 𝑦) ∈ dom 𝑃))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∈ wcel 1977 ∀wral 2896 Vcvv 3173 ∪ cuni 4372 ◡ccnv 5037 dom cdm 5038 ran crn 5039 “ cima 5041 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ↑𝑚 cmap 7744 ℝcr 9814 sigAlgebracsiga 29497 𝔅ℝcbrsiga 29571 MblFnMcmbfm 29639 Probcprb 29796 rRndVarcrrv 29829 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-pre-lttri 9889 ax-pre-lttrn 9890 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-ioo 12050 df-topgen 15927 df-top 20521 df-bases 20522 df-esum 29417 df-siga 29498 df-sigagen 29529 df-brsiga 29572 df-meas 29586 df-mbfm 29640 df-prob 29797 df-rrv 29830 |
This theorem is referenced by: rrvvf 29833 rrvfinvima 29839 0rrv 29840 coinfliprv 29871 |
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