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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 30504 | . 2 ⊢ (𝜑 → (𝑋 ∈ (rRndVar‘𝑃) ↔ 𝑋 ∈ (dom 𝑃MblFnM𝔅ℝ))) |
3 | domprobsiga 30473 | . . . 4 ⊢ (𝑃 ∈ Prob → dom 𝑃 ∈ ∪ ran sigAlgebra) | |
4 | 1, 3 | syl 17 | . . 3 ⊢ (𝜑 → dom 𝑃 ∈ ∪ ran sigAlgebra) |
5 | brsigarn 30247 | . . . 4 ⊢ 𝔅ℝ ∈ (sigAlgebra‘ℝ) | |
6 | elrnsiga 30189 | . . . 4 ⊢ (𝔅ℝ ∈ (sigAlgebra‘ℝ) → 𝔅ℝ ∈ ∪ ran sigAlgebra) | |
7 | 5, 6 | mp1i 13 | . . 3 ⊢ (𝜑 → 𝔅ℝ ∈ ∪ ran sigAlgebra) |
8 | 4, 7 | ismbfm 30314 | . 2 ⊢ (𝜑 → (𝑋 ∈ (dom 𝑃MblFnM𝔅ℝ) ↔ (𝑋 ∈ (∪ 𝔅ℝ ↑𝑚 ∪ dom 𝑃) ∧ ∀𝑦 ∈ 𝔅ℝ (◡𝑋 “ 𝑦) ∈ dom 𝑃))) |
9 | unibrsiga 30249 | . . . . . 6 ⊢ ∪ 𝔅ℝ = ℝ | |
10 | 9 | oveq1i 6660 | . . . . 5 ⊢ (∪ 𝔅ℝ ↑𝑚 ∪ dom 𝑃) = (ℝ ↑𝑚 ∪ dom 𝑃) |
11 | 10 | eleq2i 2693 | . . . 4 ⊢ (𝑋 ∈ (∪ 𝔅ℝ ↑𝑚 ∪ dom 𝑃) ↔ 𝑋 ∈ (ℝ ↑𝑚 ∪ dom 𝑃)) |
12 | reex 10027 | . . . . 5 ⊢ ℝ ∈ V | |
13 | uniexg 6955 | . . . . . 6 ⊢ (dom 𝑃 ∈ ∪ ran sigAlgebra → ∪ dom 𝑃 ∈ V) | |
14 | 4, 13 | syl 17 | . . . . 5 ⊢ (𝜑 → ∪ dom 𝑃 ∈ V) |
15 | elmapg 7870 | . . . . 5 ⊢ ((ℝ ∈ V ∧ ∪ dom 𝑃 ∈ V) → (𝑋 ∈ (ℝ ↑𝑚 ∪ dom 𝑃) ↔ 𝑋:∪ dom 𝑃⟶ℝ)) | |
16 | 12, 14, 15 | sylancr 695 | . . . 4 ⊢ (𝜑 → (𝑋 ∈ (ℝ ↑𝑚 ∪ dom 𝑃) ↔ 𝑋:∪ dom 𝑃⟶ℝ)) |
17 | 11, 16 | syl5bb 272 | . . 3 ⊢ (𝜑 → (𝑋 ∈ (∪ 𝔅ℝ ↑𝑚 ∪ dom 𝑃) ↔ 𝑋:∪ dom 𝑃⟶ℝ)) |
18 | 17 | anbi1d 741 | . 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 294 | 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 196 ∧ wa 384 ∈ wcel 1990 ∀wral 2912 Vcvv 3200 ∪ cuni 4436 ◡ccnv 5113 dom cdm 5114 ran crn 5115 “ cima 5117 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ↑𝑚 cmap 7857 ℝcr 9935 sigAlgebracsiga 30170 𝔅ℝcbrsiga 30244 MblFnMcmbfm 30312 Probcprb 30469 rRndVarcrrv 30502 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-pre-lttri 10010 ax-pre-lttrn 10011 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-ioo 12179 df-topgen 16104 df-top 20699 df-bases 20750 df-esum 30090 df-siga 30171 df-sigagen 30202 df-brsiga 30245 df-meas 30259 df-mbfm 30313 df-prob 30470 df-rrv 30503 |
This theorem is referenced by: rrvvf 30506 rrvfinvima 30512 0rrv 30513 coinfliprv 30544 |
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