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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rrvvf | Structured version Visualization version GIF version |
Description: A real-valued random variable is a function. (Contributed by Thierry Arnoux, 25-Jan-2017.) |
Ref | Expression |
---|---|
isrrvv.1 | ⊢ (𝜑 → 𝑃 ∈ Prob) |
rrvvf.1 | ⊢ (𝜑 → 𝑋 ∈ (rRndVar‘𝑃)) |
Ref | Expression |
---|---|
rrvvf | ⊢ (𝜑 → 𝑋:∪ dom 𝑃⟶ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rrvvf.1 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (rRndVar‘𝑃)) | |
2 | isrrvv.1 | . . . 4 ⊢ (𝜑 → 𝑃 ∈ Prob) | |
3 | 2 | isrrvv 33043 | . . 3 ⊢ (𝜑 → (𝑋 ∈ (rRndVar‘𝑃) ↔ (𝑋:∪ dom 𝑃⟶ℝ ∧ ∀𝑦 ∈ 𝔅ℝ (◡𝑋 “ 𝑦) ∈ dom 𝑃))) |
4 | 1, 3 | mpbid 231 | . 2 ⊢ (𝜑 → (𝑋:∪ dom 𝑃⟶ℝ ∧ ∀𝑦 ∈ 𝔅ℝ (◡𝑋 “ 𝑦) ∈ dom 𝑃)) |
5 | 4 | simpld 495 | 1 ⊢ (𝜑 → 𝑋:∪ dom 𝑃⟶ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2106 ∀wral 3064 ∪ cuni 4865 ◡ccnv 5632 dom cdm 5633 “ cima 5636 ⟶wf 6492 ‘cfv 6496 ℝcr 11050 𝔅ℝcbrsiga 32780 Probcprb 33007 rRndVarcrrv 33040 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-pre-lttri 11125 ax-pre-lttrn 11126 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-id 5531 df-po 5545 df-so 5546 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-ov 7360 df-oprab 7361 df-mpo 7362 df-1st 7921 df-2nd 7922 df-er 8648 df-map 8767 df-en 8884 df-dom 8885 df-sdom 8886 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-ioo 13268 df-topgen 17325 df-top 22243 df-bases 22296 df-esum 32627 df-siga 32708 df-sigagen 32738 df-brsiga 32781 df-meas 32795 df-mbfm 32849 df-prob 33008 df-rrv 33041 |
This theorem is referenced by: rrvfn 33045 rrvdm 33046 rrvrnss 33047 rrvf2 33048 rrvadd 33052 rrvmulc 33053 dstrvprob 33071 dstfrvel 33073 dstfrvunirn 33074 |
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