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Mirrors > Home > MPE Home > Th. List > Mathboxes > domprobmeas | Structured version Visualization version GIF version |
Description: A probability measure is a measure on its domain. (Contributed by Thierry Arnoux, 23-Dec-2016.) |
Ref | Expression |
---|---|
domprobmeas | ⊢ (𝑃 ∈ Prob → 𝑃 ∈ (measures‘dom 𝑃)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elprob 29798 | . . 3 ⊢ (𝑃 ∈ Prob ↔ (𝑃 ∈ ∪ ran measures ∧ (𝑃‘∪ dom 𝑃) = 1)) | |
2 | 1 | simplbi 475 | . 2 ⊢ (𝑃 ∈ Prob → 𝑃 ∈ ∪ ran measures) |
3 | measbasedom 29592 | . 2 ⊢ (𝑃 ∈ ∪ ran measures ↔ 𝑃 ∈ (measures‘dom 𝑃)) | |
4 | 2, 3 | sylib 207 | 1 ⊢ (𝑃 ∈ Prob → 𝑃 ∈ (measures‘dom 𝑃)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ∪ cuni 4372 dom cdm 5038 ran crn 5039 ‘cfv 5804 1c1 9816 measurescmeas 29585 Probcprb 29796 |
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 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 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-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-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 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-fv 5812 df-ov 6552 df-esum 29417 df-meas 29586 df-prob 29797 |
This theorem is referenced by: domprobsiga 29800 prob01 29802 probnul 29803 probcun 29807 probinc 29810 probmeasd 29812 totprobd 29815 cndprob01 29824 cndprobprob 29827 dstrvprob 29860 dstfrvinc 29865 dstfrvclim1 29866 |
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