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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 32060 | . . 3 ⊢ (𝑃 ∈ Prob ↔ (𝑃 ∈ ∪ ran measures ∧ (𝑃‘∪ dom 𝑃) = 1)) | |
2 | 1 | simplbi 501 | . 2 ⊢ (𝑃 ∈ Prob → 𝑃 ∈ ∪ ran measures) |
3 | measbasedom 31854 | . 2 ⊢ (𝑃 ∈ ∪ ran measures ↔ 𝑃 ∈ (measures‘dom 𝑃)) | |
4 | 2, 3 | sylib 221 | 1 ⊢ (𝑃 ∈ Prob → 𝑃 ∈ (measures‘dom 𝑃)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2110 ∪ cuni 4809 dom cdm 5540 ran crn 5541 ‘cfv 6369 1c1 10713 measurescmeas 31847 Probcprb 32058 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-op 4538 df-uni 4810 df-br 5044 df-opab 5106 df-mpt 5125 df-id 5444 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-fv 6377 df-ov 7205 df-esum 31680 df-meas 31848 df-prob 32059 |
This theorem is referenced by: domprobsiga 32062 prob01 32064 probnul 32065 probcun 32069 probinc 32072 probmeasd 32074 totprobd 32077 cndprob01 32086 cndprobprob 32089 dstrvprob 32122 dstfrvinc 32127 dstfrvclim1 32128 |
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