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Mirrors > Home > MPE Home > Th. List > Mathboxes > domprobsiga | Structured version Visualization version GIF version |
Description: The domain of a probability measure is a sigma-algebra. (Contributed by Thierry Arnoux, 23-Dec-2016.) |
Ref | Expression |
---|---|
domprobsiga | ⊢ (𝑃 ∈ Prob → dom 𝑃 ∈ ∪ ran sigAlgebra) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | domprobmeas 32373 | . 2 ⊢ (𝑃 ∈ Prob → 𝑃 ∈ (measures‘dom 𝑃)) | |
2 | measbase 32161 | . 2 ⊢ (𝑃 ∈ (measures‘dom 𝑃) → dom 𝑃 ∈ ∪ ran sigAlgebra) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝑃 ∈ Prob → dom 𝑃 ∈ ∪ ran sigAlgebra) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2110 ∪ cuni 4845 dom cdm 5590 ran crn 5591 ‘cfv 6432 sigAlgebracsiga 32072 measurescmeas 32159 Probcprb 32370 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ral 3071 df-rex 3072 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-br 5080 df-opab 5142 df-mpt 5163 df-id 5490 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-fv 6440 df-ov 7274 df-esum 31992 df-meas 32160 df-prob 32371 |
This theorem is referenced by: unveldomd 32378 nuleldmp 32380 probdif 32383 totprobd 32389 cndprobin 32397 cndprob01 32398 isrrvv 32406 0rrv 32414 rrvadd 32415 rrvmulc 32416 orrvcval4 32427 orrvcoel 32428 orrvccel 32429 |
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