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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > elprob | Structured version Visualization version GIF version |
Description: The property of being a probability measure. (Contributed by Thierry Arnoux, 8-Dec-2016.) |
Ref | Expression |
---|---|
elprob | ⊢ (𝑃 ∈ Prob ↔ (𝑃 ∈ ∪ ran measures ∧ (𝑃‘∪ dom 𝑃) = 1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . . 4 ⊢ (𝑝 = 𝑃 → 𝑝 = 𝑃) | |
2 | dmeq 5896 | . . . . 5 ⊢ (𝑝 = 𝑃 → dom 𝑝 = dom 𝑃) | |
3 | 2 | unieqd 4915 | . . . 4 ⊢ (𝑝 = 𝑃 → ∪ dom 𝑝 = ∪ dom 𝑃) |
4 | 1, 3 | fveq12d 6891 | . . 3 ⊢ (𝑝 = 𝑃 → (𝑝‘∪ dom 𝑝) = (𝑃‘∪ dom 𝑃)) |
5 | 4 | eqeq1d 2728 | . 2 ⊢ (𝑝 = 𝑃 → ((𝑝‘∪ dom 𝑝) = 1 ↔ (𝑃‘∪ dom 𝑃) = 1)) |
6 | df-prob 33937 | . 2 ⊢ Prob = {𝑝 ∈ ∪ ran measures ∣ (𝑝‘∪ dom 𝑝) = 1} | |
7 | 5, 6 | elrab2 3681 | 1 ⊢ (𝑃 ∈ Prob ↔ (𝑃 ∈ ∪ ran measures ∧ (𝑃‘∪ dom 𝑃) = 1)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∪ cuni 4902 dom cdm 5669 ran crn 5670 ‘cfv 6536 1c1 11110 measurescmeas 33723 Probcprb 33936 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-dm 5679 df-iota 6488 df-fv 6544 df-prob 33937 |
This theorem is referenced by: domprobmeas 33939 probtot 33941 probfinmeasb 33957 probfinmeasbALTV 33958 probmeasb 33959 dstrvprob 34000 |
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