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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 5757 | . . . . 5 ⊢ (𝑝 = 𝑃 → dom 𝑝 = dom 𝑃) | |
3 | 2 | unieqd 4819 | . . . 4 ⊢ (𝑝 = 𝑃 → ∪ dom 𝑝 = ∪ dom 𝑃) |
4 | 1, 3 | fveq12d 6702 | . . 3 ⊢ (𝑝 = 𝑃 → (𝑝‘∪ dom 𝑝) = (𝑃‘∪ dom 𝑃)) |
5 | 4 | eqeq1d 2738 | . 2 ⊢ (𝑝 = 𝑃 → ((𝑝‘∪ dom 𝑝) = 1 ↔ (𝑃‘∪ dom 𝑃) = 1)) |
6 | df-prob 32041 | . 2 ⊢ Prob = {𝑝 ∈ ∪ ran measures ∣ (𝑝‘∪ dom 𝑝) = 1} | |
7 | 5, 6 | elrab2 3594 | 1 ⊢ (𝑃 ∈ Prob ↔ (𝑃 ∈ ∪ ran measures ∧ (𝑃‘∪ dom 𝑃) = 1)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ∪ cuni 4805 dom cdm 5536 ran crn 5537 ‘cfv 6358 1c1 10695 measurescmeas 31829 Probcprb 32040 |
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 2018 ax-8 2114 ax-9 2122 ax-ext 2708 |
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-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-dm 5546 df-iota 6316 df-fv 6366 df-prob 32041 |
This theorem is referenced by: domprobmeas 32043 probtot 32045 probfinmeasb 32061 probfinmeasbALTV 32062 probmeasb 32063 dstrvprob 32104 |
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