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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 5904 | . . . . 5 ⊢ (𝑝 = 𝑃 → dom 𝑝 = dom 𝑃) | |
3 | 2 | unieqd 4923 | . . . 4 ⊢ (𝑝 = 𝑃 → ∪ dom 𝑝 = ∪ dom 𝑃) |
4 | 1, 3 | fveq12d 6899 | . . 3 ⊢ (𝑝 = 𝑃 → (𝑝‘∪ dom 𝑝) = (𝑃‘∪ dom 𝑃)) |
5 | 4 | eqeq1d 2735 | . 2 ⊢ (𝑝 = 𝑃 → ((𝑝‘∪ dom 𝑝) = 1 ↔ (𝑃‘∪ dom 𝑃) = 1)) |
6 | df-prob 33407 | . 2 ⊢ Prob = {𝑝 ∈ ∪ ran measures ∣ (𝑝‘∪ dom 𝑝) = 1} | |
7 | 5, 6 | elrab2 3687 | 1 ⊢ (𝑃 ∈ Prob ↔ (𝑃 ∈ ∪ ran measures ∧ (𝑃‘∪ dom 𝑃) = 1)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∪ cuni 4909 dom cdm 5677 ran crn 5678 ‘cfv 6544 1c1 11111 measurescmeas 33193 Probcprb 33406 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-dm 5687 df-iota 6496 df-fv 6552 df-prob 33407 |
This theorem is referenced by: domprobmeas 33409 probtot 33411 probfinmeasb 33427 probfinmeasbALTV 33428 probmeasb 33429 dstrvprob 33470 |
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