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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 5772 | . . . . 5 ⊢ (𝑝 = 𝑃 → dom 𝑝 = dom 𝑃) | |
3 | 2 | unieqd 4852 | . . . 4 ⊢ (𝑝 = 𝑃 → ∪ dom 𝑝 = ∪ dom 𝑃) |
4 | 1, 3 | fveq12d 6677 | . . 3 ⊢ (𝑝 = 𝑃 → (𝑝‘∪ dom 𝑝) = (𝑃‘∪ dom 𝑃)) |
5 | 4 | eqeq1d 2823 | . 2 ⊢ (𝑝 = 𝑃 → ((𝑝‘∪ dom 𝑝) = 1 ↔ (𝑃‘∪ dom 𝑃) = 1)) |
6 | df-prob 31666 | . 2 ⊢ Prob = {𝑝 ∈ ∪ ran measures ∣ (𝑝‘∪ dom 𝑝) = 1} | |
7 | 5, 6 | elrab2 3683 | 1 ⊢ (𝑃 ∈ Prob ↔ (𝑃 ∈ ∪ ran measures ∧ (𝑃‘∪ dom 𝑃) = 1)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∪ cuni 4838 dom cdm 5555 ran crn 5556 ‘cfv 6355 1c1 10538 measurescmeas 31454 Probcprb 31665 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-dm 5565 df-iota 6314 df-fv 6363 df-prob 31666 |
This theorem is referenced by: domprobmeas 31668 probtot 31670 probfinmeasb 31686 probfinmeasbALTV 31687 probmeasb 31688 dstrvprob 31729 |
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