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Mirrors > Home > MPE Home > Th. List > Mathboxes > measbasedom | Structured version Visualization version GIF version |
Description: The base set of a measure is its domain. (Contributed by Thierry Arnoux, 25-Dec-2016.) |
Ref | Expression |
---|---|
measbasedom | ⊢ (𝑀 ∈ ∪ ran measures ↔ 𝑀 ∈ (measures‘dom 𝑀)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isrnmeas 31454 | . . . 4 ⊢ (𝑀 ∈ ∪ ran measures → (dom 𝑀 ∈ ∪ ran sigAlgebra ∧ (𝑀:dom 𝑀⟶(0[,]+∞) ∧ (𝑀‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 dom 𝑀((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → (𝑀‘∪ 𝑥) = Σ*𝑦 ∈ 𝑥(𝑀‘𝑦))))) | |
2 | 1 | simprd 498 | . . 3 ⊢ (𝑀 ∈ ∪ ran measures → (𝑀:dom 𝑀⟶(0[,]+∞) ∧ (𝑀‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 dom 𝑀((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → (𝑀‘∪ 𝑥) = Σ*𝑦 ∈ 𝑥(𝑀‘𝑦)))) |
3 | dmmeas 31455 | . . . 4 ⊢ (𝑀 ∈ ∪ ran measures → dom 𝑀 ∈ ∪ ran sigAlgebra) | |
4 | ismeas 31453 | . . . 4 ⊢ (dom 𝑀 ∈ ∪ ran sigAlgebra → (𝑀 ∈ (measures‘dom 𝑀) ↔ (𝑀:dom 𝑀⟶(0[,]+∞) ∧ (𝑀‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 dom 𝑀((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → (𝑀‘∪ 𝑥) = Σ*𝑦 ∈ 𝑥(𝑀‘𝑦))))) | |
5 | 3, 4 | syl 17 | . . 3 ⊢ (𝑀 ∈ ∪ ran measures → (𝑀 ∈ (measures‘dom 𝑀) ↔ (𝑀:dom 𝑀⟶(0[,]+∞) ∧ (𝑀‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 dom 𝑀((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → (𝑀‘∪ 𝑥) = Σ*𝑦 ∈ 𝑥(𝑀‘𝑦))))) |
6 | 2, 5 | mpbird 259 | . 2 ⊢ (𝑀 ∈ ∪ ran measures → 𝑀 ∈ (measures‘dom 𝑀)) |
7 | df-meas 31450 | . . . 4 ⊢ measures = (𝑠 ∈ ∪ ran sigAlgebra ↦ {𝑚 ∣ (𝑚:𝑠⟶(0[,]+∞) ∧ (𝑚‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → (𝑚‘∪ 𝑥) = Σ*𝑦 ∈ 𝑥(𝑚‘𝑦)))}) | |
8 | 7 | funmpt2 6388 | . . 3 ⊢ Fun measures |
9 | elunirn2 30390 | . . 3 ⊢ ((Fun measures ∧ 𝑀 ∈ (measures‘dom 𝑀)) → 𝑀 ∈ ∪ ran measures) | |
10 | 8, 9 | mpan 688 | . 2 ⊢ (𝑀 ∈ (measures‘dom 𝑀) → 𝑀 ∈ ∪ ran measures) |
11 | 6, 10 | impbii 211 | 1 ⊢ (𝑀 ∈ ∪ ran measures ↔ 𝑀 ∈ (measures‘dom 𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 {cab 2799 ∀wral 3138 ∅c0 4290 𝒫 cpw 4538 ∪ cuni 4831 Disj wdisj 5023 class class class wbr 5058 dom cdm 5549 ran crn 5550 Fun wfun 6343 ⟶wf 6345 ‘cfv 6349 (class class class)co 7150 ωcom 7574 ≼ cdom 8501 0cc0 10531 +∞cpnf 10666 [,]cicc 12735 Σ*cesum 31281 sigAlgebracsiga 31362 measurescmeas 31449 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-fv 6357 df-ov 7153 df-esum 31282 df-meas 31450 |
This theorem is referenced by: truae 31497 aean 31498 mbfmbfm 31511 sibfinima 31592 sibfof 31593 domprobmeas 31663 probmeasd 31676 probfinmeasb 31681 probfinmeasbALTV 31682 probmeasb 31683 dstrvprob 31724 |
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