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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 31852 | . . . 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 499 | . . 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 31853 | . . . 4 ⊢ (𝑀 ∈ ∪ ran measures → dom 𝑀 ∈ ∪ ran sigAlgebra) | |
4 | ismeas 31851 | . . . 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 260 | . 2 ⊢ (𝑀 ∈ ∪ ran measures → 𝑀 ∈ (measures‘dom 𝑀)) |
7 | df-meas 31848 | . . . 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 6408 | . . 3 ⊢ Fun measures |
9 | elunirn2 30680 | . . 3 ⊢ ((Fun measures ∧ 𝑀 ∈ (measures‘dom 𝑀)) → 𝑀 ∈ ∪ ran measures) | |
10 | 8, 9 | mpan 690 | . 2 ⊢ (𝑀 ∈ (measures‘dom 𝑀) → 𝑀 ∈ ∪ ran measures) |
11 | 6, 10 | impbii 212 | 1 ⊢ (𝑀 ∈ ∪ ran measures ↔ 𝑀 ∈ (measures‘dom 𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 {cab 2712 ∀wral 3054 ∅c0 4227 𝒫 cpw 4503 ∪ cuni 4809 Disj wdisj 5008 class class class wbr 5043 dom cdm 5540 ran crn 5541 Fun wfun 6363 ⟶wf 6365 ‘cfv 6369 (class class class)co 7202 ωcom 7633 ≼ cdom 8613 0cc0 10712 +∞cpnf 10847 [,]cicc 12921 Σ*cesum 31679 sigAlgebracsiga 31760 measurescmeas 31847 |
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 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 |
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-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-op 4538 df-uni 4810 df-br 5044 df-opab 5106 df-mpt 5125 df-id 5444 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-fv 6377 df-ov 7205 df-esum 31680 df-meas 31848 |
This theorem is referenced by: truae 31895 aean 31896 mbfmbfm 31909 sibfinima 31990 sibfof 31991 domprobmeas 32061 probmeasd 32074 probfinmeasb 32079 probfinmeasbALTV 32080 probmeasb 32081 dstrvprob 32122 |
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