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Mirrors > Home > MPE Home > Th. List > Mathboxes > dmmeasal | Structured version Visualization version GIF version |
Description: The domain of a measure is a sigma-algebra. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
dmmeasal.m | ⊢ (𝜑 → 𝑀 ∈ Meas) |
dmmeasal.s | ⊢ 𝑆 = dom 𝑀 |
Ref | Expression |
---|---|
dmmeasal | ⊢ (𝜑 → 𝑆 ∈ SAlg) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmmeasal.s | . 2 ⊢ 𝑆 = dom 𝑀 | |
2 | dmmeasal.m | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ Meas) | |
3 | ismea 44025 | . . . . 5 ⊢ (𝑀 ∈ Meas ↔ (((𝑀:dom 𝑀⟶(0[,]+∞) ∧ dom 𝑀 ∈ SAlg) ∧ (𝑀‘∅) = 0) ∧ ∀𝑥 ∈ 𝒫 dom 𝑀((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → (𝑀‘∪ 𝑥) = (Σ^‘(𝑀 ↾ 𝑥))))) | |
4 | 2, 3 | sylib 217 | . . . 4 ⊢ (𝜑 → (((𝑀:dom 𝑀⟶(0[,]+∞) ∧ dom 𝑀 ∈ SAlg) ∧ (𝑀‘∅) = 0) ∧ ∀𝑥 ∈ 𝒫 dom 𝑀((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → (𝑀‘∪ 𝑥) = (Σ^‘(𝑀 ↾ 𝑥))))) |
5 | 4 | simplld 764 | . . 3 ⊢ (𝜑 → (𝑀:dom 𝑀⟶(0[,]+∞) ∧ dom 𝑀 ∈ SAlg)) |
6 | 5 | simprd 495 | . 2 ⊢ (𝜑 → dom 𝑀 ∈ SAlg) |
7 | 1, 6 | eqeltrid 2838 | 1 ⊢ (𝜑 → 𝑆 ∈ SAlg) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2101 ∀wral 3059 ∅c0 4259 𝒫 cpw 4536 ∪ cuni 4841 Disj wdisj 5042 class class class wbr 5077 dom cdm 5591 ↾ cres 5593 ⟶wf 6443 ‘cfv 6447 (class class class)co 7295 ωcom 7732 ≼ cdom 8751 0cc0 10899 +∞cpnf 11034 [,]cicc 13110 SAlgcsalg 43884 Σ^csumge0 43936 Meascmea 44023 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2103 ax-9 2111 ax-10 2132 ax-11 2149 ax-12 2166 ax-ext 2704 ax-rep 5212 ax-sep 5226 ax-nul 5233 ax-pr 5355 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2063 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2884 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3223 df-rab 3224 df-v 3436 df-sbc 3719 df-csb 3835 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-nul 4260 df-if 4463 df-pw 4538 df-sn 4565 df-pr 4567 df-op 4571 df-uni 4842 df-iun 4929 df-br 5078 df-opab 5140 df-mpt 5161 df-id 5491 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-iota 6399 df-fun 6449 df-fn 6450 df-f 6451 df-f1 6452 df-fo 6453 df-f1o 6454 df-fv 6455 df-mea 44024 |
This theorem is referenced by: meadjuni 44031 meassle 44037 meaunle 44038 meaiunlelem 44042 meadif 44053 meaiuninclem 44054 meaiuninc3v 44058 meaiininclem 44060 dmovnsal 44186 hoimbllem 44204 ctvonmbl 44263 vonct 44267 |
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