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Mathbox for Glauco Siliprandi |
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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 46072 | . . . . 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 766 | . . 3 ⊢ (𝜑 → (𝑀:dom 𝑀⟶(0[,]+∞) ∧ dom 𝑀 ∈ SAlg)) |
6 | 5 | simprd 494 | . 2 ⊢ (𝜑 → dom 𝑀 ∈ SAlg) |
7 | 1, 6 | eqeltrid 2830 | 1 ⊢ (𝜑 → 𝑆 ∈ SAlg) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ∀wral 3051 ∅c0 4325 𝒫 cpw 4607 ∪ cuni 4913 Disj wdisj 5118 class class class wbr 5153 dom cdm 5682 ↾ cres 5684 ⟶wf 6550 ‘cfv 6554 (class class class)co 7424 ωcom 7876 ≼ cdom 8972 0cc0 11158 +∞cpnf 11295 [,]cicc 13381 SAlgcsalg 45929 Σ^csumge0 45983 Meascmea 46070 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-mea 46071 |
This theorem is referenced by: meadjuni 46078 meassle 46084 meaunle 46085 meaiunlelem 46089 meadif 46100 meaiuninclem 46101 meaiuninc3v 46105 meaiininclem 46107 dmovnsal 46233 hoimbllem 46251 ctvonmbl 46310 vonct 46314 |
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