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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 44766 | . . . . 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 767 | . . 3 ⊢ (𝜑 → (𝑀:dom 𝑀⟶(0[,]+∞) ∧ dom 𝑀 ∈ SAlg)) |
6 | 5 | simprd 497 | . 2 ⊢ (𝜑 → dom 𝑀 ∈ SAlg) |
7 | 1, 6 | eqeltrid 2842 | 1 ⊢ (𝜑 → 𝑆 ∈ SAlg) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3065 ∅c0 4287 𝒫 cpw 4565 ∪ cuni 4870 Disj wdisj 5075 class class class wbr 5110 dom cdm 5638 ↾ cres 5640 ⟶wf 6497 ‘cfv 6501 (class class class)co 7362 ωcom 7807 ≼ cdom 8888 0cc0 11058 +∞cpnf 11193 [,]cicc 13274 SAlgcsalg 44623 Σ^csumge0 44677 Meascmea 44764 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pr 5389 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-mea 44765 |
This theorem is referenced by: meadjuni 44772 meassle 44778 meaunle 44779 meaiunlelem 44783 meadif 44794 meaiuninclem 44795 meaiuninc3v 44799 meaiininclem 44801 dmovnsal 44927 hoimbllem 44945 ctvonmbl 45004 vonct 45008 |
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