Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
||
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 42732 | . . . . 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 220 | . . . 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 498 | . 2 ⊢ (𝜑 → dom 𝑀 ∈ SAlg) |
7 | 1, 6 | eqeltrid 2917 | 1 ⊢ (𝜑 → 𝑆 ∈ SAlg) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∀wral 3138 ∅c0 4290 𝒫 cpw 4538 ∪ cuni 4837 Disj wdisj 5030 class class class wbr 5065 dom cdm 5554 ↾ cres 5556 ⟶wf 6350 ‘cfv 6354 (class class class)co 7155 ωcom 7579 ≼ cdom 8506 0cc0 10536 +∞cpnf 10671 [,]cicc 12740 SAlgcsalg 42592 Σ^csumge0 42643 Meascmea 42730 |
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-rep 5189 ax-sep 5202 ax-nul 5209 ax-pr 5329 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 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-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 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 4567 df-pr 4569 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-mea 42731 |
This theorem is referenced by: meadjuni 42738 meassle 42744 meaunle 42745 meaiunlelem 42749 meadif 42760 meaiuninclem 42761 meaiuninc3v 42765 meaiininclem 42767 dmovnsal 42893 hoimbllem 42911 ctvonmbl 42970 vonct 42974 |
Copyright terms: Public domain | W3C validator |