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 43943 | . . . . 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 2844 | 1 ⊢ (𝜑 → 𝑆 ∈ SAlg) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2109 ∀wral 3065 ∅c0 4261 𝒫 cpw 4538 ∪ cuni 4844 Disj wdisj 5043 class class class wbr 5078 dom cdm 5588 ↾ cres 5590 ⟶wf 6426 ‘cfv 6430 (class class class)co 7268 ωcom 7700 ≼ cdom 8705 0cc0 10855 +∞cpnf 10990 [,]cicc 13064 SAlgcsalg 43803 Σ^csumge0 43854 Meascmea 43941 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-rep 5213 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 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-ral 3070 df-rex 3071 df-reu 3072 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4845 df-iun 4931 df-br 5079 df-opab 5141 df-mpt 5162 df-id 5488 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-mea 43942 |
This theorem is referenced by: meadjuni 43949 meassle 43955 meaunle 43956 meaiunlelem 43960 meadif 43971 meaiuninclem 43972 meaiuninc3v 43976 meaiininclem 43978 dmovnsal 44104 hoimbllem 44122 ctvonmbl 44181 vonct 44185 |
Copyright terms: Public domain | W3C validator |