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Theorem dmmeasal 43944
Description: The domain of a measure is a sigma-algebra. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypotheses
Ref Expression
dmmeasal.m (𝜑𝑀 ∈ Meas)
dmmeasal.s 𝑆 = dom 𝑀
Assertion
Ref Expression
dmmeasal (𝜑𝑆 ∈ SAlg)

Proof of Theorem dmmeasal
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dmmeasal.s . 2 𝑆 = dom 𝑀
2 dmmeasal.m . . . . 5 (𝜑𝑀 ∈ Meas)
3 ismea 43943 . . . . 5 (𝑀 ∈ Meas ↔ (((𝑀:dom 𝑀⟶(0[,]+∞) ∧ dom 𝑀 ∈ SAlg) ∧ (𝑀‘∅) = 0) ∧ ∀𝑥 ∈ 𝒫 dom 𝑀((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → (𝑀 𝑥) = (Σ^‘(𝑀𝑥)))))
42, 3sylib 217 . . . 4 (𝜑 → (((𝑀:dom 𝑀⟶(0[,]+∞) ∧ dom 𝑀 ∈ SAlg) ∧ (𝑀‘∅) = 0) ∧ ∀𝑥 ∈ 𝒫 dom 𝑀((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → (𝑀 𝑥) = (Σ^‘(𝑀𝑥)))))
54simplld 764 . . 3 (𝜑 → (𝑀:dom 𝑀⟶(0[,]+∞) ∧ dom 𝑀 ∈ SAlg))
65simprd 495 . 2 (𝜑 → dom 𝑀 ∈ SAlg)
71, 6eqeltrid 2844 1 (𝜑𝑆 ∈ 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
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