Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dmmeasal Structured version   Visualization version   GIF version

Theorem dmmeasal 39345
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 39344 . . . . 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 207 . . . 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 787 . . 3 (𝜑 → (𝑀:dom 𝑀⟶(0[,]+∞) ∧ dom 𝑀 ∈ SAlg))
65simprd 478 . 2 (𝜑 → dom 𝑀 ∈ SAlg)
71, 6syl5eqel 2692 1 (𝜑𝑆 ∈ SAlg)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  wcel 1977  wral 2896  c0 3874  𝒫 cpw 4108   cuni 4372  Disj wdisj 4553   class class class wbr 4583  dom cdm 5038  cres 5040  wf 5800  cfv 5804  (class class class)co 6549  ωcom 6957  cdom 7839  0cc0 9815  +∞cpnf 9950  [,]cicc 12049  SAlgcsalg 39204  Σ^csumge0 39255  Meascmea 39342
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-mea 39343
This theorem is referenced by:  meadjuni  39350  meassle  39356  meaunle  39357  meaiunlelem  39361  meadif  39372  meaiuninclem  39373  meaiininclem  39376  dmovnsal  39502  hoimbllem  39520  ctvonmbl  39580  vonct  39584
  Copyright terms: Public domain W3C validator