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 46073
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 46072 . . . . 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 766 . . 3 (𝜑 → (𝑀:dom 𝑀⟶(0[,]+∞) ∧ dom 𝑀 ∈ SAlg))
65simprd 494 . 2 (𝜑 → dom 𝑀 ∈ SAlg)
71, 6eqeltrid 2830 1 (𝜑𝑆 ∈ SAlg)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1534  wcel 2099  wral 3051  c0 4325  𝒫 cpw 4607   cuni 4913  Disj wdisj 5118   class class class wbr 5153  dom cdm 5682  cres 5684  wf 6550  cfv 6554  (class class class)co 7424  ωcom 7876  cdom 8972  0cc0 11158  +∞cpnf 11295  [,]cicc 13381  SAlgcsalg 45929  Σ^csumge0 45983  Meascmea 46070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-mea 46071
This theorem is referenced by:  meadjuni  46078  meassle  46084  meaunle  46085  meaiunlelem  46089  meadif  46100  meaiuninclem  46101  meaiuninc3v  46105  meaiininclem  46107  dmovnsal  46233  hoimbllem  46251  ctvonmbl  46310  vonct  46314
  Copyright terms: Public domain W3C validator