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Theorem dmmeasal 40002
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 40001 . . . . 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 208 . . . 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 790 . . 3 (𝜑 → (𝑀:dom 𝑀⟶(0[,]+∞) ∧ dom 𝑀 ∈ SAlg))
65simprd 479 . 2 (𝜑 → dom 𝑀 ∈ SAlg)
71, 6syl5eqel 2702 1 (𝜑𝑆 ∈ SAlg)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wral 2907  c0 3896  𝒫 cpw 4135   cuni 4407  Disj wdisj 4588   class class class wbr 4618  dom cdm 5079  cres 5081  wf 5848  cfv 5852  (class class class)co 6610  ωcom 7019  cdom 7905  0cc0 9888  +∞cpnf 10023  [,]cicc 12128  SAlgcsalg 39861  Σ^csumge0 39912  Meascmea 39999
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-mea 40000
This theorem is referenced by:  meadjuni  40007  meassle  40013  meaunle  40014  meaiunlelem  40018  meadif  40029  meaiuninclem  40030  meaiininclem  40033  dmovnsal  40159  hoimbllem  40177  ctvonmbl  40236  vonct  40240
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