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Theorem dmmeasal 42948
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 42947 . . . . 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 221 . . . 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 767 . . 3 (𝜑 → (𝑀:dom 𝑀⟶(0[,]+∞) ∧ dom 𝑀 ∈ SAlg))
65simprd 499 . 2 (𝜑 → dom 𝑀 ∈ SAlg)
71, 6eqeltrid 2920 1 (𝜑𝑆 ∈ SAlg)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  wral 3132  c0 4274  𝒫 cpw 4520   cuni 4821  Disj wdisj 5014   class class class wbr 5049  dom cdm 5538  cres 5540  wf 6334  cfv 6338  (class class class)co 7140  ωcom 7565  cdom 8492  0cc0 10524  +∞cpnf 10659  [,]cicc 12729  SAlgcsalg 42807  Σ^csumge0 42858  Meascmea 42945
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5173  ax-sep 5186  ax-nul 5193  ax-pr 5313
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-ral 3137  df-rex 3138  df-reu 3139  df-rab 3141  df-v 3481  df-sbc 3758  df-csb 3866  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4822  df-iun 4904  df-br 5050  df-opab 5112  df-mpt 5130  df-id 5443  df-xp 5544  df-rel 5545  df-cnv 5546  df-co 5547  df-dm 5548  df-rn 5549  df-res 5550  df-ima 5551  df-iota 6297  df-fun 6340  df-fn 6341  df-f 6342  df-f1 6343  df-fo 6344  df-f1o 6345  df-fv 6346  df-mea 42946
This theorem is referenced by:  meadjuni  42953  meassle  42959  meaunle  42960  meaiunlelem  42964  meadif  42975  meaiuninclem  42976  meaiuninc3v  42980  meaiininclem  42982  dmovnsal  43108  hoimbllem  43126  ctvonmbl  43185  vonct  43189
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