Theorem dmmeasal 44026

Description: The domain of a measure is a sigma-algebra. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypotheses

Ref	Expression
dmmeasal.m	⊢ (𝜑 → 𝑀 ∈ Meas)
dmmeasal.s	⊢ 𝑆 = dom 𝑀

Assertion

Ref	Expression
dmmeasal	⊢ (𝜑 → 𝑆 ∈ SAlg)

Proof of Theorem dmmeasal

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dmmeasal.s	. 2 ⊢ 𝑆 = dom 𝑀
2		dmmeasal.m	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ Meas)
3		ismea 44025	. . . . 5 ⊢ (𝑀 ∈ Meas ↔ (((𝑀:dom 𝑀⟶(0[,]+∞) ∧ dom 𝑀 ∈ SAlg) ∧ (𝑀‘∅) = 0) ∧ ∀𝑥 ∈ 𝒫 dom 𝑀((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → (𝑀‘∪ 𝑥) = (Σ^‘(𝑀 ↾ 𝑥)))))
4	2, 3	sylib 217	. . . 4 ⊢ (𝜑 → (((𝑀:dom 𝑀⟶(0[,]+∞) ∧ dom 𝑀 ∈ SAlg) ∧ (𝑀‘∅) = 0) ∧ ∀𝑥 ∈ 𝒫 dom 𝑀((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → (𝑀‘∪ 𝑥) = (Σ^‘(𝑀 ↾ 𝑥)))))
5	4	simplld 764	. . 3 ⊢ (𝜑 → (𝑀:dom 𝑀⟶(0[,]+∞) ∧ dom 𝑀 ∈ SAlg))
6	5	simprd 495	. 2 ⊢ (𝜑 → dom 𝑀 ∈ SAlg)
7	1, 6	eqeltrid 2838	1 ⊢ (𝜑 → 𝑆 ∈ SAlg)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2101  ∀wral 3059  ∅c0 4259  𝒫 cpw 4536  ∪ cuni 4841  Disj wdisj 5042   class class class wbr 5077  dom cdm 5591   ↾ cres 5593  ⟶wf 6443  ‘cfv 6447  (class class class)co 7295  ωcom 7732   ≼ cdom 8751  0cc0 10899  +∞cpnf 11034  [,]cicc 13110  SAlgcsalg 43884  Σ^{^}csumge0 43936  Meascmea 44023

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2103  ax-9 2111  ax-10 2132  ax-11 2149  ax-12 2166  ax-ext 2704  ax-rep 5212  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 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2063  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2884  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3223  df-rab 3224  df-v 3436  df-sbc 3719  df-csb 3835  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4260  df-if 4463  df-pw 4538  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4842  df-iun 4929  df-br 5078  df-opab 5140  df-mpt 5161  df-id 5491  df-xp 5597  df-rel 5598  df-cnv 5599  df-co 5600  df-dm 5601  df-rn 5602  df-res 5603  df-ima 5604  df-iota 6399  df-fun 6449  df-fn 6450  df-f 6451  df-f1 6452  df-fo 6453  df-f1o 6454  df-fv 6455  df-mea 44024

This theorem is referenced by:  meadjuni  44031  meassle  44037  meaunle  44038  meaiunlelem  44042  meadif  44053  meaiuninclem  44054  meaiuninc3v  44058  meaiininclem  44060  dmovnsal  44186  hoimbllem  44204  ctvonmbl  44263  vonct  44267