Theorem dmmeasal 41603

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 41602	. . . . 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 210	. . . 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 758	. . 3 ⊢ (𝜑 → (𝑀:dom 𝑀⟶(0[,]+∞) ∧ dom 𝑀 ∈ SAlg))
6	5	simprd 491	. 2 ⊢ (𝜑 → dom 𝑀 ∈ SAlg)
7	1, 6	syl5eqel 2863	1 ⊢ (𝜑 → 𝑆 ∈ SAlg)

Syntax hints:   → wi 4   ∧ wa 386   = wceq 1601   ∈ wcel 2107  ∀wral 3090  ∅c0 4141  𝒫 cpw 4379  ∪ cuni 4673  Disj wdisj 4856   class class class wbr 4888  dom cdm 5357   ↾ cres 5359  ⟶wf 6133  ‘cfv 6137  (class class class)co 6924  ωcom 7345   ≼ cdom 8241  0cc0 10274  +∞cpnf 10410  [,]cicc 12494  SAlgcsalg 41462  Σ^{^}csumge0 41513  Meascmea 41600

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5008  ax-sep 5019  ax-nul 5027  ax-pr 5140

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-mea 41601

This theorem is referenced by:  meadjuni  41608  meassle  41614  meaunle  41615  meaiunlelem  41619  meadif  41630  meaiuninclem  41631  meaiuninc3v  41635  meaiininclem  41637  dmovnsal  41763  hoimbllem  41781  ctvonmbl  41840  vonct  41844