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	⊢ (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 46072	. . . . 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 766	. . 3 ⊢ (𝜑 → (𝑀:dom 𝑀⟶(0[,]+∞) ∧ dom 𝑀 ∈ SAlg))
6	5	simprd 494	. 2 ⊢ (𝜑 → dom 𝑀 ∈ SAlg)
7	1, 6	eqeltrid 2830	1 ⊢ (𝜑 → 𝑆 ∈ SAlg)

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