Theorem dmmeasal 46898

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 46897	. . . . 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 218	. . . 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 768	. . 3 ⊢ (𝜑 → (𝑀:dom 𝑀⟶(0[,]+∞) ∧ dom 𝑀 ∈ SAlg))
6	5	simprd 495	. 2 ⊢ (𝜑 → dom 𝑀 ∈ SAlg)
7	1, 6	eqeltrid 2841	1 ⊢ (𝜑 → 𝑆 ∈ SAlg)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∅c0 4274  𝒫 cpw 4542  ∪ cuni 4851  Disj wdisj 5053   class class class wbr 5086  dom cdm 5624   ↾ cres 5626  ⟶wf 6488  ‘cfv 6492  (class class class)co 7360  ωcom 7810   ≼ cdom 8884  0cc0 11029  +∞cpnf 11167  [,]cicc 13292  SAlgcsalg 46754  Σ^{^}csumge0 46808  Meascmea 46895

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 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-mea 46896

This theorem is referenced by:  meadjuni  46903  meassle  46909  meaunle  46910  meaiunlelem  46914  meadif  46925  meaiuninclem  46926  meaiuninc3v  46930  meaiininclem  46932  dmovnsal  47058  hoimbllem  47076  ctvonmbl  47135  vonct  47139