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Theorem measvun 34190
Description: The measure of a countable disjoint union is the sum of the measures. (Contributed by Thierry Arnoux, 26-Dec-2016.)
Assertion
Ref Expression
measvun ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝒫 𝑆 ∧ (𝐴 ≼ ω ∧ Disj 𝑥𝐴 𝑥)) → (𝑀 𝐴) = Σ*𝑥𝐴(𝑀𝑥))
Distinct variable groups:   𝑥,𝐴   𝑥,𝑀
Allowed substitution hint:   𝑆(𝑥)

Proof of Theorem measvun
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 simp2 1136 . 2 ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝒫 𝑆 ∧ (𝐴 ≼ ω ∧ Disj 𝑥𝐴 𝑥)) → 𝐴 ∈ 𝒫 𝑆)
2 measbase 34178 . . . . . 6 (𝑀 ∈ (measures‘𝑆) → 𝑆 ran sigAlgebra)
3 ismeas 34180 . . . . . 6 (𝑆 ran sigAlgebra → (𝑀 ∈ (measures‘𝑆) ↔ (𝑀:𝑆⟶(0[,]+∞) ∧ (𝑀‘∅) = 0 ∧ ∀𝑦 ∈ 𝒫 𝑆((𝑦 ≼ ω ∧ Disj 𝑥𝑦 𝑥) → (𝑀 𝑦) = Σ*𝑥𝑦(𝑀𝑥)))))
42, 3syl 17 . . . . 5 (𝑀 ∈ (measures‘𝑆) → (𝑀 ∈ (measures‘𝑆) ↔ (𝑀:𝑆⟶(0[,]+∞) ∧ (𝑀‘∅) = 0 ∧ ∀𝑦 ∈ 𝒫 𝑆((𝑦 ≼ ω ∧ Disj 𝑥𝑦 𝑥) → (𝑀 𝑦) = Σ*𝑥𝑦(𝑀𝑥)))))
54ibi 267 . . . 4 (𝑀 ∈ (measures‘𝑆) → (𝑀:𝑆⟶(0[,]+∞) ∧ (𝑀‘∅) = 0 ∧ ∀𝑦 ∈ 𝒫 𝑆((𝑦 ≼ ω ∧ Disj 𝑥𝑦 𝑥) → (𝑀 𝑦) = Σ*𝑥𝑦(𝑀𝑥))))
65simp3d 1143 . . 3 (𝑀 ∈ (measures‘𝑆) → ∀𝑦 ∈ 𝒫 𝑆((𝑦 ≼ ω ∧ Disj 𝑥𝑦 𝑥) → (𝑀 𝑦) = Σ*𝑥𝑦(𝑀𝑥)))
763ad2ant1 1132 . 2 ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝒫 𝑆 ∧ (𝐴 ≼ ω ∧ Disj 𝑥𝐴 𝑥)) → ∀𝑦 ∈ 𝒫 𝑆((𝑦 ≼ ω ∧ Disj 𝑥𝑦 𝑥) → (𝑀 𝑦) = Σ*𝑥𝑦(𝑀𝑥)))
8 simp3 1137 . 2 ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝒫 𝑆 ∧ (𝐴 ≼ ω ∧ Disj 𝑥𝐴 𝑥)) → (𝐴 ≼ ω ∧ Disj 𝑥𝐴 𝑥))
9 breq1 5151 . . . . 5 (𝑦 = 𝐴 → (𝑦 ≼ ω ↔ 𝐴 ≼ ω))
10 disjeq1 5122 . . . . 5 (𝑦 = 𝐴 → (Disj 𝑥𝑦 𝑥Disj 𝑥𝐴 𝑥))
119, 10anbi12d 632 . . . 4 (𝑦 = 𝐴 → ((𝑦 ≼ ω ∧ Disj 𝑥𝑦 𝑥) ↔ (𝐴 ≼ ω ∧ Disj 𝑥𝐴 𝑥)))
12 unieq 4923 . . . . . 6 (𝑦 = 𝐴 𝑦 = 𝐴)
1312fveq2d 6911 . . . . 5 (𝑦 = 𝐴 → (𝑀 𝑦) = (𝑀 𝐴))
14 esumeq1 34015 . . . . 5 (𝑦 = 𝐴 → Σ*𝑥𝑦(𝑀𝑥) = Σ*𝑥𝐴(𝑀𝑥))
1513, 14eqeq12d 2751 . . . 4 (𝑦 = 𝐴 → ((𝑀 𝑦) = Σ*𝑥𝑦(𝑀𝑥) ↔ (𝑀 𝐴) = Σ*𝑥𝐴(𝑀𝑥)))
1611, 15imbi12d 344 . . 3 (𝑦 = 𝐴 → (((𝑦 ≼ ω ∧ Disj 𝑥𝑦 𝑥) → (𝑀 𝑦) = Σ*𝑥𝑦(𝑀𝑥)) ↔ ((𝐴 ≼ ω ∧ Disj 𝑥𝐴 𝑥) → (𝑀 𝐴) = Σ*𝑥𝐴(𝑀𝑥))))
1716rspcv 3618 . 2 (𝐴 ∈ 𝒫 𝑆 → (∀𝑦 ∈ 𝒫 𝑆((𝑦 ≼ ω ∧ Disj 𝑥𝑦 𝑥) → (𝑀 𝑦) = Σ*𝑥𝑦(𝑀𝑥)) → ((𝐴 ≼ ω ∧ Disj 𝑥𝐴 𝑥) → (𝑀 𝐴) = Σ*𝑥𝐴(𝑀𝑥))))
181, 7, 8, 17syl3c 66 1 ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝒫 𝑆 ∧ (𝐴 ≼ ω ∧ Disj 𝑥𝐴 𝑥)) → (𝑀 𝐴) = Σ*𝑥𝐴(𝑀𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wral 3059  c0 4339  𝒫 cpw 4605   cuni 4912  Disj wdisj 5115   class class class wbr 5148  ran crn 5690  wf 6559  cfv 6563  (class class class)co 7431  ωcom 7887  cdom 8982  0cc0 11153  +∞cpnf 11290  [,]cicc 13387  Σ*cesum 34008  sigAlgebracsiga 34089  measurescmeas 34176
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-disj 5116  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-esum 34009  df-meas 34177
This theorem is referenced by:  measxun2  34191  measvunilem  34193  measssd  34196  measres  34203  measdivcst  34205  measdivcstALTV  34206  probcun  34400  totprobd  34408
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