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Theorem measvun 30870
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 1128 . 2 ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝒫 𝑆 ∧ (𝐴 ≼ ω ∧ Disj 𝑥𝐴 𝑥)) → 𝐴 ∈ 𝒫 𝑆)
2 measbase 30858 . . . . . 6 (𝑀 ∈ (measures‘𝑆) → 𝑆 ran sigAlgebra)
3 ismeas 30860 . . . . . 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 259 . . . 4 (𝑀 ∈ (measures‘𝑆) → (𝑀:𝑆⟶(0[,]+∞) ∧ (𝑀‘∅) = 0 ∧ ∀𝑦 ∈ 𝒫 𝑆((𝑦 ≼ ω ∧ Disj 𝑥𝑦 𝑥) → (𝑀 𝑦) = Σ*𝑥𝑦(𝑀𝑥))))
65simp3d 1135 . . 3 (𝑀 ∈ (measures‘𝑆) → ∀𝑦 ∈ 𝒫 𝑆((𝑦 ≼ ω ∧ Disj 𝑥𝑦 𝑥) → (𝑀 𝑦) = Σ*𝑥𝑦(𝑀𝑥)))
763ad2ant1 1124 . 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 1129 . 2 ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝒫 𝑆 ∧ (𝐴 ≼ ω ∧ Disj 𝑥𝐴 𝑥)) → (𝐴 ≼ ω ∧ Disj 𝑥𝐴 𝑥))
9 breq1 4889 . . . . 5 (𝑦 = 𝐴 → (𝑦 ≼ ω ↔ 𝐴 ≼ ω))
10 disjeq1 4861 . . . . 5 (𝑦 = 𝐴 → (Disj 𝑥𝑦 𝑥Disj 𝑥𝐴 𝑥))
119, 10anbi12d 624 . . . 4 (𝑦 = 𝐴 → ((𝑦 ≼ ω ∧ Disj 𝑥𝑦 𝑥) ↔ (𝐴 ≼ ω ∧ Disj 𝑥𝐴 𝑥)))
12 unieq 4679 . . . . . 6 (𝑦 = 𝐴 𝑦 = 𝐴)
1312fveq2d 6450 . . . . 5 (𝑦 = 𝐴 → (𝑀 𝑦) = (𝑀 𝐴))
14 esumeq1 30694 . . . . 5 (𝑦 = 𝐴 → Σ*𝑥𝑦(𝑀𝑥) = Σ*𝑥𝐴(𝑀𝑥))
1513, 14eqeq12d 2792 . . . 4 (𝑦 = 𝐴 → ((𝑀 𝑦) = Σ*𝑥𝑦(𝑀𝑥) ↔ (𝑀 𝐴) = Σ*𝑥𝐴(𝑀𝑥)))
1611, 15imbi12d 336 . . 3 (𝑦 = 𝐴 → (((𝑦 ≼ ω ∧ Disj 𝑥𝑦 𝑥) → (𝑀 𝑦) = Σ*𝑥𝑦(𝑀𝑥)) ↔ ((𝐴 ≼ ω ∧ Disj 𝑥𝐴 𝑥) → (𝑀 𝐴) = Σ*𝑥𝐴(𝑀𝑥))))
1716rspcv 3506 . 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 198  wa 386  w3a 1071   = wceq 1601  wcel 2106  wral 3089  c0 4140  𝒫 cpw 4378   cuni 4671  Disj wdisj 4854   class class class wbr 4886  ran crn 5356  wf 6131  cfv 6135  (class class class)co 6922  ωcom 7343  cdom 8239  0cc0 10272  +∞cpnf 10408  [,]cicc 12490  Σ*cesum 30687  sigAlgebracsiga 30768  measurescmeas 30856
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 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-fal 1615  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ral 3094  df-rex 3095  df-rmo 3097  df-rab 3098  df-v 3399  df-sbc 3652  df-csb 3751  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-disj 4855  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-fv 6143  df-ov 6925  df-esum 30688  df-meas 30857
This theorem is referenced by:  measxun2  30871  measvunilem  30873  measssd  30876  measres  30883  measdivcstOLD  30885  measdivcst  30886  probcun  31079  totprobd  31087
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