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Theorem measvun 33144
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 1138 . 2 ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝒫 𝑆 ∧ (𝐴 ≼ ω ∧ Disj 𝑥𝐴 𝑥)) → 𝐴 ∈ 𝒫 𝑆)
2 measbase 33132 . . . . . 6 (𝑀 ∈ (measures‘𝑆) → 𝑆 ran sigAlgebra)
3 ismeas 33134 . . . . . 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 1145 . . 3 (𝑀 ∈ (measures‘𝑆) → ∀𝑦 ∈ 𝒫 𝑆((𝑦 ≼ ω ∧ Disj 𝑥𝑦 𝑥) → (𝑀 𝑦) = Σ*𝑥𝑦(𝑀𝑥)))
763ad2ant1 1134 . 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 1139 . 2 ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝒫 𝑆 ∧ (𝐴 ≼ ω ∧ Disj 𝑥𝐴 𝑥)) → (𝐴 ≼ ω ∧ Disj 𝑥𝐴 𝑥))
9 breq1 5149 . . . . 5 (𝑦 = 𝐴 → (𝑦 ≼ ω ↔ 𝐴 ≼ ω))
10 disjeq1 5118 . . . . 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 4917 . . . . . 6 (𝑦 = 𝐴 𝑦 = 𝐴)
1312fveq2d 6891 . . . . 5 (𝑦 = 𝐴 → (𝑀 𝑦) = (𝑀 𝐴))
14 esumeq1 32969 . . . . 5 (𝑦 = 𝐴 → Σ*𝑥𝑦(𝑀𝑥) = Σ*𝑥𝐴(𝑀𝑥))
1513, 14eqeq12d 2749 . . . 4 (𝑦 = 𝐴 → ((𝑀 𝑦) = Σ*𝑥𝑦(𝑀𝑥) ↔ (𝑀 𝐴) = Σ*𝑥𝐴(𝑀𝑥)))
1611, 15imbi12d 345 . . 3 (𝑦 = 𝐴 → (((𝑦 ≼ ω ∧ Disj 𝑥𝑦 𝑥) → (𝑀 𝑦) = Σ*𝑥𝑦(𝑀𝑥)) ↔ ((𝐴 ≼ ω ∧ Disj 𝑥𝐴 𝑥) → (𝑀 𝐴) = Σ*𝑥𝐴(𝑀𝑥))))
1716rspcv 3607 . 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 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  wral 3062  c0 4320  𝒫 cpw 4600   cuni 4906  Disj wdisj 5111   class class class wbr 5146  ran crn 5675  wf 6535  cfv 6539  (class class class)co 7403  ωcom 7849  cdom 8932  0cc0 11105  +∞cpnf 11240  [,]cicc 13322  Σ*cesum 32962  sigAlgebracsiga 33043  measurescmeas 33130
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5297  ax-nul 5304  ax-pow 5361  ax-pr 5425  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-rab 3434  df-v 3477  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4527  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4907  df-disj 5112  df-br 5147  df-opab 5209  df-mpt 5230  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-iota 6491  df-fun 6541  df-fn 6542  df-f 6543  df-fv 6547  df-ov 7406  df-esum 32963  df-meas 33131
This theorem is referenced by:  measxun2  33145  measvunilem  33147  measssd  33150  measres  33157  measdivcst  33159  measdivcstALTV  33160  probcun  33354  totprobd  33362
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