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Theorem measvun 30816
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 1173 . 2 ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝒫 𝑆 ∧ (𝐴 ≼ ω ∧ Disj 𝑥𝐴 𝑥)) → 𝐴 ∈ 𝒫 𝑆)
2 measbase 30804 . . . . . 6 (𝑀 ∈ (measures‘𝑆) → 𝑆 ran sigAlgebra)
3 ismeas 30806 . . . . . 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 1180 . . 3 (𝑀 ∈ (measures‘𝑆) → ∀𝑦 ∈ 𝒫 𝑆((𝑦 ≼ ω ∧ Disj 𝑥𝑦 𝑥) → (𝑀 𝑦) = Σ*𝑥𝑦(𝑀𝑥)))
763ad2ant1 1169 . 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 1174 . 2 ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝒫 𝑆 ∧ (𝐴 ≼ ω ∧ Disj 𝑥𝐴 𝑥)) → (𝐴 ≼ ω ∧ Disj 𝑥𝐴 𝑥))
9 breq1 4875 . . . . 5 (𝑦 = 𝐴 → (𝑦 ≼ ω ↔ 𝐴 ≼ ω))
10 disjeq1 4847 . . . . 5 (𝑦 = 𝐴 → (Disj 𝑥𝑦 𝑥Disj 𝑥𝐴 𝑥))
119, 10anbi12d 626 . . . 4 (𝑦 = 𝐴 → ((𝑦 ≼ ω ∧ Disj 𝑥𝑦 𝑥) ↔ (𝐴 ≼ ω ∧ Disj 𝑥𝐴 𝑥)))
12 unieq 4665 . . . . . 6 (𝑦 = 𝐴 𝑦 = 𝐴)
1312fveq2d 6436 . . . . 5 (𝑦 = 𝐴 → (𝑀 𝑦) = (𝑀 𝐴))
14 esumeq1 30640 . . . . 5 (𝑦 = 𝐴 → Σ*𝑥𝑦(𝑀𝑥) = Σ*𝑥𝐴(𝑀𝑥))
1513, 14eqeq12d 2839 . . . 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 3521 . 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 1113   = wceq 1658  wcel 2166  wral 3116  c0 4143  𝒫 cpw 4377   cuni 4657  Disj wdisj 4840   class class class wbr 4872  ran crn 5342  wf 6118  cfv 6122  (class class class)co 6904  ωcom 7325  cdom 8219  0cc0 10251  +∞cpnf 10387  [,]cicc 12465  Σ*cesum 30633  sigAlgebracsiga 30714  measurescmeas 30802
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2390  ax-ext 2802  ax-sep 5004  ax-nul 5012  ax-pow 5064  ax-pr 5126  ax-un 7208
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-fal 1672  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2604  df-eu 2639  df-clab 2811  df-cleq 2817  df-clel 2820  df-nfc 2957  df-ral 3121  df-rex 3122  df-rmo 3124  df-rab 3125  df-v 3415  df-sbc 3662  df-csb 3757  df-dif 3800  df-un 3802  df-in 3804  df-ss 3811  df-nul 4144  df-if 4306  df-pw 4379  df-sn 4397  df-pr 4399  df-op 4403  df-uni 4658  df-disj 4841  df-br 4873  df-opab 4935  df-mpt 4952  df-id 5249  df-xp 5347  df-rel 5348  df-cnv 5349  df-co 5350  df-dm 5351  df-rn 5352  df-iota 6085  df-fun 6124  df-fn 6125  df-f 6126  df-fv 6130  df-ov 6907  df-esum 30634  df-meas 30803
This theorem is referenced by:  measxun2  30817  measvunilem  30819  measssd  30822  measres  30829  measdivcstOLD  30831  measdivcst  30832  probcun  31025  totprobd  31033
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