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Theorem measvun 33493
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 ∈ (measuresβ€˜π‘†) ∧ 𝐴 ∈ 𝒫 𝑆 ∧ (𝐴 β‰Ό Ο‰ ∧ Disj π‘₯ ∈ 𝐴 π‘₯)) β†’ (π‘€β€˜βˆͺ 𝐴) = Ξ£*π‘₯ ∈ 𝐴(π‘€β€˜π‘₯))
Distinct variable groups:   π‘₯,𝐴   π‘₯,𝑀
Allowed substitution hint:   𝑆(π‘₯)

Proof of Theorem measvun
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 simp2 1137 . 2 ((𝑀 ∈ (measuresβ€˜π‘†) ∧ 𝐴 ∈ 𝒫 𝑆 ∧ (𝐴 β‰Ό Ο‰ ∧ Disj π‘₯ ∈ 𝐴 π‘₯)) β†’ 𝐴 ∈ 𝒫 𝑆)
2 measbase 33481 . . . . . 6 (𝑀 ∈ (measuresβ€˜π‘†) β†’ 𝑆 ∈ βˆͺ ran sigAlgebra)
3 ismeas 33483 . . . . . 6 (𝑆 ∈ βˆͺ ran sigAlgebra β†’ (𝑀 ∈ (measuresβ€˜π‘†) ↔ (𝑀:π‘†βŸΆ(0[,]+∞) ∧ (π‘€β€˜βˆ…) = 0 ∧ βˆ€π‘¦ ∈ 𝒫 𝑆((𝑦 β‰Ό Ο‰ ∧ Disj π‘₯ ∈ 𝑦 π‘₯) β†’ (π‘€β€˜βˆͺ 𝑦) = Ξ£*π‘₯ ∈ 𝑦(π‘€β€˜π‘₯)))))
42, 3syl 17 . . . . 5 (𝑀 ∈ (measuresβ€˜π‘†) β†’ (𝑀 ∈ (measuresβ€˜π‘†) ↔ (𝑀:π‘†βŸΆ(0[,]+∞) ∧ (π‘€β€˜βˆ…) = 0 ∧ βˆ€π‘¦ ∈ 𝒫 𝑆((𝑦 β‰Ό Ο‰ ∧ Disj π‘₯ ∈ 𝑦 π‘₯) β†’ (π‘€β€˜βˆͺ 𝑦) = Ξ£*π‘₯ ∈ 𝑦(π‘€β€˜π‘₯)))))
54ibi 266 . . . 4 (𝑀 ∈ (measuresβ€˜π‘†) β†’ (𝑀:π‘†βŸΆ(0[,]+∞) ∧ (π‘€β€˜βˆ…) = 0 ∧ βˆ€π‘¦ ∈ 𝒫 𝑆((𝑦 β‰Ό Ο‰ ∧ Disj π‘₯ ∈ 𝑦 π‘₯) β†’ (π‘€β€˜βˆͺ 𝑦) = Ξ£*π‘₯ ∈ 𝑦(π‘€β€˜π‘₯))))
65simp3d 1144 . . 3 (𝑀 ∈ (measuresβ€˜π‘†) β†’ βˆ€π‘¦ ∈ 𝒫 𝑆((𝑦 β‰Ό Ο‰ ∧ Disj π‘₯ ∈ 𝑦 π‘₯) β†’ (π‘€β€˜βˆͺ 𝑦) = Ξ£*π‘₯ ∈ 𝑦(π‘€β€˜π‘₯)))
763ad2ant1 1133 . 2 ((𝑀 ∈ (measuresβ€˜π‘†) ∧ 𝐴 ∈ 𝒫 𝑆 ∧ (𝐴 β‰Ό Ο‰ ∧ Disj π‘₯ ∈ 𝐴 π‘₯)) β†’ βˆ€π‘¦ ∈ 𝒫 𝑆((𝑦 β‰Ό Ο‰ ∧ Disj π‘₯ ∈ 𝑦 π‘₯) β†’ (π‘€β€˜βˆͺ 𝑦) = Ξ£*π‘₯ ∈ 𝑦(π‘€β€˜π‘₯)))
8 simp3 1138 . 2 ((𝑀 ∈ (measuresβ€˜π‘†) ∧ 𝐴 ∈ 𝒫 𝑆 ∧ (𝐴 β‰Ό Ο‰ ∧ Disj π‘₯ ∈ 𝐴 π‘₯)) β†’ (𝐴 β‰Ό Ο‰ ∧ Disj π‘₯ ∈ 𝐴 π‘₯))
9 breq1 5151 . . . . 5 (𝑦 = 𝐴 β†’ (𝑦 β‰Ό Ο‰ ↔ 𝐴 β‰Ό Ο‰))
10 disjeq1 5120 . . . . 5 (𝑦 = 𝐴 β†’ (Disj π‘₯ ∈ 𝑦 π‘₯ ↔ Disj π‘₯ ∈ 𝐴 π‘₯))
119, 10anbi12d 631 . . . 4 (𝑦 = 𝐴 β†’ ((𝑦 β‰Ό Ο‰ ∧ Disj π‘₯ ∈ 𝑦 π‘₯) ↔ (𝐴 β‰Ό Ο‰ ∧ Disj π‘₯ ∈ 𝐴 π‘₯)))
12 unieq 4919 . . . . . 6 (𝑦 = 𝐴 β†’ βˆͺ 𝑦 = βˆͺ 𝐴)
1312fveq2d 6895 . . . . 5 (𝑦 = 𝐴 β†’ (π‘€β€˜βˆͺ 𝑦) = (π‘€β€˜βˆͺ 𝐴))
14 esumeq1 33318 . . . . 5 (𝑦 = 𝐴 β†’ Ξ£*π‘₯ ∈ 𝑦(π‘€β€˜π‘₯) = Ξ£*π‘₯ ∈ 𝐴(π‘€β€˜π‘₯))
1513, 14eqeq12d 2748 . . . 4 (𝑦 = 𝐴 β†’ ((π‘€β€˜βˆͺ 𝑦) = Ξ£*π‘₯ ∈ 𝑦(π‘€β€˜π‘₯) ↔ (π‘€β€˜βˆͺ 𝐴) = Ξ£*π‘₯ ∈ 𝐴(π‘€β€˜π‘₯)))
1611, 15imbi12d 344 . . 3 (𝑦 = 𝐴 β†’ (((𝑦 β‰Ό Ο‰ ∧ Disj π‘₯ ∈ 𝑦 π‘₯) β†’ (π‘€β€˜βˆͺ 𝑦) = Ξ£*π‘₯ ∈ 𝑦(π‘€β€˜π‘₯)) ↔ ((𝐴 β‰Ό Ο‰ ∧ Disj π‘₯ ∈ 𝐴 π‘₯) β†’ (π‘€β€˜βˆͺ 𝐴) = Ξ£*π‘₯ ∈ 𝐴(π‘€β€˜π‘₯))))
1716rspcv 3608 . 2 (𝐴 ∈ 𝒫 𝑆 β†’ (βˆ€π‘¦ ∈ 𝒫 𝑆((𝑦 β‰Ό Ο‰ ∧ Disj π‘₯ ∈ 𝑦 π‘₯) β†’ (π‘€β€˜βˆͺ 𝑦) = Ξ£*π‘₯ ∈ 𝑦(π‘€β€˜π‘₯)) β†’ ((𝐴 β‰Ό Ο‰ ∧ Disj π‘₯ ∈ 𝐴 π‘₯) β†’ (π‘€β€˜βˆͺ 𝐴) = Ξ£*π‘₯ ∈ 𝐴(π‘€β€˜π‘₯))))
181, 7, 8, 17syl3c 66 1 ((𝑀 ∈ (measuresβ€˜π‘†) ∧ 𝐴 ∈ 𝒫 𝑆 ∧ (𝐴 β‰Ό Ο‰ ∧ Disj π‘₯ ∈ 𝐴 π‘₯)) β†’ (π‘€β€˜βˆͺ 𝐴) = Ξ£*π‘₯ ∈ 𝐴(π‘€β€˜π‘₯))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  βˆ…c0 4322  π’« cpw 4602  βˆͺ cuni 4908  Disj wdisj 5113   class class class wbr 5148  ran crn 5677  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7411  Ο‰com 7857   β‰Ό cdom 8939  0cc0 11112  +∞cpnf 11249  [,]cicc 13331  Ξ£*cesum 33311  sigAlgebracsiga 33392  measurescmeas 33479
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-disj 5114  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-ov 7414  df-esum 33312  df-meas 33480
This theorem is referenced by:  measxun2  33494  measvunilem  33496  measssd  33499  measres  33506  measdivcst  33508  measdivcstALTV  33509  probcun  33703  totprobd  33711
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