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Theorem measvun 30077
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 1060 . 2 ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝒫 𝑆 ∧ (𝐴 ≼ ω ∧ Disj 𝑥𝐴 𝑥)) → 𝐴 ∈ 𝒫 𝑆)
2 measbase 30065 . . . . . 6 (𝑀 ∈ (measures‘𝑆) → 𝑆 ran sigAlgebra)
3 ismeas 30067 . . . . . 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 256 . . . 4 (𝑀 ∈ (measures‘𝑆) → (𝑀:𝑆⟶(0[,]+∞) ∧ (𝑀‘∅) = 0 ∧ ∀𝑦 ∈ 𝒫 𝑆((𝑦 ≼ ω ∧ Disj 𝑥𝑦 𝑥) → (𝑀 𝑦) = Σ*𝑥𝑦(𝑀𝑥))))
65simp3d 1073 . . 3 (𝑀 ∈ (measures‘𝑆) → ∀𝑦 ∈ 𝒫 𝑆((𝑦 ≼ ω ∧ Disj 𝑥𝑦 𝑥) → (𝑀 𝑦) = Σ*𝑥𝑦(𝑀𝑥)))
763ad2ant1 1080 . 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 1061 . 2 ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝒫 𝑆 ∧ (𝐴 ≼ ω ∧ Disj 𝑥𝐴 𝑥)) → (𝐴 ≼ ω ∧ Disj 𝑥𝐴 𝑥))
9 breq1 4621 . . . . 5 (𝑦 = 𝐴 → (𝑦 ≼ ω ↔ 𝐴 ≼ ω))
10 disjeq1 4595 . . . . 5 (𝑦 = 𝐴 → (Disj 𝑥𝑦 𝑥Disj 𝑥𝐴 𝑥))
119, 10anbi12d 746 . . . 4 (𝑦 = 𝐴 → ((𝑦 ≼ ω ∧ Disj 𝑥𝑦 𝑥) ↔ (𝐴 ≼ ω ∧ Disj 𝑥𝐴 𝑥)))
12 unieq 4415 . . . . . 6 (𝑦 = 𝐴 𝑦 = 𝐴)
1312fveq2d 6157 . . . . 5 (𝑦 = 𝐴 → (𝑀 𝑦) = (𝑀 𝐴))
14 esumeq1 29901 . . . . 5 (𝑦 = 𝐴 → Σ*𝑥𝑦(𝑀𝑥) = Σ*𝑥𝐴(𝑀𝑥))
1513, 14eqeq12d 2636 . . . 4 (𝑦 = 𝐴 → ((𝑀 𝑦) = Σ*𝑥𝑦(𝑀𝑥) ↔ (𝑀 𝐴) = Σ*𝑥𝐴(𝑀𝑥)))
1611, 15imbi12d 334 . . 3 (𝑦 = 𝐴 → (((𝑦 ≼ ω ∧ Disj 𝑥𝑦 𝑥) → (𝑀 𝑦) = Σ*𝑥𝑦(𝑀𝑥)) ↔ ((𝐴 ≼ ω ∧ Disj 𝑥𝐴 𝑥) → (𝑀 𝐴) = Σ*𝑥𝐴(𝑀𝑥))))
1716rspcv 3294 . 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 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  c0 3896  𝒫 cpw 4135   cuni 4407  Disj wdisj 4588   class class class wbr 4618  ran crn 5080  wf 5848  cfv 5852  (class class class)co 6610  ωcom 7019  cdom 7905  0cc0 9888  +∞cpnf 10023  [,]cicc 12128  Σ*cesum 29894  sigAlgebracsiga 29975  measurescmeas 30063
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-disj 4589  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-fv 5860  df-ov 6613  df-esum 29895  df-meas 30064
This theorem is referenced by:  measxun2  30078  measvunilem  30080  measssd  30083  measres  30090  measdivcstOLD  30092  measdivcst  30093  probcun  30285  totprobd  30293
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