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Theorem meadjuni 39350
Description: The measure of the disjoint union of a countable set is the extended sum of the measures. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypotheses
Ref Expression
meadjuni.m (𝜑𝑀 ∈ Meas)
meadjuni.s 𝑆 = dom 𝑀
meadjuni.x (𝜑𝑋𝑆)
meadjuni.cnb (𝜑𝑋 ≼ ω)
meadjuni.dj (𝜑Disj 𝑥𝑋 𝑥)
Assertion
Ref Expression
meadjuni (𝜑 → (𝑀 𝑋) = (Σ^‘(𝑀𝑋)))
Distinct variable group:   𝑥,𝑋
Allowed substitution hints:   𝜑(𝑥)   𝑆(𝑥)   𝑀(𝑥)

Proof of Theorem meadjuni
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 meadjuni.cnb . . 3 (𝜑𝑋 ≼ ω)
2 meadjuni.dj . . 3 (𝜑Disj 𝑥𝑋 𝑥)
31, 2jca 553 . 2 (𝜑 → (𝑋 ≼ ω ∧ Disj 𝑥𝑋 𝑥))
4 meadjuni.x . . . . 5 (𝜑𝑋𝑆)
5 meadjuni.s . . . . 5 𝑆 = dom 𝑀
64, 5syl6sseq 3614 . . . 4 (𝜑𝑋 ⊆ dom 𝑀)
7 meadjuni.m . . . . . . 7 (𝜑𝑀 ∈ Meas)
87, 5dmmeasal 39345 . . . . . 6 (𝜑𝑆 ∈ SAlg)
98, 4ssexd 4733 . . . . 5 (𝜑𝑋 ∈ V)
10 elpwg 4116 . . . . 5 (𝑋 ∈ V → (𝑋 ∈ 𝒫 dom 𝑀𝑋 ⊆ dom 𝑀))
119, 10syl 17 . . . 4 (𝜑 → (𝑋 ∈ 𝒫 dom 𝑀𝑋 ⊆ dom 𝑀))
126, 11mpbird 246 . . 3 (𝜑𝑋 ∈ 𝒫 dom 𝑀)
13 ismea 39344 . . . . 5 (𝑀 ∈ Meas ↔ (((𝑀:dom 𝑀⟶(0[,]+∞) ∧ dom 𝑀 ∈ SAlg) ∧ (𝑀‘∅) = 0) ∧ ∀𝑦 ∈ 𝒫 dom 𝑀((𝑦 ≼ ω ∧ Disj 𝑥𝑦 𝑥) → (𝑀 𝑦) = (Σ^‘(𝑀𝑦)))))
147, 13sylib 207 . . . 4 (𝜑 → (((𝑀:dom 𝑀⟶(0[,]+∞) ∧ dom 𝑀 ∈ SAlg) ∧ (𝑀‘∅) = 0) ∧ ∀𝑦 ∈ 𝒫 dom 𝑀((𝑦 ≼ ω ∧ Disj 𝑥𝑦 𝑥) → (𝑀 𝑦) = (Σ^‘(𝑀𝑦)))))
1514simprd 478 . . 3 (𝜑 → ∀𝑦 ∈ 𝒫 dom 𝑀((𝑦 ≼ ω ∧ Disj 𝑥𝑦 𝑥) → (𝑀 𝑦) = (Σ^‘(𝑀𝑦))))
16 breq1 4586 . . . . . 6 (𝑦 = 𝑋 → (𝑦 ≼ ω ↔ 𝑋 ≼ ω))
17 disjeq1 4560 . . . . . 6 (𝑦 = 𝑋 → (Disj 𝑥𝑦 𝑥Disj 𝑥𝑋 𝑥))
1816, 17anbi12d 743 . . . . 5 (𝑦 = 𝑋 → ((𝑦 ≼ ω ∧ Disj 𝑥𝑦 𝑥) ↔ (𝑋 ≼ ω ∧ Disj 𝑥𝑋 𝑥)))
19 unieq 4380 . . . . . . 7 (𝑦 = 𝑋 𝑦 = 𝑋)
2019fveq2d 6107 . . . . . 6 (𝑦 = 𝑋 → (𝑀 𝑦) = (𝑀 𝑋))
21 reseq2 5312 . . . . . . 7 (𝑦 = 𝑋 → (𝑀𝑦) = (𝑀𝑋))
2221fveq2d 6107 . . . . . 6 (𝑦 = 𝑋 → (Σ^‘(𝑀𝑦)) = (Σ^‘(𝑀𝑋)))
2320, 22eqeq12d 2625 . . . . 5 (𝑦 = 𝑋 → ((𝑀 𝑦) = (Σ^‘(𝑀𝑦)) ↔ (𝑀 𝑋) = (Σ^‘(𝑀𝑋))))
2418, 23imbi12d 333 . . . 4 (𝑦 = 𝑋 → (((𝑦 ≼ ω ∧ Disj 𝑥𝑦 𝑥) → (𝑀 𝑦) = (Σ^‘(𝑀𝑦))) ↔ ((𝑋 ≼ ω ∧ Disj 𝑥𝑋 𝑥) → (𝑀 𝑋) = (Σ^‘(𝑀𝑋)))))
2524rspcva 3280 . . 3 ((𝑋 ∈ 𝒫 dom 𝑀 ∧ ∀𝑦 ∈ 𝒫 dom 𝑀((𝑦 ≼ ω ∧ Disj 𝑥𝑦 𝑥) → (𝑀 𝑦) = (Σ^‘(𝑀𝑦)))) → ((𝑋 ≼ ω ∧ Disj 𝑥𝑋 𝑥) → (𝑀 𝑋) = (Σ^‘(𝑀𝑋))))
2612, 15, 25syl2anc 691 . 2 (𝜑 → ((𝑋 ≼ ω ∧ Disj 𝑥𝑋 𝑥) → (𝑀 𝑋) = (Σ^‘(𝑀𝑋))))
273, 26mpd 15 1 (𝜑 → (𝑀 𝑋) = (Σ^‘(𝑀𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977  wral 2896  Vcvv 3173  wss 3540  c0 3874  𝒫 cpw 4108   cuni 4372  Disj wdisj 4553   class class class wbr 4583  dom cdm 5038  cres 5040  wf 5800  cfv 5804  (class class class)co 6549  ωcom 6957  cdom 7839  0cc0 9815  +∞cpnf 9950  [,]cicc 12049  SAlgcsalg 39204  Σ^csumge0 39255  Meascmea 39342
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-disj 4554  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-mea 39343
This theorem is referenced by:  meadjun  39355  meadjiun  39359
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