Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  meadjuni Structured version   Visualization version   GIF version

Theorem meadjuni 43022
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 . 2 (𝜑𝑋 ≼ ω)
2 meadjuni.dj . 2 (𝜑Disj 𝑥𝑋 𝑥)
3 breq1 5055 . . . . 5 (𝑦 = 𝑋 → (𝑦 ≼ ω ↔ 𝑋 ≼ ω))
4 disjeq1 5024 . . . . 5 (𝑦 = 𝑋 → (Disj 𝑥𝑦 𝑥Disj 𝑥𝑋 𝑥))
53, 4anbi12d 633 . . . 4 (𝑦 = 𝑋 → ((𝑦 ≼ ω ∧ Disj 𝑥𝑦 𝑥) ↔ (𝑋 ≼ ω ∧ Disj 𝑥𝑋 𝑥)))
6 unieq 4835 . . . . . 6 (𝑦 = 𝑋 𝑦 = 𝑋)
76fveq2d 6665 . . . . 5 (𝑦 = 𝑋 → (𝑀 𝑦) = (𝑀 𝑋))
8 reseq2 5835 . . . . . 6 (𝑦 = 𝑋 → (𝑀𝑦) = (𝑀𝑋))
98fveq2d 6665 . . . . 5 (𝑦 = 𝑋 → (Σ^‘(𝑀𝑦)) = (Σ^‘(𝑀𝑋)))
107, 9eqeq12d 2840 . . . 4 (𝑦 = 𝑋 → ((𝑀 𝑦) = (Σ^‘(𝑀𝑦)) ↔ (𝑀 𝑋) = (Σ^‘(𝑀𝑋))))
115, 10imbi12d 348 . . 3 (𝑦 = 𝑋 → (((𝑦 ≼ ω ∧ Disj 𝑥𝑦 𝑥) → (𝑀 𝑦) = (Σ^‘(𝑀𝑦))) ↔ ((𝑋 ≼ ω ∧ Disj 𝑥𝑋 𝑥) → (𝑀 𝑋) = (Σ^‘(𝑀𝑋)))))
12 meadjuni.m . . . . 5 (𝜑𝑀 ∈ Meas)
13 ismea 43016 . . . . 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)))))
1412, 13sylib 221 . . . 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 499 . . 3 (𝜑 → ∀𝑦 ∈ 𝒫 dom 𝑀((𝑦 ≼ ω ∧ Disj 𝑥𝑦 𝑥) → (𝑀 𝑦) = (Σ^‘(𝑀𝑦))))
16 meadjuni.s . . . . . 6 𝑆 = dom 𝑀
1712, 16dmmeasal 43017 . . . . 5 (𝜑𝑆 ∈ SAlg)
18 meadjuni.x . . . . 5 (𝜑𝑋𝑆)
1917, 18ssexd 5214 . . . 4 (𝜑𝑋 ∈ V)
2018, 16sseqtrdi 4003 . . . 4 (𝜑𝑋 ⊆ dom 𝑀)
2119, 20elpwd 4530 . . 3 (𝜑𝑋 ∈ 𝒫 dom 𝑀)
2211, 15, 21rspcdva 3611 . 2 (𝜑 → ((𝑋 ≼ ω ∧ Disj 𝑥𝑋 𝑥) → (𝑀 𝑋) = (Σ^‘(𝑀𝑋))))
231, 2, 22mp2and 698 1 (𝜑 → (𝑀 𝑋) = (Σ^‘(𝑀𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  wral 3133  Vcvv 3480  wss 3919  c0 4276  𝒫 cpw 4522   cuni 4824  Disj wdisj 5017   class class class wbr 5052  dom cdm 5542  cres 5544  wf 6339  cfv 6343  (class class class)co 7149  ωcom 7574  cdom 8503  0cc0 10535  +∞cpnf 10670  [,]cicc 12738  SAlgcsalg 42876  Σ^csumge0 42927  Meascmea 43014
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pr 5317
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-disj 5018  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-mea 43015
This theorem is referenced by:  meadjun  43027  meadjiun  43031
  Copyright terms: Public domain W3C validator