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

 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
Assertion
Ref Expression
meadjuni (𝜑 → (𝑀 𝑋) = (Σ^‘(𝑀𝑋)))
Distinct variable group:   𝑥,𝑋
Allowed substitution hints:   𝜑(𝑥)   𝑆(𝑥)   𝑀(𝑥)

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
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