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Theorem meadjuni 45892
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 5155 . . . . 5 (𝑦 = 𝑋 → (𝑦 ≼ ω ↔ 𝑋 ≼ ω))
4 disjeq1 5124 . . . . 5 (𝑦 = 𝑋 → (Disj 𝑥𝑦 𝑥Disj 𝑥𝑋 𝑥))
53, 4anbi12d 630 . . . 4 (𝑦 = 𝑋 → ((𝑦 ≼ ω ∧ Disj 𝑥𝑦 𝑥) ↔ (𝑋 ≼ ω ∧ Disj 𝑥𝑋 𝑥)))
6 unieq 4923 . . . . . 6 (𝑦 = 𝑋 𝑦 = 𝑋)
76fveq2d 6906 . . . . 5 (𝑦 = 𝑋 → (𝑀 𝑦) = (𝑀 𝑋))
8 reseq2 5984 . . . . . 6 (𝑦 = 𝑋 → (𝑀𝑦) = (𝑀𝑋))
98fveq2d 6906 . . . . 5 (𝑦 = 𝑋 → (Σ^‘(𝑀𝑦)) = (Σ^‘(𝑀𝑋)))
107, 9eqeq12d 2744 . . . 4 (𝑦 = 𝑋 → ((𝑀 𝑦) = (Σ^‘(𝑀𝑦)) ↔ (𝑀 𝑋) = (Σ^‘(𝑀𝑋))))
115, 10imbi12d 343 . . 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 45886 . . . . 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 217 . . . 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 494 . . 3 (𝜑 → ∀𝑦 ∈ 𝒫 dom 𝑀((𝑦 ≼ ω ∧ Disj 𝑥𝑦 𝑥) → (𝑀 𝑦) = (Σ^‘(𝑀𝑦))))
16 meadjuni.s . . . . . 6 𝑆 = dom 𝑀
1712, 16dmmeasal 45887 . . . . 5 (𝜑𝑆 ∈ SAlg)
18 meadjuni.x . . . . 5 (𝜑𝑋𝑆)
1917, 18ssexd 5328 . . . 4 (𝜑𝑋 ∈ V)
2018, 16sseqtrdi 4032 . . . 4 (𝜑𝑋 ⊆ dom 𝑀)
2119, 20elpwd 4612 . . 3 (𝜑𝑋 ∈ 𝒫 dom 𝑀)
2211, 15, 21rspcdva 3612 . 2 (𝜑 → ((𝑋 ≼ ω ∧ Disj 𝑥𝑋 𝑥) → (𝑀 𝑋) = (Σ^‘(𝑀𝑋))))
231, 2, 22mp2and 697 1 (𝜑 → (𝑀 𝑋) = (Σ^‘(𝑀𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  wral 3058  Vcvv 3473  wss 3949  c0 4326  𝒫 cpw 4606   cuni 4912  Disj wdisj 5117   class class class wbr 5152  dom cdm 5682  cres 5684  wf 6549  cfv 6553  (class class class)co 7426  ωcom 7878  cdom 8970  0cc0 11148  +∞cpnf 11285  [,]cicc 13369  SAlgcsalg 45743  Σ^csumge0 45797  Meascmea 45884
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-disj 5118  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-mea 45885
This theorem is referenced by:  meadjun  45897  meadjiun  45901
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