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Theorem meadjuni 41191
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 501 . 2 (𝜑 → (𝑋 ≼ ω ∧ Disj 𝑥𝑋 𝑥))
4 meadjuni.x . . . . 5 (𝜑𝑋𝑆)
5 meadjuni.s . . . . 5 𝑆 = dom 𝑀
64, 5syl6sseq 3800 . . . 4 (𝜑𝑋 ⊆ dom 𝑀)
7 meadjuni.m . . . . . . 7 (𝜑𝑀 ∈ Meas)
87, 5dmmeasal 41186 . . . . . 6 (𝜑𝑆 ∈ SAlg)
98, 4ssexd 4939 . . . . 5 (𝜑𝑋 ∈ V)
10 elpwg 4305 . . . . 5 (𝑋 ∈ V → (𝑋 ∈ 𝒫 dom 𝑀𝑋 ⊆ dom 𝑀))
119, 10syl 17 . . . 4 (𝜑 → (𝑋 ∈ 𝒫 dom 𝑀𝑋 ⊆ dom 𝑀))
126, 11mpbird 247 . . 3 (𝜑𝑋 ∈ 𝒫 dom 𝑀)
13 ismea 41185 . . . . 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 208 . . . 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 483 . . 3 (𝜑 → ∀𝑦 ∈ 𝒫 dom 𝑀((𝑦 ≼ ω ∧ Disj 𝑥𝑦 𝑥) → (𝑀 𝑦) = (Σ^‘(𝑀𝑦))))
16 breq1 4789 . . . . . 6 (𝑦 = 𝑋 → (𝑦 ≼ ω ↔ 𝑋 ≼ ω))
17 disjeq1 4761 . . . . . 6 (𝑦 = 𝑋 → (Disj 𝑥𝑦 𝑥Disj 𝑥𝑋 𝑥))
1816, 17anbi12d 616 . . . . 5 (𝑦 = 𝑋 → ((𝑦 ≼ ω ∧ Disj 𝑥𝑦 𝑥) ↔ (𝑋 ≼ ω ∧ Disj 𝑥𝑋 𝑥)))
19 unieq 4582 . . . . . . 7 (𝑦 = 𝑋 𝑦 = 𝑋)
2019fveq2d 6336 . . . . . 6 (𝑦 = 𝑋 → (𝑀 𝑦) = (𝑀 𝑋))
21 reseq2 5529 . . . . . . 7 (𝑦 = 𝑋 → (𝑀𝑦) = (𝑀𝑋))
2221fveq2d 6336 . . . . . 6 (𝑦 = 𝑋 → (Σ^‘(𝑀𝑦)) = (Σ^‘(𝑀𝑋)))
2320, 22eqeq12d 2786 . . . . 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 3458 . . 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 573 . 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 196  wa 382   = wceq 1631  wcel 2145  wral 3061  Vcvv 3351  wss 3723  c0 4063  𝒫 cpw 4297   cuni 4574  Disj wdisj 4754   class class class wbr 4786  dom cdm 5249  cres 5251  wf 6027  cfv 6031  (class class class)co 6793  ωcom 7212  cdom 8107  0cc0 10138  +∞cpnf 10273  [,]cicc 12383  SAlgcsalg 41045  Σ^csumge0 41096  Meascmea 41183
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-disj 4755  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-mea 41184
This theorem is referenced by:  meadjun  41196  meadjiun  41200
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