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Theorem meadjuni 41062
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 555 . 2 (𝜑 → (𝑋 ≼ ω ∧ Disj 𝑥𝑋 𝑥))
4 meadjuni.x . . . . 5 (𝜑𝑋𝑆)
5 meadjuni.s . . . . 5 𝑆 = dom 𝑀
64, 5syl6sseq 3725 . . . 4 (𝜑𝑋 ⊆ dom 𝑀)
7 meadjuni.m . . . . . . 7 (𝜑𝑀 ∈ Meas)
87, 5dmmeasal 41057 . . . . . 6 (𝜑𝑆 ∈ SAlg)
98, 4ssexd 4881 . . . . 5 (𝜑𝑋 ∈ V)
10 elpwg 4242 . . . . 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 41056 . . . . 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 482 . . 3 (𝜑 → ∀𝑦 ∈ 𝒫 dom 𝑀((𝑦 ≼ ω ∧ Disj 𝑥𝑦 𝑥) → (𝑀 𝑦) = (Σ^‘(𝑀𝑦))))
16 breq1 4731 . . . . . 6 (𝑦 = 𝑋 → (𝑦 ≼ ω ↔ 𝑋 ≼ ω))
17 disjeq1 4703 . . . . . 6 (𝑦 = 𝑋 → (Disj 𝑥𝑦 𝑥Disj 𝑥𝑋 𝑥))
1816, 17anbi12d 749 . . . . 5 (𝑦 = 𝑋 → ((𝑦 ≼ ω ∧ Disj 𝑥𝑦 𝑥) ↔ (𝑋 ≼ ω ∧ Disj 𝑥𝑋 𝑥)))
19 unieq 4520 . . . . . . 7 (𝑦 = 𝑋 𝑦 = 𝑋)
2019fveq2d 6276 . . . . . 6 (𝑦 = 𝑋 → (𝑀 𝑦) = (𝑀 𝑋))
21 reseq2 5466 . . . . . . 7 (𝑦 = 𝑋 → (𝑀𝑦) = (𝑀𝑋))
2221fveq2d 6276 . . . . . 6 (𝑦 = 𝑋 → (Σ^‘(𝑀𝑦)) = (Σ^‘(𝑀𝑋)))
2320, 22eqeq12d 2707 . . . . 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 3379 . . 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 696 . 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 383   = wceq 1564  wcel 2071  wral 2982  Vcvv 3272  wss 3648  c0 3991  𝒫 cpw 4234   cuni 4512  Disj wdisj 4696   class class class wbr 4728  dom cdm 5186  cres 5188  wf 5965  cfv 5969  (class class class)co 6733  ωcom 7150  cdom 8038  0cc0 10017  +∞cpnf 10152  [,]cicc 12260  SAlgcsalg 40916  Σ^csumge0 40967  Meascmea 41054
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1818  ax-5 1920  ax-6 1986  ax-7 2022  ax-9 2080  ax-10 2100  ax-11 2115  ax-12 2128  ax-13 2323  ax-ext 2672  ax-rep 4847  ax-sep 4857  ax-nul 4865  ax-pr 4979
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1567  df-ex 1786  df-nf 1791  df-sb 1979  df-eu 2543  df-mo 2544  df-clab 2679  df-cleq 2685  df-clel 2688  df-nfc 2823  df-ne 2865  df-ral 2987  df-rex 2988  df-reu 2989  df-rmo 2990  df-rab 2991  df-v 3274  df-sbc 3510  df-csb 3608  df-dif 3651  df-un 3653  df-in 3655  df-ss 3662  df-nul 3992  df-if 4163  df-pw 4236  df-sn 4254  df-pr 4256  df-op 4260  df-uni 4513  df-iun 4598  df-disj 4697  df-br 4729  df-opab 4789  df-mpt 4806  df-id 5096  df-xp 5192  df-rel 5193  df-cnv 5194  df-co 5195  df-dm 5196  df-rn 5197  df-res 5198  df-ima 5199  df-iota 5932  df-fun 5971  df-fn 5972  df-f 5973  df-f1 5974  df-fo 5975  df-f1o 5976  df-fv 5977  df-mea 41055
This theorem is referenced by:  meadjun  41067  meadjiun  41071
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