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
meadjuni.m (𝜑𝑀 ∈ Meas)
meadjuni.s 𝑆 = dom 𝑀
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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