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Theorem measval 32861
Description: The value of the measures function applied on a sigma-algebra. (Contributed by Thierry Arnoux, 17-Oct-2016.)
Assertion
Ref Expression
measval (𝑆 ∈ βˆͺ ran sigAlgebra β†’ (measuresβ€˜π‘†) = {π‘š ∣ (π‘š:π‘†βŸΆ(0[,]+∞) ∧ (π‘šβ€˜βˆ…) = 0 ∧ βˆ€π‘₯ ∈ 𝒫 𝑆((π‘₯ β‰Ό Ο‰ ∧ Disj 𝑦 ∈ π‘₯ 𝑦) β†’ (π‘šβ€˜βˆͺ π‘₯) = Ξ£*𝑦 ∈ π‘₯(π‘šβ€˜π‘¦)))})
Distinct variable groups:   π‘₯,π‘š,𝑦   𝑆,π‘š,π‘₯
Allowed substitution hint:   𝑆(𝑦)

Proof of Theorem measval
Dummy variable 𝑠 is distinct from all other variables.
StepHypRef Expression
1 simp1 1137 . . . 4 ((π‘š:π‘†βŸΆ(0[,]+∞) ∧ (π‘šβ€˜βˆ…) = 0 ∧ βˆ€π‘₯ ∈ 𝒫 𝑆((π‘₯ β‰Ό Ο‰ ∧ Disj 𝑦 ∈ π‘₯ 𝑦) β†’ (π‘šβ€˜βˆͺ π‘₯) = Ξ£*𝑦 ∈ π‘₯(π‘šβ€˜π‘¦))) β†’ π‘š:π‘†βŸΆ(0[,]+∞))
21ss2abi 4027 . . 3 {π‘š ∣ (π‘š:π‘†βŸΆ(0[,]+∞) ∧ (π‘šβ€˜βˆ…) = 0 ∧ βˆ€π‘₯ ∈ 𝒫 𝑆((π‘₯ β‰Ό Ο‰ ∧ Disj 𝑦 ∈ π‘₯ 𝑦) β†’ (π‘šβ€˜βˆͺ π‘₯) = Ξ£*𝑦 ∈ π‘₯(π‘šβ€˜π‘¦)))} βŠ† {π‘š ∣ π‘š:π‘†βŸΆ(0[,]+∞)}
3 ovex 7394 . . . 4 (0[,]+∞) ∈ V
4 mapex 8777 . . . 4 ((𝑆 ∈ βˆͺ ran sigAlgebra ∧ (0[,]+∞) ∈ V) β†’ {π‘š ∣ π‘š:π‘†βŸΆ(0[,]+∞)} ∈ V)
53, 4mpan2 690 . . 3 (𝑆 ∈ βˆͺ ran sigAlgebra β†’ {π‘š ∣ π‘š:π‘†βŸΆ(0[,]+∞)} ∈ V)
6 ssexg 5284 . . 3 (({π‘š ∣ (π‘š:π‘†βŸΆ(0[,]+∞) ∧ (π‘šβ€˜βˆ…) = 0 ∧ βˆ€π‘₯ ∈ 𝒫 𝑆((π‘₯ β‰Ό Ο‰ ∧ Disj 𝑦 ∈ π‘₯ 𝑦) β†’ (π‘šβ€˜βˆͺ π‘₯) = Ξ£*𝑦 ∈ π‘₯(π‘šβ€˜π‘¦)))} βŠ† {π‘š ∣ π‘š:π‘†βŸΆ(0[,]+∞)} ∧ {π‘š ∣ π‘š:π‘†βŸΆ(0[,]+∞)} ∈ V) β†’ {π‘š ∣ (π‘š:π‘†βŸΆ(0[,]+∞) ∧ (π‘šβ€˜βˆ…) = 0 ∧ βˆ€π‘₯ ∈ 𝒫 𝑆((π‘₯ β‰Ό Ο‰ ∧ Disj 𝑦 ∈ π‘₯ 𝑦) β†’ (π‘šβ€˜βˆͺ π‘₯) = Ξ£*𝑦 ∈ π‘₯(π‘šβ€˜π‘¦)))} ∈ V)
72, 5, 6sylancr 588 . 2 (𝑆 ∈ βˆͺ ran sigAlgebra β†’ {π‘š ∣ (π‘š:π‘†βŸΆ(0[,]+∞) ∧ (π‘šβ€˜βˆ…) = 0 ∧ βˆ€π‘₯ ∈ 𝒫 𝑆((π‘₯ β‰Ό Ο‰ ∧ Disj 𝑦 ∈ π‘₯ 𝑦) β†’ (π‘šβ€˜βˆͺ π‘₯) = Ξ£*𝑦 ∈ π‘₯(π‘šβ€˜π‘¦)))} ∈ V)
8 feq2 6654 . . . . 5 (𝑠 = 𝑆 β†’ (π‘š:π‘ βŸΆ(0[,]+∞) ↔ π‘š:π‘†βŸΆ(0[,]+∞)))
9 pweq 4578 . . . . . 6 (𝑠 = 𝑆 β†’ 𝒫 𝑠 = 𝒫 𝑆)
109raleqdv 3312 . . . . 5 (𝑠 = 𝑆 β†’ (βˆ€π‘₯ ∈ 𝒫 𝑠((π‘₯ β‰Ό Ο‰ ∧ Disj 𝑦 ∈ π‘₯ 𝑦) β†’ (π‘šβ€˜βˆͺ π‘₯) = Ξ£*𝑦 ∈ π‘₯(π‘šβ€˜π‘¦)) ↔ βˆ€π‘₯ ∈ 𝒫 𝑆((π‘₯ β‰Ό Ο‰ ∧ Disj 𝑦 ∈ π‘₯ 𝑦) β†’ (π‘šβ€˜βˆͺ π‘₯) = Ξ£*𝑦 ∈ π‘₯(π‘šβ€˜π‘¦))))
118, 103anbi13d 1439 . . . 4 (𝑠 = 𝑆 β†’ ((π‘š:π‘ βŸΆ(0[,]+∞) ∧ (π‘šβ€˜βˆ…) = 0 ∧ βˆ€π‘₯ ∈ 𝒫 𝑠((π‘₯ β‰Ό Ο‰ ∧ Disj 𝑦 ∈ π‘₯ 𝑦) β†’ (π‘šβ€˜βˆͺ π‘₯) = Ξ£*𝑦 ∈ π‘₯(π‘šβ€˜π‘¦))) ↔ (π‘š:π‘†βŸΆ(0[,]+∞) ∧ (π‘šβ€˜βˆ…) = 0 ∧ βˆ€π‘₯ ∈ 𝒫 𝑆((π‘₯ β‰Ό Ο‰ ∧ Disj 𝑦 ∈ π‘₯ 𝑦) β†’ (π‘šβ€˜βˆͺ π‘₯) = Ξ£*𝑦 ∈ π‘₯(π‘šβ€˜π‘¦)))))
1211abbidv 2802 . . 3 (𝑠 = 𝑆 β†’ {π‘š ∣ (π‘š:π‘ βŸΆ(0[,]+∞) ∧ (π‘šβ€˜βˆ…) = 0 ∧ βˆ€π‘₯ ∈ 𝒫 𝑠((π‘₯ β‰Ό Ο‰ ∧ Disj 𝑦 ∈ π‘₯ 𝑦) β†’ (π‘šβ€˜βˆͺ π‘₯) = Ξ£*𝑦 ∈ π‘₯(π‘šβ€˜π‘¦)))} = {π‘š ∣ (π‘š:π‘†βŸΆ(0[,]+∞) ∧ (π‘šβ€˜βˆ…) = 0 ∧ βˆ€π‘₯ ∈ 𝒫 𝑆((π‘₯ β‰Ό Ο‰ ∧ Disj 𝑦 ∈ π‘₯ 𝑦) β†’ (π‘šβ€˜βˆͺ π‘₯) = Ξ£*𝑦 ∈ π‘₯(π‘šβ€˜π‘¦)))})
13 df-meas 32859 . . 3 measures = (𝑠 ∈ βˆͺ ran sigAlgebra ↦ {π‘š ∣ (π‘š:π‘ βŸΆ(0[,]+∞) ∧ (π‘šβ€˜βˆ…) = 0 ∧ βˆ€π‘₯ ∈ 𝒫 𝑠((π‘₯ β‰Ό Ο‰ ∧ Disj 𝑦 ∈ π‘₯ 𝑦) β†’ (π‘šβ€˜βˆͺ π‘₯) = Ξ£*𝑦 ∈ π‘₯(π‘šβ€˜π‘¦)))})
1412, 13fvmptg 6950 . 2 ((𝑆 ∈ βˆͺ ran sigAlgebra ∧ {π‘š ∣ (π‘š:π‘†βŸΆ(0[,]+∞) ∧ (π‘šβ€˜βˆ…) = 0 ∧ βˆ€π‘₯ ∈ 𝒫 𝑆((π‘₯ β‰Ό Ο‰ ∧ Disj 𝑦 ∈ π‘₯ 𝑦) β†’ (π‘šβ€˜βˆͺ π‘₯) = Ξ£*𝑦 ∈ π‘₯(π‘šβ€˜π‘¦)))} ∈ V) β†’ (measuresβ€˜π‘†) = {π‘š ∣ (π‘š:π‘†βŸΆ(0[,]+∞) ∧ (π‘šβ€˜βˆ…) = 0 ∧ βˆ€π‘₯ ∈ 𝒫 𝑆((π‘₯ β‰Ό Ο‰ ∧ Disj 𝑦 ∈ π‘₯ 𝑦) β†’ (π‘šβ€˜βˆͺ π‘₯) = Ξ£*𝑦 ∈ π‘₯(π‘šβ€˜π‘¦)))})
157, 14mpdan 686 1 (𝑆 ∈ βˆͺ ran sigAlgebra β†’ (measuresβ€˜π‘†) = {π‘š ∣ (π‘š:π‘†βŸΆ(0[,]+∞) ∧ (π‘šβ€˜βˆ…) = 0 ∧ βˆ€π‘₯ ∈ 𝒫 𝑆((π‘₯ β‰Ό Ο‰ ∧ Disj 𝑦 ∈ π‘₯ 𝑦) β†’ (π‘šβ€˜βˆͺ π‘₯) = Ξ£*𝑦 ∈ π‘₯(π‘šβ€˜π‘¦)))})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  {cab 2710  βˆ€wral 3061  Vcvv 3447   βŠ† wss 3914  βˆ…c0 4286  π’« cpw 4564  βˆͺ cuni 4869  Disj wdisj 5074   class class class wbr 5109  ran crn 5638  βŸΆwf 6496  β€˜cfv 6500  (class class class)co 7361  Ο‰com 7806   β‰Ό cdom 8887  0cc0 11059  +∞cpnf 11194  [,]cicc 13276  Ξ£*cesum 32690  sigAlgebracsiga 32771  measurescmeas 32858
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-fv 6508  df-ov 7364  df-meas 32859
This theorem is referenced by: (None)
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