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Theorem ismeas 30390
Description: The property of being a measure. (Contributed by Thierry Arnoux, 10-Sep-2016.) (Revised by Thierry Arnoux, 19-Oct-2016.)
Assertion
Ref Expression
ismeas (𝑆 ran sigAlgebra → (𝑀 ∈ (measures‘𝑆) ↔ (𝑀:𝑆⟶(0[,]+∞) ∧ (𝑀‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 𝑆((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → (𝑀 𝑥) = Σ*𝑦𝑥(𝑀𝑦)))))
Distinct variable groups:   𝑥,𝑦,𝑀   𝑥,𝑆
Allowed substitution hint:   𝑆(𝑦)

Proof of Theorem ismeas
Dummy variables 𝑚 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3243 . . 3 (𝑀 ∈ (measures‘𝑆) → 𝑀 ∈ V)
21a1i 11 . 2 (𝑆 ran sigAlgebra → (𝑀 ∈ (measures‘𝑆) → 𝑀 ∈ V))
3 simp1 1081 . . 3 ((𝑀:𝑆⟶(0[,]+∞) ∧ (𝑀‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 𝑆((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → (𝑀 𝑥) = Σ*𝑦𝑥(𝑀𝑦))) → 𝑀:𝑆⟶(0[,]+∞))
4 ovex 6718 . . . 4 (0[,]+∞) ∈ V
5 fex2 7163 . . . . . 6 ((𝑀:𝑆⟶(0[,]+∞) ∧ 𝑆 ran sigAlgebra ∧ (0[,]+∞) ∈ V) → 𝑀 ∈ V)
653expb 1285 . . . . 5 ((𝑀:𝑆⟶(0[,]+∞) ∧ (𝑆 ran sigAlgebra ∧ (0[,]+∞) ∈ V)) → 𝑀 ∈ V)
76expcom 450 . . . 4 ((𝑆 ran sigAlgebra ∧ (0[,]+∞) ∈ V) → (𝑀:𝑆⟶(0[,]+∞) → 𝑀 ∈ V))
84, 7mpan2 707 . . 3 (𝑆 ran sigAlgebra → (𝑀:𝑆⟶(0[,]+∞) → 𝑀 ∈ V))
93, 8syl5 34 . 2 (𝑆 ran sigAlgebra → ((𝑀:𝑆⟶(0[,]+∞) ∧ (𝑀‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 𝑆((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → (𝑀 𝑥) = Σ*𝑦𝑥(𝑀𝑦))) → 𝑀 ∈ V))
10 df-meas 30387 . . . 4 measures = (𝑠 ran sigAlgebra ↦ {𝑚 ∣ (𝑚:𝑠⟶(0[,]+∞) ∧ (𝑚‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → (𝑚 𝑥) = Σ*𝑦𝑥(𝑚𝑦)))})
11 vex 3234 . . . . . 6 𝑠 ∈ V
12 mapex 7905 . . . . . 6 ((𝑠 ∈ V ∧ (0[,]+∞) ∈ V) → {𝑚𝑚:𝑠⟶(0[,]+∞)} ∈ V)
1311, 4, 12mp2an 708 . . . . 5 {𝑚𝑚:𝑠⟶(0[,]+∞)} ∈ V
14 simp1 1081 . . . . . 6 ((𝑚:𝑠⟶(0[,]+∞) ∧ (𝑚‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → (𝑚 𝑥) = Σ*𝑦𝑥(𝑚𝑦))) → 𝑚:𝑠⟶(0[,]+∞))
1514ss2abi 3707 . . . . 5 {𝑚 ∣ (𝑚:𝑠⟶(0[,]+∞) ∧ (𝑚‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → (𝑚 𝑥) = Σ*𝑦𝑥(𝑚𝑦)))} ⊆ {𝑚𝑚:𝑠⟶(0[,]+∞)}
1613, 15ssexi 4836 . . . 4 {𝑚 ∣ (𝑚:𝑠⟶(0[,]+∞) ∧ (𝑚‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → (𝑚 𝑥) = Σ*𝑦𝑥(𝑚𝑦)))} ∈ V
17 simpr 476 . . . . . 6 ((𝑠 = 𝑆𝑚 = 𝑀) → 𝑚 = 𝑀)
18 simpl 472 . . . . . 6 ((𝑠 = 𝑆𝑚 = 𝑀) → 𝑠 = 𝑆)
1917, 18feq12d 6071 . . . . 5 ((𝑠 = 𝑆𝑚 = 𝑀) → (𝑚:𝑠⟶(0[,]+∞) ↔ 𝑀:𝑆⟶(0[,]+∞)))
20 fveq1 6228 . . . . . . 7 (𝑚 = 𝑀 → (𝑚‘∅) = (𝑀‘∅))
2120eqeq1d 2653 . . . . . 6 (𝑚 = 𝑀 → ((𝑚‘∅) = 0 ↔ (𝑀‘∅) = 0))
2221adantl 481 . . . . 5 ((𝑠 = 𝑆𝑚 = 𝑀) → ((𝑚‘∅) = 0 ↔ (𝑀‘∅) = 0))
2318pweqd 4196 . . . . . 6 ((𝑠 = 𝑆𝑚 = 𝑀) → 𝒫 𝑠 = 𝒫 𝑆)
24 fveq1 6228 . . . . . . . . 9 (𝑚 = 𝑀 → (𝑚 𝑥) = (𝑀 𝑥))
25 fveq1 6228 . . . . . . . . . 10 (𝑚 = 𝑀 → (𝑚𝑦) = (𝑀𝑦))
2625esumeq2sdv 30229 . . . . . . . . 9 (𝑚 = 𝑀 → Σ*𝑦𝑥(𝑚𝑦) = Σ*𝑦𝑥(𝑀𝑦))
2724, 26eqeq12d 2666 . . . . . . . 8 (𝑚 = 𝑀 → ((𝑚 𝑥) = Σ*𝑦𝑥(𝑚𝑦) ↔ (𝑀 𝑥) = Σ*𝑦𝑥(𝑀𝑦)))
2827imbi2d 329 . . . . . . 7 (𝑚 = 𝑀 → (((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → (𝑚 𝑥) = Σ*𝑦𝑥(𝑚𝑦)) ↔ ((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → (𝑀 𝑥) = Σ*𝑦𝑥(𝑀𝑦))))
2928adantl 481 . . . . . 6 ((𝑠 = 𝑆𝑚 = 𝑀) → (((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → (𝑚 𝑥) = Σ*𝑦𝑥(𝑚𝑦)) ↔ ((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → (𝑀 𝑥) = Σ*𝑦𝑥(𝑀𝑦))))
3023, 29raleqbidv 3182 . . . . 5 ((𝑠 = 𝑆𝑚 = 𝑀) → (∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → (𝑚 𝑥) = Σ*𝑦𝑥(𝑚𝑦)) ↔ ∀𝑥 ∈ 𝒫 𝑆((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → (𝑀 𝑥) = Σ*𝑦𝑥(𝑀𝑦))))
3119, 22, 303anbi123d 1439 . . . 4 ((𝑠 = 𝑆𝑚 = 𝑀) → ((𝑚:𝑠⟶(0[,]+∞) ∧ (𝑚‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → (𝑚 𝑥) = Σ*𝑦𝑥(𝑚𝑦))) ↔ (𝑀:𝑆⟶(0[,]+∞) ∧ (𝑀‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 𝑆((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → (𝑀 𝑥) = Σ*𝑦𝑥(𝑀𝑦)))))
3210, 16, 31abfmpel 29583 . . 3 ((𝑆 ran sigAlgebra ∧ 𝑀 ∈ V) → (𝑀 ∈ (measures‘𝑆) ↔ (𝑀:𝑆⟶(0[,]+∞) ∧ (𝑀‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 𝑆((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → (𝑀 𝑥) = Σ*𝑦𝑥(𝑀𝑦)))))
3332ex 449 . 2 (𝑆 ran sigAlgebra → (𝑀 ∈ V → (𝑀 ∈ (measures‘𝑆) ↔ (𝑀:𝑆⟶(0[,]+∞) ∧ (𝑀‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 𝑆((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → (𝑀 𝑥) = Σ*𝑦𝑥(𝑀𝑦))))))
342, 9, 33pm5.21ndd 368 1 (𝑆 ran sigAlgebra → (𝑀 ∈ (measures‘𝑆) ↔ (𝑀:𝑆⟶(0[,]+∞) ∧ (𝑀‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 𝑆((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → (𝑀 𝑥) = Σ*𝑦𝑥(𝑀𝑦)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  {cab 2637  wral 2941  Vcvv 3231  c0 3948  𝒫 cpw 4191   cuni 4468  Disj wdisj 4652   class class class wbr 4685  ran crn 5144  wf 5922  cfv 5926  (class class class)co 6690  ωcom 7107  cdom 7995  0cc0 9974  +∞cpnf 10109  [,]cicc 12216  Σ*cesum 30217  sigAlgebracsiga 30298  measurescmeas 30386
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-ov 6693  df-esum 30218  df-meas 30387
This theorem is referenced by:  measbasedom  30393  measfrge0  30394  measvnul  30397  measvun  30400  measinb  30412  measres  30413  measdivcstOLD  30415  measdivcst  30416  cntmeas  30417  volmeas  30422  ddemeas  30427  omsmeas  30513  dstrvprob  30661
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