Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ismeas Structured version   Visualization version   GIF version

Theorem ismeas 33686
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 3485 . . 3 (𝑀 ∈ (measures‘𝑆) → 𝑀 ∈ V)
21a1i 11 . 2 (𝑆 ran sigAlgebra → (𝑀 ∈ (measures‘𝑆) → 𝑀 ∈ V))
3 simp1 1133 . . 3 ((𝑀:𝑆⟶(0[,]+∞) ∧ (𝑀‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 𝑆((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → (𝑀 𝑥) = Σ*𝑦𝑥(𝑀𝑦))) → 𝑀:𝑆⟶(0[,]+∞))
4 ovex 7434 . . . 4 (0[,]+∞) ∈ V
5 fex2 7917 . . . . . 6 ((𝑀:𝑆⟶(0[,]+∞) ∧ 𝑆 ran sigAlgebra ∧ (0[,]+∞) ∈ V) → 𝑀 ∈ V)
653expb 1117 . . . . 5 ((𝑀:𝑆⟶(0[,]+∞) ∧ (𝑆 ran sigAlgebra ∧ (0[,]+∞) ∈ V)) → 𝑀 ∈ V)
76expcom 413 . . . 4 ((𝑆 ran sigAlgebra ∧ (0[,]+∞) ∈ V) → (𝑀:𝑆⟶(0[,]+∞) → 𝑀 ∈ V))
84, 7mpan2 688 . . 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 33683 . . . 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 3470 . . . . . 6 𝑠 ∈ V
12 mapex 8822 . . . . . 6 ((𝑠 ∈ V ∧ (0[,]+∞) ∈ V) → {𝑚𝑚:𝑠⟶(0[,]+∞)} ∈ V)
1311, 4, 12mp2an 689 . . . . 5 {𝑚𝑚:𝑠⟶(0[,]+∞)} ∈ V
14 simp1 1133 . . . . . 6 ((𝑚:𝑠⟶(0[,]+∞) ∧ (𝑚‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → (𝑚 𝑥) = Σ*𝑦𝑥(𝑚𝑦))) → 𝑚:𝑠⟶(0[,]+∞))
1514ss2abi 4055 . . . . 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 5312 . . . 4 {𝑚 ∣ (𝑚:𝑠⟶(0[,]+∞) ∧ (𝑚‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → (𝑚 𝑥) = Σ*𝑦𝑥(𝑚𝑦)))} ∈ V
17 simpr 484 . . . . . 6 ((𝑠 = 𝑆𝑚 = 𝑀) → 𝑚 = 𝑀)
18 simpl 482 . . . . . 6 ((𝑠 = 𝑆𝑚 = 𝑀) → 𝑠 = 𝑆)
1917, 18feq12d 6695 . . . . 5 ((𝑠 = 𝑆𝑚 = 𝑀) → (𝑚:𝑠⟶(0[,]+∞) ↔ 𝑀:𝑆⟶(0[,]+∞)))
20 fveq1 6880 . . . . . . 7 (𝑚 = 𝑀 → (𝑚‘∅) = (𝑀‘∅))
2120eqeq1d 2726 . . . . . 6 (𝑚 = 𝑀 → ((𝑚‘∅) = 0 ↔ (𝑀‘∅) = 0))
2221adantl 481 . . . . 5 ((𝑠 = 𝑆𝑚 = 𝑀) → ((𝑚‘∅) = 0 ↔ (𝑀‘∅) = 0))
2318pweqd 4611 . . . . . 6 ((𝑠 = 𝑆𝑚 = 𝑀) → 𝒫 𝑠 = 𝒫 𝑆)
24 fveq1 6880 . . . . . . . . 9 (𝑚 = 𝑀 → (𝑚 𝑥) = (𝑀 𝑥))
25 fveq1 6880 . . . . . . . . . 10 (𝑚 = 𝑀 → (𝑚𝑦) = (𝑀𝑦))
2625esumeq2sdv 33526 . . . . . . . . 9 (𝑚 = 𝑀 → Σ*𝑦𝑥(𝑚𝑦) = Σ*𝑦𝑥(𝑀𝑦))
2724, 26eqeq12d 2740 . . . . . . . 8 (𝑚 = 𝑀 → ((𝑚 𝑥) = Σ*𝑦𝑥(𝑚𝑦) ↔ (𝑀 𝑥) = Σ*𝑦𝑥(𝑀𝑦)))
2827imbi2d 340 . . . . . . 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 3334 . . . . 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 1432 . . . 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 32349 . . 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 412 . 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 379 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 205  wa 395  w3a 1084   = wceq 1533  wcel 2098  {cab 2701  wral 3053  Vcvv 3466  c0 4314  𝒫 cpw 4594   cuni 4899  Disj wdisj 5103   class class class wbr 5138  ran crn 5667  wf 6529  cfv 6533  (class class class)co 7401  ωcom 7848  cdom 8933  0cc0 11106  +∞cpnf 11242  [,]cicc 13324  Σ*cesum 33514  sigAlgebracsiga 33595  measurescmeas 33682
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-fv 6541  df-ov 7404  df-esum 33515  df-meas 33683
This theorem is referenced by:  measbasedom  33689  measfrge0  33690  measvnul  33693  measvun  33696  measinb  33708  measres  33709  measdivcst  33711  measdivcstALTV  33712  cntmeas  33713  volmeas  33718  ddemeas  33723  omsmeas  33811  dstrvprob  33959
  Copyright terms: Public domain W3C validator