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

Theorem measbase 30038
Description: The base set of a measure is a sigma-algebra. (Contributed by Thierry Arnoux, 25-Dec-2016.)
Assertion
Ref Expression
measbase (𝑀 ∈ (measures‘𝑆) → 𝑆 ran sigAlgebra)

Proof of Theorem measbase
Dummy variables 𝑥 𝑚 𝑦 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elfvdm 6177 . 2 (𝑀 ∈ (measures‘𝑆) → 𝑆 ∈ dom measures)
2 vex 3189 . . . . 5 𝑠 ∈ V
3 ovex 6632 . . . . 5 (0[,]+∞) ∈ V
4 mapex 7808 . . . . 5 ((𝑠 ∈ V ∧ (0[,]+∞) ∈ V) → {𝑚𝑚:𝑠⟶(0[,]+∞)} ∈ V)
52, 3, 4mp2an 707 . . . 4 {𝑚𝑚:𝑠⟶(0[,]+∞)} ∈ V
6 simp1 1059 . . . . 5 ((𝑚:𝑠⟶(0[,]+∞) ∧ (𝑚‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → (𝑚 𝑥) = Σ*𝑦𝑥(𝑚𝑦))) → 𝑚:𝑠⟶(0[,]+∞))
76ss2abi 3653 . . . 4 {𝑚 ∣ (𝑚:𝑠⟶(0[,]+∞) ∧ (𝑚‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → (𝑚 𝑥) = Σ*𝑦𝑥(𝑚𝑦)))} ⊆ {𝑚𝑚:𝑠⟶(0[,]+∞)}
85, 7ssexi 4763 . . 3 {𝑚 ∣ (𝑚:𝑠⟶(0[,]+∞) ∧ (𝑚‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → (𝑚 𝑥) = Σ*𝑦𝑥(𝑚𝑦)))} ∈ V
9 df-meas 30037 . . 3 measures = (𝑠 ran sigAlgebra ↦ {𝑚 ∣ (𝑚:𝑠⟶(0[,]+∞) ∧ (𝑚‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → (𝑚 𝑥) = Σ*𝑦𝑥(𝑚𝑦)))})
108, 9dmmpti 5980 . 2 dom measures = ran sigAlgebra
111, 10syl6eleq 2708 1 (𝑀 ∈ (measures‘𝑆) → 𝑆 ran sigAlgebra)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  {cab 2607  wral 2907  Vcvv 3186  c0 3891  𝒫 cpw 4130   cuni 4402  Disj wdisj 4583   class class class wbr 4613  dom cdm 5074  ran crn 5075  wf 5843  cfv 5847  (class class class)co 6604  ωcom 7012  cdom 7897  0cc0 9880  +∞cpnf 10015  [,]cicc 12120  Σ*cesum 29867  sigAlgebracsiga 29948  measurescmeas 30036
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fv 5855  df-ov 6607  df-meas 30037
This theorem is referenced by:  measfrge0  30044  measvnul  30047  measvun  30050  measxun2  30051  measun  30052  measvuni  30055  measssd  30056  measunl  30057  measiuns  30058  measiun  30059  meascnbl  30060  measinblem  30061  measinb  30062  measinb2  30064  measdivcstOLD  30065  measdivcst  30066  aean  30085  mbfmbfm  30098  domprobsiga  30251  prob01  30253  probfinmeasbOLD  30268  probfinmeasb  30269  probmeasb  30270
  Copyright terms: Public domain W3C validator