Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  measfrge0 Structured version   Visualization version   GIF version

Theorem measfrge0 31361
Description: A measure is a function over its base to the positive extended reals. (Contributed by Thierry Arnoux, 26-Dec-2016.)
Assertion
Ref Expression
measfrge0 (𝑀 ∈ (measures‘𝑆) → 𝑀:𝑆⟶(0[,]+∞))

Proof of Theorem measfrge0
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 measbase 31355 . . . 4 (𝑀 ∈ (measures‘𝑆) → 𝑆 ran sigAlgebra)
2 ismeas 31357 . . . 4 (𝑆 ran sigAlgebra → (𝑀 ∈ (measures‘𝑆) ↔ (𝑀:𝑆⟶(0[,]+∞) ∧ (𝑀‘∅) = 0 ∧ ∀𝑦 ∈ 𝒫 𝑆((𝑦 ≼ ω ∧ Disj 𝑥𝑦 𝑥) → (𝑀 𝑦) = Σ*𝑥𝑦(𝑀𝑥)))))
31, 2syl 17 . . 3 (𝑀 ∈ (measures‘𝑆) → (𝑀 ∈ (measures‘𝑆) ↔ (𝑀:𝑆⟶(0[,]+∞) ∧ (𝑀‘∅) = 0 ∧ ∀𝑦 ∈ 𝒫 𝑆((𝑦 ≼ ω ∧ Disj 𝑥𝑦 𝑥) → (𝑀 𝑦) = Σ*𝑥𝑦(𝑀𝑥)))))
43ibi 268 . 2 (𝑀 ∈ (measures‘𝑆) → (𝑀:𝑆⟶(0[,]+∞) ∧ (𝑀‘∅) = 0 ∧ ∀𝑦 ∈ 𝒫 𝑆((𝑦 ≼ ω ∧ Disj 𝑥𝑦 𝑥) → (𝑀 𝑦) = Σ*𝑥𝑦(𝑀𝑥))))
54simp1d 1134 1 (𝑀 ∈ (measures‘𝑆) → 𝑀:𝑆⟶(0[,]+∞))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  wral 3135  c0 4288  𝒫 cpw 4535   cuni 4830  Disj wdisj 5022   class class class wbr 5057  ran crn 5549  wf 6344  cfv 6348  (class class class)co 7145  ωcom 7569  cdom 8495  0cc0 10525  +∞cpnf 10660  [,]cicc 12729  Σ*cesum 31185  sigAlgebracsiga 31266  measurescmeas 31353
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-ov 7148  df-esum 31186  df-meas 31354
This theorem is referenced by:  measfn  31362  measvxrge0  31363  meascnbl  31377  measres  31380  measdivcst  31382  measdivcstALTV  31383
  Copyright terms: Public domain W3C validator