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Theorem measfrge0 30596
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 30590 . . . 4 (𝑀 ∈ (measures‘𝑆) → 𝑆 ran sigAlgebra)
2 ismeas 30592 . . . 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 256 . 2 (𝑀 ∈ (measures‘𝑆) → (𝑀:𝑆⟶(0[,]+∞) ∧ (𝑀‘∅) = 0 ∧ ∀𝑦 ∈ 𝒫 𝑆((𝑦 ≼ ω ∧ Disj 𝑥𝑦 𝑥) → (𝑀 𝑦) = Σ*𝑥𝑦(𝑀𝑥))))
54simp1d 1137 1 (𝑀 ∈ (measures‘𝑆) → 𝑀:𝑆⟶(0[,]+∞))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1632  wcel 2139  wral 3050  c0 4058  𝒫 cpw 4302   cuni 4588  Disj wdisj 4772   class class class wbr 4804  ran crn 5267  wf 6045  cfv 6049  (class class class)co 6814  ωcom 7231  cdom 8121  0cc0 10148  +∞cpnf 10283  [,]cicc 12391  Σ*cesum 30419  sigAlgebracsiga 30500  measurescmeas 30588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fv 6057  df-ov 6817  df-esum 30420  df-meas 30589
This theorem is referenced by:  measfn  30597  measvxrge0  30598  meascnbl  30612  measres  30615  measdivcstOLD  30617  measdivcst  30618
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