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Theorem measfrge0 32866
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 ∈ (measuresβ€˜π‘†) β†’ 𝑀:π‘†βŸΆ(0[,]+∞))

Proof of Theorem measfrge0
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 measbase 32860 . . . 4 (𝑀 ∈ (measuresβ€˜π‘†) β†’ 𝑆 ∈ βˆͺ ran sigAlgebra)
2 ismeas 32862 . . . 4 (𝑆 ∈ βˆͺ ran sigAlgebra β†’ (𝑀 ∈ (measuresβ€˜π‘†) ↔ (𝑀:π‘†βŸΆ(0[,]+∞) ∧ (π‘€β€˜βˆ…) = 0 ∧ βˆ€π‘¦ ∈ 𝒫 𝑆((𝑦 β‰Ό Ο‰ ∧ Disj π‘₯ ∈ 𝑦 π‘₯) β†’ (π‘€β€˜βˆͺ 𝑦) = Ξ£*π‘₯ ∈ 𝑦(π‘€β€˜π‘₯)))))
31, 2syl 17 . . 3 (𝑀 ∈ (measuresβ€˜π‘†) β†’ (𝑀 ∈ (measuresβ€˜π‘†) ↔ (𝑀:π‘†βŸΆ(0[,]+∞) ∧ (π‘€β€˜βˆ…) = 0 ∧ βˆ€π‘¦ ∈ 𝒫 𝑆((𝑦 β‰Ό Ο‰ ∧ Disj π‘₯ ∈ 𝑦 π‘₯) β†’ (π‘€β€˜βˆͺ 𝑦) = Ξ£*π‘₯ ∈ 𝑦(π‘€β€˜π‘₯)))))
43ibi 267 . 2 (𝑀 ∈ (measuresβ€˜π‘†) β†’ (𝑀:π‘†βŸΆ(0[,]+∞) ∧ (π‘€β€˜βˆ…) = 0 ∧ βˆ€π‘¦ ∈ 𝒫 𝑆((𝑦 β‰Ό Ο‰ ∧ Disj π‘₯ ∈ 𝑦 π‘₯) β†’ (π‘€β€˜βˆͺ 𝑦) = Ξ£*π‘₯ ∈ 𝑦(π‘€β€˜π‘₯))))
54simp1d 1143 1 (𝑀 ∈ (measuresβ€˜π‘†) β†’ 𝑀:π‘†βŸΆ(0[,]+∞))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061  βˆ…c0 4286  π’« cpw 4564  βˆͺ cuni 4869  Disj wdisj 5074   class class class wbr 5109  ran crn 5638  βŸΆwf 6496  β€˜cfv 6500  (class class class)co 7361  Ο‰com 7806   β‰Ό cdom 8887  0cc0 11059  +∞cpnf 11194  [,]cicc 13276  Ξ£*cesum 32690  sigAlgebracsiga 32771  measurescmeas 32858
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-fv 6508  df-ov 7364  df-esum 32691  df-meas 32859
This theorem is referenced by:  measfn  32867  measvxrge0  32868  meascnbl  32882  measres  32885  measdivcst  32887  measdivcstALTV  32888
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