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Theorem meaf 41191
Description: A measure is a function that maps to nonnegative extended reals. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypotheses
Ref Expression
meaf.m (𝜑𝑀 ∈ Meas)
meaf.s 𝑆 = dom 𝑀
Assertion
Ref Expression
meaf (𝜑𝑀:𝑆⟶(0[,]+∞))

Proof of Theorem meaf
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 meaf.m . . . . 5 (𝜑𝑀 ∈ Meas)
2 ismea 41189 . . . . 5 (𝑀 ∈ Meas ↔ (((𝑀:dom 𝑀⟶(0[,]+∞) ∧ dom 𝑀 ∈ SAlg) ∧ (𝑀‘∅) = 0) ∧ ∀𝑥 ∈ 𝒫 dom 𝑀((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → (𝑀 𝑥) = (Σ^‘(𝑀𝑥)))))
31, 2sylib 208 . . . 4 (𝜑 → (((𝑀:dom 𝑀⟶(0[,]+∞) ∧ dom 𝑀 ∈ SAlg) ∧ (𝑀‘∅) = 0) ∧ ∀𝑥 ∈ 𝒫 dom 𝑀((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → (𝑀 𝑥) = (Σ^‘(𝑀𝑥)))))
43simpld 477 . . 3 (𝜑 → ((𝑀:dom 𝑀⟶(0[,]+∞) ∧ dom 𝑀 ∈ SAlg) ∧ (𝑀‘∅) = 0))
54simplld 808 . 2 (𝜑𝑀:dom 𝑀⟶(0[,]+∞))
6 meaf.s . . . 4 𝑆 = dom 𝑀
76a1i 11 . . 3 (𝜑𝑆 = dom 𝑀)
87feq2d 6192 . 2 (𝜑 → (𝑀:𝑆⟶(0[,]+∞) ↔ 𝑀:dom 𝑀⟶(0[,]+∞)))
95, 8mpbird 247 1 (𝜑𝑀:𝑆⟶(0[,]+∞))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  wral 3050  c0 4058  𝒫 cpw 4302   cuni 4588  Disj wdisj 4772   class class class wbr 4804  dom cdm 5266  cres 5268  wf 6045  cfv 6049  (class class class)co 6814  ωcom 7231  cdom 8121  0cc0 10148  +∞cpnf 10283  [,]cicc 12391  SAlgcsalg 41049  Σ^csumge0 41100  Meascmea 41187
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-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  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-reu 3057  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-iun 4674  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-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-mea 41188
This theorem is referenced by:  meacl  41196  meadjun  41200  meadjiunlem  41203  meadjiun  41204
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