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Theorem measvnul 30243
Description: The measure of the empty set is always zero. (Contributed by Thierry Arnoux, 26-Dec-2016.)
Assertion
Ref Expression
measvnul (𝑀 ∈ (measures‘𝑆) → (𝑀‘∅) = 0)

Proof of Theorem measvnul
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 measbase 30234 . . . 4 (𝑀 ∈ (measures‘𝑆) → 𝑆 ran sigAlgebra)
2 ismeas 30236 . . . 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))))
54simp2d 1072 1 (𝑀 ∈ (measures‘𝑆) → (𝑀‘∅) = 0)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1481  wcel 1988  wral 2909  c0 3907  𝒫 cpw 4149   cuni 4427  Disj wdisj 4611   class class class wbr 4644  ran crn 5105  wf 5872  cfv 5876  (class class class)co 6635  ωcom 7050  cdom 7938  0cc0 9921  +∞cpnf 10056  [,]cicc 12163  Σ*cesum 30063  sigAlgebracsiga 30144  measurescmeas 30232
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-fal 1487  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-fv 5884  df-ov 6638  df-esum 30064  df-meas 30233
This theorem is referenced by:  measxun2  30247  measvunilem0  30250  measssd  30252  measinb  30258  measres  30259  measdivcstOLD  30261  measdivcst  30262  truae  30280  probnul  30450
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