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Theorem ismeas 34200
Description: The property of being a measure. (Contributed by Thierry Arnoux, 10-Sep-2016.) (Revised by Thierry Arnoux, 19-Oct-2016.)
Assertion
Ref Expression
ismeas (𝑆 ran sigAlgebra → (𝑀 ∈ (measures‘𝑆) ↔ (𝑀:𝑆⟶(0[,]+∞) ∧ (𝑀‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 𝑆((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → (𝑀 𝑥) = Σ*𝑦𝑥(𝑀𝑦)))))
Distinct variable groups:   𝑥,𝑦,𝑀   𝑥,𝑆
Allowed substitution hint:   𝑆(𝑦)

Proof of Theorem ismeas
Dummy variables 𝑚 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3501 . . 3 (𝑀 ∈ (measures‘𝑆) → 𝑀 ∈ V)
21a1i 11 . 2 (𝑆 ran sigAlgebra → (𝑀 ∈ (measures‘𝑆) → 𝑀 ∈ V))
3 simp1 1137 . . 3 ((𝑀:𝑆⟶(0[,]+∞) ∧ (𝑀‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 𝑆((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → (𝑀 𝑥) = Σ*𝑦𝑥(𝑀𝑦))) → 𝑀:𝑆⟶(0[,]+∞))
4 ovex 7464 . . . 4 (0[,]+∞) ∈ V
5 fex2 7958 . . . . . 6 ((𝑀:𝑆⟶(0[,]+∞) ∧ 𝑆 ran sigAlgebra ∧ (0[,]+∞) ∈ V) → 𝑀 ∈ V)
653expb 1121 . . . . 5 ((𝑀:𝑆⟶(0[,]+∞) ∧ (𝑆 ran sigAlgebra ∧ (0[,]+∞) ∈ V)) → 𝑀 ∈ V)
76expcom 413 . . . 4 ((𝑆 ran sigAlgebra ∧ (0[,]+∞) ∈ V) → (𝑀:𝑆⟶(0[,]+∞) → 𝑀 ∈ V))
84, 7mpan2 691 . . 3 (𝑆 ran sigAlgebra → (𝑀:𝑆⟶(0[,]+∞) → 𝑀 ∈ V))
93, 8syl5 34 . 2 (𝑆 ran sigAlgebra → ((𝑀:𝑆⟶(0[,]+∞) ∧ (𝑀‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 𝑆((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → (𝑀 𝑥) = Σ*𝑦𝑥(𝑀𝑦))) → 𝑀 ∈ V))
10 df-meas 34197 . . . 4 measures = (𝑠 ran sigAlgebra ↦ {𝑚 ∣ (𝑚:𝑠⟶(0[,]+∞) ∧ (𝑚‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → (𝑚 𝑥) = Σ*𝑦𝑥(𝑚𝑦)))})
11 vex 3484 . . . . . 6 𝑠 ∈ V
12 mapex 7963 . . . . . 6 ((𝑠 ∈ V ∧ (0[,]+∞) ∈ V) → {𝑚𝑚:𝑠⟶(0[,]+∞)} ∈ V)
1311, 4, 12mp2an 692 . . . . 5 {𝑚𝑚:𝑠⟶(0[,]+∞)} ∈ V
14 simp1 1137 . . . . . 6 ((𝑚:𝑠⟶(0[,]+∞) ∧ (𝑚‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → (𝑚 𝑥) = Σ*𝑦𝑥(𝑚𝑦))) → 𝑚:𝑠⟶(0[,]+∞))
1514ss2abi 4067 . . . . 5 {𝑚 ∣ (𝑚:𝑠⟶(0[,]+∞) ∧ (𝑚‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → (𝑚 𝑥) = Σ*𝑦𝑥(𝑚𝑦)))} ⊆ {𝑚𝑚:𝑠⟶(0[,]+∞)}
1613, 15ssexi 5322 . . . 4 {𝑚 ∣ (𝑚:𝑠⟶(0[,]+∞) ∧ (𝑚‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → (𝑚 𝑥) = Σ*𝑦𝑥(𝑚𝑦)))} ∈ V
17 simpr 484 . . . . . 6 ((𝑠 = 𝑆𝑚 = 𝑀) → 𝑚 = 𝑀)
18 simpl 482 . . . . . 6 ((𝑠 = 𝑆𝑚 = 𝑀) → 𝑠 = 𝑆)
1917, 18feq12d 6724 . . . . 5 ((𝑠 = 𝑆𝑚 = 𝑀) → (𝑚:𝑠⟶(0[,]+∞) ↔ 𝑀:𝑆⟶(0[,]+∞)))
20 fveq1 6905 . . . . . . 7 (𝑚 = 𝑀 → (𝑚‘∅) = (𝑀‘∅))
2120eqeq1d 2739 . . . . . 6 (𝑚 = 𝑀 → ((𝑚‘∅) = 0 ↔ (𝑀‘∅) = 0))
2221adantl 481 . . . . 5 ((𝑠 = 𝑆𝑚 = 𝑀) → ((𝑚‘∅) = 0 ↔ (𝑀‘∅) = 0))
2318pweqd 4617 . . . . . 6 ((𝑠 = 𝑆𝑚 = 𝑀) → 𝒫 𝑠 = 𝒫 𝑆)
24 fveq1 6905 . . . . . . . . 9 (𝑚 = 𝑀 → (𝑚 𝑥) = (𝑀 𝑥))
25 fveq1 6905 . . . . . . . . . 10 (𝑚 = 𝑀 → (𝑚𝑦) = (𝑀𝑦))
2625esumeq2sdv 34040 . . . . . . . . 9 (𝑚 = 𝑀 → Σ*𝑦𝑥(𝑚𝑦) = Σ*𝑦𝑥(𝑀𝑦))
2724, 26eqeq12d 2753 . . . . . . . 8 (𝑚 = 𝑀 → ((𝑚 𝑥) = Σ*𝑦𝑥(𝑚𝑦) ↔ (𝑀 𝑥) = Σ*𝑦𝑥(𝑀𝑦)))
2827imbi2d 340 . . . . . . 7 (𝑚 = 𝑀 → (((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → (𝑚 𝑥) = Σ*𝑦𝑥(𝑚𝑦)) ↔ ((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → (𝑀 𝑥) = Σ*𝑦𝑥(𝑀𝑦))))
2928adantl 481 . . . . . 6 ((𝑠 = 𝑆𝑚 = 𝑀) → (((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → (𝑚 𝑥) = Σ*𝑦𝑥(𝑚𝑦)) ↔ ((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → (𝑀 𝑥) = Σ*𝑦𝑥(𝑀𝑦))))
3023, 29raleqbidv 3346 . . . . 5 ((𝑠 = 𝑆𝑚 = 𝑀) → (∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → (𝑚 𝑥) = Σ*𝑦𝑥(𝑚𝑦)) ↔ ∀𝑥 ∈ 𝒫 𝑆((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → (𝑀 𝑥) = Σ*𝑦𝑥(𝑀𝑦))))
3119, 22, 303anbi123d 1438 . . . 4 ((𝑠 = 𝑆𝑚 = 𝑀) → ((𝑚:𝑠⟶(0[,]+∞) ∧ (𝑚‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → (𝑚 𝑥) = Σ*𝑦𝑥(𝑚𝑦))) ↔ (𝑀:𝑆⟶(0[,]+∞) ∧ (𝑀‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 𝑆((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → (𝑀 𝑥) = Σ*𝑦𝑥(𝑀𝑦)))))
3210, 16, 31abfmpel 32665 . . 3 ((𝑆 ran sigAlgebra ∧ 𝑀 ∈ V) → (𝑀 ∈ (measures‘𝑆) ↔ (𝑀:𝑆⟶(0[,]+∞) ∧ (𝑀‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 𝑆((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → (𝑀 𝑥) = Σ*𝑦𝑥(𝑀𝑦)))))
3332ex 412 . 2 (𝑆 ran sigAlgebra → (𝑀 ∈ V → (𝑀 ∈ (measures‘𝑆) ↔ (𝑀:𝑆⟶(0[,]+∞) ∧ (𝑀‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 𝑆((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → (𝑀 𝑥) = Σ*𝑦𝑥(𝑀𝑦))))))
342, 9, 33pm5.21ndd 379 1 (𝑆 ran sigAlgebra → (𝑀 ∈ (measures‘𝑆) ↔ (𝑀:𝑆⟶(0[,]+∞) ∧ (𝑀‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 𝑆((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → (𝑀 𝑥) = Σ*𝑦𝑥(𝑀𝑦)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  {cab 2714  wral 3061  Vcvv 3480  c0 4333  𝒫 cpw 4600   cuni 4907  Disj wdisj 5110   class class class wbr 5143  ran crn 5686  wf 6557  cfv 6561  (class class class)co 7431  ωcom 7887  cdom 8983  0cc0 11155  +∞cpnf 11292  [,]cicc 13390  Σ*cesum 34028  sigAlgebracsiga 34109  measurescmeas 34196
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-esum 34029  df-meas 34197
This theorem is referenced by:  measbasedom  34203  measfrge0  34204  measvnul  34207  measvun  34210  measinb  34222  measres  34223  measdivcst  34225  measdivcstALTV  34226  cntmeas  34227  volmeas  34232  ddemeas  34237  omsmeas  34325  dstrvprob  34474
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