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Theorem ismeas 31543
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 3498 . . 3 (𝑀 ∈ (measures‘𝑆) → 𝑀 ∈ V)
21a1i 11 . 2 (𝑆 ran sigAlgebra → (𝑀 ∈ (measures‘𝑆) → 𝑀 ∈ V))
3 simp1 1133 . . 3 ((𝑀:𝑆⟶(0[,]+∞) ∧ (𝑀‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 𝑆((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → (𝑀 𝑥) = Σ*𝑦𝑥(𝑀𝑦))) → 𝑀:𝑆⟶(0[,]+∞))
4 ovex 7184 . . . 4 (0[,]+∞) ∈ V
5 fex2 7635 . . . . . 6 ((𝑀:𝑆⟶(0[,]+∞) ∧ 𝑆 ran sigAlgebra ∧ (0[,]+∞) ∈ V) → 𝑀 ∈ V)
653expb 1117 . . . . 5 ((𝑀:𝑆⟶(0[,]+∞) ∧ (𝑆 ran sigAlgebra ∧ (0[,]+∞) ∈ V)) → 𝑀 ∈ V)
76expcom 417 . . . 4 ((𝑆 ran sigAlgebra ∧ (0[,]+∞) ∈ V) → (𝑀:𝑆⟶(0[,]+∞) → 𝑀 ∈ V))
84, 7mpan2 690 . . 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 31540 . . . 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 3483 . . . . . 6 𝑠 ∈ V
12 mapex 8410 . . . . . 6 ((𝑠 ∈ V ∧ (0[,]+∞) ∈ V) → {𝑚𝑚:𝑠⟶(0[,]+∞)} ∈ V)
1311, 4, 12mp2an 691 . . . . 5 {𝑚𝑚:𝑠⟶(0[,]+∞)} ∈ V
14 simp1 1133 . . . . . 6 ((𝑚:𝑠⟶(0[,]+∞) ∧ (𝑚‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → (𝑚 𝑥) = Σ*𝑦𝑥(𝑚𝑦))) → 𝑚:𝑠⟶(0[,]+∞))
1514ss2abi 4029 . . . . 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 5213 . . . 4 {𝑚 ∣ (𝑚:𝑠⟶(0[,]+∞) ∧ (𝑚‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → (𝑚 𝑥) = Σ*𝑦𝑥(𝑚𝑦)))} ∈ V
17 simpr 488 . . . . . 6 ((𝑠 = 𝑆𝑚 = 𝑀) → 𝑚 = 𝑀)
18 simpl 486 . . . . . 6 ((𝑠 = 𝑆𝑚 = 𝑀) → 𝑠 = 𝑆)
1917, 18feq12d 6493 . . . . 5 ((𝑠 = 𝑆𝑚 = 𝑀) → (𝑚:𝑠⟶(0[,]+∞) ↔ 𝑀:𝑆⟶(0[,]+∞)))
20 fveq1 6662 . . . . . . 7 (𝑚 = 𝑀 → (𝑚‘∅) = (𝑀‘∅))
2120eqeq1d 2826 . . . . . 6 (𝑚 = 𝑀 → ((𝑚‘∅) = 0 ↔ (𝑀‘∅) = 0))
2221adantl 485 . . . . 5 ((𝑠 = 𝑆𝑚 = 𝑀) → ((𝑚‘∅) = 0 ↔ (𝑀‘∅) = 0))
2318pweqd 4541 . . . . . 6 ((𝑠 = 𝑆𝑚 = 𝑀) → 𝒫 𝑠 = 𝒫 𝑆)
24 fveq1 6662 . . . . . . . . 9 (𝑚 = 𝑀 → (𝑚 𝑥) = (𝑀 𝑥))
25 fveq1 6662 . . . . . . . . . 10 (𝑚 = 𝑀 → (𝑚𝑦) = (𝑀𝑦))
2625esumeq2sdv 31383 . . . . . . . . 9 (𝑚 = 𝑀 → Σ*𝑦𝑥(𝑚𝑦) = Σ*𝑦𝑥(𝑀𝑦))
2724, 26eqeq12d 2840 . . . . . . . 8 (𝑚 = 𝑀 → ((𝑚 𝑥) = Σ*𝑦𝑥(𝑚𝑦) ↔ (𝑀 𝑥) = Σ*𝑦𝑥(𝑀𝑦)))
2827imbi2d 344 . . . . . . 7 (𝑚 = 𝑀 → (((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → (𝑚 𝑥) = Σ*𝑦𝑥(𝑚𝑦)) ↔ ((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → (𝑀 𝑥) = Σ*𝑦𝑥(𝑀𝑦))))
2928adantl 485 . . . . . 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 3392 . . . . 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 1433 . . . 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 30419 . . 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 416 . 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 384 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 209  wa 399  w3a 1084   = wceq 1538  wcel 2115  {cab 2802  wral 3133  Vcvv 3480  c0 4276  𝒫 cpw 4522   cuni 4824  Disj wdisj 5018   class class class wbr 5053  ran crn 5544  wf 6341  cfv 6345  (class class class)co 7151  ωcom 7576  cdom 8505  0cc0 10537  +∞cpnf 10672  [,]cicc 12740  Σ*cesum 31371  sigAlgebracsiga 31452  measurescmeas 31539
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7457
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-iota 6304  df-fun 6347  df-fn 6348  df-f 6349  df-fv 6353  df-ov 7154  df-esum 31372  df-meas 31540
This theorem is referenced by:  measbasedom  31546  measfrge0  31547  measvnul  31550  measvun  31553  measinb  31565  measres  31566  measdivcst  31568  measdivcstALTV  31569  cntmeas  31570  volmeas  31575  ddemeas  31580  omsmeas  31666  dstrvprob  31814
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