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Theorem ismeas 31075
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 3455 . . 3 (𝑀 ∈ (measures‘𝑆) → 𝑀 ∈ V)
21a1i 11 . 2 (𝑆 ran sigAlgebra → (𝑀 ∈ (measures‘𝑆) → 𝑀 ∈ V))
3 simp1 1129 . . 3 ((𝑀:𝑆⟶(0[,]+∞) ∧ (𝑀‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 𝑆((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → (𝑀 𝑥) = Σ*𝑦𝑥(𝑀𝑦))) → 𝑀:𝑆⟶(0[,]+∞))
4 ovex 7048 . . . 4 (0[,]+∞) ∈ V
5 fex2 7494 . . . . . 6 ((𝑀:𝑆⟶(0[,]+∞) ∧ 𝑆 ran sigAlgebra ∧ (0[,]+∞) ∈ V) → 𝑀 ∈ V)
653expb 1113 . . . . 5 ((𝑀:𝑆⟶(0[,]+∞) ∧ (𝑆 ran sigAlgebra ∧ (0[,]+∞) ∈ V)) → 𝑀 ∈ V)
76expcom 414 . . . 4 ((𝑆 ran sigAlgebra ∧ (0[,]+∞) ∈ V) → (𝑀:𝑆⟶(0[,]+∞) → 𝑀 ∈ V))
84, 7mpan2 687 . . 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 31072 . . . 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 3440 . . . . . 6 𝑠 ∈ V
12 mapex 8262 . . . . . 6 ((𝑠 ∈ V ∧ (0[,]+∞) ∈ V) → {𝑚𝑚:𝑠⟶(0[,]+∞)} ∈ V)
1311, 4, 12mp2an 688 . . . . 5 {𝑚𝑚:𝑠⟶(0[,]+∞)} ∈ V
14 simp1 1129 . . . . . 6 ((𝑚:𝑠⟶(0[,]+∞) ∧ (𝑚‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → (𝑚 𝑥) = Σ*𝑦𝑥(𝑚𝑦))) → 𝑚:𝑠⟶(0[,]+∞))
1514ss2abi 3964 . . . . 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 5117 . . . 4 {𝑚 ∣ (𝑚:𝑠⟶(0[,]+∞) ∧ (𝑚‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → (𝑚 𝑥) = Σ*𝑦𝑥(𝑚𝑦)))} ∈ V
17 simpr 485 . . . . . 6 ((𝑠 = 𝑆𝑚 = 𝑀) → 𝑚 = 𝑀)
18 simpl 483 . . . . . 6 ((𝑠 = 𝑆𝑚 = 𝑀) → 𝑠 = 𝑆)
1917, 18feq12d 6370 . . . . 5 ((𝑠 = 𝑆𝑚 = 𝑀) → (𝑚:𝑠⟶(0[,]+∞) ↔ 𝑀:𝑆⟶(0[,]+∞)))
20 fveq1 6537 . . . . . . 7 (𝑚 = 𝑀 → (𝑚‘∅) = (𝑀‘∅))
2120eqeq1d 2797 . . . . . 6 (𝑚 = 𝑀 → ((𝑚‘∅) = 0 ↔ (𝑀‘∅) = 0))
2221adantl 482 . . . . 5 ((𝑠 = 𝑆𝑚 = 𝑀) → ((𝑚‘∅) = 0 ↔ (𝑀‘∅) = 0))
2318pweqd 4458 . . . . . 6 ((𝑠 = 𝑆𝑚 = 𝑀) → 𝒫 𝑠 = 𝒫 𝑆)
24 fveq1 6537 . . . . . . . . 9 (𝑚 = 𝑀 → (𝑚 𝑥) = (𝑀 𝑥))
25 fveq1 6537 . . . . . . . . . 10 (𝑚 = 𝑀 → (𝑚𝑦) = (𝑀𝑦))
2625esumeq2sdv 30915 . . . . . . . . 9 (𝑚 = 𝑀 → Σ*𝑦𝑥(𝑚𝑦) = Σ*𝑦𝑥(𝑀𝑦))
2724, 26eqeq12d 2810 . . . . . . . 8 (𝑚 = 𝑀 → ((𝑚 𝑥) = Σ*𝑦𝑥(𝑚𝑦) ↔ (𝑀 𝑥) = Σ*𝑦𝑥(𝑀𝑦)))
2827imbi2d 342 . . . . . . 7 (𝑚 = 𝑀 → (((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → (𝑚 𝑥) = Σ*𝑦𝑥(𝑚𝑦)) ↔ ((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → (𝑀 𝑥) = Σ*𝑦𝑥(𝑀𝑦))))
2928adantl 482 . . . . . 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 3361 . . . . 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 1428 . . . 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 30090 . . 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 413 . 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 381 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 207  wa 396  w3a 1080   = wceq 1522  wcel 2081  {cab 2775  wral 3105  Vcvv 3437  c0 4211  𝒫 cpw 4453   cuni 4745  Disj wdisj 4930   class class class wbr 4962  ran crn 5444  wf 6221  cfv 6225  (class class class)co 7016  ωcom 7436  cdom 8355  0cc0 10383  +∞cpnf 10518  [,]cicc 12591  Σ*cesum 30903  sigAlgebracsiga 30984  measurescmeas 31071
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-fal 1535  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-br 4963  df-opab 5025  df-mpt 5042  df-id 5348  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-fv 6233  df-ov 7019  df-esum 30904  df-meas 31072
This theorem is referenced by:  measbasedom  31078  measfrge0  31079  measvnul  31082  measvun  31085  measinb  31097  measres  31098  measdivcst  31100  measdivcstALTV  31101  cntmeas  31102  volmeas  31107  ddemeas  31112  omsmeas  31198  dstrvprob  31346
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