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Theorem measres 30883
Description: Building a measure restricted to a smaller sigma-algebra. (Contributed by Thierry Arnoux, 25-Dec-2016.)
Assertion
Ref Expression
measres ((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ran sigAlgebra ∧ 𝑇𝑆) → (𝑀𝑇) ∈ (measures‘𝑇))

Proof of Theorem measres
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp2 1128 . 2 ((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ran sigAlgebra ∧ 𝑇𝑆) → 𝑇 ran sigAlgebra)
2 measfrge0 30864 . . . . 5 (𝑀 ∈ (measures‘𝑆) → 𝑀:𝑆⟶(0[,]+∞))
323ad2ant1 1124 . . . 4 ((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ran sigAlgebra ∧ 𝑇𝑆) → 𝑀:𝑆⟶(0[,]+∞))
4 simp3 1129 . . . 4 ((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ran sigAlgebra ∧ 𝑇𝑆) → 𝑇𝑆)
53, 4fssresd 6321 . . 3 ((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ran sigAlgebra ∧ 𝑇𝑆) → (𝑀𝑇):𝑇⟶(0[,]+∞))
6 0elsiga 30775 . . . . 5 (𝑇 ran sigAlgebra → ∅ ∈ 𝑇)
7 fvres 6465 . . . . 5 (∅ ∈ 𝑇 → ((𝑀𝑇)‘∅) = (𝑀‘∅))
81, 6, 73syl 18 . . . 4 ((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ran sigAlgebra ∧ 𝑇𝑆) → ((𝑀𝑇)‘∅) = (𝑀‘∅))
9 measvnul 30867 . . . . 5 (𝑀 ∈ (measures‘𝑆) → (𝑀‘∅) = 0)
1093ad2ant1 1124 . . . 4 ((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ran sigAlgebra ∧ 𝑇𝑆) → (𝑀‘∅) = 0)
118, 10eqtrd 2814 . . 3 ((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ran sigAlgebra ∧ 𝑇𝑆) → ((𝑀𝑇)‘∅) = 0)
12 simp11 1217 . . . . . . 7 (((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ran sigAlgebra ∧ 𝑇𝑆) ∧ 𝑥 ∈ 𝒫 𝑇 ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) → 𝑀 ∈ (measures‘𝑆))
13 simp13 1219 . . . . . . . 8 (((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ran sigAlgebra ∧ 𝑇𝑆) ∧ 𝑥 ∈ 𝒫 𝑇 ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) → 𝑇𝑆)
14 simp2 1128 . . . . . . . 8 (((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ran sigAlgebra ∧ 𝑇𝑆) ∧ 𝑥 ∈ 𝒫 𝑇 ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) → 𝑥 ∈ 𝒫 𝑇)
15 sspwb 5149 . . . . . . . . 9 (𝑇𝑆 ↔ 𝒫 𝑇 ⊆ 𝒫 𝑆)
16 ssel2 3816 . . . . . . . . 9 ((𝒫 𝑇 ⊆ 𝒫 𝑆𝑥 ∈ 𝒫 𝑇) → 𝑥 ∈ 𝒫 𝑆)
1715, 16sylanb 576 . . . . . . . 8 ((𝑇𝑆𝑥 ∈ 𝒫 𝑇) → 𝑥 ∈ 𝒫 𝑆)
1813, 14, 17syl2anc 579 . . . . . . 7 (((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ran sigAlgebra ∧ 𝑇𝑆) ∧ 𝑥 ∈ 𝒫 𝑇 ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) → 𝑥 ∈ 𝒫 𝑆)
19 simp3 1129 . . . . . . 7 (((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ran sigAlgebra ∧ 𝑇𝑆) ∧ 𝑥 ∈ 𝒫 𝑇 ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) → (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦))
20 measvun 30870 . . . . . . 7 ((𝑀 ∈ (measures‘𝑆) ∧ 𝑥 ∈ 𝒫 𝑆 ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) → (𝑀 𝑥) = Σ*𝑦𝑥(𝑀𝑦))
2112, 18, 19, 20syl3anc 1439 . . . . . 6 (((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ran sigAlgebra ∧ 𝑇𝑆) ∧ 𝑥 ∈ 𝒫 𝑇 ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) → (𝑀 𝑥) = Σ*𝑦𝑥(𝑀𝑦))
2213ad2ant1 1124 . . . . . . . 8 (((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ran sigAlgebra ∧ 𝑇𝑆) ∧ 𝑥 ∈ 𝒫 𝑇 ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) → 𝑇 ran sigAlgebra)
23 simp3l 1215 . . . . . . . 8 (((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ran sigAlgebra ∧ 𝑇𝑆) ∧ 𝑥 ∈ 𝒫 𝑇 ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) → 𝑥 ≼ ω)
24 sigaclcu 30778 . . . . . . . 8 ((𝑇 ran sigAlgebra ∧ 𝑥 ∈ 𝒫 𝑇𝑥 ≼ ω) → 𝑥𝑇)
2522, 14, 23, 24syl3anc 1439 . . . . . . 7 (((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ran sigAlgebra ∧ 𝑇𝑆) ∧ 𝑥 ∈ 𝒫 𝑇 ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) → 𝑥𝑇)
26 fvres 6465 . . . . . . 7 ( 𝑥𝑇 → ((𝑀𝑇)‘ 𝑥) = (𝑀 𝑥))
2725, 26syl 17 . . . . . 6 (((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ran sigAlgebra ∧ 𝑇𝑆) ∧ 𝑥 ∈ 𝒫 𝑇 ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) → ((𝑀𝑇)‘ 𝑥) = (𝑀 𝑥))
28 elpwi 4389 . . . . . . . . . . 11 (𝑥 ∈ 𝒫 𝑇𝑥𝑇)
2928sselda 3821 . . . . . . . . . 10 ((𝑥 ∈ 𝒫 𝑇𝑦𝑥) → 𝑦𝑇)
3029adantll 704 . . . . . . . . 9 ((((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ran sigAlgebra ∧ 𝑇𝑆) ∧ 𝑥 ∈ 𝒫 𝑇) ∧ 𝑦𝑥) → 𝑦𝑇)
31 fvres 6465 . . . . . . . . 9 (𝑦𝑇 → ((𝑀𝑇)‘𝑦) = (𝑀𝑦))
3230, 31syl 17 . . . . . . . 8 ((((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ran sigAlgebra ∧ 𝑇𝑆) ∧ 𝑥 ∈ 𝒫 𝑇) ∧ 𝑦𝑥) → ((𝑀𝑇)‘𝑦) = (𝑀𝑦))
3332esumeq2dv 30698 . . . . . . 7 (((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ran sigAlgebra ∧ 𝑇𝑆) ∧ 𝑥 ∈ 𝒫 𝑇) → Σ*𝑦𝑥((𝑀𝑇)‘𝑦) = Σ*𝑦𝑥(𝑀𝑦))
34333adant3 1123 . . . . . 6 (((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ran sigAlgebra ∧ 𝑇𝑆) ∧ 𝑥 ∈ 𝒫 𝑇 ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) → Σ*𝑦𝑥((𝑀𝑇)‘𝑦) = Σ*𝑦𝑥(𝑀𝑦))
3521, 27, 343eqtr4d 2824 . . . . 5 (((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ran sigAlgebra ∧ 𝑇𝑆) ∧ 𝑥 ∈ 𝒫 𝑇 ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) → ((𝑀𝑇)‘ 𝑥) = Σ*𝑦𝑥((𝑀𝑇)‘𝑦))
36353expia 1111 . . . 4 (((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ran sigAlgebra ∧ 𝑇𝑆) ∧ 𝑥 ∈ 𝒫 𝑇) → ((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → ((𝑀𝑇)‘ 𝑥) = Σ*𝑦𝑥((𝑀𝑇)‘𝑦)))
3736ralrimiva 3148 . . 3 ((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ran sigAlgebra ∧ 𝑇𝑆) → ∀𝑥 ∈ 𝒫 𝑇((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → ((𝑀𝑇)‘ 𝑥) = Σ*𝑦𝑥((𝑀𝑇)‘𝑦)))
385, 11, 373jca 1119 . 2 ((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ran sigAlgebra ∧ 𝑇𝑆) → ((𝑀𝑇):𝑇⟶(0[,]+∞) ∧ ((𝑀𝑇)‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 𝑇((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → ((𝑀𝑇)‘ 𝑥) = Σ*𝑦𝑥((𝑀𝑇)‘𝑦))))
39 ismeas 30860 . . 3 (𝑇 ran sigAlgebra → ((𝑀𝑇) ∈ (measures‘𝑇) ↔ ((𝑀𝑇):𝑇⟶(0[,]+∞) ∧ ((𝑀𝑇)‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 𝑇((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → ((𝑀𝑇)‘ 𝑥) = Σ*𝑦𝑥((𝑀𝑇)‘𝑦)))))
4039biimprd 240 . 2 (𝑇 ran sigAlgebra → (((𝑀𝑇):𝑇⟶(0[,]+∞) ∧ ((𝑀𝑇)‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 𝑇((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → ((𝑀𝑇)‘ 𝑥) = Σ*𝑦𝑥((𝑀𝑇)‘𝑦))) → (𝑀𝑇) ∈ (measures‘𝑇)))
411, 38, 40sylc 65 1 ((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ran sigAlgebra ∧ 𝑇𝑆) → (𝑀𝑇) ∈ (measures‘𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  w3a 1071   = wceq 1601  wcel 2107  wral 3090  wss 3792  c0 4141  𝒫 cpw 4379   cuni 4671  Disj wdisj 4854   class class class wbr 4886  ran crn 5356  cres 5357  wf 6131  cfv 6135  (class class class)co 6922  ωcom 7343  cdom 8239  0cc0 10272  +∞cpnf 10408  [,]cicc 12490  Σ*cesum 30687  sigAlgebracsiga 30768  measurescmeas 30856
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-fal 1615  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-disj 4855  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-fv 6143  df-ov 6925  df-esum 30688  df-siga 30769  df-meas 30857
This theorem is referenced by:  measinb2  30884
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