Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  measres Structured version   Visualization version   GIF version

Theorem measres 32821
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 1137 . 2 ((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ran sigAlgebra ∧ 𝑇𝑆) → 𝑇 ran sigAlgebra)
2 measfrge0 32802 . . . . 5 (𝑀 ∈ (measures‘𝑆) → 𝑀:𝑆⟶(0[,]+∞))
323ad2ant1 1133 . . . 4 ((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ran sigAlgebra ∧ 𝑇𝑆) → 𝑀:𝑆⟶(0[,]+∞))
4 simp3 1138 . . . 4 ((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ran sigAlgebra ∧ 𝑇𝑆) → 𝑇𝑆)
53, 4fssresd 6709 . . 3 ((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ran sigAlgebra ∧ 𝑇𝑆) → (𝑀𝑇):𝑇⟶(0[,]+∞))
6 0elsiga 32713 . . . . 5 (𝑇 ran sigAlgebra → ∅ ∈ 𝑇)
7 fvres 6861 . . . . 5 (∅ ∈ 𝑇 → ((𝑀𝑇)‘∅) = (𝑀‘∅))
81, 6, 73syl 18 . . . 4 ((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ran sigAlgebra ∧ 𝑇𝑆) → ((𝑀𝑇)‘∅) = (𝑀‘∅))
9 measvnul 32805 . . . . 5 (𝑀 ∈ (measures‘𝑆) → (𝑀‘∅) = 0)
1093ad2ant1 1133 . . . 4 ((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ran sigAlgebra ∧ 𝑇𝑆) → (𝑀‘∅) = 0)
118, 10eqtrd 2776 . . 3 ((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ran sigAlgebra ∧ 𝑇𝑆) → ((𝑀𝑇)‘∅) = 0)
12 simp11 1203 . . . . . . 7 (((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ran sigAlgebra ∧ 𝑇𝑆) ∧ 𝑥 ∈ 𝒫 𝑇 ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) → 𝑀 ∈ (measures‘𝑆))
13 simp13 1205 . . . . . . . 8 (((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ran sigAlgebra ∧ 𝑇𝑆) ∧ 𝑥 ∈ 𝒫 𝑇 ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) → 𝑇𝑆)
14 simp2 1137 . . . . . . . 8 (((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ran sigAlgebra ∧ 𝑇𝑆) ∧ 𝑥 ∈ 𝒫 𝑇 ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) → 𝑥 ∈ 𝒫 𝑇)
15 sspw 4571 . . . . . . . . 9 (𝑇𝑆 → 𝒫 𝑇 ⊆ 𝒫 𝑆)
1615sselda 3944 . . . . . . . 8 ((𝑇𝑆𝑥 ∈ 𝒫 𝑇) → 𝑥 ∈ 𝒫 𝑆)
1713, 14, 16syl2anc 584 . . . . . . 7 (((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ran sigAlgebra ∧ 𝑇𝑆) ∧ 𝑥 ∈ 𝒫 𝑇 ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) → 𝑥 ∈ 𝒫 𝑆)
18 simp3 1138 . . . . . . 7 (((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ran sigAlgebra ∧ 𝑇𝑆) ∧ 𝑥 ∈ 𝒫 𝑇 ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) → (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦))
19 measvun 32808 . . . . . . 7 ((𝑀 ∈ (measures‘𝑆) ∧ 𝑥 ∈ 𝒫 𝑆 ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) → (𝑀 𝑥) = Σ*𝑦𝑥(𝑀𝑦))
2012, 17, 18, 19syl3anc 1371 . . . . . 6 (((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ran sigAlgebra ∧ 𝑇𝑆) ∧ 𝑥 ∈ 𝒫 𝑇 ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) → (𝑀 𝑥) = Σ*𝑦𝑥(𝑀𝑦))
2113ad2ant1 1133 . . . . . . . 8 (((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ran sigAlgebra ∧ 𝑇𝑆) ∧ 𝑥 ∈ 𝒫 𝑇 ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) → 𝑇 ran sigAlgebra)
22 simp3l 1201 . . . . . . . 8 (((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ran sigAlgebra ∧ 𝑇𝑆) ∧ 𝑥 ∈ 𝒫 𝑇 ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) → 𝑥 ≼ ω)
23 sigaclcu 32716 . . . . . . . 8 ((𝑇 ran sigAlgebra ∧ 𝑥 ∈ 𝒫 𝑇𝑥 ≼ ω) → 𝑥𝑇)
2421, 14, 22, 23syl3anc 1371 . . . . . . 7 (((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ran sigAlgebra ∧ 𝑇𝑆) ∧ 𝑥 ∈ 𝒫 𝑇 ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) → 𝑥𝑇)
2524fvresd 6862 . . . . . 6 (((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ran sigAlgebra ∧ 𝑇𝑆) ∧ 𝑥 ∈ 𝒫 𝑇 ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) → ((𝑀𝑇)‘ 𝑥) = (𝑀 𝑥))
26 elpwi 4567 . . . . . . . . . . 11 (𝑥 ∈ 𝒫 𝑇𝑥𝑇)
2726sselda 3944 . . . . . . . . . 10 ((𝑥 ∈ 𝒫 𝑇𝑦𝑥) → 𝑦𝑇)
2827adantll 712 . . . . . . . . 9 ((((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ran sigAlgebra ∧ 𝑇𝑆) ∧ 𝑥 ∈ 𝒫 𝑇) ∧ 𝑦𝑥) → 𝑦𝑇)
2928fvresd 6862 . . . . . . . 8 ((((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ran sigAlgebra ∧ 𝑇𝑆) ∧ 𝑥 ∈ 𝒫 𝑇) ∧ 𝑦𝑥) → ((𝑀𝑇)‘𝑦) = (𝑀𝑦))
3029esumeq2dv 32637 . . . . . . 7 (((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ran sigAlgebra ∧ 𝑇𝑆) ∧ 𝑥 ∈ 𝒫 𝑇) → Σ*𝑦𝑥((𝑀𝑇)‘𝑦) = Σ*𝑦𝑥(𝑀𝑦))
31303adant3 1132 . . . . . 6 (((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ran sigAlgebra ∧ 𝑇𝑆) ∧ 𝑥 ∈ 𝒫 𝑇 ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) → Σ*𝑦𝑥((𝑀𝑇)‘𝑦) = Σ*𝑦𝑥(𝑀𝑦))
3220, 25, 313eqtr4d 2786 . . . . 5 (((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ran sigAlgebra ∧ 𝑇𝑆) ∧ 𝑥 ∈ 𝒫 𝑇 ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) → ((𝑀𝑇)‘ 𝑥) = Σ*𝑦𝑥((𝑀𝑇)‘𝑦))
33323expia 1121 . . . 4 (((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ran sigAlgebra ∧ 𝑇𝑆) ∧ 𝑥 ∈ 𝒫 𝑇) → ((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → ((𝑀𝑇)‘ 𝑥) = Σ*𝑦𝑥((𝑀𝑇)‘𝑦)))
3433ralrimiva 3143 . . 3 ((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ran sigAlgebra ∧ 𝑇𝑆) → ∀𝑥 ∈ 𝒫 𝑇((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → ((𝑀𝑇)‘ 𝑥) = Σ*𝑦𝑥((𝑀𝑇)‘𝑦)))
355, 11, 343jca 1128 . 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))))
36 ismeas 32798 . . 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)))))
3736biimprd 247 . 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)))
381, 35, 37sylc 65 1 ((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ran sigAlgebra ∧ 𝑇𝑆) → (𝑀𝑇) ∈ (measures‘𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087   = wceq 1541  wcel 2106  wral 3064  wss 3910  c0 4282  𝒫 cpw 4560   cuni 4865  Disj wdisj 5070   class class class wbr 5105  ran crn 5634  cres 5635  wf 6492  cfv 6496  (class class class)co 7357  ωcom 7802  cdom 8881  0cc0 11051  +∞cpnf 11186  [,]cicc 13267  Σ*cesum 32626  sigAlgebracsiga 32707  measurescmeas 32794
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-disj 5071  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-fv 6504  df-ov 7360  df-esum 32627  df-siga 32708  df-meas 32795
This theorem is referenced by:  measinb2  32822
  Copyright terms: Public domain W3C validator