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Theorem measres 31499
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 1134 . 2 ((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ran sigAlgebra ∧ 𝑇𝑆) → 𝑇 ran sigAlgebra)
2 measfrge0 31480 . . . . 5 (𝑀 ∈ (measures‘𝑆) → 𝑀:𝑆⟶(0[,]+∞))
323ad2ant1 1130 . . . 4 ((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ran sigAlgebra ∧ 𝑇𝑆) → 𝑀:𝑆⟶(0[,]+∞))
4 simp3 1135 . . . 4 ((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ran sigAlgebra ∧ 𝑇𝑆) → 𝑇𝑆)
53, 4fssresd 6526 . . 3 ((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ran sigAlgebra ∧ 𝑇𝑆) → (𝑀𝑇):𝑇⟶(0[,]+∞))
6 0elsiga 31391 . . . . 5 (𝑇 ran sigAlgebra → ∅ ∈ 𝑇)
7 fvres 6670 . . . . 5 (∅ ∈ 𝑇 → ((𝑀𝑇)‘∅) = (𝑀‘∅))
81, 6, 73syl 18 . . . 4 ((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ran sigAlgebra ∧ 𝑇𝑆) → ((𝑀𝑇)‘∅) = (𝑀‘∅))
9 measvnul 31483 . . . . 5 (𝑀 ∈ (measures‘𝑆) → (𝑀‘∅) = 0)
1093ad2ant1 1130 . . . 4 ((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ran sigAlgebra ∧ 𝑇𝑆) → (𝑀‘∅) = 0)
118, 10eqtrd 2859 . . 3 ((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ran sigAlgebra ∧ 𝑇𝑆) → ((𝑀𝑇)‘∅) = 0)
12 simp11 1200 . . . . . . 7 (((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ran sigAlgebra ∧ 𝑇𝑆) ∧ 𝑥 ∈ 𝒫 𝑇 ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) → 𝑀 ∈ (measures‘𝑆))
13 simp13 1202 . . . . . . . 8 (((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ran sigAlgebra ∧ 𝑇𝑆) ∧ 𝑥 ∈ 𝒫 𝑇 ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) → 𝑇𝑆)
14 simp2 1134 . . . . . . . 8 (((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ran sigAlgebra ∧ 𝑇𝑆) ∧ 𝑥 ∈ 𝒫 𝑇 ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) → 𝑥 ∈ 𝒫 𝑇)
15 sspw 4533 . . . . . . . . 9 (𝑇𝑆 → 𝒫 𝑇 ⊆ 𝒫 𝑆)
1615sselda 3951 . . . . . . . 8 ((𝑇𝑆𝑥 ∈ 𝒫 𝑇) → 𝑥 ∈ 𝒫 𝑆)
1713, 14, 16syl2anc 587 . . . . . . 7 (((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ran sigAlgebra ∧ 𝑇𝑆) ∧ 𝑥 ∈ 𝒫 𝑇 ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) → 𝑥 ∈ 𝒫 𝑆)
18 simp3 1135 . . . . . . 7 (((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ran sigAlgebra ∧ 𝑇𝑆) ∧ 𝑥 ∈ 𝒫 𝑇 ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) → (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦))
19 measvun 31486 . . . . . . 7 ((𝑀 ∈ (measures‘𝑆) ∧ 𝑥 ∈ 𝒫 𝑆 ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) → (𝑀 𝑥) = Σ*𝑦𝑥(𝑀𝑦))
2012, 17, 18, 19syl3anc 1368 . . . . . 6 (((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ran sigAlgebra ∧ 𝑇𝑆) ∧ 𝑥 ∈ 𝒫 𝑇 ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) → (𝑀 𝑥) = Σ*𝑦𝑥(𝑀𝑦))
2113ad2ant1 1130 . . . . . . . 8 (((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ran sigAlgebra ∧ 𝑇𝑆) ∧ 𝑥 ∈ 𝒫 𝑇 ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) → 𝑇 ran sigAlgebra)
22 simp3l 1198 . . . . . . . 8 (((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ran sigAlgebra ∧ 𝑇𝑆) ∧ 𝑥 ∈ 𝒫 𝑇 ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) → 𝑥 ≼ ω)
23 sigaclcu 31394 . . . . . . . 8 ((𝑇 ran sigAlgebra ∧ 𝑥 ∈ 𝒫 𝑇𝑥 ≼ ω) → 𝑥𝑇)
2421, 14, 22, 23syl3anc 1368 . . . . . . 7 (((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ran sigAlgebra ∧ 𝑇𝑆) ∧ 𝑥 ∈ 𝒫 𝑇 ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) → 𝑥𝑇)
2524fvresd 6671 . . . . . 6 (((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ran sigAlgebra ∧ 𝑇𝑆) ∧ 𝑥 ∈ 𝒫 𝑇 ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) → ((𝑀𝑇)‘ 𝑥) = (𝑀 𝑥))
26 elpwi 4529 . . . . . . . . . . 11 (𝑥 ∈ 𝒫 𝑇𝑥𝑇)
2726sselda 3951 . . . . . . . . . 10 ((𝑥 ∈ 𝒫 𝑇𝑦𝑥) → 𝑦𝑇)
2827adantll 713 . . . . . . . . 9 ((((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ran sigAlgebra ∧ 𝑇𝑆) ∧ 𝑥 ∈ 𝒫 𝑇) ∧ 𝑦𝑥) → 𝑦𝑇)
2928fvresd 6671 . . . . . . . 8 ((((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ran sigAlgebra ∧ 𝑇𝑆) ∧ 𝑥 ∈ 𝒫 𝑇) ∧ 𝑦𝑥) → ((𝑀𝑇)‘𝑦) = (𝑀𝑦))
3029esumeq2dv 31315 . . . . . . 7 (((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ran sigAlgebra ∧ 𝑇𝑆) ∧ 𝑥 ∈ 𝒫 𝑇) → Σ*𝑦𝑥((𝑀𝑇)‘𝑦) = Σ*𝑦𝑥(𝑀𝑦))
31303adant3 1129 . . . . . 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 2869 . . . . 5 (((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ran sigAlgebra ∧ 𝑇𝑆) ∧ 𝑥 ∈ 𝒫 𝑇 ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) → ((𝑀𝑇)‘ 𝑥) = Σ*𝑦𝑥((𝑀𝑇)‘𝑦))
33323expia 1118 . . . 4 (((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ran sigAlgebra ∧ 𝑇𝑆) ∧ 𝑥 ∈ 𝒫 𝑇) → ((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → ((𝑀𝑇)‘ 𝑥) = Σ*𝑦𝑥((𝑀𝑇)‘𝑦)))
3433ralrimiva 3176 . . 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 1125 . 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 31476 . . 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 251 . 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 399  w3a 1084   = wceq 1538  wcel 2115  wral 3132  wss 3918  c0 4274  𝒫 cpw 4520   cuni 4819  Disj wdisj 5012   class class class wbr 5047  ran crn 5537  cres 5538  wf 6332  cfv 6336  (class class class)co 7138  ωcom 7563  cdom 8490  0cc0 10522  +∞cpnf 10657  [,]cicc 12727  Σ*cesum 31304  sigAlgebracsiga 31385  measurescmeas 31472
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 5184  ax-nul 5191  ax-pow 5247  ax-pr 5311  ax-un 7444
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 3137  df-rex 3138  df-rmo 3140  df-rab 3141  df-v 3481  df-sbc 3758  df-csb 3866  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4820  df-disj 5013  df-br 5048  df-opab 5110  df-mpt 5128  df-id 5441  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-fv 6344  df-ov 7141  df-esum 31305  df-siga 31386  df-meas 31473
This theorem is referenced by:  measinb2  31500
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