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Theorem measres 34055
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 34036 . . . . 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 6769 . . 3 ((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ran sigAlgebra ∧ 𝑇𝑆) → (𝑀𝑇):𝑇⟶(0[,]+∞))
6 0elsiga 33947 . . . . 5 (𝑇 ran sigAlgebra → ∅ ∈ 𝑇)
7 fvres 6920 . . . . 5 (∅ ∈ 𝑇 → ((𝑀𝑇)‘∅) = (𝑀‘∅))
81, 6, 73syl 18 . . . 4 ((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ran sigAlgebra ∧ 𝑇𝑆) → ((𝑀𝑇)‘∅) = (𝑀‘∅))
9 measvnul 34039 . . . . 5 (𝑀 ∈ (measures‘𝑆) → (𝑀‘∅) = 0)
1093ad2ant1 1130 . . . 4 ((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ran sigAlgebra ∧ 𝑇𝑆) → (𝑀‘∅) = 0)
118, 10eqtrd 2766 . . 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 4618 . . . . . . . . 9 (𝑇𝑆 → 𝒫 𝑇 ⊆ 𝒫 𝑆)
1615sselda 3979 . . . . . . . 8 ((𝑇𝑆𝑥 ∈ 𝒫 𝑇) → 𝑥 ∈ 𝒫 𝑆)
1713, 14, 16syl2anc 582 . . . . . . 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 34042 . . . . . . 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 33950 . . . . . . . 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 6921 . . . . . 6 (((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ran sigAlgebra ∧ 𝑇𝑆) ∧ 𝑥 ∈ 𝒫 𝑇 ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) → ((𝑀𝑇)‘ 𝑥) = (𝑀 𝑥))
26 elpwi 4614 . . . . . . . . . . 11 (𝑥 ∈ 𝒫 𝑇𝑥𝑇)
2726sselda 3979 . . . . . . . . . 10 ((𝑥 ∈ 𝒫 𝑇𝑦𝑥) → 𝑦𝑇)
2827adantll 712 . . . . . . . . 9 ((((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ran sigAlgebra ∧ 𝑇𝑆) ∧ 𝑥 ∈ 𝒫 𝑇) ∧ 𝑦𝑥) → 𝑦𝑇)
2928fvresd 6921 . . . . . . . 8 ((((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ran sigAlgebra ∧ 𝑇𝑆) ∧ 𝑥 ∈ 𝒫 𝑇) ∧ 𝑦𝑥) → ((𝑀𝑇)‘𝑦) = (𝑀𝑦))
3029esumeq2dv 33871 . . . . . . 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 2776 . . . . 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 3136 . . 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 34032 . . 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 394  w3a 1084   = wceq 1534  wcel 2099  wral 3051  wss 3947  c0 4325  𝒫 cpw 4607   cuni 4913  Disj wdisj 5118   class class class wbr 5153  ran crn 5683  cres 5684  wf 6550  cfv 6554  (class class class)co 7424  ωcom 7876  cdom 8972  0cc0 11158  +∞cpnf 11295  [,]cicc 13381  Σ*cesum 33860  sigAlgebracsiga 33941  measurescmeas 34028
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-disj 5119  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-fv 6562  df-ov 7427  df-esum 33861  df-siga 33942  df-meas 34029
This theorem is referenced by:  measinb2  34056
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