Theorem measres 34258

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 ∧ 𝑇 ⊆ 𝑆) → (𝑀 ↾ 𝑇) ∈ (measures‘𝑇))

Proof of Theorem measres

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp2 1137	. 2 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆) → 𝑇 ∈ ∪ ran sigAlgebra)
2		measfrge0 34239	. . . . 5 ⊢ (𝑀 ∈ (measures‘𝑆) → 𝑀:𝑆⟶(0[,]+∞))
3	2	3ad2ant1 1133	. . . 4 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆) → 𝑀:𝑆⟶(0[,]+∞))
4		simp3 1138	. . . 4 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆) → 𝑇 ⊆ 𝑆)
5	3, 4	fssresd 6750	. . 3 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆) → (𝑀 ↾ 𝑇):𝑇⟶(0[,]+∞))
6		0elsiga 34150	. . . . 5 ⊢ (𝑇 ∈ ∪ ran sigAlgebra → ∅ ∈ 𝑇)
7		fvres 6900	. . . . 5 ⊢ (∅ ∈ 𝑇 → ((𝑀 ↾ 𝑇)‘∅) = (𝑀‘∅))
8	1, 6, 7	3syl 18	. . . 4 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆) → ((𝑀 ↾ 𝑇)‘∅) = (𝑀‘∅))
9		measvnul 34242	. . . . 5 ⊢ (𝑀 ∈ (measures‘𝑆) → (𝑀‘∅) = 0)
10	9	3ad2ant1 1133	. . . 4 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆) → (𝑀‘∅) = 0)
11	8, 10	eqtrd 2771	. . 3 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆) → ((𝑀 ↾ 𝑇)‘∅) = 0)
12		simp11 1204	. . . . . . 7 ⊢ (((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆) ∧ 𝑥 ∈ 𝒫 𝑇 ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) → 𝑀 ∈ (measures‘𝑆))
13		simp13 1206	. . . . . . . 8 ⊢ (((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆) ∧ 𝑥 ∈ 𝒫 𝑇 ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) → 𝑇 ⊆ 𝑆)
14		simp2 1137	. . . . . . . 8 ⊢ (((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆) ∧ 𝑥 ∈ 𝒫 𝑇 ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) → 𝑥 ∈ 𝒫 𝑇)
15		sspw 4591	. . . . . . . . 9 ⊢ (𝑇 ⊆ 𝑆 → 𝒫 𝑇 ⊆ 𝒫 𝑆)
16	15	sselda 3963	. . . . . . . 8 ⊢ ((𝑇 ⊆ 𝑆 ∧ 𝑥 ∈ 𝒫 𝑇) → 𝑥 ∈ 𝒫 𝑆)
17	13, 14, 16	syl2anc 584	. . . . . . 7 ⊢ (((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆) ∧ 𝑥 ∈ 𝒫 𝑇 ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) → 𝑥 ∈ 𝒫 𝑆)
18		simp3 1138	. . . . . . 7 ⊢ (((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆) ∧ 𝑥 ∈ 𝒫 𝑇 ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) → (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦))
19		measvun 34245	. . . . . . 7 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝑥 ∈ 𝒫 𝑆 ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) → (𝑀‘∪ 𝑥) = Σ*𝑦 ∈ 𝑥(𝑀‘𝑦))
20	12, 17, 18, 19	syl3anc 1373	. . . . . 6 ⊢ (((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆) ∧ 𝑥 ∈ 𝒫 𝑇 ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) → (𝑀‘∪ 𝑥) = Σ*𝑦 ∈ 𝑥(𝑀‘𝑦))
21	1	3ad2ant1 1133	. . . . . . . 8 ⊢ (((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆) ∧ 𝑥 ∈ 𝒫 𝑇 ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) → 𝑇 ∈ ∪ ran sigAlgebra)
22		simp3l 1202	. . . . . . . 8 ⊢ (((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆) ∧ 𝑥 ∈ 𝒫 𝑇 ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) → 𝑥 ≼ ω)
23		sigaclcu 34153	. . . . . . . 8 ⊢ ((𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑥 ∈ 𝒫 𝑇 ∧ 𝑥 ≼ ω) → ∪ 𝑥 ∈ 𝑇)
24	21, 14, 22, 23	syl3anc 1373	. . . . . . 7 ⊢ (((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆) ∧ 𝑥 ∈ 𝒫 𝑇 ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) → ∪ 𝑥 ∈ 𝑇)
25	24	fvresd 6901	. . . . . 6 ⊢ (((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆) ∧ 𝑥 ∈ 𝒫 𝑇 ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) → ((𝑀 ↾ 𝑇)‘∪ 𝑥) = (𝑀‘∪ 𝑥))
26		elpwi 4587	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝒫 𝑇 → 𝑥 ⊆ 𝑇)
27	26	sselda 3963	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝒫 𝑇 ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑇)
28	27	adantll 714	. . . . . . . . 9 ⊢ ((((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆) ∧ 𝑥 ∈ 𝒫 𝑇) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑇)
29	28	fvresd 6901	. . . . . . . 8 ⊢ ((((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆) ∧ 𝑥 ∈ 𝒫 𝑇) ∧ 𝑦 ∈ 𝑥) → ((𝑀 ↾ 𝑇)‘𝑦) = (𝑀‘𝑦))
30	29	esumeq2dv 34074	. . . . . . 7 ⊢ (((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆) ∧ 𝑥 ∈ 𝒫 𝑇) → Σ*𝑦 ∈ 𝑥((𝑀 ↾ 𝑇)‘𝑦) = Σ*𝑦 ∈ 𝑥(𝑀‘𝑦))
31	30	3adant3 1132	. . . . . 6 ⊢ (((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆) ∧ 𝑥 ∈ 𝒫 𝑇 ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) → Σ*𝑦 ∈ 𝑥((𝑀 ↾ 𝑇)‘𝑦) = Σ*𝑦 ∈ 𝑥(𝑀‘𝑦))
32	20, 25, 31	3eqtr4d 2781	. . . . 5 ⊢ (((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆) ∧ 𝑥 ∈ 𝒫 𝑇 ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) → ((𝑀 ↾ 𝑇)‘∪ 𝑥) = Σ*𝑦 ∈ 𝑥((𝑀 ↾ 𝑇)‘𝑦))
33	32	3expia 1121	. . . 4 ⊢ (((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆) ∧ 𝑥 ∈ 𝒫 𝑇) → ((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ((𝑀 ↾ 𝑇)‘∪ 𝑥) = Σ*𝑦 ∈ 𝑥((𝑀 ↾ 𝑇)‘𝑦)))
34	33	ralrimiva 3133	. . 3 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆) → ∀𝑥 ∈ 𝒫 𝑇((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ((𝑀 ↾ 𝑇)‘∪ 𝑥) = Σ*𝑦 ∈ 𝑥((𝑀 ↾ 𝑇)‘𝑦)))
35	5, 11, 34	3jca 1128	. 2 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆) → ((𝑀 ↾ 𝑇):𝑇⟶(0[,]+∞) ∧ ((𝑀 ↾ 𝑇)‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 𝑇((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ((𝑀 ↾ 𝑇)‘∪ 𝑥) = Σ*𝑦 ∈ 𝑥((𝑀 ↾ 𝑇)‘𝑦))))
36		ismeas 34235	. . 3 ⊢ (𝑇 ∈ ∪ ran sigAlgebra → ((𝑀 ↾ 𝑇) ∈ (measures‘𝑇) ↔ ((𝑀 ↾ 𝑇):𝑇⟶(0[,]+∞) ∧ ((𝑀 ↾ 𝑇)‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 𝑇((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ((𝑀 ↾ 𝑇)‘∪ 𝑥) = Σ*𝑦 ∈ 𝑥((𝑀 ↾ 𝑇)‘𝑦)))))
37	36	biimprd 248	. 2 ⊢ (𝑇 ∈ ∪ ran sigAlgebra → (((𝑀 ↾ 𝑇):𝑇⟶(0[,]+∞) ∧ ((𝑀 ↾ 𝑇)‘∅) = 0 ∧ ∀𝑥 ∈ 𝒫 𝑇((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ((𝑀 ↾ 𝑇)‘∪ 𝑥) = Σ*𝑦 ∈ 𝑥((𝑀 ↾ 𝑇)‘𝑦))) → (𝑀 ↾ 𝑇) ∈ (measures‘𝑇)))
38	1, 35, 37	sylc 65	1 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆) → (𝑀 ↾ 𝑇) ∈ (measures‘𝑇))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3052   ⊆ wss 3931  ∅c0 4313  𝒫 cpw 4580  ∪ cuni 4888  Disj wdisj 5091   class class class wbr 5124  ran crn 5660   ↾ cres 5661  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410  ωcom 7866   ≼ cdom 8962  0cc0 11134  +∞cpnf 11271  [,]cicc 13370  Σ^*cesum 34063  sigAlgebracsiga 34144  measurescmeas 34231

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rmo 3364  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-disj 5092  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-ov 7413  df-esum 34064  df-siga 34145  df-meas 34232

This theorem is referenced by:  measinb2  34259