Theorem measres 32090

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 1135	. 2 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆) → 𝑇 ∈ ∪ ran sigAlgebra)
2		measfrge0 32071	. . . . 5 ⊢ (𝑀 ∈ (measures‘𝑆) → 𝑀:𝑆⟶(0[,]+∞))
3	2	3ad2ant1 1131	. . . 4 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆) → 𝑀:𝑆⟶(0[,]+∞))
4		simp3 1136	. . . 4 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆) → 𝑇 ⊆ 𝑆)
5	3, 4	fssresd 6625	. . 3 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆) → (𝑀 ↾ 𝑇):𝑇⟶(0[,]+∞))
6		0elsiga 31982	. . . . 5 ⊢ (𝑇 ∈ ∪ ran sigAlgebra → ∅ ∈ 𝑇)
7		fvres 6775	. . . . 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 32074	. . . . 5 ⊢ (𝑀 ∈ (measures‘𝑆) → (𝑀‘∅) = 0)
10	9	3ad2ant1 1131	. . . 4 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆) → (𝑀‘∅) = 0)
11	8, 10	eqtrd 2778	. . 3 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆) → ((𝑀 ↾ 𝑇)‘∅) = 0)
12		simp11 1201	. . . . . . 7 ⊢ (((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆) ∧ 𝑥 ∈ 𝒫 𝑇 ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) → 𝑀 ∈ (measures‘𝑆))
13		simp13 1203	. . . . . . . 8 ⊢ (((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆) ∧ 𝑥 ∈ 𝒫 𝑇 ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) → 𝑇 ⊆ 𝑆)
14		simp2 1135	. . . . . . . 8 ⊢ (((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆) ∧ 𝑥 ∈ 𝒫 𝑇 ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) → 𝑥 ∈ 𝒫 𝑇)
15		sspw 4543	. . . . . . . . 9 ⊢ (𝑇 ⊆ 𝑆 → 𝒫 𝑇 ⊆ 𝒫 𝑆)
16	15	sselda 3917	. . . . . . . 8 ⊢ ((𝑇 ⊆ 𝑆 ∧ 𝑥 ∈ 𝒫 𝑇) → 𝑥 ∈ 𝒫 𝑆)
17	13, 14, 16	syl2anc 583	. . . . . . 7 ⊢ (((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆) ∧ 𝑥 ∈ 𝒫 𝑇 ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) → 𝑥 ∈ 𝒫 𝑆)
18		simp3 1136	. . . . . . 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 32077	. . . . . . 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 1369	. . . . . 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 1131	. . . . . . . 8 ⊢ (((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆) ∧ 𝑥 ∈ 𝒫 𝑇 ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) → 𝑇 ∈ ∪ ran sigAlgebra)
22		simp3l 1199	. . . . . . . 8 ⊢ (((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆) ∧ 𝑥 ∈ 𝒫 𝑇 ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) → 𝑥 ≼ ω)
23		sigaclcu 31985	. . . . . . . 8 ⊢ ((𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑥 ∈ 𝒫 𝑇 ∧ 𝑥 ≼ ω) → ∪ 𝑥 ∈ 𝑇)
24	21, 14, 22, 23	syl3anc 1369	. . . . . . 7 ⊢ (((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆) ∧ 𝑥 ∈ 𝒫 𝑇 ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) → ∪ 𝑥 ∈ 𝑇)
25	24	fvresd 6776	. . . . . 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 4539	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝒫 𝑇 → 𝑥 ⊆ 𝑇)
27	26	sselda 3917	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝒫 𝑇 ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑇)
28	27	adantll 710	. . . . . . . . 9 ⊢ ((((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆) ∧ 𝑥 ∈ 𝒫 𝑇) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑇)
29	28	fvresd 6776	. . . . . . . 8 ⊢ ((((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆) ∧ 𝑥 ∈ 𝒫 𝑇) ∧ 𝑦 ∈ 𝑥) → ((𝑀 ↾ 𝑇)‘𝑦) = (𝑀‘𝑦))
30	29	esumeq2dv 31906	. . . . . . 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 1130	. . . . . 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 2788	. . . . 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 1119	. . . 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 3107	. . 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 1126	. 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 32067	. . 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 247	. 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 1085   = wceq 1539   ∈ wcel 2108  ∀wral 3063   ⊆ wss 3883  ∅c0 4253  𝒫 cpw 4530  ∪ cuni 4836  Disj wdisj 5035   class class class wbr 5070  ran crn 5581   ↾ cres 5582  ⟶wf 6414  ‘cfv 6418  (class class class)co 7255  ωcom 7687   ≼ cdom 8689  0cc0 10802  +∞cpnf 10937  [,]cicc 13011  Σ^*cesum 31895  sigAlgebracsiga 31976  measurescmeas 32063

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-disj 5036  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-fv 6426  df-ov 7258  df-esum 31896  df-siga 31977  df-meas 32064

This theorem is referenced by:  measinb2  32091