Theorem measres 34734

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 1155	. 2 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆) → 𝑇 ∈ ∪ ran sigAlgebra)
2		measfrge0 34715	. . . . 5 ⊢ (𝑀 ∈ (measures‘𝑆) → 𝑀:𝑆⟶(0[,]+∞))
3	2	3ad2ant1 1151	. . . 4 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆) → 𝑀:𝑆⟶(0[,]+∞))
4		simp3 1156	. . . 4 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆) → 𝑇 ⊆ 𝑆)
5	3, 4	fssresd 6743	. . 3 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆) → (𝑀 ↾ 𝑇):𝑇⟶(0[,]+∞))
6		0elsiga 34625	. . . . 5 ⊢ (𝑇 ∈ ∪ ran sigAlgebra → ∅ ∈ 𝑇)
7		fvres 6898	. . . . 5 ⊢ (∅ ∈ 𝑇 → ((𝑀 ↾ 𝑇)‘∅) = (𝑀‘∅))
8	1, 6, 7	3syl 19	. . . 4 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆) → ((𝑀 ↾ 𝑇)‘∅) = (𝑀‘∅))
9		measvnul 34718	. . . . 5 ⊢ (𝑀 ∈ (measures‘𝑆) → (𝑀‘∅) = 0)
10	9	3ad2ant1 1151	. . . 4 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆) → (𝑀‘∅) = 0)
11	8, 10	eqtrd 2795	. . 3 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆) → ((𝑀 ↾ 𝑇)‘∅) = 0)
12		simp11 1222	. . . . . . 7 ⊢ (((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆) ∧ 𝑥 ∈ 𝒫 𝑇 ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) → 𝑀 ∈ (measures‘𝑆))
13		simp13 1224	. . . . . . . 8 ⊢ (((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆) ∧ 𝑥 ∈ 𝒫 𝑇 ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) → 𝑇 ⊆ 𝑆)
14		simp2 1155	. . . . . . . 8 ⊢ (((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆) ∧ 𝑥 ∈ 𝒫 𝑇 ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) → 𝑥 ∈ 𝒫 𝑇)
15		sspw 4568	. . . . . . . . 9 ⊢ (𝑇 ⊆ 𝑆 → 𝒫 𝑇 ⊆ 𝒫 𝑆)
16	15	sselda 3931	. . . . . . . 8 ⊢ ((𝑇 ⊆ 𝑆 ∧ 𝑥 ∈ 𝒫 𝑇) → 𝑥 ∈ 𝒫 𝑆)
17	13, 14, 16	syl2anc 596	. . . . . . 7 ⊢ (((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆) ∧ 𝑥 ∈ 𝒫 𝑇 ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) → 𝑥 ∈ 𝒫 𝑆)
18		simp3 1156	. . . . . . 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 34721	. . . . . . 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 1398	. . . . . 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 1151	. . . . . . . 8 ⊢ (((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆) ∧ 𝑥 ∈ 𝒫 𝑇 ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) → 𝑇 ∈ ∪ ran sigAlgebra)
22		simp3l 1220	. . . . . . . 8 ⊢ (((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆) ∧ 𝑥 ∈ 𝒫 𝑇 ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) → 𝑥 ≼ ω)
23		sigaclcu 34628	. . . . . . . 8 ⊢ ((𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑥 ∈ 𝒫 𝑇 ∧ 𝑥 ≼ ω) → ∪ 𝑥 ∈ 𝑇)
24	21, 14, 22, 23	syl3anc 1398	. . . . . . 7 ⊢ (((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆) ∧ 𝑥 ∈ 𝒫 𝑇 ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) → ∪ 𝑥 ∈ 𝑇)
25	24	fvresd 6899	. . . . . 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 4564	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝒫 𝑇 → 𝑥 ⊆ 𝑇)
27	26	sselda 3931	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝒫 𝑇 ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑇)
28	27	adantll 727	. . . . . . . . 9 ⊢ ((((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆) ∧ 𝑥 ∈ 𝒫 𝑇) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑇)
29	28	fvresd 6899	. . . . . . . 8 ⊢ ((((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆) ∧ 𝑥 ∈ 𝒫 𝑇) ∧ 𝑦 ∈ 𝑥) → ((𝑀 ↾ 𝑇)‘𝑦) = (𝑀‘𝑦))
30	29	esumeq2dv 34549	. . . . . . 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 1150	. . . . . 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 2805	. . . . 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 1139	. . . 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 3154	. . 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 1146	. 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 34711	. . 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 251	. 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 66	1 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆) → (𝑀 ↾ 𝑇) ∈ (measures‘𝑇))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3076   ⊆ wss 3899  ∅c0 4279  𝒫 cpw 4557  ∪ cuni 4867  Disj wdisj 5070   class class class wbr 5103  ran crn 5656   ↾ cres 5657  ⟶wf 6529  ‘cfv 6533  (class class class)co 7414  ωcom 7863   ≼ cdom 8951  0cc0 11125  +∞cpnf 11265  [,]cicc 13402  Σ^*cesum 34538  sigAlgebracsiga 34619  measurescmeas 34707

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-disj 5071  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-fv 6541  df-ov 7417  df-esum 34539  df-siga 34620  df-meas 34708

This theorem is used by:  measinb2  34735