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 ∧ 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 1134	. 2 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆) → 𝑇 ∈ ∪ ran sigAlgebra)
2		measfrge0 34036	. . . . 5 ⊢ (𝑀 ∈ (measures‘𝑆) → 𝑀:𝑆⟶(0[,]+∞))
3	2	3ad2ant1 1130	. . . 4 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆) → 𝑀:𝑆⟶(0[,]+∞))
4		simp3 1135	. . . 4 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆) → 𝑇 ⊆ 𝑆)
5	3, 4	fssresd 6769	. . 3 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆) → (𝑀 ↾ 𝑇):𝑇⟶(0[,]+∞))
6		0elsiga 33947	. . . . 5 ⊢ (𝑇 ∈ ∪ ran sigAlgebra → ∅ ∈ 𝑇)
7		fvres 6920	. . . . 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 34039	. . . . 5 ⊢ (𝑀 ∈ (measures‘𝑆) → (𝑀‘∅) = 0)
10	9	3ad2ant1 1130	. . . 4 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆) → (𝑀‘∅) = 0)
11	8, 10	eqtrd 2766	. . 3 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆) → ((𝑀 ↾ 𝑇)‘∅) = 0)
12		simp11 1200	. . . . . . 7 ⊢ (((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆) ∧ 𝑥 ∈ 𝒫 𝑇 ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) → 𝑀 ∈ (measures‘𝑆))
13		simp13 1202	. . . . . . . 8 ⊢ (((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆) ∧ 𝑥 ∈ 𝒫 𝑇 ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) → 𝑇 ⊆ 𝑆)
14		simp2 1134	. . . . . . . 8 ⊢ (((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆) ∧ 𝑥 ∈ 𝒫 𝑇 ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) → 𝑥 ∈ 𝒫 𝑇)
15		sspw 4618	. . . . . . . . 9 ⊢ (𝑇 ⊆ 𝑆 → 𝒫 𝑇 ⊆ 𝒫 𝑆)
16	15	sselda 3979	. . . . . . . 8 ⊢ ((𝑇 ⊆ 𝑆 ∧ 𝑥 ∈ 𝒫 𝑇) → 𝑥 ∈ 𝒫 𝑆)
17	13, 14, 16	syl2anc 582	. . . . . . 7 ⊢ (((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆) ∧ 𝑥 ∈ 𝒫 𝑇 ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) → 𝑥 ∈ 𝒫 𝑆)
18		simp3 1135	. . . . . . 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 34042	. . . . . . 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 1368	. . . . . 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 1130	. . . . . . . 8 ⊢ (((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆) ∧ 𝑥 ∈ 𝒫 𝑇 ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) → 𝑇 ∈ ∪ ran sigAlgebra)
22		simp3l 1198	. . . . . . . 8 ⊢ (((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆) ∧ 𝑥 ∈ 𝒫 𝑇 ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) → 𝑥 ≼ ω)
23		sigaclcu 33950	. . . . . . . 8 ⊢ ((𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑥 ∈ 𝒫 𝑇 ∧ 𝑥 ≼ ω) → ∪ 𝑥 ∈ 𝑇)
24	21, 14, 22, 23	syl3anc 1368	. . . . . . 7 ⊢ (((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆) ∧ 𝑥 ∈ 𝒫 𝑇 ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) → ∪ 𝑥 ∈ 𝑇)
25	24	fvresd 6921	. . . . . 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 4614	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝒫 𝑇 → 𝑥 ⊆ 𝑇)
27	26	sselda 3979	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝒫 𝑇 ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑇)
28	27	adantll 712	. . . . . . . . 9 ⊢ ((((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆) ∧ 𝑥 ∈ 𝒫 𝑇) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑇)
29	28	fvresd 6921	. . . . . . . 8 ⊢ ((((𝑀 ∈ (measures‘𝑆) ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ 𝑇 ⊆ 𝑆) ∧ 𝑥 ∈ 𝒫 𝑇) ∧ 𝑦 ∈ 𝑥) → ((𝑀 ↾ 𝑇)‘𝑦) = (𝑀‘𝑦))
30	29	esumeq2dv 33871	. . . . . . 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 1129	. . . . . 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 2776	. . . . 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 1118	. . . 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 3136	. . 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 1125	. 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 34032	. . 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 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