measdivcstALTV - Mathbox for Thierry Arnoux

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Theorem measdivcstALTV 33218

Description: Alternate version of measdivcst 33217. (Contributed by Thierry Arnoux, 25-Dec-2016.) (New usage is discouraged.)

**Assertion**
Ref	Expression
measdivcstALTV	⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ ℝ⁺) → (𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /_𝑒 𝐴)) ∈ (measures‘𝑆))

Distinct variable groups: 𝑥,𝐴 𝑥,𝑀 𝑥,𝑆

Proof of Theorem measdivcstALTV Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		funmpt 6586	. . . . . 6 ⊢ Fun (𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /_𝑒 𝐴))
2		ovex 7441	. . . . . . . 8 ⊢ ((𝑀‘𝑥) /_𝑒 𝐴) ∈ V
3	2	rgenw 3065	. . . . . . 7 ⊢ ∀𝑥 ∈ 𝑆 ((𝑀‘𝑥) /_𝑒 𝐴) ∈ V
4		dmmptg 6241	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝑆 ((𝑀‘𝑥) /_𝑒 𝐴) ∈ V → dom (𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /_𝑒 𝐴)) = 𝑆)
5	3, 4	ax-mp 5	. . . . . 6 ⊢ dom (𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /_𝑒 𝐴)) = 𝑆
6		df-fn 6546	. . . . . 6 ⊢ ((𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /_𝑒 𝐴)) Fn 𝑆 ↔ (Fun (𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /_𝑒 𝐴)) ∧ dom (𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /_𝑒 𝐴)) = 𝑆))
7	1, 5, 6	mpbir2an 709	. . . . 5 ⊢ (𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /_𝑒 𝐴)) Fn 𝑆
8	7	a1i 11	. . . 4 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ ℝ⁺) → (𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /_𝑒 𝐴)) Fn 𝑆)
9		vex 3478	. . . . . . 7 ⊢ 𝑦 ∈ V
10		eqid 2732	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /_𝑒 𝐴)) = (𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /_𝑒 𝐴))
11	10	elrnmpt 5955	. . . . . . 7 ⊢ (𝑦 ∈ V → (𝑦 ∈ ran (𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /_𝑒 𝐴)) ↔ ∃𝑥 ∈ 𝑆 𝑦 = ((𝑀‘𝑥) /_𝑒 𝐴)))
12	9, 11	ax-mp 5	. . . . . 6 ⊢ (𝑦 ∈ ran (𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /_𝑒 𝐴)) ↔ ∃𝑥 ∈ 𝑆 𝑦 = ((𝑀‘𝑥) /_𝑒 𝐴))
13		measfrge0 33196	. . . . . . . . . . 11 ⊢ (𝑀 ∈ (measures‘𝑆) → 𝑀:𝑆⟶(0[,]+∞))
14		ffvelcdm 7083	. . . . . . . . . . 11 ⊢ ((𝑀:𝑆⟶(0[,]+∞) ∧ 𝑥 ∈ 𝑆) → (𝑀‘𝑥) ∈ (0[,]+∞))
15	13, 14	sylan 580	. . . . . . . . . 10 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝑥 ∈ 𝑆) → (𝑀‘𝑥) ∈ (0[,]+∞))
16	15	adantlr 713	. . . . . . . . 9 ⊢ (((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ ℝ⁺) ∧ 𝑥 ∈ 𝑆) → (𝑀‘𝑥) ∈ (0[,]+∞))
17		simplr 767	. . . . . . . . 9 ⊢ (((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ ℝ⁺) ∧ 𝑥 ∈ 𝑆) → 𝐴 ∈ ℝ⁺)
18	16, 17	xrpxdivcld 32096	. . . . . . . 8 ⊢ (((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ ℝ⁺) ∧ 𝑥 ∈ 𝑆) → ((𝑀‘𝑥) /_𝑒 𝐴) ∈ (0[,]+∞))
19		eleq1a 2828	. . . . . . . 8 ⊢ (((𝑀‘𝑥) /_𝑒 𝐴) ∈ (0[,]+∞) → (𝑦 = ((𝑀‘𝑥) /_𝑒 𝐴) → 𝑦 ∈ (0[,]+∞)))
20	18, 19	syl 17	. . . . . . 7 ⊢ (((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ ℝ⁺) ∧ 𝑥 ∈ 𝑆) → (𝑦 = ((𝑀‘𝑥) /_𝑒 𝐴) → 𝑦 ∈ (0[,]+∞)))
21	20	rexlimdva 3155	. . . . . 6 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ ℝ⁺) → (∃𝑥 ∈ 𝑆 𝑦 = ((𝑀‘𝑥) /_𝑒 𝐴) → 𝑦 ∈ (0[,]+∞)))
22	12, 21	biimtrid 241	. . . . 5 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ ℝ⁺) → (𝑦 ∈ ran (𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /_𝑒 𝐴)) → 𝑦 ∈ (0[,]+∞)))
23	22	ssrdv 3988	. . . 4 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ ℝ⁺) → ran (𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /_𝑒 𝐴)) ⊆ (0[,]+∞))
24		df-f 6547	. . . 4 ⊢ ((𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /_𝑒 𝐴)):𝑆⟶(0[,]+∞) ↔ ((𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /_𝑒 𝐴)) Fn 𝑆 ∧ ran (𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /_𝑒 𝐴)) ⊆ (0[,]+∞)))
25	8, 23, 24	sylanbrc 583	. . 3 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ ℝ⁺) → (𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /_𝑒 𝐴)):𝑆⟶(0[,]+∞))
26		measbase 33190	. . . . . . . 8 ⊢ (𝑀 ∈ (measures‘𝑆) → 𝑆 ∈ ∪ ran sigAlgebra)
27		0elsiga 33107	. . . . . . . 8 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → ∅ ∈ 𝑆)
28	26, 27	syl 17	. . . . . . 7 ⊢ (𝑀 ∈ (measures‘𝑆) → ∅ ∈ 𝑆)
29	28	adantr 481	. . . . . 6 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ ℝ⁺) → ∅ ∈ 𝑆)
30		ovex 7441	. . . . . 6 ⊢ ((𝑀‘∅) /_𝑒 𝐴) ∈ V
31	29, 30	jctir 521	. . . . 5 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ ℝ⁺) → (∅ ∈ 𝑆 ∧ ((𝑀‘∅) /_𝑒 𝐴) ∈ V))
32		fveq2 6891	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝑀‘𝑥) = (𝑀‘∅))
33	32	oveq1d 7423	. . . . . 6 ⊢ (𝑥 = ∅ → ((𝑀‘𝑥) /_𝑒 𝐴) = ((𝑀‘∅) /_𝑒 𝐴))
34	33, 10	fvmptg 6996	. . . . 5 ⊢ ((∅ ∈ 𝑆 ∧ ((𝑀‘∅) /_𝑒 𝐴) ∈ V) → ((𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /_𝑒 𝐴))‘∅) = ((𝑀‘∅) /_𝑒 𝐴))
35	31, 34	syl 17	. . . 4 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ ℝ⁺) → ((𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /_𝑒 𝐴))‘∅) = ((𝑀‘∅) /_𝑒 𝐴))
36		measvnul 33199	. . . . . 6 ⊢ (𝑀 ∈ (measures‘𝑆) → (𝑀‘∅) = 0)
37	36	oveq1d 7423	. . . . 5 ⊢ (𝑀 ∈ (measures‘𝑆) → ((𝑀‘∅) /_𝑒 𝐴) = (0 /_𝑒 𝐴))
38		xdiv0rp 32091	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (0 /_𝑒 𝐴) = 0)
39	37, 38	sylan9eq 2792	. . . 4 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ ℝ⁺) → ((𝑀‘∅) /_𝑒 𝐴) = 0)
40	35, 39	eqtrd 2772	. . 3 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ ℝ⁺) → ((𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /_𝑒 𝐴))‘∅) = 0)
41		simpll 765	. . . . . 6 ⊢ ((((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝒫 𝑆) ∧ (𝑦 ≼ ω ∧ Disj 𝑧 ∈ 𝑦 𝑧)) → (𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ ℝ⁺))
42		simplr 767	. . . . . . 7 ⊢ ((((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝒫 𝑆) ∧ (𝑦 ≼ ω ∧ Disj 𝑧 ∈ 𝑦 𝑧)) → 𝑦 ∈ 𝒫 𝑆)
43		simprl 769	. . . . . . 7 ⊢ ((((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝒫 𝑆) ∧ (𝑦 ≼ ω ∧ Disj 𝑧 ∈ 𝑦 𝑧)) → 𝑦 ≼ ω)
44		simprr 771	. . . . . . 7 ⊢ ((((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝒫 𝑆) ∧ (𝑦 ≼ ω ∧ Disj 𝑧 ∈ 𝑦 𝑧)) → Disj 𝑧 ∈ 𝑦 𝑧)
45	42, 43, 44	3jca 1128	. . . . . 6 ⊢ ((((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝒫 𝑆) ∧ (𝑦 ≼ ω ∧ Disj 𝑧 ∈ 𝑦 𝑧)) → (𝑦 ∈ 𝒫 𝑆 ∧ 𝑦 ≼ ω ∧ Disj 𝑧 ∈ 𝑦 𝑧))
46	9	a1i 11	. . . . . . . . 9 ⊢ (((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝒫 𝑆) → 𝑦 ∈ V)
47		simplll 773	. . . . . . . . . 10 ⊢ ((((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝒫 𝑆) ∧ 𝑧 ∈ 𝑦) → 𝑀 ∈ (measures‘𝑆))
48		simplr 767	. . . . . . . . . . 11 ⊢ ((((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝒫 𝑆) ∧ 𝑧 ∈ 𝑦) → 𝑦 ∈ 𝒫 𝑆)
49		simpr 485	. . . . . . . . . . 11 ⊢ ((((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝒫 𝑆) ∧ 𝑧 ∈ 𝑦) → 𝑧 ∈ 𝑦)
50		elpwg 4605	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ V → (𝑦 ∈ 𝒫 𝑆 ↔ 𝑦 ⊆ 𝑆))
51	9, 50	ax-mp 5	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝒫 𝑆 ↔ 𝑦 ⊆ 𝑆)
52		ssel2 3977	. . . . . . . . . . . 12 ⊢ ((𝑦 ⊆ 𝑆 ∧ 𝑧 ∈ 𝑦) → 𝑧 ∈ 𝑆)
53	51, 52	sylanb 581	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ 𝒫 𝑆 ∧ 𝑧 ∈ 𝑦) → 𝑧 ∈ 𝑆)
54	48, 49, 53	syl2anc 584	. . . . . . . . . 10 ⊢ ((((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝒫 𝑆) ∧ 𝑧 ∈ 𝑦) → 𝑧 ∈ 𝑆)
55		measvxrge0 33198	. . . . . . . . . 10 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝑧 ∈ 𝑆) → (𝑀‘𝑧) ∈ (0[,]+∞))
56	47, 54, 55	syl2anc 584	. . . . . . . . 9 ⊢ ((((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝒫 𝑆) ∧ 𝑧 ∈ 𝑦) → (𝑀‘𝑧) ∈ (0[,]+∞))
57		simplr 767	. . . . . . . . 9 ⊢ (((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝒫 𝑆) → 𝐴 ∈ ℝ⁺)
58	46, 56, 57	esumdivc 33076	. . . . . . . 8 ⊢ (((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝒫 𝑆) → (Σ^𝑧 ∈ 𝑦(𝑀‘𝑧) /_𝑒 𝐴) = Σ^𝑧 ∈ 𝑦((𝑀‘𝑧) /_𝑒 𝐴))
59	58	3ad2antr1 1188	. . . . . . 7 ⊢ (((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ ℝ⁺) ∧ (𝑦 ∈ 𝒫 𝑆 ∧ 𝑦 ≼ ω ∧ Disj 𝑧 ∈ 𝑦 𝑧)) → (Σ^𝑧 ∈ 𝑦(𝑀‘𝑧) /_𝑒 𝐴) = Σ^𝑧 ∈ 𝑦((𝑀‘𝑧) /_𝑒 𝐴))
60	26	ad2antrr 724	. . . . . . . . . 10 ⊢ (((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ ℝ⁺) ∧ (𝑦 ∈ 𝒫 𝑆 ∧ 𝑦 ≼ ω ∧ Disj 𝑧 ∈ 𝑦 𝑧)) → 𝑆 ∈ ∪ ran sigAlgebra)
61		simpr1 1194	. . . . . . . . . 10 ⊢ (((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ ℝ⁺) ∧ (𝑦 ∈ 𝒫 𝑆 ∧ 𝑦 ≼ ω ∧ Disj 𝑧 ∈ 𝑦 𝑧)) → 𝑦 ∈ 𝒫 𝑆)
62		simpr2 1195	. . . . . . . . . 10 ⊢ (((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ ℝ⁺) ∧ (𝑦 ∈ 𝒫 𝑆 ∧ 𝑦 ≼ ω ∧ Disj 𝑧 ∈ 𝑦 𝑧)) → 𝑦 ≼ ω)
63		sigaclcu 33110	. . . . . . . . . 10 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝑦 ∈ 𝒫 𝑆 ∧ 𝑦 ≼ ω) → ∪ 𝑦 ∈ 𝑆)
64	60, 61, 62, 63	syl3anc 1371	. . . . . . . . 9 ⊢ (((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ ℝ⁺) ∧ (𝑦 ∈ 𝒫 𝑆 ∧ 𝑦 ≼ ω ∧ Disj 𝑧 ∈ 𝑦 𝑧)) → ∪ 𝑦 ∈ 𝑆)
65		fveq2 6891	. . . . . . . . . . 11 ⊢ (𝑥 = ∪ 𝑦 → (𝑀‘𝑥) = (𝑀‘∪ 𝑦))
66	65	oveq1d 7423	. . . . . . . . . 10 ⊢ (𝑥 = ∪ 𝑦 → ((𝑀‘𝑥) /_𝑒 𝐴) = ((𝑀‘∪ 𝑦) /_𝑒 𝐴))
67	66, 10, 2	fvmpt3i 7003	. . . . . . . . 9 ⊢ (∪ 𝑦 ∈ 𝑆 → ((𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /_𝑒 𝐴))‘∪ 𝑦) = ((𝑀‘∪ 𝑦) /_𝑒 𝐴))
68	64, 67	syl 17	. . . . . . . 8 ⊢ (((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ ℝ⁺) ∧ (𝑦 ∈ 𝒫 𝑆 ∧ 𝑦 ≼ ω ∧ Disj 𝑧 ∈ 𝑦 𝑧)) → ((𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /_𝑒 𝐴))‘∪ 𝑦) = ((𝑀‘∪ 𝑦) /_𝑒 𝐴))
69		simpll 765	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ ℝ⁺) ∧ (𝑦 ∈ 𝒫 𝑆 ∧ 𝑦 ≼ ω ∧ Disj 𝑧 ∈ 𝑦 𝑧)) → 𝑀 ∈ (measures‘𝑆))
70	69, 61	jca 512	. . . . . . . . . 10 ⊢ (((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ ℝ⁺) ∧ (𝑦 ∈ 𝒫 𝑆 ∧ 𝑦 ≼ ω ∧ Disj 𝑧 ∈ 𝑦 𝑧)) → (𝑀 ∈ (measures‘𝑆) ∧ 𝑦 ∈ 𝒫 𝑆))
71		simpr3 1196	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ ℝ⁺) ∧ (𝑦 ∈ 𝒫 𝑆 ∧ 𝑦 ≼ ω ∧ Disj 𝑧 ∈ 𝑦 𝑧)) → Disj 𝑧 ∈ 𝑦 𝑧)
72	62, 71	jca 512	. . . . . . . . . 10 ⊢ (((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ ℝ⁺) ∧ (𝑦 ∈ 𝒫 𝑆 ∧ 𝑦 ≼ ω ∧ Disj 𝑧 ∈ 𝑦 𝑧)) → (𝑦 ≼ ω ∧ Disj 𝑧 ∈ 𝑦 𝑧))
73		measvun 33202	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝑦 ∈ 𝒫 𝑆 ∧ (𝑦 ≼ ω ∧ Disj 𝑧 ∈ 𝑦 𝑧)) → (𝑀‘∪ 𝑦) = Σ^*𝑧 ∈ 𝑦(𝑀‘𝑧))
74	73	3expia 1121	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝑦 ∈ 𝒫 𝑆) → ((𝑦 ≼ ω ∧ Disj 𝑧 ∈ 𝑦 𝑧) → (𝑀‘∪ 𝑦) = Σ^*𝑧 ∈ 𝑦(𝑀‘𝑧)))
75	74	ralrimiva 3146	. . . . . . . . . . 11 ⊢ (𝑀 ∈ (measures‘𝑆) → ∀𝑦 ∈ 𝒫 𝑆((𝑦 ≼ ω ∧ Disj 𝑧 ∈ 𝑦 𝑧) → (𝑀‘∪ 𝑦) = Σ^*𝑧 ∈ 𝑦(𝑀‘𝑧)))
76	75	r19.21bi 3248	. . . . . . . . . 10 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝑦 ∈ 𝒫 𝑆) → ((𝑦 ≼ ω ∧ Disj 𝑧 ∈ 𝑦 𝑧) → (𝑀‘∪ 𝑦) = Σ^*𝑧 ∈ 𝑦(𝑀‘𝑧)))
77	70, 72, 76	sylc 65	. . . . . . . . 9 ⊢ (((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ ℝ⁺) ∧ (𝑦 ∈ 𝒫 𝑆 ∧ 𝑦 ≼ ω ∧ Disj 𝑧 ∈ 𝑦 𝑧)) → (𝑀‘∪ 𝑦) = Σ^*𝑧 ∈ 𝑦(𝑀‘𝑧))
78	77	oveq1d 7423	. . . . . . . 8 ⊢ (((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ ℝ⁺) ∧ (𝑦 ∈ 𝒫 𝑆 ∧ 𝑦 ≼ ω ∧ Disj 𝑧 ∈ 𝑦 𝑧)) → ((𝑀‘∪ 𝑦) /_𝑒 𝐴) = (Σ^*𝑧 ∈ 𝑦(𝑀‘𝑧) /_𝑒 𝐴))
79	68, 78	eqtrd 2772	. . . . . . 7 ⊢ (((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ ℝ⁺) ∧ (𝑦 ∈ 𝒫 𝑆 ∧ 𝑦 ≼ ω ∧ Disj 𝑧 ∈ 𝑦 𝑧)) → ((𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /_𝑒 𝐴))‘∪ 𝑦) = (Σ^*𝑧 ∈ 𝑦(𝑀‘𝑧) /_𝑒 𝐴))
80		fveq2 6891	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (𝑀‘𝑥) = (𝑀‘𝑧))
81	80	oveq1d 7423	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → ((𝑀‘𝑥) /_𝑒 𝐴) = ((𝑀‘𝑧) /_𝑒 𝐴))
82	81, 10, 2	fvmpt3i 7003	. . . . . . . . . 10 ⊢ (𝑧 ∈ 𝑆 → ((𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /_𝑒 𝐴))‘𝑧) = ((𝑀‘𝑧) /_𝑒 𝐴))
83	53, 82	syl 17	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝒫 𝑆 ∧ 𝑧 ∈ 𝑦) → ((𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /_𝑒 𝐴))‘𝑧) = ((𝑀‘𝑧) /_𝑒 𝐴))
84	83	esumeq2dv 33031	. . . . . . . 8 ⊢ (𝑦 ∈ 𝒫 𝑆 → Σ^𝑧 ∈ 𝑦((𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /_𝑒 𝐴))‘𝑧) = Σ^𝑧 ∈ 𝑦((𝑀‘𝑧) /_𝑒 𝐴))
85	61, 84	syl 17	. . . . . . 7 ⊢ (((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ ℝ⁺) ∧ (𝑦 ∈ 𝒫 𝑆 ∧ 𝑦 ≼ ω ∧ Disj 𝑧 ∈ 𝑦 𝑧)) → Σ^𝑧 ∈ 𝑦((𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /_𝑒 𝐴))‘𝑧) = Σ^𝑧 ∈ 𝑦((𝑀‘𝑧) /_𝑒 𝐴))
86	59, 79, 85	3eqtr4d 2782	. . . . . 6 ⊢ (((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ ℝ⁺) ∧ (𝑦 ∈ 𝒫 𝑆 ∧ 𝑦 ≼ ω ∧ Disj 𝑧 ∈ 𝑦 𝑧)) → ((𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /_𝑒 𝐴))‘∪ 𝑦) = Σ^*𝑧 ∈ 𝑦((𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /_𝑒 𝐴))‘𝑧))
87	41, 45, 86	syl2anc 584	. . . . 5 ⊢ ((((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝒫 𝑆) ∧ (𝑦 ≼ ω ∧ Disj 𝑧 ∈ 𝑦 𝑧)) → ((𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /_𝑒 𝐴))‘∪ 𝑦) = Σ^*𝑧 ∈ 𝑦((𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /_𝑒 𝐴))‘𝑧))
88	87	ex 413	. . . 4 ⊢ (((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝒫 𝑆) → ((𝑦 ≼ ω ∧ Disj 𝑧 ∈ 𝑦 𝑧) → ((𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /_𝑒 𝐴))‘∪ 𝑦) = Σ^*𝑧 ∈ 𝑦((𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /_𝑒 𝐴))‘𝑧)))
89	88	ralrimiva 3146	. . 3 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ ℝ⁺) → ∀𝑦 ∈ 𝒫 𝑆((𝑦 ≼ ω ∧ Disj 𝑧 ∈ 𝑦 𝑧) → ((𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /_𝑒 𝐴))‘∪ 𝑦) = Σ^*𝑧 ∈ 𝑦((𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /_𝑒 𝐴))‘𝑧)))
90	25, 40, 89	3jca 1128	. 2 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ ℝ⁺) → ((𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /_𝑒 𝐴)):𝑆⟶(0[,]+∞) ∧ ((𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /_𝑒 𝐴))‘∅) = 0 ∧ ∀𝑦 ∈ 𝒫 𝑆((𝑦 ≼ ω ∧ Disj 𝑧 ∈ 𝑦 𝑧) → ((𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /_𝑒 𝐴))‘∪ 𝑦) = Σ^*𝑧 ∈ 𝑦((𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /_𝑒 𝐴))‘𝑧))))
91		ismeas 33192	. . . . 5 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → ((𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /_𝑒 𝐴)) ∈ (measures‘𝑆) ↔ ((𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /_𝑒 𝐴)):𝑆⟶(0[,]+∞) ∧ ((𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /_𝑒 𝐴))‘∅) = 0 ∧ ∀𝑦 ∈ 𝒫 𝑆((𝑦 ≼ ω ∧ Disj 𝑧 ∈ 𝑦 𝑧) → ((𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /_𝑒 𝐴))‘∪ 𝑦) = Σ^*𝑧 ∈ 𝑦((𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /_𝑒 𝐴))‘𝑧)))))
92	26, 91	syl 17	. . . 4 ⊢ (𝑀 ∈ (measures‘𝑆) → ((𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /_𝑒 𝐴)) ∈ (measures‘𝑆) ↔ ((𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /_𝑒 𝐴)):𝑆⟶(0[,]+∞) ∧ ((𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /_𝑒 𝐴))‘∅) = 0 ∧ ∀𝑦 ∈ 𝒫 𝑆((𝑦 ≼ ω ∧ Disj 𝑧 ∈ 𝑦 𝑧) → ((𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /_𝑒 𝐴))‘∪ 𝑦) = Σ^*𝑧 ∈ 𝑦((𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /_𝑒 𝐴))‘𝑧)))))
93	92	biimprd 247	. . 3 ⊢ (𝑀 ∈ (measures‘𝑆) → (((𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /_𝑒 𝐴)):𝑆⟶(0[,]+∞) ∧ ((𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /_𝑒 𝐴))‘∅) = 0 ∧ ∀𝑦 ∈ 𝒫 𝑆((𝑦 ≼ ω ∧ Disj 𝑧 ∈ 𝑦 𝑧) → ((𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /_𝑒 𝐴))‘∪ 𝑦) = Σ^*𝑧 ∈ 𝑦((𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /_𝑒 𝐴))‘𝑧))) → (𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /_𝑒 𝐴)) ∈ (measures‘𝑆)))
94	93	adantr 481	. 2 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ ℝ⁺) → (((𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /_𝑒 𝐴)):𝑆⟶(0[,]+∞) ∧ ((𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /_𝑒 𝐴))‘∅) = 0 ∧ ∀𝑦 ∈ 𝒫 𝑆((𝑦 ≼ ω ∧ Disj 𝑧 ∈ 𝑦 𝑧) → ((𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /_𝑒 𝐴))‘∪ 𝑦) = Σ^*𝑧 ∈ 𝑦((𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /_𝑒 𝐴))‘𝑧))) → (𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /_𝑒 𝐴)) ∈ (measures‘𝑆)))
95	90, 94	mpd 15	1 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ ℝ⁺) → (𝑥 ∈ 𝑆 ↦ ((𝑀‘𝑥) /_𝑒 𝐴)) ∈ (measures‘𝑆))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∀wral 3061 ∃wrex 3070 Vcvv 3474 ⊆ wss 3948 ∅c0 4322 𝒫 cpw 4602 ∪ cuni 4908 Disj wdisj 5113 class class class wbr 5148 ↦ cmpt 5231 dom cdm 5676 ran crn 5677 Fun wfun 6537 Fn wfn 6538 ⟶wf 6539 ‘cfv 6543 (class class class)co 7408 ωcom 7854 ≼ cdom 8936 0cc0 11109 +∞cpnf 11244 ℝ⁺crp 12973 [,]cicc 13326 /_𝑒 cxdiv 32078 Σ^*cesum 33020 sigAlgebracsiga 33101 measurescmeas 33188

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-disj 5114 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-of 7669 df-om 7855 df-1st 7974 df-2nd 7975 df-supp 8146 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-1o 8465 df-er 8702 df-map 8821 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-fsupp 9361 df-fi 9405 df-sup 9436 df-inf 9437 df-oi 9504 df-card 9933 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-div 11871 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12472 df-z 12558 df-dec 12677 df-uz 12822 df-q 12932 df-rp 12974 df-xneg 13091 df-xadd 13092 df-xmul 13093 df-ioo 13327 df-ioc 13328 df-ico 13329 df-icc 13330 df-fz 13484 df-fzo 13627 df-seq 13966 df-hash 14290 df-struct 17079 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-ress 17173 df-plusg 17209 df-mulr 17210 df-tset 17215 df-ple 17216 df-ds 17218 df-rest 17367 df-topn 17368 df-0g 17386 df-gsum 17387 df-topgen 17388 df-ordt 17446 df-xrs 17447 df-mre 17529 df-mrc 17530 df-acs 17532 df-ps 18518 df-tsr 18519 df-mgm 18560 df-sgrp 18609 df-mnd 18625 df-mhm 18670 df-submnd 18671 df-cntz 19180 df-cmn 19649 df-fbas 20940 df-fg 20941 df-top 22395 df-topon 22412 df-topsp 22434 df-bases 22448 df-ntr 22523 df-nei 22601 df-cn 22730 df-cnp 22731 df-haus 22818 df-fil 23349 df-fm 23441 df-flim 23442 df-flf 23443 df-tsms 23630 df-xdiv 32079 df-esum 33021 df-siga 33102 df-meas 33189

This theorem is referenced by: probfinmeasbALTV 33423 probmeasb 33424

W3C validator