voliune - Mathbox for Thierry Arnoux

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Theorem voliune 31598

Description: The Lebesgue measure function is countably additive. This formulation on the extended reals, allows for +∞ for the measure of any set in the sum. Cf. ovoliun 24109 and voliun 24158. (Contributed by Thierry Arnoux, 16-Oct-2017.)

**Assertion**
Ref	Expression
voliune	⊢ ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) → (vol‘∪ 𝑛 ∈ ℕ 𝐴) = Σ^*𝑛 ∈ ℕ(vol‘𝐴))

Proof of Theorem voliune Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		r19.26 3137	. . . . 5 ⊢ (∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ↔ (∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ))
2		eqid 2798	. . . . . 6 ⊢ seq1( + , (𝑛 ∈ ℕ ↦ (vol‘𝐴))) = seq1( + , (𝑛 ∈ ℕ ↦ (vol‘𝐴)))
3		eqid 2798	. . . . . 6 ⊢ (𝑛 ∈ ℕ ↦ (vol‘𝐴)) = (𝑛 ∈ ℕ ↦ (vol‘𝐴))
4	2, 3	voliun 24158	. . . . 5 ⊢ ((∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ Disj 𝑛 ∈ ℕ 𝐴) → (vol‘∪ 𝑛 ∈ ℕ 𝐴) = sup(ran seq1( + , (𝑛 ∈ ℕ ↦ (vol‘𝐴))), ℝ^*, < ))
5	1, 4	sylanbr 585	. . . 4 ⊢ (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ) ∧ Disj 𝑛 ∈ ℕ 𝐴) → (vol‘∪ 𝑛 ∈ ℕ 𝐴) = sup(ran seq1( + , (𝑛 ∈ ℕ ↦ (vol‘𝐴))), ℝ^*, < ))
6	5	an32s 651	. . 3 ⊢ (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ) → (vol‘∪ 𝑛 ∈ ℕ 𝐴) = sup(ran seq1( + , (𝑛 ∈ ℕ ↦ (vol‘𝐴))), ℝ^*, < ))
7		nfra1 3183	. . . . . . 7 ⊢ Ⅎ𝑛∀𝑛 ∈ ℕ 𝐴 ∈ dom vol
8		nfra1 3183	. . . . . . 7 ⊢ Ⅎ𝑛∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ
9	7, 8	nfan 1900	. . . . . 6 ⊢ Ⅎ𝑛(∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ)
10		simpr 488	. . . . . . . . . 10 ⊢ ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
11		rspa 3171	. . . . . . . . . . 11 ⊢ ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ 𝑛 ∈ ℕ) → 𝐴 ∈ dom vol)
12		volf 24133	. . . . . . . . . . . 12 ⊢ vol:dom vol⟶(0[,]+∞)
13	12	ffvelrni 6827	. . . . . . . . . . 11 ⊢ (𝐴 ∈ dom vol → (vol‘𝐴) ∈ (0[,]+∞))
14	11, 13	syl 17	. . . . . . . . . 10 ⊢ ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ 𝑛 ∈ ℕ) → (vol‘𝐴) ∈ (0[,]+∞))
15	3	fvmpt2 6756	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ (vol‘𝐴) ∈ (0[,]+∞)) → ((𝑛 ∈ ℕ ↦ (vol‘𝐴))‘𝑛) = (vol‘𝐴))
16	10, 14, 15	syl2anc 587	. . . . . . . . 9 ⊢ ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ 𝑛 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (vol‘𝐴))‘𝑛) = (vol‘𝐴))
17	16	adantlr 714	. . . . . . . 8 ⊢ (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (vol‘𝐴))‘𝑛) = (vol‘𝐴))
18	17	ex 416	. . . . . . 7 ⊢ ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ) → (𝑛 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (vol‘𝐴))‘𝑛) = (vol‘𝐴)))
19	9, 18	ralrimi 3180	. . . . . 6 ⊢ ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ) → ∀𝑛 ∈ ℕ ((𝑛 ∈ ℕ ↦ (vol‘𝐴))‘𝑛) = (vol‘𝐴))
20	9, 19	esumeq2d 31406	. . . . 5 ⊢ ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ) → Σ^𝑛 ∈ ℕ((𝑛 ∈ ℕ ↦ (vol‘𝐴))‘𝑛) = Σ^𝑛 ∈ ℕ(vol‘𝐴))
21		simpr 488	. . . . . . . . 9 ⊢ ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ) → ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ)
22	21	r19.21bi 3173	. . . . . . . 8 ⊢ (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (vol‘𝐴) ∈ ℝ)
23	14	adantlr 714	. . . . . . . . 9 ⊢ (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (vol‘𝐴) ∈ (0[,]+∞))
24		0xr 10677	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ^*
25		pnfxr 10684	. . . . . . . . . . 11 ⊢ +∞ ∈ ℝ^*
26		elicc1 12770	. . . . . . . . . . 11 ⊢ ((0 ∈ ℝ^* ∧ +∞ ∈ ℝ^) → ((vol‘𝐴) ∈ (0[,]+∞) ↔ ((vol‘𝐴) ∈ ℝ^ ∧ 0 ≤ (vol‘𝐴) ∧ (vol‘𝐴) ≤ +∞)))
27	24, 25, 26	mp2an 691	. . . . . . . . . 10 ⊢ ((vol‘𝐴) ∈ (0[,]+∞) ↔ ((vol‘𝐴) ∈ ℝ^* ∧ 0 ≤ (vol‘𝐴) ∧ (vol‘𝐴) ≤ +∞))
28	27	simp2bi 1143	. . . . . . . . 9 ⊢ ((vol‘𝐴) ∈ (0[,]+∞) → 0 ≤ (vol‘𝐴))
29	23, 28	syl 17	. . . . . . . 8 ⊢ (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → 0 ≤ (vol‘𝐴))
30		ltpnf 12503	. . . . . . . . 9 ⊢ ((vol‘𝐴) ∈ ℝ → (vol‘𝐴) < +∞)
31	22, 30	syl 17	. . . . . . . 8 ⊢ (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (vol‘𝐴) < +∞)
32		0re 10632	. . . . . . . . 9 ⊢ 0 ∈ ℝ
33		elico2 12789	. . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ +∞ ∈ ℝ^*) → ((vol‘𝐴) ∈ (0[,)+∞) ↔ ((vol‘𝐴) ∈ ℝ ∧ 0 ≤ (vol‘𝐴) ∧ (vol‘𝐴) < +∞)))
34	32, 25, 33	mp2an 691	. . . . . . . 8 ⊢ ((vol‘𝐴) ∈ (0[,)+∞) ↔ ((vol‘𝐴) ∈ ℝ ∧ 0 ≤ (vol‘𝐴) ∧ (vol‘𝐴) < +∞))
35	22, 29, 31, 34	syl3anbrc 1340	. . . . . . 7 ⊢ (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (vol‘𝐴) ∈ (0[,)+∞))
36	9, 35, 3	fmptdf 6858	. . . . . 6 ⊢ ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ) → (𝑛 ∈ ℕ ↦ (vol‘𝐴)):ℕ⟶(0[,)+∞))
37		nfmpt1 5128	. . . . . . 7 ⊢ Ⅎ𝑛(𝑛 ∈ ℕ ↦ (vol‘𝐴))
38	37	esumfsupre 31440	. . . . . 6 ⊢ ((𝑛 ∈ ℕ ↦ (vol‘𝐴)):ℕ⟶(0[,)+∞) → Σ^𝑛 ∈ ℕ((𝑛 ∈ ℕ ↦ (vol‘𝐴))‘𝑛) = sup(ran seq1( + , (𝑛 ∈ ℕ ↦ (vol‘𝐴))), ℝ^, < ))
39	36, 38	syl 17	. . . . 5 ⊢ ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ) → Σ^𝑛 ∈ ℕ((𝑛 ∈ ℕ ↦ (vol‘𝐴))‘𝑛) = sup(ran seq1( + , (𝑛 ∈ ℕ ↦ (vol‘𝐴))), ℝ^, < ))
40	20, 39	eqtr3d 2835	. . . 4 ⊢ ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ) → Σ^𝑛 ∈ ℕ(vol‘𝐴) = sup(ran seq1( + , (𝑛 ∈ ℕ ↦ (vol‘𝐴))), ℝ^, < ))
41	40	adantlr 714	. . 3 ⊢ (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ) → Σ^𝑛 ∈ ℕ(vol‘𝐴) = sup(ran seq1( + , (𝑛 ∈ ℕ ↦ (vol‘𝐴))), ℝ^, < ))
42	6, 41	eqtr4d 2836	. 2 ⊢ (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ) → (vol‘∪ 𝑛 ∈ ℕ 𝐴) = Σ^*𝑛 ∈ ℕ(vol‘𝐴))
43		simpr 488	. . . . . . . 8 ⊢ (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) ∧ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞) → ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞)
44		nfv 1915	. . . . . . . . 9 ⊢ Ⅎ𝑘(vol‘𝐴) = +∞
45		nfcv 2955	. . . . . . . . . . 11 ⊢ Ⅎ𝑛vol
46		nfcsb1v 3852	. . . . . . . . . . 11 ⊢ Ⅎ𝑛⦋𝑘 / 𝑛⦌𝐴
47	45, 46	nffv 6655	. . . . . . . . . 10 ⊢ Ⅎ𝑛(vol‘⦋𝑘 / 𝑛⦌𝐴)
48	47	nfeq1 2970	. . . . . . . . 9 ⊢ Ⅎ𝑛(vol‘⦋𝑘 / 𝑛⦌𝐴) = +∞
49		csbeq1a 3842	. . . . . . . . . 10 ⊢ (𝑛 = 𝑘 → 𝐴 = ⦋𝑘 / 𝑛⦌𝐴)
50	49	fveqeq2d 6653	. . . . . . . . 9 ⊢ (𝑛 = 𝑘 → ((vol‘𝐴) = +∞ ↔ (vol‘⦋𝑘 / 𝑛⦌𝐴) = +∞))
51	44, 48, 50	cbvrexw 3388	. . . . . . . 8 ⊢ (∃𝑛 ∈ ℕ (vol‘𝐴) = +∞ ↔ ∃𝑘 ∈ ℕ (vol‘⦋𝑘 / 𝑛⦌𝐴) = +∞)
52	43, 51	sylib 221	. . . . . . 7 ⊢ (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) ∧ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞) → ∃𝑘 ∈ ℕ (vol‘⦋𝑘 / 𝑛⦌𝐴) = +∞)
53	46	nfel1 2971	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛⦋𝑘 / 𝑛⦌𝐴 ∈ dom vol
54	49	eleq1d 2874	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑘 → (𝐴 ∈ dom vol ↔ ⦋𝑘 / 𝑛⦌𝐴 ∈ dom vol))
55	53, 54	rspc 3559	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → (∀𝑛 ∈ ℕ 𝐴 ∈ dom vol → ⦋𝑘 / 𝑛⦌𝐴 ∈ dom vol))
56	55	impcom 411	. . . . . . . . . . 11 ⊢ ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ 𝑘 ∈ ℕ) → ⦋𝑘 / 𝑛⦌𝐴 ∈ dom vol)
57		iunmbl 24157	. . . . . . . . . . . 12 ⊢ (∀𝑛 ∈ ℕ 𝐴 ∈ dom vol → ∪ 𝑛 ∈ ℕ 𝐴 ∈ dom vol)
58	57	adantr 484	. . . . . . . . . . 11 ⊢ ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ 𝑘 ∈ ℕ) → ∪ 𝑛 ∈ ℕ 𝐴 ∈ dom vol)
59		nfcv 2955	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛ℕ
60		nfcv 2955	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛𝑘
61	59, 60, 46, 49	ssiun2sf 30323	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → ⦋𝑘 / 𝑛⦌𝐴 ⊆ ∪ 𝑛 ∈ ℕ 𝐴)
62	61	adantl 485	. . . . . . . . . . 11 ⊢ ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ 𝑘 ∈ ℕ) → ⦋𝑘 / 𝑛⦌𝐴 ⊆ ∪ 𝑛 ∈ ℕ 𝐴)
63		volss 24137	. . . . . . . . . . 11 ⊢ ((⦋𝑘 / 𝑛⦌𝐴 ∈ dom vol ∧ ∪ 𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ⦋𝑘 / 𝑛⦌𝐴 ⊆ ∪ 𝑛 ∈ ℕ 𝐴) → (vol‘⦋𝑘 / 𝑛⦌𝐴) ≤ (vol‘∪ 𝑛 ∈ ℕ 𝐴))
64	56, 58, 62, 63	syl3anc 1368	. . . . . . . . . 10 ⊢ ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ 𝑘 ∈ ℕ) → (vol‘⦋𝑘 / 𝑛⦌𝐴) ≤ (vol‘∪ 𝑛 ∈ ℕ 𝐴))
65	64	adantlr 714	. . . . . . . . 9 ⊢ (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) ∧ 𝑘 ∈ ℕ) → (vol‘⦋𝑘 / 𝑛⦌𝐴) ≤ (vol‘∪ 𝑛 ∈ ℕ 𝐴))
66	65	adantlr 714	. . . . . . . 8 ⊢ ((((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) ∧ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞) ∧ 𝑘 ∈ ℕ) → (vol‘⦋𝑘 / 𝑛⦌𝐴) ≤ (vol‘∪ 𝑛 ∈ ℕ 𝐴))
67	66	ralrimiva 3149	. . . . . . 7 ⊢ (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) ∧ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞) → ∀𝑘 ∈ ℕ (vol‘⦋𝑘 / 𝑛⦌𝐴) ≤ (vol‘∪ 𝑛 ∈ ℕ 𝐴))
68		r19.29r 3217	. . . . . . 7 ⊢ ((∃𝑘 ∈ ℕ (vol‘⦋𝑘 / 𝑛⦌𝐴) = +∞ ∧ ∀𝑘 ∈ ℕ (vol‘⦋𝑘 / 𝑛⦌𝐴) ≤ (vol‘∪ 𝑛 ∈ ℕ 𝐴)) → ∃𝑘 ∈ ℕ ((vol‘⦋𝑘 / 𝑛⦌𝐴) = +∞ ∧ (vol‘⦋𝑘 / 𝑛⦌𝐴) ≤ (vol‘∪ 𝑛 ∈ ℕ 𝐴)))
69	52, 67, 68	syl2anc 587	. . . . . 6 ⊢ (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) ∧ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞) → ∃𝑘 ∈ ℕ ((vol‘⦋𝑘 / 𝑛⦌𝐴) = +∞ ∧ (vol‘⦋𝑘 / 𝑛⦌𝐴) ≤ (vol‘∪ 𝑛 ∈ ℕ 𝐴)))
70		breq1 5033	. . . . . . . 8 ⊢ ((vol‘⦋𝑘 / 𝑛⦌𝐴) = +∞ → ((vol‘⦋𝑘 / 𝑛⦌𝐴) ≤ (vol‘∪ 𝑛 ∈ ℕ 𝐴) ↔ +∞ ≤ (vol‘∪ 𝑛 ∈ ℕ 𝐴)))
71	70	biimpa 480	. . . . . . 7 ⊢ (((vol‘⦋𝑘 / 𝑛⦌𝐴) = +∞ ∧ (vol‘⦋𝑘 / 𝑛⦌𝐴) ≤ (vol‘∪ 𝑛 ∈ ℕ 𝐴)) → +∞ ≤ (vol‘∪ 𝑛 ∈ ℕ 𝐴))
72	71	reximi 3206	. . . . . 6 ⊢ (∃𝑘 ∈ ℕ ((vol‘⦋𝑘 / 𝑛⦌𝐴) = +∞ ∧ (vol‘⦋𝑘 / 𝑛⦌𝐴) ≤ (vol‘∪ 𝑛 ∈ ℕ 𝐴)) → ∃𝑘 ∈ ℕ +∞ ≤ (vol‘∪ 𝑛 ∈ ℕ 𝐴))
73	69, 72	syl 17	. . . . 5 ⊢ (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) ∧ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞) → ∃𝑘 ∈ ℕ +∞ ≤ (vol‘∪ 𝑛 ∈ ℕ 𝐴))
74		1nn 11636	. . . . . 6 ⊢ 1 ∈ ℕ
75		ne0i 4250	. . . . . 6 ⊢ (1 ∈ ℕ → ℕ ≠ ∅)
76		r19.9rzv 4403	. . . . . 6 ⊢ (ℕ ≠ ∅ → (+∞ ≤ (vol‘∪ 𝑛 ∈ ℕ 𝐴) ↔ ∃𝑘 ∈ ℕ +∞ ≤ (vol‘∪ 𝑛 ∈ ℕ 𝐴)))
77	74, 75, 76	mp2b 10	. . . . 5 ⊢ (+∞ ≤ (vol‘∪ 𝑛 ∈ ℕ 𝐴) ↔ ∃𝑘 ∈ ℕ +∞ ≤ (vol‘∪ 𝑛 ∈ ℕ 𝐴))
78	73, 77	sylibr 237	. . . 4 ⊢ (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) ∧ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞) → +∞ ≤ (vol‘∪ 𝑛 ∈ ℕ 𝐴))
79		iccssxr 12808	. . . . . . . 8 ⊢ (0[,]+∞) ⊆ ℝ^*
80	12	ffvelrni 6827	. . . . . . . 8 ⊢ (∪ 𝑛 ∈ ℕ 𝐴 ∈ dom vol → (vol‘∪ 𝑛 ∈ ℕ 𝐴) ∈ (0[,]+∞))
81	79, 80	sseldi 3913	. . . . . . 7 ⊢ (∪ 𝑛 ∈ ℕ 𝐴 ∈ dom vol → (vol‘∪ 𝑛 ∈ ℕ 𝐴) ∈ ℝ^*)
82	57, 81	syl 17	. . . . . 6 ⊢ (∀𝑛 ∈ ℕ 𝐴 ∈ dom vol → (vol‘∪ 𝑛 ∈ ℕ 𝐴) ∈ ℝ^*)
83	82	ad2antrr 725	. . . . 5 ⊢ (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) ∧ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞) → (vol‘∪ 𝑛 ∈ ℕ 𝐴) ∈ ℝ^*)
84		xgepnf 12546	. . . . 5 ⊢ ((vol‘∪ 𝑛 ∈ ℕ 𝐴) ∈ ℝ^* → (+∞ ≤ (vol‘∪ 𝑛 ∈ ℕ 𝐴) ↔ (vol‘∪ 𝑛 ∈ ℕ 𝐴) = +∞))
85	83, 84	syl 17	. . . 4 ⊢ (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) ∧ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞) → (+∞ ≤ (vol‘∪ 𝑛 ∈ ℕ 𝐴) ↔ (vol‘∪ 𝑛 ∈ ℕ 𝐴) = +∞))
86	78, 85	mpbid 235	. . 3 ⊢ (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) ∧ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞) → (vol‘∪ 𝑛 ∈ ℕ 𝐴) = +∞)
87		nfdisj1 5009	. . . . . 6 ⊢ Ⅎ𝑛Disj 𝑛 ∈ ℕ 𝐴
88	7, 87	nfan 1900	. . . . 5 ⊢ Ⅎ𝑛(∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴)
89		nfre1 3265	. . . . 5 ⊢ Ⅎ𝑛∃𝑛 ∈ ℕ (vol‘𝐴) = +∞
90	88, 89	nfan 1900	. . . 4 ⊢ Ⅎ𝑛((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) ∧ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞)
91		nnex 11631	. . . . 5 ⊢ ℕ ∈ V
92	91	a1i 11	. . . 4 ⊢ (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) ∧ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞) → ℕ ∈ V)
93	14	3ad2antr3 1187	. . . . 5 ⊢ ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ (Disj 𝑛 ∈ ℕ 𝐴 ∧ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞ ∧ 𝑛 ∈ ℕ)) → (vol‘𝐴) ∈ (0[,]+∞))
94	93	3anassrs 1357	. . . 4 ⊢ ((((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) ∧ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞) ∧ 𝑛 ∈ ℕ) → (vol‘𝐴) ∈ (0[,]+∞))
95	90, 92, 94, 43	esumpinfval 31442	. . 3 ⊢ (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) ∧ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞) → Σ^*𝑛 ∈ ℕ(vol‘𝐴) = +∞)
96	86, 95	eqtr4d 2836	. 2 ⊢ (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) ∧ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞) → (vol‘∪ 𝑛 ∈ ℕ 𝐴) = Σ^*𝑛 ∈ ℕ(vol‘𝐴))
97		exmid 892	. . . . 5 ⊢ (∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ ∨ ¬ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ)
98		rexnal 3201	. . . . . 6 ⊢ (∃𝑛 ∈ ℕ ¬ (vol‘𝐴) ∈ ℝ ↔ ¬ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ)
99	98	orbi2i 910	. . . . 5 ⊢ ((∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ ∨ ∃𝑛 ∈ ℕ ¬ (vol‘𝐴) ∈ ℝ) ↔ (∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ ∨ ¬ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ))
100	97, 99	mpbir 234	. . . 4 ⊢ (∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ ∨ ∃𝑛 ∈ ℕ ¬ (vol‘𝐴) ∈ ℝ)
101		r19.29 3216	. . . . . . 7 ⊢ ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∃𝑛 ∈ ℕ ¬ (vol‘𝐴) ∈ ℝ) → ∃𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ ¬ (vol‘𝐴) ∈ ℝ))
102		xrge0nre 12831	. . . . . . . . 9 ⊢ (((vol‘𝐴) ∈ (0[,]+∞) ∧ ¬ (vol‘𝐴) ∈ ℝ) → (vol‘𝐴) = +∞)
103	13, 102	sylan 583	. . . . . . . 8 ⊢ ((𝐴 ∈ dom vol ∧ ¬ (vol‘𝐴) ∈ ℝ) → (vol‘𝐴) = +∞)
104	103	reximi 3206	. . . . . . 7 ⊢ (∃𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ ¬ (vol‘𝐴) ∈ ℝ) → ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞)
105	101, 104	syl 17	. . . . . 6 ⊢ ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∃𝑛 ∈ ℕ ¬ (vol‘𝐴) ∈ ℝ) → ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞)
106	105	ex 416	. . . . 5 ⊢ (∀𝑛 ∈ ℕ 𝐴 ∈ dom vol → (∃𝑛 ∈ ℕ ¬ (vol‘𝐴) ∈ ℝ → ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞))
107	106	orim2d 964	. . . 4 ⊢ (∀𝑛 ∈ ℕ 𝐴 ∈ dom vol → ((∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ ∨ ∃𝑛 ∈ ℕ ¬ (vol‘𝐴) ∈ ℝ) → (∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ ∨ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞)))
108	100, 107	mpi 20	. . 3 ⊢ (∀𝑛 ∈ ℕ 𝐴 ∈ dom vol → (∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ ∨ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞))
109	108	adantr 484	. 2 ⊢ ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) → (∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ ∨ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞))
110	42, 96, 109	mpjaodan 956	1 ⊢ ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) → (vol‘∪ 𝑛 ∈ ℕ 𝐴) = Σ^*𝑛 ∈ ℕ(vol‘𝐴))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∨ wo 844 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∀wral 3106 ∃wrex 3107 Vcvv 3441 ⦋csb 3828 ⊆ wss 3881 ∅c0 4243 ∪ ciun 4881 Disj wdisj 4995 class class class wbr 5030 ↦ cmpt 5110 dom cdm 5519 ran crn 5520 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 supcsup 8888 ℝcr 10525 0cc0 10526 1c1 10527 + caddc 10529 +∞cpnf 10661 ℝ^*cxr 10663 < clt 10664 ≤ cle 10665 ℕcn 11625 [,)cico 12728 [,]cicc 12729 seqcseq 13364 volcvol 24067 Σ^*cesum 31396

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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cc 9846 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 ax-mulf 10606

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-disj 4996 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-map 8391 df-pm 8392 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-fi 8859 df-sup 8890 df-inf 8891 df-oi 8958 df-dju 9314 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-xnn0 11956 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ioo 12730 df-ioc 12731 df-ico 12732 df-icc 12733 df-fz 12886 df-fzo 13029 df-fl 13157 df-mod 13233 df-seq 13365 df-exp 13426 df-fac 13630 df-bc 13659 df-hash 13687 df-shft 14418 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-limsup 14820 df-clim 14837 df-rlim 14838 df-sum 15035 df-ef 15413 df-sin 15415 df-cos 15416 df-pi 15418 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-hom 16581 df-cco 16582 df-rest 16688 df-topn 16689 df-0g 16707 df-gsum 16708 df-topgen 16709 df-pt 16710 df-prds 16713 df-ordt 16766 df-xrs 16767 df-qtop 16772 df-imas 16773 df-xps 16775 df-mre 16849 df-mrc 16850 df-acs 16852 df-ps 17802 df-tsr 17803 df-plusf 17843 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-mhm 17948 df-submnd 17949 df-grp 18098 df-minusg 18099 df-sbg 18100 df-mulg 18217 df-subg 18268 df-cntz 18439 df-cmn 18900 df-abl 18901 df-mgp 19233 df-ur 19245 df-ring 19292 df-cring 19293 df-subrg 19526 df-abv 19581 df-lmod 19629 df-scaf 19630 df-sra 19937 df-rgmod 19938 df-psmet 20083 df-xmet 20084 df-met 20085 df-bl 20086 df-mopn 20087 df-fbas 20088 df-fg 20089 df-cnfld 20092 df-top 21499 df-topon 21516 df-topsp 21538 df-bases 21551 df-cld 21624 df-ntr 21625 df-cls 21626 df-nei 21703 df-lp 21741 df-perf 21742 df-cn 21832 df-cnp 21833 df-haus 21920 df-tx 22167 df-hmeo 22360 df-fil 22451 df-fm 22543 df-flim 22544 df-flf 22545 df-tmd 22677 df-tgp 22678 df-tsms 22732 df-trg 22765 df-xms 22927 df-ms 22928 df-tms 22929 df-nm 23189 df-ngp 23190 df-nrg 23192 df-nlm 23193 df-ii 23482 df-cncf 23483 df-ovol 24068 df-vol 24069 df-limc 24469 df-dv 24470 df-log 25148 df-esum 31397

This theorem is referenced by: volmeas 31600

W3C validator