voliune - Mathbox for Thierry Arnoux

	Mathbox for Thierry Arnoux		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > voliune		Structured version Visualization version GIF version

Theorem voliune 34260

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 25458 and voliun 25507. (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 3098	. . . . 5 ⊢ (∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ↔ (∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ))
2		eqid 2735	. . . . . 6 ⊢ seq1( + , (𝑛 ∈ ℕ ↦ (vol‘𝐴))) = seq1( + , (𝑛 ∈ ℕ ↦ (vol‘𝐴)))
3		eqid 2735	. . . . . 6 ⊢ (𝑛 ∈ ℕ ↦ (vol‘𝐴)) = (𝑛 ∈ ℕ ↦ (vol‘𝐴))
4	2, 3	voliun 25507	. . . . 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 582	. . . 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 652	. . 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 3266	. . . . . . 7 ⊢ Ⅎ𝑛∀𝑛 ∈ ℕ 𝐴 ∈ dom vol
8		nfra1 3266	. . . . . . 7 ⊢ Ⅎ𝑛∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ
9	7, 8	nfan 1899	. . . . . 6 ⊢ Ⅎ𝑛(∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ)
10		simpr 484	. . . . . . . . . 10 ⊢ ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
11		rspa 3231	. . . . . . . . . . 11 ⊢ ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ 𝑛 ∈ ℕ) → 𝐴 ∈ dom vol)
12		volf 25482	. . . . . . . . . . . 12 ⊢ vol:dom vol⟶(0[,]+∞)
13	12	ffvelcdmi 7073	. . . . . . . . . . 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 6997	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ (vol‘𝐴) ∈ (0[,]+∞)) → ((𝑛 ∈ ℕ ↦ (vol‘𝐴))‘𝑛) = (vol‘𝐴))
16	10, 14, 15	syl2anc 584	. . . . . . . . 9 ⊢ ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ 𝑛 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (vol‘𝐴))‘𝑛) = (vol‘𝐴))
17	16	adantlr 715	. . . . . . . 8 ⊢ (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (vol‘𝐴))‘𝑛) = (vol‘𝐴))
18	17	ex 412	. . . . . . 7 ⊢ ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ) → (𝑛 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (vol‘𝐴))‘𝑛) = (vol‘𝐴)))
19	9, 18	ralrimi 3240	. . . . . 6 ⊢ ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ) → ∀𝑛 ∈ ℕ ((𝑛 ∈ ℕ ↦ (vol‘𝐴))‘𝑛) = (vol‘𝐴))
20	9, 19	esumeq2d 34068	. . . . 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 484	. . . . . . . . 9 ⊢ ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ) → ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ)
22	21	r19.21bi 3234	. . . . . . . 8 ⊢ (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (vol‘𝐴) ∈ ℝ)
23	14	adantlr 715	. . . . . . . . 9 ⊢ (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (vol‘𝐴) ∈ (0[,]+∞))
24		0xr 11282	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ^*
25		pnfxr 11289	. . . . . . . . . . 11 ⊢ +∞ ∈ ℝ^*
26		elicc1 13406	. . . . . . . . . . 11 ⊢ ((0 ∈ ℝ^* ∧ +∞ ∈ ℝ^) → ((vol‘𝐴) ∈ (0[,]+∞) ↔ ((vol‘𝐴) ∈ ℝ^ ∧ 0 ≤ (vol‘𝐴) ∧ (vol‘𝐴) ≤ +∞)))
27	24, 25, 26	mp2an 692	. . . . . . . . . 10 ⊢ ((vol‘𝐴) ∈ (0[,]+∞) ↔ ((vol‘𝐴) ∈ ℝ^* ∧ 0 ≤ (vol‘𝐴) ∧ (vol‘𝐴) ≤ +∞))
28	27	simp2bi 1146	. . . . . . . . 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 13136	. . . . . . . . 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 11237	. . . . . . . . 9 ⊢ 0 ∈ ℝ
33		elico2 13427	. . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ +∞ ∈ ℝ^*) → ((vol‘𝐴) ∈ (0[,)+∞) ↔ ((vol‘𝐴) ∈ ℝ ∧ 0 ≤ (vol‘𝐴) ∧ (vol‘𝐴) < +∞)))
34	32, 25, 33	mp2an 692	. . . . . . . 8 ⊢ ((vol‘𝐴) ∈ (0[,)+∞) ↔ ((vol‘𝐴) ∈ ℝ ∧ 0 ≤ (vol‘𝐴) ∧ (vol‘𝐴) < +∞))
35	22, 29, 31, 34	syl3anbrc 1344	. . . . . . 7 ⊢ (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (vol‘𝐴) ∈ (0[,)+∞))
36	9, 35, 3	fmptdf 7107	. . . . . 6 ⊢ ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ) → (𝑛 ∈ ℕ ↦ (vol‘𝐴)):ℕ⟶(0[,)+∞))
37		nfmpt1 5220	. . . . . . 7 ⊢ Ⅎ𝑛(𝑛 ∈ ℕ ↦ (vol‘𝐴))
38	37	esumfsupre 34102	. . . . . 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 2772	. . . 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 715	. . 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 2773	. 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 484	. . . . . . . 8 ⊢ (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) ∧ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞) → ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞)
44		nfv 1914	. . . . . . . . 9 ⊢ Ⅎ𝑘(vol‘𝐴) = +∞
45		nfcv 2898	. . . . . . . . . . 11 ⊢ Ⅎ𝑛vol
46		nfcsb1v 3898	. . . . . . . . . . 11 ⊢ Ⅎ𝑛⦋𝑘 / 𝑛⦌𝐴
47	45, 46	nffv 6886	. . . . . . . . . 10 ⊢ Ⅎ𝑛(vol‘⦋𝑘 / 𝑛⦌𝐴)
48	47	nfeq1 2914	. . . . . . . . 9 ⊢ Ⅎ𝑛(vol‘⦋𝑘 / 𝑛⦌𝐴) = +∞
49		csbeq1a 3888	. . . . . . . . . 10 ⊢ (𝑛 = 𝑘 → 𝐴 = ⦋𝑘 / 𝑛⦌𝐴)
50	49	fveqeq2d 6884	. . . . . . . . 9 ⊢ (𝑛 = 𝑘 → ((vol‘𝐴) = +∞ ↔ (vol‘⦋𝑘 / 𝑛⦌𝐴) = +∞))
51	44, 48, 50	cbvrexw 3287	. . . . . . . 8 ⊢ (∃𝑛 ∈ ℕ (vol‘𝐴) = +∞ ↔ ∃𝑘 ∈ ℕ (vol‘⦋𝑘 / 𝑛⦌𝐴) = +∞)
52	43, 51	sylib 218	. . . . . . 7 ⊢ (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) ∧ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞) → ∃𝑘 ∈ ℕ (vol‘⦋𝑘 / 𝑛⦌𝐴) = +∞)
53	46	nfel1 2915	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛⦋𝑘 / 𝑛⦌𝐴 ∈ dom vol
54	49	eleq1d 2819	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑘 → (𝐴 ∈ dom vol ↔ ⦋𝑘 / 𝑛⦌𝐴 ∈ dom vol))
55	53, 54	rspc 3589	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → (∀𝑛 ∈ ℕ 𝐴 ∈ dom vol → ⦋𝑘 / 𝑛⦌𝐴 ∈ dom vol))
56	55	impcom 407	. . . . . . . . . . 11 ⊢ ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ 𝑘 ∈ ℕ) → ⦋𝑘 / 𝑛⦌𝐴 ∈ dom vol)
57		iunmbl 25506	. . . . . . . . . . . 12 ⊢ (∀𝑛 ∈ ℕ 𝐴 ∈ dom vol → ∪ 𝑛 ∈ ℕ 𝐴 ∈ dom vol)
58	57	adantr 480	. . . . . . . . . . 11 ⊢ ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ 𝑘 ∈ ℕ) → ∪ 𝑛 ∈ ℕ 𝐴 ∈ dom vol)
59		nfcv 2898	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛ℕ
60		nfcv 2898	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛𝑘
61	59, 60, 46, 49	ssiun2sf 32540	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → ⦋𝑘 / 𝑛⦌𝐴 ⊆ ∪ 𝑛 ∈ ℕ 𝐴)
62	61	adantl 481	. . . . . . . . . . 11 ⊢ ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ 𝑘 ∈ ℕ) → ⦋𝑘 / 𝑛⦌𝐴 ⊆ ∪ 𝑛 ∈ ℕ 𝐴)
63		volss 25486	. . . . . . . . . . 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 1373	. . . . . . . . . 10 ⊢ ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ 𝑘 ∈ ℕ) → (vol‘⦋𝑘 / 𝑛⦌𝐴) ≤ (vol‘∪ 𝑛 ∈ ℕ 𝐴))
65	64	adantlr 715	. . . . . . . . 9 ⊢ (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) ∧ 𝑘 ∈ ℕ) → (vol‘⦋𝑘 / 𝑛⦌𝐴) ≤ (vol‘∪ 𝑛 ∈ ℕ 𝐴))
66	65	adantlr 715	. . . . . . . 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 3132	. . . . . . 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 3103	. . . . . . 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 584	. . . . . 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 5122	. . . . . . . 8 ⊢ ((vol‘⦋𝑘 / 𝑛⦌𝐴) = +∞ → ((vol‘⦋𝑘 / 𝑛⦌𝐴) ≤ (vol‘∪ 𝑛 ∈ ℕ 𝐴) ↔ +∞ ≤ (vol‘∪ 𝑛 ∈ ℕ 𝐴)))
71	70	biimpa 476	. . . . . . 7 ⊢ (((vol‘⦋𝑘 / 𝑛⦌𝐴) = +∞ ∧ (vol‘⦋𝑘 / 𝑛⦌𝐴) ≤ (vol‘∪ 𝑛 ∈ ℕ 𝐴)) → +∞ ≤ (vol‘∪ 𝑛 ∈ ℕ 𝐴))
72	71	reximi 3074	. . . . . 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 12251	. . . . . 6 ⊢ 1 ∈ ℕ
75		ne0i 4316	. . . . . 6 ⊢ (1 ∈ ℕ → ℕ ≠ ∅)
76		r19.9rzv 4475	. . . . . 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 234	. . . 4 ⊢ (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) ∧ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞) → +∞ ≤ (vol‘∪ 𝑛 ∈ ℕ 𝐴))
79		iccssxr 13447	. . . . . . . 8 ⊢ (0[,]+∞) ⊆ ℝ^*
80	12	ffvelcdmi 7073	. . . . . . . 8 ⊢ (∪ 𝑛 ∈ ℕ 𝐴 ∈ dom vol → (vol‘∪ 𝑛 ∈ ℕ 𝐴) ∈ (0[,]+∞))
81	79, 80	sselid 3956	. . . . . . 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 726	. . . . 5 ⊢ (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) ∧ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞) → (vol‘∪ 𝑛 ∈ ℕ 𝐴) ∈ ℝ^*)
84		xgepnf 13181	. . . . 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 232	. . 3 ⊢ (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) ∧ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞) → (vol‘∪ 𝑛 ∈ ℕ 𝐴) = +∞)
87		nfdisj1 5100	. . . . . 6 ⊢ Ⅎ𝑛Disj 𝑛 ∈ ℕ 𝐴
88	7, 87	nfan 1899	. . . . 5 ⊢ Ⅎ𝑛(∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴)
89		nfre1 3267	. . . . 5 ⊢ Ⅎ𝑛∃𝑛 ∈ ℕ (vol‘𝐴) = +∞
90	88, 89	nfan 1899	. . . 4 ⊢ Ⅎ𝑛((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) ∧ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞)
91		nnex 12246	. . . . 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 1191	. . . . 5 ⊢ ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ (Disj 𝑛 ∈ ℕ 𝐴 ∧ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞ ∧ 𝑛 ∈ ℕ)) → (vol‘𝐴) ∈ (0[,]+∞))
94	93	3anassrs 1361	. . . 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 34104	. . 3 ⊢ (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) ∧ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞) → Σ^*𝑛 ∈ ℕ(vol‘𝐴) = +∞)
96	86, 95	eqtr4d 2773	. 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 894	. . . . 5 ⊢ (∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ ∨ ¬ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ)
98		rexnal 3089	. . . . . 6 ⊢ (∃𝑛 ∈ ℕ ¬ (vol‘𝐴) ∈ ℝ ↔ ¬ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ)
99	98	orbi2i 912	. . . . 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 231	. . . 4 ⊢ (∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ ∨ ∃𝑛 ∈ ℕ ¬ (vol‘𝐴) ∈ ℝ)
101		r19.29 3101	. . . . . . 7 ⊢ ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∃𝑛 ∈ ℕ ¬ (vol‘𝐴) ∈ ℝ) → ∃𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ ¬ (vol‘𝐴) ∈ ℝ))
102		xrge0nre 13470	. . . . . . . . 9 ⊢ (((vol‘𝐴) ∈ (0[,]+∞) ∧ ¬ (vol‘𝐴) ∈ ℝ) → (vol‘𝐴) = +∞)
103	13, 102	sylan 580	. . . . . . . 8 ⊢ ((𝐴 ∈ dom vol ∧ ¬ (vol‘𝐴) ∈ ℝ) → (vol‘𝐴) = +∞)
104	103	reximi 3074	. . . . . . 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 412	. . . . 5 ⊢ (∀𝑛 ∈ ℕ 𝐴 ∈ dom vol → (∃𝑛 ∈ ℕ ¬ (vol‘𝐴) ∈ ℝ → ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞))
107	106	orim2d 968	. . . 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 480	. 2 ⊢ ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) → (∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ ∨ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞))
110	42, 96, 109	mpjaodan 960	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 206 ∧ wa 395 ∨ wo 847 ∧ w3a 1086 = wceq 1540 ∈ wcel 2108 ≠ wne 2932 ∀wral 3051 ∃wrex 3060 Vcvv 3459 ⦋csb 3874 ⊆ wss 3926 ∅c0 4308 ∪ ciun 4967 Disj wdisj 5086 class class class wbr 5119 ↦ cmpt 5201 dom cdm 5654 ran crn 5655 ⟶wf 6527 ‘cfv 6531 (class class class)co 7405 supcsup 9452 ℝcr 11128 0cc0 11129 1c1 11130 + caddc 11132 +∞cpnf 11266 ℝ^*cxr 11268 < clt 11269 ≤ cle 11270 ℕcn 12240 [,)cico 13364 [,]cicc 13365 seqcseq 14019 volcvol 25416 Σ^*cesum 34058

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729 ax-inf2 9655 ax-cc 10449 ax-cnex 11185 ax-resscn 11186 ax-1cn 11187 ax-icn 11188 ax-addcl 11189 ax-addrcl 11190 ax-mulcl 11191 ax-mulrcl 11192 ax-mulcom 11193 ax-addass 11194 ax-mulass 11195 ax-distr 11196 ax-i2m1 11197 ax-1ne0 11198 ax-1rid 11199 ax-rnegex 11200 ax-rrecex 11201 ax-cnre 11202 ax-pre-lttri 11203 ax-pre-lttrn 11204 ax-pre-ltadd 11205 ax-pre-mulgt0 11206 ax-pre-sup 11207 ax-addf 11208 ax-mulf 11209

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-tp 4606 df-op 4608 df-uni 4884 df-int 4923 df-iun 4969 df-iin 4970 df-disj 5087 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-se 5607 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-isom 6540 df-riota 7362 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7671 df-om 7862 df-1st 7988 df-2nd 7989 df-supp 8160 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-oadd 8484 df-er 8719 df-map 8842 df-pm 8843 df-ixp 8912 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-fsupp 9374 df-fi 9423 df-sup 9454 df-inf 9455 df-oi 9524 df-dju 9915 df-card 9953 df-pnf 11271 df-mnf 11272 df-xr 11273 df-ltxr 11274 df-le 11275 df-sub 11468 df-neg 11469 df-div 11895 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12502 df-xnn0 12575 df-z 12589 df-dec 12709 df-uz 12853 df-q 12965 df-rp 13009 df-xneg 13128 df-xadd 13129 df-xmul 13130 df-ioo 13366 df-ioc 13367 df-ico 13368 df-icc 13369 df-fz 13525 df-fzo 13672 df-fl 13809 df-mod 13887 df-seq 14020 df-exp 14080 df-fac 14292 df-bc 14321 df-hash 14349 df-shft 15086 df-cj 15118 df-re 15119 df-im 15120 df-sqrt 15254 df-abs 15255 df-limsup 15487 df-clim 15504 df-rlim 15505 df-sum 15703 df-ef 16083 df-sin 16085 df-cos 16086 df-pi 16088 df-struct 17166 df-sets 17183 df-slot 17201 df-ndx 17213 df-base 17229 df-ress 17252 df-plusg 17284 df-mulr 17285 df-starv 17286 df-sca 17287 df-vsca 17288 df-ip 17289 df-tset 17290 df-ple 17291 df-ds 17293 df-unif 17294 df-hom 17295 df-cco 17296 df-rest 17436 df-topn 17437 df-0g 17455 df-gsum 17456 df-topgen 17457 df-pt 17458 df-prds 17461 df-ordt 17515 df-xrs 17516 df-qtop 17521 df-imas 17522 df-xps 17524 df-mre 17598 df-mrc 17599 df-acs 17601 df-ps 18576 df-tsr 18577 df-plusf 18617 df-mgm 18618 df-sgrp 18697 df-mnd 18713 df-mhm 18761 df-submnd 18762 df-grp 18919 df-minusg 18920 df-sbg 18921 df-mulg 19051 df-subg 19106 df-cntz 19300 df-cmn 19763 df-abl 19764 df-mgp 20101 df-rng 20113 df-ur 20142 df-ring 20195 df-cring 20196 df-subrng 20506 df-subrg 20530 df-abv 20769 df-lmod 20819 df-scaf 20820 df-sra 21131 df-rgmod 21132 df-psmet 21307 df-xmet 21308 df-met 21309 df-bl 21310 df-mopn 21311 df-fbas 21312 df-fg 21313 df-cnfld 21316 df-top 22832 df-topon 22849 df-topsp 22871 df-bases 22884 df-cld 22957 df-ntr 22958 df-cls 22959 df-nei 23036 df-lp 23074 df-perf 23075 df-cn 23165 df-cnp 23166 df-haus 23253 df-tx 23500 df-hmeo 23693 df-fil 23784 df-fm 23876 df-flim 23877 df-flf 23878 df-tmd 24010 df-tgp 24011 df-tsms 24065 df-trg 24098 df-xms 24259 df-ms 24260 df-tms 24261 df-nm 24521 df-ngp 24522 df-nrg 24524 df-nlm 24525 df-ii 24821 df-cncf 24822 df-ovol 25417 df-vol 25418 df-limc 25819 df-dv 25820 df-log 26517 df-esum 34059

This theorem is referenced by: volmeas 34262

W3C validator