voliune - Mathbox for Thierry Arnoux

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Theorem voliune 33222

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 25021 and voliun 25070. (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 3111	. . . . 5 ⊢ (∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ↔ (∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ))
2		eqid 2732	. . . . . 6 ⊢ seq1( + , (𝑛 ∈ ℕ ↦ (vol‘𝐴))) = seq1( + , (𝑛 ∈ ℕ ↦ (vol‘𝐴)))
3		eqid 2732	. . . . . 6 ⊢ (𝑛 ∈ ℕ ↦ (vol‘𝐴)) = (𝑛 ∈ ℕ ↦ (vol‘𝐴))
4	2, 3	voliun 25070	. . . . 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 650	. . 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 3281	. . . . . . 7 ⊢ Ⅎ𝑛∀𝑛 ∈ ℕ 𝐴 ∈ dom vol
8		nfra1 3281	. . . . . . 7 ⊢ Ⅎ𝑛∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ
9	7, 8	nfan 1902	. . . . . 6 ⊢ Ⅎ𝑛(∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ)
10		simpr 485	. . . . . . . . . 10 ⊢ ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
11		rspa 3245	. . . . . . . . . . 11 ⊢ ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ 𝑛 ∈ ℕ) → 𝐴 ∈ dom vol)
12		volf 25045	. . . . . . . . . . . 12 ⊢ vol:dom vol⟶(0[,]+∞)
13	12	ffvelcdmi 7085	. . . . . . . . . . 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 7009	. . . . . . . . . 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 713	. . . . . . . 8 ⊢ (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (vol‘𝐴))‘𝑛) = (vol‘𝐴))
18	17	ex 413	. . . . . . 7 ⊢ ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ) → (𝑛 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (vol‘𝐴))‘𝑛) = (vol‘𝐴)))
19	9, 18	ralrimi 3254	. . . . . 6 ⊢ ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ) → ∀𝑛 ∈ ℕ ((𝑛 ∈ ℕ ↦ (vol‘𝐴))‘𝑛) = (vol‘𝐴))
20	9, 19	esumeq2d 33030	. . . . 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 485	. . . . . . . . 9 ⊢ ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ) → ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ)
22	21	r19.21bi 3248	. . . . . . . 8 ⊢ (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (vol‘𝐴) ∈ ℝ)
23	14	adantlr 713	. . . . . . . . 9 ⊢ (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (vol‘𝐴) ∈ (0[,]+∞))
24		0xr 11260	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ^*
25		pnfxr 11267	. . . . . . . . . . 11 ⊢ +∞ ∈ ℝ^*
26		elicc1 13367	. . . . . . . . . . 11 ⊢ ((0 ∈ ℝ^* ∧ +∞ ∈ ℝ^) → ((vol‘𝐴) ∈ (0[,]+∞) ↔ ((vol‘𝐴) ∈ ℝ^ ∧ 0 ≤ (vol‘𝐴) ∧ (vol‘𝐴) ≤ +∞)))
27	24, 25, 26	mp2an 690	. . . . . . . . . 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 13099	. . . . . . . . 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 11215	. . . . . . . . 9 ⊢ 0 ∈ ℝ
33		elico2 13387	. . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ +∞ ∈ ℝ^*) → ((vol‘𝐴) ∈ (0[,)+∞) ↔ ((vol‘𝐴) ∈ ℝ ∧ 0 ≤ (vol‘𝐴) ∧ (vol‘𝐴) < +∞)))
34	32, 25, 33	mp2an 690	. . . . . . . 8 ⊢ ((vol‘𝐴) ∈ (0[,)+∞) ↔ ((vol‘𝐴) ∈ ℝ ∧ 0 ≤ (vol‘𝐴) ∧ (vol‘𝐴) < +∞))
35	22, 29, 31, 34	syl3anbrc 1343	. . . . . . 7 ⊢ (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (vol‘𝐴) ∈ (0[,)+∞))
36	9, 35, 3	fmptdf 7116	. . . . . 6 ⊢ ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ) → (𝑛 ∈ ℕ ↦ (vol‘𝐴)):ℕ⟶(0[,)+∞))
37		nfmpt1 5256	. . . . . . 7 ⊢ Ⅎ𝑛(𝑛 ∈ ℕ ↦ (vol‘𝐴))
38	37	esumfsupre 33064	. . . . . 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 2774	. . . 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 713	. . 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 2775	. 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 485	. . . . . . . 8 ⊢ (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) ∧ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞) → ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞)
44		nfv 1917	. . . . . . . . 9 ⊢ Ⅎ𝑘(vol‘𝐴) = +∞
45		nfcv 2903	. . . . . . . . . . 11 ⊢ Ⅎ𝑛vol
46		nfcsb1v 3918	. . . . . . . . . . 11 ⊢ Ⅎ𝑛⦋𝑘 / 𝑛⦌𝐴
47	45, 46	nffv 6901	. . . . . . . . . 10 ⊢ Ⅎ𝑛(vol‘⦋𝑘 / 𝑛⦌𝐴)
48	47	nfeq1 2918	. . . . . . . . 9 ⊢ Ⅎ𝑛(vol‘⦋𝑘 / 𝑛⦌𝐴) = +∞
49		csbeq1a 3907	. . . . . . . . . 10 ⊢ (𝑛 = 𝑘 → 𝐴 = ⦋𝑘 / 𝑛⦌𝐴)
50	49	fveqeq2d 6899	. . . . . . . . 9 ⊢ (𝑛 = 𝑘 → ((vol‘𝐴) = +∞ ↔ (vol‘⦋𝑘 / 𝑛⦌𝐴) = +∞))
51	44, 48, 50	cbvrexw 3304	. . . . . . . 8 ⊢ (∃𝑛 ∈ ℕ (vol‘𝐴) = +∞ ↔ ∃𝑘 ∈ ℕ (vol‘⦋𝑘 / 𝑛⦌𝐴) = +∞)
52	43, 51	sylib 217	. . . . . . 7 ⊢ (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) ∧ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞) → ∃𝑘 ∈ ℕ (vol‘⦋𝑘 / 𝑛⦌𝐴) = +∞)
53	46	nfel1 2919	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛⦋𝑘 / 𝑛⦌𝐴 ∈ dom vol
54	49	eleq1d 2818	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑘 → (𝐴 ∈ dom vol ↔ ⦋𝑘 / 𝑛⦌𝐴 ∈ dom vol))
55	53, 54	rspc 3600	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → (∀𝑛 ∈ ℕ 𝐴 ∈ dom vol → ⦋𝑘 / 𝑛⦌𝐴 ∈ dom vol))
56	55	impcom 408	. . . . . . . . . . 11 ⊢ ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ 𝑘 ∈ ℕ) → ⦋𝑘 / 𝑛⦌𝐴 ∈ dom vol)
57		iunmbl 25069	. . . . . . . . . . . 12 ⊢ (∀𝑛 ∈ ℕ 𝐴 ∈ dom vol → ∪ 𝑛 ∈ ℕ 𝐴 ∈ dom vol)
58	57	adantr 481	. . . . . . . . . . 11 ⊢ ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ 𝑘 ∈ ℕ) → ∪ 𝑛 ∈ ℕ 𝐴 ∈ dom vol)
59		nfcv 2903	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛ℕ
60		nfcv 2903	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛𝑘
61	59, 60, 46, 49	ssiun2sf 31786	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → ⦋𝑘 / 𝑛⦌𝐴 ⊆ ∪ 𝑛 ∈ ℕ 𝐴)
62	61	adantl 482	. . . . . . . . . . 11 ⊢ ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ 𝑘 ∈ ℕ) → ⦋𝑘 / 𝑛⦌𝐴 ⊆ ∪ 𝑛 ∈ ℕ 𝐴)
63		volss 25049	. . . . . . . . . . 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 1371	. . . . . . . . . 10 ⊢ ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ 𝑘 ∈ ℕ) → (vol‘⦋𝑘 / 𝑛⦌𝐴) ≤ (vol‘∪ 𝑛 ∈ ℕ 𝐴))
65	64	adantlr 713	. . . . . . . . 9 ⊢ (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) ∧ 𝑘 ∈ ℕ) → (vol‘⦋𝑘 / 𝑛⦌𝐴) ≤ (vol‘∪ 𝑛 ∈ ℕ 𝐴))
66	65	adantlr 713	. . . . . . . 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 3146	. . . . . . 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 3116	. . . . . . 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 5151	. . . . . . . 8 ⊢ ((vol‘⦋𝑘 / 𝑛⦌𝐴) = +∞ → ((vol‘⦋𝑘 / 𝑛⦌𝐴) ≤ (vol‘∪ 𝑛 ∈ ℕ 𝐴) ↔ +∞ ≤ (vol‘∪ 𝑛 ∈ ℕ 𝐴)))
71	70	biimpa 477	. . . . . . 7 ⊢ (((vol‘⦋𝑘 / 𝑛⦌𝐴) = +∞ ∧ (vol‘⦋𝑘 / 𝑛⦌𝐴) ≤ (vol‘∪ 𝑛 ∈ ℕ 𝐴)) → +∞ ≤ (vol‘∪ 𝑛 ∈ ℕ 𝐴))
72	71	reximi 3084	. . . . . 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 12222	. . . . . 6 ⊢ 1 ∈ ℕ
75		ne0i 4334	. . . . . 6 ⊢ (1 ∈ ℕ → ℕ ≠ ∅)
76		r19.9rzv 4499	. . . . . 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 233	. . . 4 ⊢ (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) ∧ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞) → +∞ ≤ (vol‘∪ 𝑛 ∈ ℕ 𝐴))
79		iccssxr 13406	. . . . . . . 8 ⊢ (0[,]+∞) ⊆ ℝ^*
80	12	ffvelcdmi 7085	. . . . . . . 8 ⊢ (∪ 𝑛 ∈ ℕ 𝐴 ∈ dom vol → (vol‘∪ 𝑛 ∈ ℕ 𝐴) ∈ (0[,]+∞))
81	79, 80	sselid 3980	. . . . . . 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 724	. . . . 5 ⊢ (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) ∧ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞) → (vol‘∪ 𝑛 ∈ ℕ 𝐴) ∈ ℝ^*)
84		xgepnf 13143	. . . . 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 231	. . 3 ⊢ (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) ∧ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞) → (vol‘∪ 𝑛 ∈ ℕ 𝐴) = +∞)
87		nfdisj1 5127	. . . . . 6 ⊢ Ⅎ𝑛Disj 𝑛 ∈ ℕ 𝐴
88	7, 87	nfan 1902	. . . . 5 ⊢ Ⅎ𝑛(∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴)
89		nfre1 3282	. . . . 5 ⊢ Ⅎ𝑛∃𝑛 ∈ ℕ (vol‘𝐴) = +∞
90	88, 89	nfan 1902	. . . 4 ⊢ Ⅎ𝑛((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) ∧ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞)
91		nnex 12217	. . . . 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 1190	. . . . 5 ⊢ ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ (Disj 𝑛 ∈ ℕ 𝐴 ∧ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞ ∧ 𝑛 ∈ ℕ)) → (vol‘𝐴) ∈ (0[,]+∞))
94	93	3anassrs 1360	. . . 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 33066	. . 3 ⊢ (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) ∧ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞) → Σ^*𝑛 ∈ ℕ(vol‘𝐴) = +∞)
96	86, 95	eqtr4d 2775	. 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 893	. . . . 5 ⊢ (∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ ∨ ¬ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ)
98		rexnal 3100	. . . . . 6 ⊢ (∃𝑛 ∈ ℕ ¬ (vol‘𝐴) ∈ ℝ ↔ ¬ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ)
99	98	orbi2i 911	. . . . 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 230	. . . 4 ⊢ (∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ ∨ ∃𝑛 ∈ ℕ ¬ (vol‘𝐴) ∈ ℝ)
101		r19.29 3114	. . . . . . 7 ⊢ ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∃𝑛 ∈ ℕ ¬ (vol‘𝐴) ∈ ℝ) → ∃𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ ¬ (vol‘𝐴) ∈ ℝ))
102		xrge0nre 13429	. . . . . . . . 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 3084	. . . . . . 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 413	. . . . 5 ⊢ (∀𝑛 ∈ ℕ 𝐴 ∈ dom vol → (∃𝑛 ∈ ℕ ¬ (vol‘𝐴) ∈ ℝ → ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞))
107	106	orim2d 965	. . . 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 481	. 2 ⊢ ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) → (∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ ∨ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞))
110	42, 96, 109	mpjaodan 957	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 205 ∧ wa 396 ∨ wo 845 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 Vcvv 3474 ⦋csb 3893 ⊆ wss 3948 ∅c0 4322 ∪ ciun 4997 Disj wdisj 5113 class class class wbr 5148 ↦ cmpt 5231 dom cdm 5676 ran crn 5677 ⟶wf 6539 ‘cfv 6543 (class class class)co 7408 supcsup 9434 ℝcr 11108 0cc0 11109 1c1 11110 + caddc 11112 +∞cpnf 11244 ℝ^*cxr 11246 < clt 11247 ≤ cle 11248 ℕcn 12211 [,)cico 13325 [,]cicc 13326 seqcseq 13965 volcvol 24979 Σ^*cesum 33020

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-inf2 9635 ax-cc 10429 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 ax-addf 11188 ax-mulf 11189

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-2o 8466 df-oadd 8469 df-er 8702 df-map 8821 df-pm 8822 df-ixp 8891 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-dju 9895 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-xnn0 12544 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-fl 13756 df-mod 13834 df-seq 13966 df-exp 14027 df-fac 14233 df-bc 14262 df-hash 14290 df-shft 15013 df-cj 15045 df-re 15046 df-im 15047 df-sqrt 15181 df-abs 15182 df-limsup 15414 df-clim 15431 df-rlim 15432 df-sum 15632 df-ef 16010 df-sin 16012 df-cos 16013 df-pi 16015 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-starv 17211 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-hom 17220 df-cco 17221 df-rest 17367 df-topn 17368 df-0g 17386 df-gsum 17387 df-topgen 17388 df-pt 17389 df-prds 17392 df-ordt 17446 df-xrs 17447 df-qtop 17452 df-imas 17453 df-xps 17455 df-mre 17529 df-mrc 17530 df-acs 17532 df-ps 18518 df-tsr 18519 df-plusf 18559 df-mgm 18560 df-sgrp 18609 df-mnd 18625 df-mhm 18670 df-submnd 18671 df-grp 18821 df-minusg 18822 df-sbg 18823 df-mulg 18950 df-subg 19002 df-cntz 19180 df-cmn 19649 df-abl 19650 df-mgp 19987 df-ur 20004 df-ring 20057 df-cring 20058 df-subrg 20316 df-abv 20424 df-lmod 20472 df-scaf 20473 df-sra 20784 df-rgmod 20785 df-psmet 20935 df-xmet 20936 df-met 20937 df-bl 20938 df-mopn 20939 df-fbas 20940 df-fg 20941 df-cnfld 20944 df-top 22395 df-topon 22412 df-topsp 22434 df-bases 22448 df-cld 22522 df-ntr 22523 df-cls 22524 df-nei 22601 df-lp 22639 df-perf 22640 df-cn 22730 df-cnp 22731 df-haus 22818 df-tx 23065 df-hmeo 23258 df-fil 23349 df-fm 23441 df-flim 23442 df-flf 23443 df-tmd 23575 df-tgp 23576 df-tsms 23630 df-trg 23663 df-xms 23825 df-ms 23826 df-tms 23827 df-nm 24090 df-ngp 24091 df-nrg 24093 df-nlm 24094 df-ii 24392 df-cncf 24393 df-ovol 24980 df-vol 24981 df-limc 25382 df-dv 25383 df-log 26064 df-esum 33021

This theorem is referenced by: volmeas 33224

W3C validator