voliune - Mathbox for Thierry Arnoux

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Theorem voliune 34388

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 25466 and voliun 25515. (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 3097	. . . . 5 ⊢ (∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ↔ (∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ))
2		eqid 2737	. . . . . 6 ⊢ seq1( + , (𝑛 ∈ ℕ ↦ (vol‘𝐴))) = seq1( + , (𝑛 ∈ ℕ ↦ (vol‘𝐴)))
3		eqid 2737	. . . . . 6 ⊢ (𝑛 ∈ ℕ ↦ (vol‘𝐴)) = (𝑛 ∈ ℕ ↦ (vol‘𝐴))
4	2, 3	voliun 25515	. . . . 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 583	. . . 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 653	. . 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 3261	. . . . . . 7 ⊢ Ⅎ𝑛∀𝑛 ∈ ℕ 𝐴 ∈ dom vol
8		nfra1 3261	. . . . . . 7 ⊢ Ⅎ𝑛∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ
9	7, 8	nfan 1901	. . . . . 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 3226	. . . . . . . . . . 11 ⊢ ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ 𝑛 ∈ ℕ) → 𝐴 ∈ dom vol)
12		volf 25490	. . . . . . . . . . . 12 ⊢ vol:dom vol⟶(0[,]+∞)
13	12	ffvelcdmi 7030	. . . . . . . . . . 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 6954	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ (vol‘𝐴) ∈ (0[,]+∞)) → ((𝑛 ∈ ℕ ↦ (vol‘𝐴))‘𝑛) = (vol‘𝐴))
16	10, 14, 15	syl2anc 585	. . . . . . . . 9 ⊢ ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ 𝑛 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (vol‘𝐴))‘𝑛) = (vol‘𝐴))
17	16	adantlr 716	. . . . . . . 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 3235	. . . . . 6 ⊢ ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ) → ∀𝑛 ∈ ℕ ((𝑛 ∈ ℕ ↦ (vol‘𝐴))‘𝑛) = (vol‘𝐴))
20	9, 19	esumeq2d 34196	. . . . 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 3229	. . . . . . . 8 ⊢ (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (vol‘𝐴) ∈ ℝ)
23	14	adantlr 716	. . . . . . . . 9 ⊢ (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (vol‘𝐴) ∈ (0[,]+∞))
24		0xr 11183	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ^*
25		pnfxr 11190	. . . . . . . . . . 11 ⊢ +∞ ∈ ℝ^*
26		elicc1 13309	. . . . . . . . . . 11 ⊢ ((0 ∈ ℝ^* ∧ +∞ ∈ ℝ^) → ((vol‘𝐴) ∈ (0[,]+∞) ↔ ((vol‘𝐴) ∈ ℝ^ ∧ 0 ≤ (vol‘𝐴) ∧ (vol‘𝐴) ≤ +∞)))
27	24, 25, 26	mp2an 693	. . . . . . . . . 10 ⊢ ((vol‘𝐴) ∈ (0[,]+∞) ↔ ((vol‘𝐴) ∈ ℝ^* ∧ 0 ≤ (vol‘𝐴) ∧ (vol‘𝐴) ≤ +∞))
28	27	simp2bi 1147	. . . . . . . . 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 13038	. . . . . . . . 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 11138	. . . . . . . . 9 ⊢ 0 ∈ ℝ
33		elico2 13330	. . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ +∞ ∈ ℝ^*) → ((vol‘𝐴) ∈ (0[,)+∞) ↔ ((vol‘𝐴) ∈ ℝ ∧ 0 ≤ (vol‘𝐴) ∧ (vol‘𝐴) < +∞)))
34	32, 25, 33	mp2an 693	. . . . . . . 8 ⊢ ((vol‘𝐴) ∈ (0[,)+∞) ↔ ((vol‘𝐴) ∈ ℝ ∧ 0 ≤ (vol‘𝐴) ∧ (vol‘𝐴) < +∞))
35	22, 29, 31, 34	syl3anbrc 1345	. . . . . . 7 ⊢ (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (vol‘𝐴) ∈ (0[,)+∞))
36	9, 35, 3	fmptdf 7064	. . . . . 6 ⊢ ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ) → (𝑛 ∈ ℕ ↦ (vol‘𝐴)):ℕ⟶(0[,)+∞))
37		nfmpt1 5198	. . . . . . 7 ⊢ Ⅎ𝑛(𝑛 ∈ ℕ ↦ (vol‘𝐴))
38	37	esumfsupre 34230	. . . . . 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 716	. . 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 484	. . . . . . . 8 ⊢ (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) ∧ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞) → ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞)
44		nfv 1916	. . . . . . . . 9 ⊢ Ⅎ𝑘(vol‘𝐴) = +∞
45		nfcv 2899	. . . . . . . . . . 11 ⊢ Ⅎ𝑛vol
46		nfcsb1v 3874	. . . . . . . . . . 11 ⊢ Ⅎ𝑛⦋𝑘 / 𝑛⦌𝐴
47	45, 46	nffv 6845	. . . . . . . . . 10 ⊢ Ⅎ𝑛(vol‘⦋𝑘 / 𝑛⦌𝐴)
48	47	nfeq1 2915	. . . . . . . . 9 ⊢ Ⅎ𝑛(vol‘⦋𝑘 / 𝑛⦌𝐴) = +∞
49		csbeq1a 3864	. . . . . . . . . 10 ⊢ (𝑛 = 𝑘 → 𝐴 = ⦋𝑘 / 𝑛⦌𝐴)
50	49	fveqeq2d 6843	. . . . . . . . 9 ⊢ (𝑛 = 𝑘 → ((vol‘𝐴) = +∞ ↔ (vol‘⦋𝑘 / 𝑛⦌𝐴) = +∞))
51	44, 48, 50	cbvrexw 3280	. . . . . . . 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 2916	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛⦋𝑘 / 𝑛⦌𝐴 ∈ dom vol
54	49	eleq1d 2822	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑘 → (𝐴 ∈ dom vol ↔ ⦋𝑘 / 𝑛⦌𝐴 ∈ dom vol))
55	53, 54	rspc 3565	. . . . . . . . . . . 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 25514	. . . . . . . . . . . 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 2899	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛ℕ
60		nfcv 2899	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛𝑘
61	59, 60, 46, 49	ssiun2sf 32637	. . . . . . . . . . . 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 25494	. . . . . . . . . . 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 1374	. . . . . . . . . 10 ⊢ ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ 𝑘 ∈ ℕ) → (vol‘⦋𝑘 / 𝑛⦌𝐴) ≤ (vol‘∪ 𝑛 ∈ ℕ 𝐴))
65	64	adantlr 716	. . . . . . . . 9 ⊢ (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) ∧ 𝑘 ∈ ℕ) → (vol‘⦋𝑘 / 𝑛⦌𝐴) ≤ (vol‘∪ 𝑛 ∈ ℕ 𝐴))
66	65	adantlr 716	. . . . . . . 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 3129	. . . . . . 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 3101	. . . . . . 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 585	. . . . . 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 5102	. . . . . . . 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 3075	. . . . . 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 12160	. . . . . 6 ⊢ 1 ∈ ℕ
75		ne0i 4294	. . . . . 6 ⊢ (1 ∈ ℕ → ℕ ≠ ∅)
76		r19.9rzv 4459	. . . . . 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 13350	. . . . . . . 8 ⊢ (0[,]+∞) ⊆ ℝ^*
80	12	ffvelcdmi 7030	. . . . . . . 8 ⊢ (∪ 𝑛 ∈ ℕ 𝐴 ∈ dom vol → (vol‘∪ 𝑛 ∈ ℕ 𝐴) ∈ (0[,]+∞))
81	79, 80	sselid 3932	. . . . . . 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 727	. . . . 5 ⊢ (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) ∧ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞) → (vol‘∪ 𝑛 ∈ ℕ 𝐴) ∈ ℝ^*)
84		xgepnf 13084	. . . . 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 5080	. . . . . 6 ⊢ Ⅎ𝑛Disj 𝑛 ∈ ℕ 𝐴
88	7, 87	nfan 1901	. . . . 5 ⊢ Ⅎ𝑛(∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴)
89		nfre1 3262	. . . . 5 ⊢ Ⅎ𝑛∃𝑛 ∈ ℕ (vol‘𝐴) = +∞
90	88, 89	nfan 1901	. . . 4 ⊢ Ⅎ𝑛((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) ∧ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞)
91		nnex 12155	. . . . 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 1192	. . . . 5 ⊢ ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ (Disj 𝑛 ∈ ℕ 𝐴 ∧ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞ ∧ 𝑛 ∈ ℕ)) → (vol‘𝐴) ∈ (0[,]+∞))
94	93	3anassrs 1362	. . . 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 34232	. . 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 895	. . . . 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 913	. . . . 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 3100	. . . . . . 7 ⊢ ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∃𝑛 ∈ ℕ ¬ (vol‘𝐴) ∈ ℝ) → ∃𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ ¬ (vol‘𝐴) ∈ ℝ))
102		xrge0nre 13373	. . . . . . . . 9 ⊢ (((vol‘𝐴) ∈ (0[,]+∞) ∧ ¬ (vol‘𝐴) ∈ ℝ) → (vol‘𝐴) = +∞)
103	13, 102	sylan 581	. . . . . . . 8 ⊢ ((𝐴 ∈ dom vol ∧ ¬ (vol‘𝐴) ∈ ℝ) → (vol‘𝐴) = +∞)
104	103	reximi 3075	. . . . . . 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 969	. . . 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 961	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 848 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 ∀wral 3052 ∃wrex 3061 Vcvv 3441 ⦋csb 3850 ⊆ wss 3902 ∅c0 4286 ∪ ciun 4947 Disj wdisj 5066 class class class wbr 5099 ↦ cmpt 5180 dom cdm 5625 ran crn 5626 ⟶wf 6489 ‘cfv 6493 (class class class)co 7360 supcsup 9347 ℝcr 11029 0cc0 11030 1c1 11031 + caddc 11033 +∞cpnf 11167 ℝ^*cxr 11169 < clt 11170 ≤ cle 11171 ℕcn 12149 [,)cico 13267 [,]cicc 13268 seqcseq 13928 volcvol 25424 Σ^*cesum 34186

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5225 ax-sep 5242 ax-nul 5252 ax-pow 5311 ax-pr 5378 ax-un 7682 ax-inf2 9554 ax-cc 10349 ax-cnex 11086 ax-resscn 11087 ax-1cn 11088 ax-icn 11089 ax-addcl 11090 ax-addrcl 11091 ax-mulcl 11092 ax-mulrcl 11093 ax-mulcom 11094 ax-addass 11095 ax-mulass 11096 ax-distr 11097 ax-i2m1 11098 ax-1ne0 11099 ax-1rid 11100 ax-rnegex 11101 ax-rrecex 11102 ax-cnre 11103 ax-pre-lttri 11104 ax-pre-lttrn 11105 ax-pre-ltadd 11106 ax-pre-mulgt0 11107 ax-pre-sup 11108 ax-addf 11109 ax-mulf 11110

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3351 df-reu 3352 df-rab 3401 df-v 3443 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4287 df-if 4481 df-pw 4557 df-sn 4582 df-pr 4584 df-tp 4586 df-op 4588 df-uni 4865 df-int 4904 df-iun 4949 df-iin 4950 df-disj 5067 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-se 5579 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-of 7624 df-om 7811 df-1st 7935 df-2nd 7936 df-supp 8105 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-2o 8400 df-oadd 8403 df-er 8637 df-map 8769 df-pm 8770 df-ixp 8840 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-fsupp 9269 df-fi 9318 df-sup 9349 df-inf 9350 df-oi 9419 df-dju 9817 df-card 9855 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12150 df-2 12212 df-3 12213 df-4 12214 df-5 12215 df-6 12216 df-7 12217 df-8 12218 df-9 12219 df-n0 12406 df-xnn0 12479 df-z 12493 df-dec 12612 df-uz 12756 df-q 12866 df-rp 12910 df-xneg 13030 df-xadd 13031 df-xmul 13032 df-ioo 13269 df-ioc 13270 df-ico 13271 df-icc 13272 df-fz 13428 df-fzo 13575 df-fl 13716 df-mod 13794 df-seq 13929 df-exp 13989 df-fac 14201 df-bc 14230 df-hash 14258 df-shft 14994 df-cj 15026 df-re 15027 df-im 15028 df-sqrt 15162 df-abs 15163 df-limsup 15398 df-clim 15415 df-rlim 15416 df-sum 15614 df-ef 15994 df-sin 15996 df-cos 15997 df-pi 15999 df-struct 17078 df-sets 17095 df-slot 17113 df-ndx 17125 df-base 17141 df-ress 17162 df-plusg 17194 df-mulr 17195 df-starv 17196 df-sca 17197 df-vsca 17198 df-ip 17199 df-tset 17200 df-ple 17201 df-ds 17203 df-unif 17204 df-hom 17205 df-cco 17206 df-rest 17346 df-topn 17347 df-0g 17365 df-gsum 17366 df-topgen 17367 df-pt 17368 df-prds 17371 df-ordt 17426 df-xrs 17427 df-qtop 17432 df-imas 17433 df-xps 17435 df-mre 17509 df-mrc 17510 df-acs 17512 df-ps 18493 df-tsr 18494 df-plusf 18568 df-mgm 18569 df-sgrp 18648 df-mnd 18664 df-mhm 18712 df-submnd 18713 df-grp 18870 df-minusg 18871 df-sbg 18872 df-mulg 19002 df-subg 19057 df-cntz 19250 df-cmn 19715 df-abl 19716 df-mgp 20080 df-rng 20092 df-ur 20121 df-ring 20174 df-cring 20175 df-subrng 20483 df-subrg 20507 df-abv 20746 df-lmod 20817 df-scaf 20818 df-sra 21129 df-rgmod 21130 df-psmet 21305 df-xmet 21306 df-met 21307 df-bl 21308 df-mopn 21309 df-fbas 21310 df-fg 21311 df-cnfld 21314 df-top 22842 df-topon 22859 df-topsp 22881 df-bases 22894 df-cld 22967 df-ntr 22968 df-cls 22969 df-nei 23046 df-lp 23084 df-perf 23085 df-cn 23175 df-cnp 23176 df-haus 23263 df-tx 23510 df-hmeo 23703 df-fil 23794 df-fm 23886 df-flim 23887 df-flf 23888 df-tmd 24020 df-tgp 24021 df-tsms 24075 df-trg 24108 df-xms 24268 df-ms 24269 df-tms 24270 df-nm 24530 df-ngp 24531 df-nrg 24533 df-nlm 24534 df-ii 24830 df-cncf 24831 df-ovol 25425 df-vol 25426 df-limc 25827 df-dv 25828 df-log 26525 df-esum 34187

This theorem is referenced by: volmeas 34390

W3C validator