voliune - Mathbox for Thierry Arnoux

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Theorem voliune 34628

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 25675 and voliun 25724. (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 3124	. . . . 5 ⊢ (∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ↔ (∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ))
2		eqid 2762	. . . . . 6 ⊢ seq1( + , (𝑛 ∈ ℕ ↦ (vol‘𝐴))) = seq1( + , (𝑛 ∈ ℕ ↦ (vol‘𝐴)))
3		eqid 2762	. . . . . 6 ⊢ (𝑛 ∈ ℕ ↦ (vol‘𝐴)) = (𝑛 ∈ ℕ ↦ (vol‘𝐴))
4	2, 3	voliun 25724	. . . . 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 593	. . . 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 664	. . 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 3288	. . . . . . 7 ⊢ Ⅎ𝑛∀𝑛 ∈ ℕ 𝐴 ∈ dom vol
8		nfra1 3288	. . . . . . 7 ⊢ Ⅎ𝑛∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ
9	7, 8	nfan 1928	. . . . . 6 ⊢ Ⅎ𝑛(∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ)
10		simpr 489	. . . . . . . . . 10 ⊢ ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
11		rspa 3253	. . . . . . . . . . 11 ⊢ ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ 𝑛 ∈ ℕ) → 𝐴 ∈ dom vol)
12		volf 25699	. . . . . . . . . . . 12 ⊢ vol:dom vol⟶(0[,]+∞)
13	12	ffvelcdmi 7078	. . . . . . . . . . 11 ⊢ (𝐴 ∈ dom vol → (vol‘𝐴) ∈ (0[,]+∞))
14	11, 13	syl 18	. . . . . . . . . 10 ⊢ ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ 𝑛 ∈ ℕ) → (vol‘𝐴) ∈ (0[,]+∞))
15	3	fvmpt2 7001	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ (vol‘𝐴) ∈ (0[,]+∞)) → ((𝑛 ∈ ℕ ↦ (vol‘𝐴))‘𝑛) = (vol‘𝐴))
16	10, 14, 15	syl2anc 595	. . . . . . . . 9 ⊢ ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ 𝑛 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (vol‘𝐴))‘𝑛) = (vol‘𝐴))
17	16	adantlr 727	. . . . . . . 8 ⊢ (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (vol‘𝐴))‘𝑛) = (vol‘𝐴))
18	17	ex 417	. . . . . . 7 ⊢ ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ) → (𝑛 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (vol‘𝐴))‘𝑛) = (vol‘𝐴)))
19	9, 18	ralrimi 3262	. . . . . 6 ⊢ ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ) → ∀𝑛 ∈ ℕ ((𝑛 ∈ ℕ ↦ (vol‘𝐴))‘𝑛) = (vol‘𝐴))
20	9, 19	esumeq2d 34436	. . . . 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 489	. . . . . . . . 9 ⊢ ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ) → ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ)
22	21	r19.21bi 3256	. . . . . . . 8 ⊢ (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (vol‘𝐴) ∈ ℝ)
23	14	adantlr 727	. . . . . . . . 9 ⊢ (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (vol‘𝐴) ∈ (0[,]+∞))
24		0xr 11262	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ^*
25		pnfxr 11269	. . . . . . . . . . 11 ⊢ +∞ ∈ ℝ^*
26		elicc1 13422	. . . . . . . . . . 11 ⊢ ((0 ∈ ℝ^* ∧ +∞ ∈ ℝ^) → ((vol‘𝐴) ∈ (0[,]+∞) ↔ ((vol‘𝐴) ∈ ℝ^ ∧ 0 ≤ (vol‘𝐴) ∧ (vol‘𝐴) ≤ +∞)))
27	24, 25, 26	mp2an 704	. . . . . . . . . 10 ⊢ ((vol‘𝐴) ∈ (0[,]+∞) ↔ ((vol‘𝐴) ∈ ℝ^* ∧ 0 ≤ (vol‘𝐴) ∧ (vol‘𝐴) ≤ +∞))
28	27	simp2bi 1163	. . . . . . . . 9 ⊢ ((vol‘𝐴) ∈ (0[,]+∞) → 0 ≤ (vol‘𝐴))
29	23, 28	syl 18	. . . . . . . 8 ⊢ (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → 0 ≤ (vol‘𝐴))
30		ltpnf 13151	. . . . . . . . 9 ⊢ ((vol‘𝐴) ∈ ℝ → (vol‘𝐴) < +∞)
31	22, 30	syl 18	. . . . . . . 8 ⊢ (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (vol‘𝐴) < +∞)
32		0re 11216	. . . . . . . . 9 ⊢ 0 ∈ ℝ
33		elico2 13443	. . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ +∞ ∈ ℝ^*) → ((vol‘𝐴) ∈ (0[,)+∞) ↔ ((vol‘𝐴) ∈ ℝ ∧ 0 ≤ (vol‘𝐴) ∧ (vol‘𝐴) < +∞)))
34	32, 25, 33	mp2an 704	. . . . . . . 8 ⊢ ((vol‘𝐴) ∈ (0[,)+∞) ↔ ((vol‘𝐴) ∈ ℝ ∧ 0 ≤ (vol‘𝐴) ∧ (vol‘𝐴) < +∞))
35	22, 29, 31, 34	syl3anbrc 1361	. . . . . . 7 ⊢ (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (vol‘𝐴) ∈ (0[,)+∞))
36	9, 35, 3	fmptdf 7112	. . . . . 6 ⊢ ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ) → (𝑛 ∈ ℕ ↦ (vol‘𝐴)):ℕ⟶(0[,)+∞))
37		nfmpt1 5209	. . . . . . 7 ⊢ Ⅎ𝑛(𝑛 ∈ ℕ ↦ (vol‘𝐴))
38	37	esumfsupre 34470	. . . . . 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 18	. . . . 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 2799	. . . 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 727	. . 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 2800	. 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		nfv 1943	. . . . . . . . 9 ⊢ Ⅎ𝑘(vol‘𝐴) = +∞
44		nfcv 2924	. . . . . . . . . . 11 ⊢ Ⅎ𝑛vol
45		nfcsb1v 3876	. . . . . . . . . . 11 ⊢ Ⅎ𝑛⦋𝑘 / 𝑛⦌𝐴
46	44, 45	nffv 6891	. . . . . . . . . 10 ⊢ Ⅎ𝑛(vol‘⦋𝑘 / 𝑛⦌𝐴)
47	46	nfeq1 2939	. . . . . . . . 9 ⊢ Ⅎ𝑛(vol‘⦋𝑘 / 𝑛⦌𝐴) = +∞
48		csbeq1a 3866	. . . . . . . . . 10 ⊢ (𝑛 = 𝑘 → 𝐴 = ⦋𝑘 / 𝑛⦌𝐴)
49	48	fveqeq2d 6889	. . . . . . . . 9 ⊢ (𝑛 = 𝑘 → ((vol‘𝐴) = +∞ ↔ (vol‘⦋𝑘 / 𝑛⦌𝐴) = +∞))
50	43, 47, 49	cbvrexw 3307	. . . . . . . 8 ⊢ (∃𝑛 ∈ ℕ (vol‘𝐴) = +∞ ↔ ∃𝑘 ∈ ℕ (vol‘⦋𝑘 / 𝑛⦌𝐴) = +∞)
51	50	bilani 509	. . . . . . 7 ⊢ (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) ∧ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞) → ∃𝑘 ∈ ℕ (vol‘⦋𝑘 / 𝑛⦌𝐴) = +∞)
52	45	nfel1 2940	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛⦋𝑘 / 𝑛⦌𝐴 ∈ dom vol
53	48	eleq1d 2847	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑘 → (𝐴 ∈ dom vol ↔ ⦋𝑘 / 𝑛⦌𝐴 ∈ dom vol))
54	52, 53	rspc 3568	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → (∀𝑛 ∈ ℕ 𝐴 ∈ dom vol → ⦋𝑘 / 𝑛⦌𝐴 ∈ dom vol))
55	54	impcom 412	. . . . . . . . . . 11 ⊢ ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ 𝑘 ∈ ℕ) → ⦋𝑘 / 𝑛⦌𝐴 ∈ dom vol)
56		iunmbl 25723	. . . . . . . . . . . 12 ⊢ (∀𝑛 ∈ ℕ 𝐴 ∈ dom vol → ∪ 𝑛 ∈ ℕ 𝐴 ∈ dom vol)
57	56	adantr 485	. . . . . . . . . . 11 ⊢ ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ 𝑘 ∈ ℕ) → ∪ 𝑛 ∈ ℕ 𝐴 ∈ dom vol)
58		nfcv 2924	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛ℕ
59		nfcv 2924	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛𝑘
60	58, 59, 45, 48	ssiun2sf 32915	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → ⦋𝑘 / 𝑛⦌𝐴 ⊆ ∪ 𝑛 ∈ ℕ 𝐴)
61	60	adantl 486	. . . . . . . . . . 11 ⊢ ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ 𝑘 ∈ ℕ) → ⦋𝑘 / 𝑛⦌𝐴 ⊆ ∪ 𝑛 ∈ ℕ 𝐴)
62		volss 25703	. . . . . . . . . . 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))
63	55, 57, 61, 62	syl3anc 1397	. . . . . . . . . 10 ⊢ ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ 𝑘 ∈ ℕ) → (vol‘⦋𝑘 / 𝑛⦌𝐴) ≤ (vol‘∪ 𝑛 ∈ ℕ 𝐴))
64	63	adantlr 727	. . . . . . . . 9 ⊢ (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) ∧ 𝑘 ∈ ℕ) → (vol‘⦋𝑘 / 𝑛⦌𝐴) ≤ (vol‘∪ 𝑛 ∈ ℕ 𝐴))
65	64	adantlr 727	. . . . . . . 8 ⊢ ((((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) ∧ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞) ∧ 𝑘 ∈ ℕ) → (vol‘⦋𝑘 / 𝑛⦌𝐴) ≤ (vol‘∪ 𝑛 ∈ ℕ 𝐴))
66	65	ralrimiva 3156	. . . . . . 7 ⊢ (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) ∧ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞) → ∀𝑘 ∈ ℕ (vol‘⦋𝑘 / 𝑛⦌𝐴) ≤ (vol‘∪ 𝑛 ∈ ℕ 𝐴))
67		r19.29r 3128	. . . . . . 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)))
68	51, 66, 67	syl2anc 595	. . . . . 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)))
69		breq1 5111	. . . . . . . 8 ⊢ ((vol‘⦋𝑘 / 𝑛⦌𝐴) = +∞ → ((vol‘⦋𝑘 / 𝑛⦌𝐴) ≤ (vol‘∪ 𝑛 ∈ ℕ 𝐴) ↔ +∞ ≤ (vol‘∪ 𝑛 ∈ ℕ 𝐴)))
70	69	biimpa 481	. . . . . . 7 ⊢ (((vol‘⦋𝑘 / 𝑛⦌𝐴) = +∞ ∧ (vol‘⦋𝑘 / 𝑛⦌𝐴) ≤ (vol‘∪ 𝑛 ∈ ℕ 𝐴)) → +∞ ≤ (vol‘∪ 𝑛 ∈ ℕ 𝐴))
71	70	reximi 3102	. . . . . 6 ⊢ (∃𝑘 ∈ ℕ ((vol‘⦋𝑘 / 𝑛⦌𝐴) = +∞ ∧ (vol‘⦋𝑘 / 𝑛⦌𝐴) ≤ (vol‘∪ 𝑛 ∈ ℕ 𝐴)) → ∃𝑘 ∈ ℕ +∞ ≤ (vol‘∪ 𝑛 ∈ ℕ 𝐴))
72	68, 71	syl 18	. . . . 5 ⊢ (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) ∧ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞) → ∃𝑘 ∈ ℕ +∞ ≤ (vol‘∪ 𝑛 ∈ ℕ 𝐴))
73		1nn 12250	. . . . . 6 ⊢ 1 ∈ ℕ
74		ne0i 4293	. . . . . 6 ⊢ (1 ∈ ℕ → ℕ ≠ ∅)
75		r19.9rzv 4465	. . . . . 6 ⊢ (ℕ ≠ ∅ → (+∞ ≤ (vol‘∪ 𝑛 ∈ ℕ 𝐴) ↔ ∃𝑘 ∈ ℕ +∞ ≤ (vol‘∪ 𝑛 ∈ ℕ 𝐴)))
76	73, 74, 75	mp2b 10	. . . . 5 ⊢ (+∞ ≤ (vol‘∪ 𝑛 ∈ ℕ 𝐴) ↔ ∃𝑘 ∈ ℕ +∞ ≤ (vol‘∪ 𝑛 ∈ ℕ 𝐴))
77	72, 76	sylibr 237	. . . 4 ⊢ (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) ∧ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞) → +∞ ≤ (vol‘∪ 𝑛 ∈ ℕ 𝐴))
78		iccssxr 13463	. . . . . . . 8 ⊢ (0[,]+∞) ⊆ ℝ^*
79	12	ffvelcdmi 7078	. . . . . . . 8 ⊢ (∪ 𝑛 ∈ ℕ 𝐴 ∈ dom vol → (vol‘∪ 𝑛 ∈ ℕ 𝐴) ∈ (0[,]+∞))
80	78, 79	sselid 3934	. . . . . . 7 ⊢ (∪ 𝑛 ∈ ℕ 𝐴 ∈ dom vol → (vol‘∪ 𝑛 ∈ ℕ 𝐴) ∈ ℝ^*)
81	56, 80	syl 18	. . . . . 6 ⊢ (∀𝑛 ∈ ℕ 𝐴 ∈ dom vol → (vol‘∪ 𝑛 ∈ ℕ 𝐴) ∈ ℝ^*)
82	81	ad2antrr 738	. . . . 5 ⊢ (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) ∧ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞) → (vol‘∪ 𝑛 ∈ ℕ 𝐴) ∈ ℝ^*)
83		xgepnf 13197	. . . . 5 ⊢ ((vol‘∪ 𝑛 ∈ ℕ 𝐴) ∈ ℝ^* → (+∞ ≤ (vol‘∪ 𝑛 ∈ ℕ 𝐴) ↔ (vol‘∪ 𝑛 ∈ ℕ 𝐴) = +∞))
84	82, 83	syl 18	. . . 4 ⊢ (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) ∧ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞) → (+∞ ≤ (vol‘∪ 𝑛 ∈ ℕ 𝐴) ↔ (vol‘∪ 𝑛 ∈ ℕ 𝐴) = +∞))
85	77, 84	mpbid 235	. . 3 ⊢ (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) ∧ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞) → (vol‘∪ 𝑛 ∈ ℕ 𝐴) = +∞)
86		nfdisj1 5089	. . . . . 6 ⊢ Ⅎ𝑛Disj 𝑛 ∈ ℕ 𝐴
87	7, 86	nfan 1928	. . . . 5 ⊢ Ⅎ𝑛(∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴)
88		nfre1 3289	. . . . 5 ⊢ Ⅎ𝑛∃𝑛 ∈ ℕ (vol‘𝐴) = +∞
89	87, 88	nfan 1928	. . . 4 ⊢ Ⅎ𝑛((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) ∧ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞)
90		nnex 12245	. . . . 5 ⊢ ℕ ∈ V
91	90	a1i 11	. . . 4 ⊢ (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) ∧ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞) → ℕ ∈ V)
92	14	3ad2antr3 1208	. . . . 5 ⊢ ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ (Disj 𝑛 ∈ ℕ 𝐴 ∧ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞ ∧ 𝑛 ∈ ℕ)) → (vol‘𝐴) ∈ (0[,]+∞))
93	92	3anassrs 1380	. . . 4 ⊢ ((((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) ∧ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞) ∧ 𝑛 ∈ ℕ) → (vol‘𝐴) ∈ (0[,]+∞))
94		simpr 489	. . . 4 ⊢ (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) ∧ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞) → ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞)
95	89, 91, 93, 94	esumpinfval 34472	. . 3 ⊢ (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) ∧ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞) → Σ^*𝑛 ∈ ℕ(vol‘𝐴) = +∞)
96	85, 95	eqtr4d 2800	. 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 907	. . . . 5 ⊢ (∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ ∨ ¬ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ)
98		rexnal 3116	. . . . . 6 ⊢ (∃𝑛 ∈ ℕ ¬ (vol‘𝐴) ∈ ℝ ↔ ¬ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ)
99	98	orbi2i 925	. . . . 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 3127	. . . . . . 7 ⊢ ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∃𝑛 ∈ ℕ ¬ (vol‘𝐴) ∈ ℝ) → ∃𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ ¬ (vol‘𝐴) ∈ ℝ))
102		xrge0nre 13486	. . . . . . . . 9 ⊢ (((vol‘𝐴) ∈ (0[,]+∞) ∧ ¬ (vol‘𝐴) ∈ ℝ) → (vol‘𝐴) = +∞)
103	13, 102	sylan 591	. . . . . . . 8 ⊢ ((𝐴 ∈ dom vol ∧ ¬ (vol‘𝐴) ∈ ℝ) → (vol‘𝐴) = +∞)
104	103	reximi 3102	. . . . . . 7 ⊢ (∃𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ ¬ (vol‘𝐴) ∈ ℝ) → ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞)
105	101, 104	syl 18	. . . . . 6 ⊢ ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∃𝑛 ∈ ℕ ¬ (vol‘𝐴) ∈ ℝ) → ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞)
106	105	ex 417	. . . . 5 ⊢ (∀𝑛 ∈ ℕ 𝐴 ∈ dom vol → (∃𝑛 ∈ ℕ ¬ (vol‘𝐴) ∈ ℝ → ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞))
107	106	orim2d 981	. . . 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 21	. . 3 ⊢ (∀𝑛 ∈ ℕ 𝐴 ∈ dom vol → (∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ ∨ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞))
109	108	adantr 485	. 2 ⊢ ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) → (∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ ∨ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞))
110	42, 96, 109	mpjaodan 972	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

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 ∨ wo 860 ∧ w3a 1102 = wceq 1569 ∈ wcel 2142 ≠ wne 2957 ∀wral 3078 ∃wrex 3088 Vcvv 3454 ⦋csb 3852 ⊆ wss 3904 ∅c0 4285 ∪ ciun 4955 Disj wdisj 5075 class class class wbr 5108 ↦ cmpt 5191 dom cdm 5660 ran crn 5661 ⟶wf 6532 ‘cfv 6536 (class class class)co 7412 supcsup 9398 ℝcr 11105 0cc0 11106 1c1 11107 + caddc 11109 +∞cpnf 11246 ℝ^*cxr 11248 < clt 11249 ≤ cle 11250 ℕcn 12239 [,)cico 13380 [,]cicc 13381 seqcseq 14044 volcvol 25633 Σ^*cesum 34426

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1824 ax-4 1838 ax-5 1939 ax-6 1996 ax-7 2037 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5237 ax-sep 5256 ax-nul 5268 ax-pow 5335 ax-pr 5403 ax-un 7734 ax-inf2 9608 ax-cc 10425 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185 ax-mulf 11186

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1103 df-3an 1104 df-tru 1572 df-fal 1582 df-ex 1809 df-nf 1813 df-sb 2096 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3368 df-reu 3369 df-rab 3416 df-v 3456 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4595 df-uni 4872 df-int 4912 df-iun 4957 df-iin 4958 df-disj 5076 df-br 5109 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5555 df-eprel 5560 df-po 5568 df-so 5569 df-fr 5613 df-se 5614 df-we 5615 df-xp 5666 df-rel 5667 df-cnv 5668 df-co 5669 df-dm 5670 df-rn 5671 df-res 5672 df-ima 5673 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7676 df-om 7861 df-1st 7984 df-2nd 7985 df-supp 8155 df-frecs 8276 df-wrecs 8307 df-recs 8356 df-rdg 8395 df-1o 8451 df-2o 8452 df-oadd 8455 df-er 8692 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9320 df-fi 9369 df-sup 9400 df-inf 9401 df-oi 9470 df-dju 9894 df-card 9932 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11449 df-neg 11450 df-div 11878 df-nn 12240 df-2 12309 df-3 12310 df-4 12311 df-5 12312 df-6 12313 df-7 12314 df-8 12315 df-9 12316 df-n0 12511 df-xnn0 12584 df-z 12598 df-dec 12718 df-uz 12869 df-q 12979 df-rp 13023 df-xneg 13143 df-xadd 13144 df-xmul 13145 df-ioo 13382 df-ioc 13383 df-ico 13384 df-icc 13385 df-fz 13542 df-fzo 13690 df-fl 13832 df-mod 13910 df-seq 14045 df-exp 14105 df-fac 14317 df-bc 14346 df-hash 14374 df-shft 15111 df-cj 15157 df-re 15158 df-im 15159 df-sqrt 15293 df-abs 15294 df-limsup 15529 df-clim 15546 df-rlim 15547 df-sum 15745 df-ef 16127 df-sin 16129 df-cos 16130 df-pi 16132 df-struct 17213 df-sets 17230 df-slot 17248 df-ndx 17260 df-base 17276 df-ress 17297 df-plusg 17329 df-mulr 17330 df-starv 17331 df-sca 17332 df-vsca 17333 df-ip 17334 df-tset 17335 df-ple 17336 df-ds 17338 df-unif 17339 df-hom 17340 df-cco 17341 df-rest 17481 df-topn 17482 df-0g 17500 df-gsum 17501 df-topgen 17502 df-pt 17503 df-prds 17506 df-ordt 17561 df-xrs 17562 df-qtop 17567 df-imas 17568 df-xps 17570 df-mre 17644 df-mrc 17645 df-acs 17647 df-ps 18628 df-tsr 18629 df-plusf 18703 df-mgm 18704 df-sgrp 18783 df-mnd 18799 df-mhm 18847 df-submnd 18848 df-grp 19009 df-minusg 19010 df-sbg 19011 df-mulg 19140 df-subg 19195 df-cntz 19393 df-cmn 19858 df-abl 19859 df-mgp 20223 df-rng 20237 df-ur 20270 df-ring 20323 df-cring 20324 df-subrng 20656 df-subrg 20680 df-abv 20923 df-lmod 20994 df-scaf 20995 df-sra 21305 df-rgmod 21306 df-psmet 21525 df-xmet 21526 df-met 21527 df-bl 21528 df-mopn 21529 df-fbas 21530 df-fg 21531 df-cnfld 21534 df-top 23062 df-topon 23079 df-topsp 23101 df-bases 23114 df-cld 23187 df-ntr 23188 df-cls 23189 df-nei 23266 df-lp 23304 df-perf 23305 df-cn 23395 df-cnp 23396 df-haus 23483 df-tx 23730 df-hmeo 23923 df-fil 24014 df-fm 24106 df-flim 24107 df-flf 24108 df-tmd 24240 df-tgp 24241 df-tsms 24295 df-trg 24328 df-xms 24488 df-ms 24489 df-tms 24490 df-nm 24750 df-ngp 24751 df-nrg 24753 df-nlm 24754 df-ii 25047 df-cncf 25048 df-ovol 25634 df-vol 25635 df-limc 26036 df-dv 26037 df-log 26732 df-esum 34427

This theorem is used by: volmeas 34630

W3C validator