Theorem ovoliunnul 25843

Description: A countable union of nullsets is null. (Contributed by Mario Carneiro, 8-Apr-2015.)

Assertion

Ref	Expression
ovoliunnul	⊢ ((𝐴 ≼ ℕ ∧ ∀𝑛 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) → (vol*‘∪ 𝑛 ∈ 𝐴 𝐵) = 0)

Distinct variable group:   𝐴,𝑛

Allowed substitution hint:   𝐵(𝑛)

Proof of Theorem ovoliunnul

Dummy variables 𝑓 𝑘 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		iuneq1 4971	. . . . . 6 ⊢ (𝐴 = ∅ → ∪ 𝑛 ∈ 𝐴 𝐵 = ∪ 𝑛 ∈ ∅ 𝐵)
2		0iun 5024	. . . . . 6 ⊢ ∪ 𝑛 ∈ ∅ 𝐵 = ∅
3	1, 2	eqtrdi 2814	. . . . 5 ⊢ (𝐴 = ∅ → ∪ 𝑛 ∈ 𝐴 𝐵 = ∅)
4	3	fveq2d 6900	. . . 4 ⊢ (𝐴 = ∅ → (vol*‘∪ 𝑛 ∈ 𝐴 𝐵) = (vol*‘∅))
5		ovol0 25829	. . . 4 ⊢ (vol*‘∅) = 0
6	4, 5	eqtrdi 2814	. . 3 ⊢ (𝐴 = ∅ → (vol*‘∪ 𝑛 ∈ 𝐴 𝐵) = 0)
7	6	a1i 11	. 2 ⊢ ((𝐴 ≼ ℕ ∧ ∀𝑛 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) → (𝐴 = ∅ → (vol*‘∪ 𝑛 ∈ 𝐴 𝐵) = 0))
8		reldom 8993	. . . . . 6 ⊢ Rel ≼
9	8	brrelex1i 5711	. . . . 5 ⊢ (𝐴 ≼ ℕ → 𝐴 ∈ V)
10	9	adantr 486	. . . 4 ⊢ ((𝐴 ≼ ℕ ∧ ∀𝑛 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) → 𝐴 ∈ V)
11		0sdomg 9139	. . . 4 ⊢ (𝐴 ∈ V → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅))
12	10, 11	syl 18	. . 3 ⊢ ((𝐴 ≼ ℕ ∧ ∀𝑛 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅))
13		fodomr 9161	. . . . . 6 ⊢ ((∅ ≺ 𝐴 ∧ 𝐴 ≼ ℕ) → ∃𝑓 𝑓:ℕ–onto→𝐴)
14	13	expcom 419	. . . . 5 ⊢ (𝐴 ≼ ℕ → (∅ ≺ 𝐴 → ∃𝑓 𝑓:ℕ–onto→𝐴))
15	14	adantr 486	. . . 4 ⊢ ((𝐴 ≼ ℕ ∧ ∀𝑛 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) → (∅ ≺ 𝐴 → ∃𝑓 𝑓:ℕ–onto→𝐴))
16		eliun 4958	. . . . . . . . . 10 ⊢ (𝑥 ∈ ∪ 𝑛 ∈ 𝐴 𝐵 ↔ ∃𝑛 ∈ 𝐴 𝑥 ∈ 𝐵)
17		nfv 1948	. . . . . . . . . . 11 ⊢ Ⅎ𝑛 𝑓:ℕ–onto→𝐴
18		nfcv 2926	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛ℕ
19		nfcsb1v 3875	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛⦋(𝑓‘𝑘) / 𝑛⦌𝐵
20	18, 19	nfiun 4985	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛∪ 𝑘 ∈ ℕ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵
21	20	nfcri 2918	. . . . . . . . . . 11 ⊢ Ⅎ𝑛 𝑥 ∈ ∪ 𝑘 ∈ ℕ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵
22		foelrn 7118	. . . . . . . . . . . . 13 ⊢ ((𝑓:ℕ–onto→𝐴 ∧ 𝑛 ∈ 𝐴) → ∃𝑘 ∈ ℕ 𝑛 = (𝑓‘𝑘))
23	22	ex 418	. . . . . . . . . . . 12 ⊢ (𝑓:ℕ–onto→𝐴 → (𝑛 ∈ 𝐴 → ∃𝑘 ∈ ℕ 𝑛 = (𝑓‘𝑘)))
24		csbeq1a 3865	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = (𝑓‘𝑘) → 𝐵 = ⦋(𝑓‘𝑘) / 𝑛⦌𝐵)
25	24	adantl 487	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓:ℕ–onto→𝐴 ∧ 𝑛 = (𝑓‘𝑘)) → 𝐵 = ⦋(𝑓‘𝑘) / 𝑛⦌𝐵)
26	25	eleq2d 2850	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓:ℕ–onto→𝐴 ∧ 𝑛 = (𝑓‘𝑘)) → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵))
27	26	biimpd 232	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓:ℕ–onto→𝐴 ∧ 𝑛 = (𝑓‘𝑘)) → (𝑥 ∈ 𝐵 → 𝑥 ∈ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵))
28	27	impancom 457	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓:ℕ–onto→𝐴 ∧ 𝑥 ∈ 𝐵) → (𝑛 = (𝑓‘𝑘) → 𝑥 ∈ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵))
29	28	reximdv 3181	. . . . . . . . . . . . . . 15 ⊢ ((𝑓:ℕ–onto→𝐴 ∧ 𝑥 ∈ 𝐵) → (∃𝑘 ∈ ℕ 𝑛 = (𝑓‘𝑘) → ∃𝑘 ∈ ℕ 𝑥 ∈ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵))
30		eliun 4958	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ∪ 𝑘 ∈ ℕ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵 ↔ ∃𝑘 ∈ ℕ 𝑥 ∈ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵)
31	29, 30	imbitrrdi 255	. . . . . . . . . . . . . 14 ⊢ ((𝑓:ℕ–onto→𝐴 ∧ 𝑥 ∈ 𝐵) → (∃𝑘 ∈ ℕ 𝑛 = (𝑓‘𝑘) → 𝑥 ∈ ∪ 𝑘 ∈ ℕ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵))
32	31	ex 418	. . . . . . . . . . . . 13 ⊢ (𝑓:ℕ–onto→𝐴 → (𝑥 ∈ 𝐵 → (∃𝑘 ∈ ℕ 𝑛 = (𝑓‘𝑘) → 𝑥 ∈ ∪ 𝑘 ∈ ℕ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵)))
33	32	com23 87	. . . . . . . . . . . 12 ⊢ (𝑓:ℕ–onto→𝐴 → (∃𝑘 ∈ ℕ 𝑛 = (𝑓‘𝑘) → (𝑥 ∈ 𝐵 → 𝑥 ∈ ∪ 𝑘 ∈ ℕ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵)))
34	23, 33	syld 48	. . . . . . . . . . 11 ⊢ (𝑓:ℕ–onto→𝐴 → (𝑛 ∈ 𝐴 → (𝑥 ∈ 𝐵 → 𝑥 ∈ ∪ 𝑘 ∈ ℕ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵)))
35	17, 21, 34	rexlimd 3273	. . . . . . . . . 10 ⊢ (𝑓:ℕ–onto→𝐴 → (∃𝑛 ∈ 𝐴 𝑥 ∈ 𝐵 → 𝑥 ∈ ∪ 𝑘 ∈ ℕ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵))
36	16, 35	biimtrid 245	. . . . . . . . 9 ⊢ (𝑓:ℕ–onto→𝐴 → (𝑥 ∈ ∪ 𝑛 ∈ 𝐴 𝐵 → 𝑥 ∈ ∪ 𝑘 ∈ ℕ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵))
37	36	ssrdv 3919	. . . . . . . 8 ⊢ (𝑓:ℕ–onto→𝐴 → ∪ 𝑛 ∈ 𝐴 𝐵 ⊆ ∪ 𝑘 ∈ ℕ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵)
38	37	adantl 487	. . . . . . 7 ⊢ (((𝐴 ≼ ℕ ∧ ∀𝑛 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto→𝐴) → ∪ 𝑛 ∈ 𝐴 𝐵 ⊆ ∪ 𝑘 ∈ ℕ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵)
39		fof 6807	. . . . . . . . . . . . 13 ⊢ (𝑓:ℕ–onto→𝐴 → 𝑓:ℕ⟶𝐴)
40	39	adantl 487	. . . . . . . . . . . 12 ⊢ (((𝐴 ≼ ℕ ∧ ∀𝑛 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto→𝐴) → 𝑓:ℕ⟶𝐴)
41	40	ffvelcdmda 7095	. . . . . . . . . . 11 ⊢ ((((𝐴 ≼ ℕ ∧ ∀𝑛 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto→𝐴) ∧ 𝑘 ∈ ℕ) → (𝑓‘𝑘) ∈ 𝐴)
42		simpllr 788	. . . . . . . . . . 11 ⊢ ((((𝐴 ≼ ℕ ∧ ∀𝑛 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto→𝐴) ∧ 𝑘 ∈ ℕ) → ∀𝑛 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0))
43		nfcv 2926	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛ℝ
44	19, 43	nfss 3906	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛⦋(𝑓‘𝑘) / 𝑛⦌𝐵 ⊆ ℝ
45		nfcv 2926	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛vol*
46	45, 19	nffv 6906	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛(vol*‘⦋(𝑓‘𝑘) / 𝑛⦌𝐵)
47	46	nfeq1 2941	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛(vol*‘⦋(𝑓‘𝑘) / 𝑛⦌𝐵) = 0
48	44, 47	nfan 1933	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛(⦋(𝑓‘𝑘) / 𝑛⦌𝐵 ⊆ ℝ ∧ (vol*‘⦋(𝑓‘𝑘) / 𝑛⦌𝐵) = 0)
49	24	sseq1d 3944	. . . . . . . . . . . . 13 ⊢ (𝑛 = (𝑓‘𝑘) → (𝐵 ⊆ ℝ ↔ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵 ⊆ ℝ))
50	24	fveqeq2d 6904	. . . . . . . . . . . . 13 ⊢ (𝑛 = (𝑓‘𝑘) → ((vol*‘𝐵) = 0 ↔ (vol*‘⦋(𝑓‘𝑘) / 𝑛⦌𝐵) = 0))
51	49, 50	anbi12d 644	. . . . . . . . . . . 12 ⊢ (𝑛 = (𝑓‘𝑘) → ((𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ↔ (⦋(𝑓‘𝑘) / 𝑛⦌𝐵 ⊆ ℝ ∧ (vol*‘⦋(𝑓‘𝑘) / 𝑛⦌𝐵) = 0)))
52	48, 51	rspc 3568	. . . . . . . . . . 11 ⊢ ((𝑓‘𝑘) ∈ 𝐴 → (∀𝑛 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → (⦋(𝑓‘𝑘) / 𝑛⦌𝐵 ⊆ ℝ ∧ (vol*‘⦋(𝑓‘𝑘) / 𝑛⦌𝐵) = 0)))
53	41, 42, 52	sylc 66	. . . . . . . . . 10 ⊢ ((((𝐴 ≼ ℕ ∧ ∀𝑛 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto→𝐴) ∧ 𝑘 ∈ ℕ) → (⦋(𝑓‘𝑘) / 𝑛⦌𝐵 ⊆ ℝ ∧ (vol*‘⦋(𝑓‘𝑘) / 𝑛⦌𝐵) = 0))
54	53	simpld 500	. . . . . . . . 9 ⊢ ((((𝐴 ≼ ℕ ∧ ∀𝑛 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto→𝐴) ∧ 𝑘 ∈ ℕ) → ⦋(𝑓‘𝑘) / 𝑛⦌𝐵 ⊆ ℝ)
55	54	ralrimiva 3158	. . . . . . . 8 ⊢ (((𝐴 ≼ ℕ ∧ ∀𝑛 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto→𝐴) → ∀𝑘 ∈ ℕ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵 ⊆ ℝ)
56		iunss 5006	. . . . . . . 8 ⊢ (∪ 𝑘 ∈ ℕ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵 ⊆ ℝ ↔ ∀𝑘 ∈ ℕ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵 ⊆ ℝ)
57	55, 56	sylibr 237	. . . . . . 7 ⊢ (((𝐴 ≼ ℕ ∧ ∀𝑛 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto→𝐴) → ∪ 𝑘 ∈ ℕ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵 ⊆ ℝ)
58		eqid 2763	. . . . . . . . . 10 ⊢ seq1( + , (𝑘 ∈ ℕ ↦ (vol*‘⦋(𝑓‘𝑘) / 𝑛⦌𝐵))) = seq1( + , (𝑘 ∈ ℕ ↦ (vol*‘⦋(𝑓‘𝑘) / 𝑛⦌𝐵)))
59		eqid 2763	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ ↦ (vol*‘⦋(𝑓‘𝑘) / 𝑛⦌𝐵)) = (𝑘 ∈ ℕ ↦ (vol*‘⦋(𝑓‘𝑘) / 𝑛⦌𝐵))
60	53	simprd 501	. . . . . . . . . . 11 ⊢ ((((𝐴 ≼ ℕ ∧ ∀𝑛 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto→𝐴) ∧ 𝑘 ∈ ℕ) → (vol*‘⦋(𝑓‘𝑘) / 𝑛⦌𝐵) = 0)
61		0re 11323	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ
62	60, 61	eqeltrdi 2872	. . . . . . . . . 10 ⊢ ((((𝐴 ≼ ℕ ∧ ∀𝑛 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto→𝐴) ∧ 𝑘 ∈ ℕ) → (vol*‘⦋(𝑓‘𝑘) / 𝑛⦌𝐵) ∈ ℝ)
63	60	mpteq2dva 5201	. . . . . . . . . . . . 13 ⊢ (((𝐴 ≼ ℕ ∧ ∀𝑛 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto→𝐴) → (𝑘 ∈ ℕ ↦ (vol*‘⦋(𝑓‘𝑘) / 𝑛⦌𝐵)) = (𝑘 ∈ ℕ ↦ 0))
64		fconstmpt 5717	. . . . . . . . . . . . . 14 ⊢ (ℕ × {0}) = (𝑘 ∈ ℕ ↦ 0)
65		nnuz 13017	. . . . . . . . . . . . . . 15 ⊢ ℕ = (ℤ≥‘1)
66	65	xpeq1i 5681	. . . . . . . . . . . . . 14 ⊢ (ℕ × {0}) = ((ℤ≥‘1) × {0})
67	64, 66	eqtr3i 2788	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ ↦ 0) = ((ℤ≥‘1) × {0})
68	63, 67	eqtrdi 2814	. . . . . . . . . . . 12 ⊢ (((𝐴 ≼ ℕ ∧ ∀𝑛 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto→𝐴) → (𝑘 ∈ ℕ ↦ (vol*‘⦋(𝑓‘𝑘) / 𝑛⦌𝐵)) = ((ℤ≥‘1) × {0}))
69	68	seqeq3d 14165	. . . . . . . . . . 11 ⊢ (((𝐴 ≼ ℕ ∧ ∀𝑛 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto→𝐴) → seq1( + , (𝑘 ∈ ℕ ↦ (vol*‘⦋(𝑓‘𝑘) / 𝑛⦌𝐵))) = seq1( + , ((ℤ≥‘1) × {0})))
70		1z 12739	. . . . . . . . . . . 12 ⊢ 1 ∈ ℤ
71		serclim0 15759	. . . . . . . . . . . 12 ⊢ (1 ∈ ℤ → seq1( + , ((ℤ≥‘1) × {0})) ⇝ 0)
72		seqex 14159	. . . . . . . . . . . . 13 ⊢ seq1( + , ((ℤ≥‘1) × {0})) ∈ V
73		c0ex 11313	. . . . . . . . . . . . 13 ⊢ 0 ∈ V
74	72, 73	breldm 5895	. . . . . . . . . . . 12 ⊢ (seq1( + , ((ℤ≥‘1) × {0})) ⇝ 0 → seq1( + , ((ℤ≥‘1) × {0})) ∈ dom ⇝ )
75	70, 71, 74	mp2b 10	. . . . . . . . . . 11 ⊢ seq1( + , ((ℤ≥‘1) × {0})) ∈ dom ⇝
76	69, 75	eqeltrdi 2872	. . . . . . . . . 10 ⊢ (((𝐴 ≼ ℕ ∧ ∀𝑛 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto→𝐴) → seq1( + , (𝑘 ∈ ℕ ↦ (vol*‘⦋(𝑓‘𝑘) / 𝑛⦌𝐵))) ∈ dom ⇝ )
77	58, 59, 54, 62, 76	ovoliun2 25842	. . . . . . . . 9 ⊢ (((𝐴 ≼ ℕ ∧ ∀𝑛 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto→𝐴) → (vol*‘∪ 𝑘 ∈ ℕ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵) ≤ Σ𝑘 ∈ ℕ (vol*‘⦋(𝑓‘𝑘) / 𝑛⦌𝐵))
78	60	sumeq2dv 15884	. . . . . . . . . 10 ⊢ (((𝐴 ≼ ℕ ∧ ∀𝑛 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto→𝐴) → Σ𝑘 ∈ ℕ (vol*‘⦋(𝑓‘𝑘) / 𝑛⦌𝐵) = Σ𝑘 ∈ ℕ 0)
79	65	eqimssi 3973	. . . . . . . . . . . 12 ⊢ ℕ ⊆ (ℤ≥‘1)
80	79	orci 879	. . . . . . . . . . 11 ⊢ (ℕ ⊆ (ℤ≥‘1) ∨ ℕ ∈ Fin)
81		sumz 15903	. . . . . . . . . . 11 ⊢ ((ℕ ⊆ (ℤ≥‘1) ∨ ℕ ∈ Fin) → Σ𝑘 ∈ ℕ 0 = 0)
82	80, 81	ax-mp 5	. . . . . . . . . 10 ⊢ Σ𝑘 ∈ ℕ 0 = 0
83	78, 82	eqtrdi 2814	. . . . . . . . 9 ⊢ (((𝐴 ≼ ℕ ∧ ∀𝑛 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto→𝐴) → Σ𝑘 ∈ ℕ (vol*‘⦋(𝑓‘𝑘) / 𝑛⦌𝐵) = 0)
84	77, 83	breqtrd 5134	. . . . . . . 8 ⊢ (((𝐴 ≼ ℕ ∧ ∀𝑛 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto→𝐴) → (vol*‘∪ 𝑘 ∈ ℕ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵) ≤ 0)
85		ovolge0 25817	. . . . . . . . 9 ⊢ (∪ 𝑘 ∈ ℕ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵 ⊆ ℝ → 0 ≤ (vol*‘∪ 𝑘 ∈ ℕ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵))
86	57, 85	syl 18	. . . . . . . 8 ⊢ (((𝐴 ≼ ℕ ∧ ∀𝑛 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto→𝐴) → 0 ≤ (vol*‘∪ 𝑘 ∈ ℕ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵))
87		ovolcl 25814	. . . . . . . . . 10 ⊢ (∪ 𝑘 ∈ ℕ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵 ⊆ ℝ → (vol*‘∪ 𝑘 ∈ ℕ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵) ∈ ℝ*)
88	57, 87	syl 18	. . . . . . . . 9 ⊢ (((𝐴 ≼ ℕ ∧ ∀𝑛 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto→𝐴) → (vol*‘∪ 𝑘 ∈ ℕ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵) ∈ ℝ*)
89		0xr 11369	. . . . . . . . 9 ⊢ 0 ∈ ℝ*
90		xrletri3 13296	. . . . . . . . 9 ⊢ (((vol*‘∪ 𝑘 ∈ ℕ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵) ∈ ℝ* ∧ 0 ∈ ℝ*) → ((vol*‘∪ 𝑘 ∈ ℕ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵) = 0 ↔ ((vol*‘∪ 𝑘 ∈ ℕ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵) ≤ 0 ∧ 0 ≤ (vol*‘∪ 𝑘 ∈ ℕ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵))))
91	88, 89, 90	sylancl 598	. . . . . . . 8 ⊢ (((𝐴 ≼ ℕ ∧ ∀𝑛 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto→𝐴) → ((vol*‘∪ 𝑘 ∈ ℕ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵) = 0 ↔ ((vol*‘∪ 𝑘 ∈ ℕ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵) ≤ 0 ∧ 0 ≤ (vol*‘∪ 𝑘 ∈ ℕ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵))))
92	84, 86, 91	mpbir2and 726	. . . . . . 7 ⊢ (((𝐴 ≼ ℕ ∧ ∀𝑛 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto→𝐴) → (vol*‘∪ 𝑘 ∈ ℕ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵) = 0)
93		ovolssnul 25823	. . . . . . 7 ⊢ ((∪ 𝑛 ∈ 𝐴 𝐵 ⊆ ∪ 𝑘 ∈ ℕ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵 ∧ ∪ 𝑘 ∈ ℕ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵 ⊆ ℝ ∧ (vol*‘∪ 𝑘 ∈ ℕ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵) = 0) → (vol*‘∪ 𝑛 ∈ 𝐴 𝐵) = 0)
94	38, 57, 92, 93	syl3anc 1398	. . . . . 6 ⊢ (((𝐴 ≼ ℕ ∧ ∀𝑛 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto→𝐴) → (vol*‘∪ 𝑛 ∈ 𝐴 𝐵) = 0)
95	94	ex 418	. . . . 5 ⊢ ((𝐴 ≼ ℕ ∧ ∀𝑛 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) → (𝑓:ℕ–onto→𝐴 → (vol*‘∪ 𝑛 ∈ 𝐴 𝐵) = 0))
96	95	exlimdv 1967	. . . 4 ⊢ ((𝐴 ≼ ℕ ∧ ∀𝑛 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) → (∃𝑓 𝑓:ℕ–onto→𝐴 → (vol*‘∪ 𝑛 ∈ 𝐴 𝐵) = 0))
97	15, 96	syld 48	. . 3 ⊢ ((𝐴 ≼ ℕ ∧ ∀𝑛 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) → (∅ ≺ 𝐴 → (vol*‘∪ 𝑛 ∈ 𝐴 𝐵) = 0))
98	12, 97	sylbird 263	. 2 ⊢ ((𝐴 ≼ ℕ ∧ ∀𝑛 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) → (𝐴 ≠ ∅ → (vol*‘∪ 𝑛 ∈ 𝐴 𝐵) = 0))
99	7, 98	pm2.61dne 3045	1 ⊢ ((𝐴 ≼ ℕ ∧ ∀𝑛 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) → (vol*‘∪ 𝑛 ∈ 𝐴 𝐵) = 0)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1569  ∃wex 1811   ∈ wcel 2146   ≠ wne 2959  ∀wral 3080  ∃wrex 3090  Vcvv 3454  ⦋csb 3851   ⊆ wss 3900  ∅c0 4281  {csn 4588  ∪ ciun 4954   class class class wbr 5106   ↦ cmpt 5189   × cxp 5653  dom cdm 5655  ⟶wf 6543  –onto→wfo 6545  ‘cfv 6547   ≼ cdom 8985   ≺ csdm 8986  Fincfn 8987  ℝcr 11212  0cc0 11213  1c1 11214   + caddc 11216  ℝ^*cxr 11355   ≤ cle 11357  ℕcn 12348  ℤcz 12706  ℤ_≥cuz 12978  seqcseq 14157   ⇝ cli 15666  Σcsu 15868  vol*covol 25798

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7761  ax-inf2 9655  ax-cc 10526  ax-cnex 11269  ax-resscn 11270  ax-1cn 11271  ax-icn 11272  ax-addcl 11273  ax-addrcl 11274  ax-mulcl 11275  ax-mulrcl 11276  ax-mulcom 11277  ax-addass 11278  ax-mulass 11279  ax-distr 11280  ax-i2m1 11281  ax-1ne0 11282  ax-1rid 11283  ax-rnegex 11284  ax-rrecex 11285  ax-cnre 11286  ax-pre-lttri 11287  ax-pre-lttrn 11288  ax-pre-ltadd 11289  ax-pre-mulgt0 11290  ax-pre-sup 11291

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6311  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6501  df-fun 6549  df-fn 6550  df-f 6551  df-f1 6552  df-fo 6553  df-f1o 6554  df-fv 6555  df-isom 6556  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-of 7703  df-om 7888  df-1st 8011  df-2nd 8012  df-frecs 8313  df-wrecs 8344  df-recs 8393  df-rdg 8432  df-1o 8490  df-2o 8491  df-er 8731  df-map 8863  df-pm 8864  df-en 8988  df-dom 8989  df-sdom 8990  df-fin 8991  df-sup 9447  df-inf 9448  df-oi 9517  df-dju 9995  df-card 10033  df-pnf 11358  df-mnf 11359  df-xr 11360  df-ltxr 11361  df-le 11362  df-sub 11556  df-neg 11557  df-div 11987  df-nn 12349  df-2 12418  df-3 12419  df-n0 12620  df-z 12707  df-uz 12979  df-q 13089  df-rp 13134  df-xadd 13255  df-ioo 13493  df-ico 13495  df-icc 13496  df-fz 13653  df-fzo 13802  df-fl 13945  df-seq 14158  df-exp 14218  df-hash 14488  df-cj 15281  df-re 15282  df-im 15283  df-sqrt 15417  df-abs 15418  df-clim 15670  df-rlim 15671  df-sum 15869  df-xmet 21686  df-met 21687  df-ovol 25800

This theorem is used by: (None)