Theorem ovoliunnul 25415

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 4975	. . . . . 6 ⊢ (𝐴 = ∅ → ∪ 𝑛 ∈ 𝐴 𝐵 = ∪ 𝑛 ∈ ∅ 𝐵)
2		0iun 5030	. . . . . 6 ⊢ ∪ 𝑛 ∈ ∅ 𝐵 = ∅
3	1, 2	eqtrdi 2781	. . . . 5 ⊢ (𝐴 = ∅ → ∪ 𝑛 ∈ 𝐴 𝐵 = ∅)
4	3	fveq2d 6865	. . . 4 ⊢ (𝐴 = ∅ → (vol*‘∪ 𝑛 ∈ 𝐴 𝐵) = (vol*‘∅))
5		ovol0 25401	. . . 4 ⊢ (vol*‘∅) = 0
6	4, 5	eqtrdi 2781	. . 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 8927	. . . . . 6 ⊢ Rel ≼
9	8	brrelex1i 5697	. . . . 5 ⊢ (𝐴 ≼ ℕ → 𝐴 ∈ V)
10	9	adantr 480	. . . 4 ⊢ ((𝐴 ≼ ℕ ∧ ∀𝑛 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) → 𝐴 ∈ V)
11		0sdomg 9076	. . . 4 ⊢ (𝐴 ∈ V → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅))
12	10, 11	syl 17	. . 3 ⊢ ((𝐴 ≼ ℕ ∧ ∀𝑛 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅))
13		fodomr 9098	. . . . . 6 ⊢ ((∅ ≺ 𝐴 ∧ 𝐴 ≼ ℕ) → ∃𝑓 𝑓:ℕ–onto→𝐴)
14	13	expcom 413	. . . . 5 ⊢ (𝐴 ≼ ℕ → (∅ ≺ 𝐴 → ∃𝑓 𝑓:ℕ–onto→𝐴))
15	14	adantr 480	. . . 4 ⊢ ((𝐴 ≼ ℕ ∧ ∀𝑛 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) → (∅ ≺ 𝐴 → ∃𝑓 𝑓:ℕ–onto→𝐴))
16		eliun 4962	. . . . . . . . . 10 ⊢ (𝑥 ∈ ∪ 𝑛 ∈ 𝐴 𝐵 ↔ ∃𝑛 ∈ 𝐴 𝑥 ∈ 𝐵)
17		nfv 1914	. . . . . . . . . . 11 ⊢ Ⅎ𝑛 𝑓:ℕ–onto→𝐴
18		nfcv 2892	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛ℕ
19		nfcsb1v 3889	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛⦋(𝑓‘𝑘) / 𝑛⦌𝐵
20	18, 19	nfiun 4990	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛∪ 𝑘 ∈ ℕ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵
21	20	nfcri 2884	. . . . . . . . . . 11 ⊢ Ⅎ𝑛 𝑥 ∈ ∪ 𝑘 ∈ ℕ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵
22		foelrn 7082	. . . . . . . . . . . . 13 ⊢ ((𝑓:ℕ–onto→𝐴 ∧ 𝑛 ∈ 𝐴) → ∃𝑘 ∈ ℕ 𝑛 = (𝑓‘𝑘))
23	22	ex 412	. . . . . . . . . . . 12 ⊢ (𝑓:ℕ–onto→𝐴 → (𝑛 ∈ 𝐴 → ∃𝑘 ∈ ℕ 𝑛 = (𝑓‘𝑘)))
24		csbeq1a 3879	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = (𝑓‘𝑘) → 𝐵 = ⦋(𝑓‘𝑘) / 𝑛⦌𝐵)
25	24	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓:ℕ–onto→𝐴 ∧ 𝑛 = (𝑓‘𝑘)) → 𝐵 = ⦋(𝑓‘𝑘) / 𝑛⦌𝐵)
26	25	eleq2d 2815	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓:ℕ–onto→𝐴 ∧ 𝑛 = (𝑓‘𝑘)) → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵))
27	26	biimpd 229	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓:ℕ–onto→𝐴 ∧ 𝑛 = (𝑓‘𝑘)) → (𝑥 ∈ 𝐵 → 𝑥 ∈ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵))
28	27	impancom 451	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓:ℕ–onto→𝐴 ∧ 𝑥 ∈ 𝐵) → (𝑛 = (𝑓‘𝑘) → 𝑥 ∈ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵))
29	28	reximdv 3149	. . . . . . . . . . . . . . 15 ⊢ ((𝑓:ℕ–onto→𝐴 ∧ 𝑥 ∈ 𝐵) → (∃𝑘 ∈ ℕ 𝑛 = (𝑓‘𝑘) → ∃𝑘 ∈ ℕ 𝑥 ∈ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵))
30		eliun 4962	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ∪ 𝑘 ∈ ℕ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵 ↔ ∃𝑘 ∈ ℕ 𝑥 ∈ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵)
31	29, 30	imbitrrdi 252	. . . . . . . . . . . . . 14 ⊢ ((𝑓:ℕ–onto→𝐴 ∧ 𝑥 ∈ 𝐵) → (∃𝑘 ∈ ℕ 𝑛 = (𝑓‘𝑘) → 𝑥 ∈ ∪ 𝑘 ∈ ℕ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵))
32	31	ex 412	. . . . . . . . . . . . 13 ⊢ (𝑓:ℕ–onto→𝐴 → (𝑥 ∈ 𝐵 → (∃𝑘 ∈ ℕ 𝑛 = (𝑓‘𝑘) → 𝑥 ∈ ∪ 𝑘 ∈ ℕ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵)))
33	32	com23 86	. . . . . . . . . . . 12 ⊢ (𝑓:ℕ–onto→𝐴 → (∃𝑘 ∈ ℕ 𝑛 = (𝑓‘𝑘) → (𝑥 ∈ 𝐵 → 𝑥 ∈ ∪ 𝑘 ∈ ℕ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵)))
34	23, 33	syld 47	. . . . . . . . . . 11 ⊢ (𝑓:ℕ–onto→𝐴 → (𝑛 ∈ 𝐴 → (𝑥 ∈ 𝐵 → 𝑥 ∈ ∪ 𝑘 ∈ ℕ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵)))
35	17, 21, 34	rexlimd 3245	. . . . . . . . . 10 ⊢ (𝑓:ℕ–onto→𝐴 → (∃𝑛 ∈ 𝐴 𝑥 ∈ 𝐵 → 𝑥 ∈ ∪ 𝑘 ∈ ℕ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵))
36	16, 35	biimtrid 242	. . . . . . . . 9 ⊢ (𝑓:ℕ–onto→𝐴 → (𝑥 ∈ ∪ 𝑛 ∈ 𝐴 𝐵 → 𝑥 ∈ ∪ 𝑘 ∈ ℕ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵))
37	36	ssrdv 3955	. . . . . . . 8 ⊢ (𝑓:ℕ–onto→𝐴 → ∪ 𝑛 ∈ 𝐴 𝐵 ⊆ ∪ 𝑘 ∈ ℕ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵)
38	37	adantl 481	. . . . . . 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 6775	. . . . . . . . . . . . 13 ⊢ (𝑓:ℕ–onto→𝐴 → 𝑓:ℕ⟶𝐴)
40	39	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝐴 ≼ ℕ ∧ ∀𝑛 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto→𝐴) → 𝑓:ℕ⟶𝐴)
41	40	ffvelcdmda 7059	. . . . . . . . . . 11 ⊢ ((((𝐴 ≼ ℕ ∧ ∀𝑛 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto→𝐴) ∧ 𝑘 ∈ ℕ) → (𝑓‘𝑘) ∈ 𝐴)
42		simpllr 775	. . . . . . . . . . 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 2892	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛ℝ
44	19, 43	nfss 3942	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛⦋(𝑓‘𝑘) / 𝑛⦌𝐵 ⊆ ℝ
45		nfcv 2892	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛vol*
46	45, 19	nffv 6871	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛(vol*‘⦋(𝑓‘𝑘) / 𝑛⦌𝐵)
47	46	nfeq1 2908	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛(vol*‘⦋(𝑓‘𝑘) / 𝑛⦌𝐵) = 0
48	44, 47	nfan 1899	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛(⦋(𝑓‘𝑘) / 𝑛⦌𝐵 ⊆ ℝ ∧ (vol*‘⦋(𝑓‘𝑘) / 𝑛⦌𝐵) = 0)
49	24	sseq1d 3981	. . . . . . . . . . . . 13 ⊢ (𝑛 = (𝑓‘𝑘) → (𝐵 ⊆ ℝ ↔ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵 ⊆ ℝ))
50	24	fveqeq2d 6869	. . . . . . . . . . . . 13 ⊢ (𝑛 = (𝑓‘𝑘) → ((vol*‘𝐵) = 0 ↔ (vol*‘⦋(𝑓‘𝑘) / 𝑛⦌𝐵) = 0))
51	49, 50	anbi12d 632	. . . . . . . . . . . 12 ⊢ (𝑛 = (𝑓‘𝑘) → ((𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ↔ (⦋(𝑓‘𝑘) / 𝑛⦌𝐵 ⊆ ℝ ∧ (vol*‘⦋(𝑓‘𝑘) / 𝑛⦌𝐵) = 0)))
52	48, 51	rspc 3579	. . . . . . . . . . 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 65	. . . . . . . . . 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 494	. . . . . . . . 9 ⊢ ((((𝐴 ≼ ℕ ∧ ∀𝑛 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto→𝐴) ∧ 𝑘 ∈ ℕ) → ⦋(𝑓‘𝑘) / 𝑛⦌𝐵 ⊆ ℝ)
55	54	ralrimiva 3126	. . . . . . . 8 ⊢ (((𝐴 ≼ ℕ ∧ ∀𝑛 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto→𝐴) → ∀𝑘 ∈ ℕ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵 ⊆ ℝ)
56		iunss 5012	. . . . . . . 8 ⊢ (∪ 𝑘 ∈ ℕ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵 ⊆ ℝ ↔ ∀𝑘 ∈ ℕ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵 ⊆ ℝ)
57	55, 56	sylibr 234	. . . . . . 7 ⊢ (((𝐴 ≼ ℕ ∧ ∀𝑛 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto→𝐴) → ∪ 𝑘 ∈ ℕ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵 ⊆ ℝ)
58		eqid 2730	. . . . . . . . . 10 ⊢ seq1( + , (𝑘 ∈ ℕ ↦ (vol*‘⦋(𝑓‘𝑘) / 𝑛⦌𝐵))) = seq1( + , (𝑘 ∈ ℕ ↦ (vol*‘⦋(𝑓‘𝑘) / 𝑛⦌𝐵)))
59		eqid 2730	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ ↦ (vol*‘⦋(𝑓‘𝑘) / 𝑛⦌𝐵)) = (𝑘 ∈ ℕ ↦ (vol*‘⦋(𝑓‘𝑘) / 𝑛⦌𝐵))
60	53	simprd 495	. . . . . . . . . . 11 ⊢ ((((𝐴 ≼ ℕ ∧ ∀𝑛 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto→𝐴) ∧ 𝑘 ∈ ℕ) → (vol*‘⦋(𝑓‘𝑘) / 𝑛⦌𝐵) = 0)
61		0re 11183	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ
62	60, 61	eqeltrdi 2837	. . . . . . . . . 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 5203	. . . . . . . . . . . . 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 5703	. . . . . . . . . . . . . 14 ⊢ (ℕ × {0}) = (𝑘 ∈ ℕ ↦ 0)
65		nnuz 12843	. . . . . . . . . . . . . . 15 ⊢ ℕ = (ℤ≥‘1)
66	65	xpeq1i 5667	. . . . . . . . . . . . . 14 ⊢ (ℕ × {0}) = ((ℤ≥‘1) × {0})
67	64, 66	eqtr3i 2755	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ ↦ 0) = ((ℤ≥‘1) × {0})
68	63, 67	eqtrdi 2781	. . . . . . . . . . . 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 13981	. . . . . . . . . . 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 12570	. . . . . . . . . . . 12 ⊢ 1 ∈ ℤ
71		serclim0 15550	. . . . . . . . . . . 12 ⊢ (1 ∈ ℤ → seq1( + , ((ℤ≥‘1) × {0})) ⇝ 0)
72		seqex 13975	. . . . . . . . . . . . 13 ⊢ seq1( + , ((ℤ≥‘1) × {0})) ∈ V
73		c0ex 11175	. . . . . . . . . . . . 13 ⊢ 0 ∈ V
74	72, 73	breldm 5875	. . . . . . . . . . . 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 2837	. . . . . . . . . 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 25414	. . . . . . . . 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 15675	. . . . . . . . . 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 4010	. . . . . . . . . . . 12 ⊢ ℕ ⊆ (ℤ≥‘1)
80	79	orci 865	. . . . . . . . . . 11 ⊢ (ℕ ⊆ (ℤ≥‘1) ∨ ℕ ∈ Fin)
81		sumz 15695	. . . . . . . . . . 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 2781	. . . . . . . . 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 5136	. . . . . . . 8 ⊢ (((𝐴 ≼ ℕ ∧ ∀𝑛 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto→𝐴) → (vol*‘∪ 𝑘 ∈ ℕ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵) ≤ 0)
85		ovolge0 25389	. . . . . . . . 9 ⊢ (∪ 𝑘 ∈ ℕ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵 ⊆ ℝ → 0 ≤ (vol*‘∪ 𝑘 ∈ ℕ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵))
86	57, 85	syl 17	. . . . . . . 8 ⊢ (((𝐴 ≼ ℕ ∧ ∀𝑛 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto→𝐴) → 0 ≤ (vol*‘∪ 𝑘 ∈ ℕ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵))
87		ovolcl 25386	. . . . . . . . . 10 ⊢ (∪ 𝑘 ∈ ℕ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵 ⊆ ℝ → (vol*‘∪ 𝑘 ∈ ℕ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵) ∈ ℝ*)
88	57, 87	syl 17	. . . . . . . . 9 ⊢ (((𝐴 ≼ ℕ ∧ ∀𝑛 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto→𝐴) → (vol*‘∪ 𝑘 ∈ ℕ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵) ∈ ℝ*)
89		0xr 11228	. . . . . . . . 9 ⊢ 0 ∈ ℝ*
90		xrletri3 13121	. . . . . . . . 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 586	. . . . . . . 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 713	. . . . . . 7 ⊢ (((𝐴 ≼ ℕ ∧ ∀𝑛 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto→𝐴) → (vol*‘∪ 𝑘 ∈ ℕ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵) = 0)
93		ovolssnul 25395	. . . . . . 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 1373	. . . . . 6 ⊢ (((𝐴 ≼ ℕ ∧ ∀𝑛 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto→𝐴) → (vol*‘∪ 𝑛 ∈ 𝐴 𝐵) = 0)
95	94	ex 412	. . . . 5 ⊢ ((𝐴 ≼ ℕ ∧ ∀𝑛 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) → (𝑓:ℕ–onto→𝐴 → (vol*‘∪ 𝑛 ∈ 𝐴 𝐵) = 0))
96	95	exlimdv 1933	. . . 4 ⊢ ((𝐴 ≼ ℕ ∧ ∀𝑛 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) → (∃𝑓 𝑓:ℕ–onto→𝐴 → (vol*‘∪ 𝑛 ∈ 𝐴 𝐵) = 0))
97	15, 96	syld 47	. . 3 ⊢ ((𝐴 ≼ ℕ ∧ ∀𝑛 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) → (∅ ≺ 𝐴 → (vol*‘∪ 𝑛 ∈ 𝐴 𝐵) = 0))
98	12, 97	sylbird 260	. 2 ⊢ ((𝐴 ≼ ℕ ∧ ∀𝑛 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) → (𝐴 ≠ ∅ → (vol*‘∪ 𝑛 ∈ 𝐴 𝐵) = 0))
99	7, 98	pm2.61dne 3012	1 ⊢ ((𝐴 ≼ ℕ ∧ ∀𝑛 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) → (vol*‘∪ 𝑛 ∈ 𝐴 𝐵) = 0)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1540  ∃wex 1779   ∈ wcel 2109   ≠ wne 2926  ∀wral 3045  ∃wrex 3054  Vcvv 3450  ⦋csb 3865   ⊆ wss 3917  ∅c0 4299  {csn 4592  ∪ ciun 4958   class class class wbr 5110   ↦ cmpt 5191   × cxp 5639  dom cdm 5641  ⟶wf 6510  –onto→wfo 6512  ‘cfv 6514   ≼ cdom 8919   ≺ csdm 8920  Fincfn 8921  ℝcr 11074  0cc0 11075  1c1 11076   + caddc 11078  ℝ^*cxr 11214   ≤ cle 11216  ℕcn 12193  ℤcz 12536  ℤ_≥cuz 12800  seqcseq 13973   ⇝ cli 15457  Σcsu 15659  vol*covol 25370

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-inf2 9601  ax-cc 10395  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152  ax-pre-sup 11153

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 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-se 5595  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-isom 6523  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-of 7656  df-om 7846  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-2o 8438  df-er 8674  df-map 8804  df-pm 8805  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-sup 9400  df-inf 9401  df-oi 9470  df-dju 9861  df-card 9899  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-div 11843  df-nn 12194  df-2 12256  df-3 12257  df-n0 12450  df-z 12537  df-uz 12801  df-q 12915  df-rp 12959  df-xadd 13080  df-ioo 13317  df-ico 13319  df-icc 13320  df-fz 13476  df-fzo 13623  df-fl 13761  df-seq 13974  df-exp 14034  df-hash 14303  df-cj 15072  df-re 15073  df-im 15074  df-sqrt 15208  df-abs 15209  df-clim 15461  df-rlim 15462  df-sum 15660  df-xmet 21264  df-met 21265  df-ovol 25372

This theorem is referenced by: (None)