Theorem ovoliunnul 25487

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 4951	. . . . . 6 ⊢ (𝐴 = ∅ → ∪ 𝑛 ∈ 𝐴 𝐵 = ∪ 𝑛 ∈ ∅ 𝐵)
2		0iun 5006	. . . . . 6 ⊢ ∪ 𝑛 ∈ ∅ 𝐵 = ∅
3	1, 2	eqtrdi 2788	. . . . 5 ⊢ (𝐴 = ∅ → ∪ 𝑛 ∈ 𝐴 𝐵 = ∅)
4	3	fveq2d 6839	. . . 4 ⊢ (𝐴 = ∅ → (vol*‘∪ 𝑛 ∈ 𝐴 𝐵) = (vol*‘∅))
5		ovol0 25473	. . . 4 ⊢ (vol*‘∅) = 0
6	4, 5	eqtrdi 2788	. . 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 8893	. . . . . 6 ⊢ Rel ≼
9	8	brrelex1i 5681	. . . . 5 ⊢ (𝐴 ≼ ℕ → 𝐴 ∈ V)
10	9	adantr 480	. . . 4 ⊢ ((𝐴 ≼ ℕ ∧ ∀𝑛 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) → 𝐴 ∈ V)
11		0sdomg 9038	. . . 4 ⊢ (𝐴 ∈ V → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅))
12	10, 11	syl 17	. . 3 ⊢ ((𝐴 ≼ ℕ ∧ ∀𝑛 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅))
13		fodomr 9060	. . . . . 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 4938	. . . . . . . . . 10 ⊢ (𝑥 ∈ ∪ 𝑛 ∈ 𝐴 𝐵 ↔ ∃𝑛 ∈ 𝐴 𝑥 ∈ 𝐵)
17		nfv 1916	. . . . . . . . . . 11 ⊢ Ⅎ𝑛 𝑓:ℕ–onto→𝐴
18		nfcv 2899	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛ℕ
19		nfcsb1v 3862	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛⦋(𝑓‘𝑘) / 𝑛⦌𝐵
20	18, 19	nfiun 4966	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛∪ 𝑘 ∈ ℕ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵
21	20	nfcri 2891	. . . . . . . . . . 11 ⊢ Ⅎ𝑛 𝑥 ∈ ∪ 𝑘 ∈ ℕ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵
22		foelrn 7054	. . . . . . . . . . . . 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 3852	. . . . . . . . . . . . . . . . . . . 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 2823	. . . . . . . . . . . . . . . . . 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 3153	. . . . . . . . . . . . . . 15 ⊢ ((𝑓:ℕ–onto→𝐴 ∧ 𝑥 ∈ 𝐵) → (∃𝑘 ∈ ℕ 𝑛 = (𝑓‘𝑘) → ∃𝑘 ∈ ℕ 𝑥 ∈ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵))
30		eliun 4938	. . . . . . . . . . . . . . 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 3928	. . . . . . . 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 6747	. . . . . . . . . . . . 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 7031	. . . . . . . . . . 11 ⊢ ((((𝐴 ≼ ℕ ∧ ∀𝑛 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto→𝐴) ∧ 𝑘 ∈ ℕ) → (𝑓‘𝑘) ∈ 𝐴)
42		simpllr 776	. . . . . . . . . . 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 2899	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛ℝ
44	19, 43	nfss 3915	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛⦋(𝑓‘𝑘) / 𝑛⦌𝐵 ⊆ ℝ
45		nfcv 2899	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛vol*
46	45, 19	nffv 6845	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛(vol*‘⦋(𝑓‘𝑘) / 𝑛⦌𝐵)
47	46	nfeq1 2915	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛(vol*‘⦋(𝑓‘𝑘) / 𝑛⦌𝐵) = 0
48	44, 47	nfan 1901	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛(⦋(𝑓‘𝑘) / 𝑛⦌𝐵 ⊆ ℝ ∧ (vol*‘⦋(𝑓‘𝑘) / 𝑛⦌𝐵) = 0)
49	24	sseq1d 3954	. . . . . . . . . . . . 13 ⊢ (𝑛 = (𝑓‘𝑘) → (𝐵 ⊆ ℝ ↔ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵 ⊆ ℝ))
50	24	fveqeq2d 6843	. . . . . . . . . . . . 13 ⊢ (𝑛 = (𝑓‘𝑘) → ((vol*‘𝐵) = 0 ↔ (vol*‘⦋(𝑓‘𝑘) / 𝑛⦌𝐵) = 0))
51	49, 50	anbi12d 633	. . . . . . . . . . . 12 ⊢ (𝑛 = (𝑓‘𝑘) → ((𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ↔ (⦋(𝑓‘𝑘) / 𝑛⦌𝐵 ⊆ ℝ ∧ (vol*‘⦋(𝑓‘𝑘) / 𝑛⦌𝐵) = 0)))
52	48, 51	rspc 3553	. . . . . . . . . . 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 3130	. . . . . . . 8 ⊢ (((𝐴 ≼ ℕ ∧ ∀𝑛 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto→𝐴) → ∀𝑘 ∈ ℕ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵 ⊆ ℝ)
56		iunss 4988	. . . . . . . 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 2737	. . . . . . . . . 10 ⊢ seq1( + , (𝑘 ∈ ℕ ↦ (vol*‘⦋(𝑓‘𝑘) / 𝑛⦌𝐵))) = seq1( + , (𝑘 ∈ ℕ ↦ (vol*‘⦋(𝑓‘𝑘) / 𝑛⦌𝐵)))
59		eqid 2737	. . . . . . . . . 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 11140	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ
62	60, 61	eqeltrdi 2845	. . . . . . . . . 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 5179	. . . . . . . . . . . . 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 5687	. . . . . . . . . . . . . 14 ⊢ (ℕ × {0}) = (𝑘 ∈ ℕ ↦ 0)
65		nnuz 12821	. . . . . . . . . . . . . . 15 ⊢ ℕ = (ℤ≥‘1)
66	65	xpeq1i 5651	. . . . . . . . . . . . . 14 ⊢ (ℕ × {0}) = ((ℤ≥‘1) × {0})
67	64, 66	eqtr3i 2762	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ ↦ 0) = ((ℤ≥‘1) × {0})
68	63, 67	eqtrdi 2788	. . . . . . . . . . . 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 13965	. . . . . . . . . . 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 12551	. . . . . . . . . . . 12 ⊢ 1 ∈ ℤ
71		serclim0 15533	. . . . . . . . . . . 12 ⊢ (1 ∈ ℤ → seq1( + , ((ℤ≥‘1) × {0})) ⇝ 0)
72		seqex 13959	. . . . . . . . . . . . 13 ⊢ seq1( + , ((ℤ≥‘1) × {0})) ∈ V
73		c0ex 11132	. . . . . . . . . . . . 13 ⊢ 0 ∈ V
74	72, 73	breldm 5858	. . . . . . . . . . . 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 2845	. . . . . . . . . 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 25486	. . . . . . . . 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 15658	. . . . . . . . . 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 3983	. . . . . . . . . . . 12 ⊢ ℕ ⊆ (ℤ≥‘1)
80	79	orci 866	. . . . . . . . . . 11 ⊢ (ℕ ⊆ (ℤ≥‘1) ∨ ℕ ∈ Fin)
81		sumz 15678	. . . . . . . . . . 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 2788	. . . . . . . . 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 5112	. . . . . . . 8 ⊢ (((𝐴 ≼ ℕ ∧ ∀𝑛 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto→𝐴) → (vol*‘∪ 𝑘 ∈ ℕ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵) ≤ 0)
85		ovolge0 25461	. . . . . . . . 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 25458	. . . . . . . . . 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 11186	. . . . . . . . 9 ⊢ 0 ∈ ℝ*
90		xrletri3 13099	. . . . . . . . 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 587	. . . . . . . 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 714	. . . . . . 7 ⊢ (((𝐴 ≼ ℕ ∧ ∀𝑛 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto→𝐴) → (vol*‘∪ 𝑘 ∈ ℕ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵) = 0)
93		ovolssnul 25467	. . . . . . 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 1374	. . . . . 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 1935	. . . 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 3019	1 ⊢ ((𝐴 ≼ ℕ ∧ ∀𝑛 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) → (vol*‘∪ 𝑛 ∈ 𝐴 𝐵) = 0)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   = wceq 1542  ∃wex 1781   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  ∃wrex 3062  Vcvv 3430  ⦋csb 3838   ⊆ wss 3890  ∅c0 4274  {csn 4568  ∪ ciun 4934   class class class wbr 5086   ↦ cmpt 5167   × cxp 5623  dom cdm 5625  ⟶wf 6489  –onto→wfo 6491  ‘cfv 6493   ≼ cdom 8885   ≺ csdm 8886  Fincfn 8887  ℝcr 11031  0cc0 11032  1c1 11033   + caddc 11035  ℝ^*cxr 11172   ≤ cle 11174  ℕcn 12168  ℤcz 12518  ℤ_≥cuz 12782  seqcseq 13957   ⇝ cli 15440  Σcsu 15642  vol*covol 25442

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 5213  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683  ax-inf2 9556  ax-cc 10351  ax-cnex 11088  ax-resscn 11089  ax-1cn 11090  ax-icn 11091  ax-addcl 11092  ax-addrcl 11093  ax-mulcl 11094  ax-mulrcl 11095  ax-mulcom 11096  ax-addass 11097  ax-mulass 11098  ax-distr 11099  ax-i2m1 11100  ax-1ne0 11101  ax-1rid 11102  ax-rnegex 11103  ax-rrecex 11104  ax-cnre 11105  ax-pre-lttri 11106  ax-pre-lttrn 11107  ax-pre-ltadd 11108  ax-pre-mulgt0 11109  ax-pre-sup 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 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  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 7318  df-ov 7364  df-oprab 7365  df-mpo 7366  df-of 7625  df-om 7812  df-1st 7936  df-2nd 7937  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-2o 8400  df-er 8637  df-map 8769  df-pm 8770  df-en 8888  df-dom 8889  df-sdom 8890  df-fin 8891  df-sup 9349  df-inf 9350  df-oi 9419  df-dju 9819  df-card 9857  df-pnf 11175  df-mnf 11176  df-xr 11177  df-ltxr 11178  df-le 11179  df-sub 11373  df-neg 11374  df-div 11802  df-nn 12169  df-2 12238  df-3 12239  df-n0 12432  df-z 12519  df-uz 12783  df-q 12893  df-rp 12937  df-xadd 13058  df-ioo 13296  df-ico 13298  df-icc 13299  df-fz 13456  df-fzo 13603  df-fl 13745  df-seq 13958  df-exp 14018  df-hash 14287  df-cj 15055  df-re 15056  df-im 15057  df-sqrt 15191  df-abs 15192  df-clim 15444  df-rlim 15445  df-sum 15643  df-xmet 21340  df-met 21341  df-ovol 25444

This theorem is referenced by: (None)