Theorem ovoliunnul 25555

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 5012	. . . . . 6 ⊢ (𝐴 = ∅ → ∪ 𝑛 ∈ 𝐴 𝐵 = ∪ 𝑛 ∈ ∅ 𝐵)
2		0iun 5067	. . . . . 6 ⊢ ∪ 𝑛 ∈ ∅ 𝐵 = ∅
3	1, 2	eqtrdi 2790	. . . . 5 ⊢ (𝐴 = ∅ → ∪ 𝑛 ∈ 𝐴 𝐵 = ∅)
4	3	fveq2d 6910	. . . 4 ⊢ (𝐴 = ∅ → (vol*‘∪ 𝑛 ∈ 𝐴 𝐵) = (vol*‘∅))
5		ovol0 25541	. . . 4 ⊢ (vol*‘∅) = 0
6	4, 5	eqtrdi 2790	. . 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 8989	. . . . . 6 ⊢ Rel ≼
9	8	brrelex1i 5744	. . . . 5 ⊢ (𝐴 ≼ ℕ → 𝐴 ∈ V)
10	9	adantr 480	. . . 4 ⊢ ((𝐴 ≼ ℕ ∧ ∀𝑛 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) → 𝐴 ∈ V)
11		0sdomg 9142	. . . 4 ⊢ (𝐴 ∈ V → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅))
12	10, 11	syl 17	. . 3 ⊢ ((𝐴 ≼ ℕ ∧ ∀𝑛 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅))
13		fodomr 9166	. . . . . 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 4999	. . . . . . . . . 10 ⊢ (𝑥 ∈ ∪ 𝑛 ∈ 𝐴 𝐵 ↔ ∃𝑛 ∈ 𝐴 𝑥 ∈ 𝐵)
17		nfv 1911	. . . . . . . . . . 11 ⊢ Ⅎ𝑛 𝑓:ℕ–onto→𝐴
18		nfcv 2902	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛ℕ
19		nfcsb1v 3932	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛⦋(𝑓‘𝑘) / 𝑛⦌𝐵
20	18, 19	nfiun 5027	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛∪ 𝑘 ∈ ℕ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵
21	20	nfcri 2894	. . . . . . . . . . 11 ⊢ Ⅎ𝑛 𝑥 ∈ ∪ 𝑘 ∈ ℕ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵
22		foelrn 7126	. . . . . . . . . . . . 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 3921	. . . . . . . . . . . . . . . . . . . 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 2824	. . . . . . . . . . . . . . . . . 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 3167	. . . . . . . . . . . . . . 15 ⊢ ((𝑓:ℕ–onto→𝐴 ∧ 𝑥 ∈ 𝐵) → (∃𝑘 ∈ ℕ 𝑛 = (𝑓‘𝑘) → ∃𝑘 ∈ ℕ 𝑥 ∈ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵))
30		eliun 4999	. . . . . . . . . . . . . . 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 3263	. . . . . . . . . 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 4000	. . . . . . . 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 6820	. . . . . . . . . . . . 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 7103	. . . . . . . . . . 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 2902	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛ℝ
44	19, 43	nfss 3987	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛⦋(𝑓‘𝑘) / 𝑛⦌𝐵 ⊆ ℝ
45		nfcv 2902	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛vol*
46	45, 19	nffv 6916	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛(vol*‘⦋(𝑓‘𝑘) / 𝑛⦌𝐵)
47	46	nfeq1 2918	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛(vol*‘⦋(𝑓‘𝑘) / 𝑛⦌𝐵) = 0
48	44, 47	nfan 1896	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛(⦋(𝑓‘𝑘) / 𝑛⦌𝐵 ⊆ ℝ ∧ (vol*‘⦋(𝑓‘𝑘) / 𝑛⦌𝐵) = 0)
49	24	sseq1d 4026	. . . . . . . . . . . . 13 ⊢ (𝑛 = (𝑓‘𝑘) → (𝐵 ⊆ ℝ ↔ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵 ⊆ ℝ))
50	24	fveqeq2d 6914	. . . . . . . . . . . . 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 3609	. . . . . . . . . . 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 3143	. . . . . . . 8 ⊢ (((𝐴 ≼ ℕ ∧ ∀𝑛 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto→𝐴) → ∀𝑘 ∈ ℕ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵 ⊆ ℝ)
56		iunss 5049	. . . . . . . 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 2734	. . . . . . . . . 10 ⊢ seq1( + , (𝑘 ∈ ℕ ↦ (vol*‘⦋(𝑓‘𝑘) / 𝑛⦌𝐵))) = seq1( + , (𝑘 ∈ ℕ ↦ (vol*‘⦋(𝑓‘𝑘) / 𝑛⦌𝐵)))
59		eqid 2734	. . . . . . . . . 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 11260	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ
62	60, 61	eqeltrdi 2846	. . . . . . . . . 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 5247	. . . . . . . . . . . . 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 5750	. . . . . . . . . . . . . 14 ⊢ (ℕ × {0}) = (𝑘 ∈ ℕ ↦ 0)
65		nnuz 12918	. . . . . . . . . . . . . . 15 ⊢ ℕ = (ℤ≥‘1)
66	65	xpeq1i 5714	. . . . . . . . . . . . . 14 ⊢ (ℕ × {0}) = ((ℤ≥‘1) × {0})
67	64, 66	eqtr3i 2764	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ ↦ 0) = ((ℤ≥‘1) × {0})
68	63, 67	eqtrdi 2790	. . . . . . . . . . . 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 14046	. . . . . . . . . . 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 12644	. . . . . . . . . . . 12 ⊢ 1 ∈ ℤ
71		serclim0 15609	. . . . . . . . . . . 12 ⊢ (1 ∈ ℤ → seq1( + , ((ℤ≥‘1) × {0})) ⇝ 0)
72		seqex 14040	. . . . . . . . . . . . 13 ⊢ seq1( + , ((ℤ≥‘1) × {0})) ∈ V
73		c0ex 11252	. . . . . . . . . . . . 13 ⊢ 0 ∈ V
74	72, 73	breldm 5921	. . . . . . . . . . . 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 2846	. . . . . . . . . 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 25554	. . . . . . . . 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 15734	. . . . . . . . . 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 4055	. . . . . . . . . . . 12 ⊢ ℕ ⊆ (ℤ≥‘1)
80	79	orci 865	. . . . . . . . . . 11 ⊢ (ℕ ⊆ (ℤ≥‘1) ∨ ℕ ∈ Fin)
81		sumz 15754	. . . . . . . . . . 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 2790	. . . . . . . . 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 5173	. . . . . . . 8 ⊢ (((𝐴 ≼ ℕ ∧ ∀𝑛 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto→𝐴) → (vol*‘∪ 𝑘 ∈ ℕ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵) ≤ 0)
85		ovolge0 25529	. . . . . . . . 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 25526	. . . . . . . . . 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 11305	. . . . . . . . 9 ⊢ 0 ∈ ℝ*
90		xrletri3 13192	. . . . . . . . 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 25535	. . . . . . 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 1370	. . . . . 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 1930	. . . 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 3025	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 1536  ∃wex 1775   ∈ wcel 2105   ≠ wne 2937  ∀wral 3058  ∃wrex 3067  Vcvv 3477  ⦋csb 3907   ⊆ wss 3962  ∅c0 4338  {csn 4630  ∪ ciun 4995   class class class wbr 5147   ↦ cmpt 5230   × cxp 5686  dom cdm 5688  ⟶wf 6558  –onto→wfo 6560  ‘cfv 6562   ≼ cdom 8981   ≺ csdm 8982  Fincfn 8983  ℝcr 11151  0cc0 11152  1c1 11153   + caddc 11155  ℝ^*cxr 11291   ≤ cle 11293  ℕcn 12263  ℤcz 12610  ℤ_≥cuz 12875  seqcseq 14038   ⇝ cli 15516  Σcsu 15718  vol*covol 25510

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-inf2 9678  ax-cc 10472  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229  ax-pre-sup 11230

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-of 7696  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-2o 8505  df-er 8743  df-map 8866  df-pm 8867  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-sup 9479  df-inf 9480  df-oi 9547  df-dju 9938  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-div 11918  df-nn 12264  df-2 12326  df-3 12327  df-n0 12524  df-z 12611  df-uz 12876  df-q 12988  df-rp 13032  df-xadd 13152  df-ioo 13387  df-ico 13389  df-icc 13390  df-fz 13544  df-fzo 13691  df-fl 13828  df-seq 14039  df-exp 14099  df-hash 14366  df-cj 15134  df-re 15135  df-im 15136  df-sqrt 15270  df-abs 15271  df-clim 15520  df-rlim 15521  df-sum 15719  df-xmet 21374  df-met 21375  df-ovol 25512

This theorem is referenced by: (None)