Theorem ovoliunnul 25031

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 5013	. . . . . 6 ⊢ (𝐴 = ∅ → ∪ 𝑛 ∈ 𝐴 𝐵 = ∪ 𝑛 ∈ ∅ 𝐵)
2		0iun 5066	. . . . . 6 ⊢ ∪ 𝑛 ∈ ∅ 𝐵 = ∅
3	1, 2	eqtrdi 2788	. . . . 5 ⊢ (𝐴 = ∅ → ∪ 𝑛 ∈ 𝐴 𝐵 = ∅)
4	3	fveq2d 6895	. . . 4 ⊢ (𝐴 = ∅ → (vol*‘∪ 𝑛 ∈ 𝐴 𝐵) = (vol*‘∅))
5		ovol0 25017	. . . 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 8947	. . . . . 6 ⊢ Rel ≼
9	8	brrelex1i 5732	. . . . 5 ⊢ (𝐴 ≼ ℕ → 𝐴 ∈ V)
10	9	adantr 481	. . . 4 ⊢ ((𝐴 ≼ ℕ ∧ ∀𝑛 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) → 𝐴 ∈ V)
11		0sdomg 9106	. . . 4 ⊢ (𝐴 ∈ V → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅))
12	10, 11	syl 17	. . 3 ⊢ ((𝐴 ≼ ℕ ∧ ∀𝑛 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅))
13		fodomr 9130	. . . . . 6 ⊢ ((∅ ≺ 𝐴 ∧ 𝐴 ≼ ℕ) → ∃𝑓 𝑓:ℕ–onto→𝐴)
14	13	expcom 414	. . . . 5 ⊢ (𝐴 ≼ ℕ → (∅ ≺ 𝐴 → ∃𝑓 𝑓:ℕ–onto→𝐴))
15	14	adantr 481	. . . 4 ⊢ ((𝐴 ≼ ℕ ∧ ∀𝑛 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) → (∅ ≺ 𝐴 → ∃𝑓 𝑓:ℕ–onto→𝐴))
16		eliun 5001	. . . . . . . . . 10 ⊢ (𝑥 ∈ ∪ 𝑛 ∈ 𝐴 𝐵 ↔ ∃𝑛 ∈ 𝐴 𝑥 ∈ 𝐵)
17		nfv 1917	. . . . . . . . . . 11 ⊢ Ⅎ𝑛 𝑓:ℕ–onto→𝐴
18		nfcv 2903	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛ℕ
19		nfcsb1v 3918	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛⦋(𝑓‘𝑘) / 𝑛⦌𝐵
20	18, 19	nfiun 5027	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛∪ 𝑘 ∈ ℕ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵
21	20	nfcri 2890	. . . . . . . . . . 11 ⊢ Ⅎ𝑛 𝑥 ∈ ∪ 𝑘 ∈ ℕ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵
22		foelrn 7108	. . . . . . . . . . . . 13 ⊢ ((𝑓:ℕ–onto→𝐴 ∧ 𝑛 ∈ 𝐴) → ∃𝑘 ∈ ℕ 𝑛 = (𝑓‘𝑘))
23	22	ex 413	. . . . . . . . . . . 12 ⊢ (𝑓:ℕ–onto→𝐴 → (𝑛 ∈ 𝐴 → ∃𝑘 ∈ ℕ 𝑛 = (𝑓‘𝑘)))
24		csbeq1a 3907	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = (𝑓‘𝑘) → 𝐵 = ⦋(𝑓‘𝑘) / 𝑛⦌𝐵)
25	24	adantl 482	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓:ℕ–onto→𝐴 ∧ 𝑛 = (𝑓‘𝑘)) → 𝐵 = ⦋(𝑓‘𝑘) / 𝑛⦌𝐵)
26	25	eleq2d 2819	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓:ℕ–onto→𝐴 ∧ 𝑛 = (𝑓‘𝑘)) → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵))
27	26	biimpd 228	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓:ℕ–onto→𝐴 ∧ 𝑛 = (𝑓‘𝑘)) → (𝑥 ∈ 𝐵 → 𝑥 ∈ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵))
28	27	impancom 452	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓:ℕ–onto→𝐴 ∧ 𝑥 ∈ 𝐵) → (𝑛 = (𝑓‘𝑘) → 𝑥 ∈ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵))
29	28	reximdv 3170	. . . . . . . . . . . . . . 15 ⊢ ((𝑓:ℕ–onto→𝐴 ∧ 𝑥 ∈ 𝐵) → (∃𝑘 ∈ ℕ 𝑛 = (𝑓‘𝑘) → ∃𝑘 ∈ ℕ 𝑥 ∈ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵))
30		eliun 5001	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ∪ 𝑘 ∈ ℕ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵 ↔ ∃𝑘 ∈ ℕ 𝑥 ∈ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵)
31	29, 30	imbitrrdi 251	. . . . . . . . . . . . . 14 ⊢ ((𝑓:ℕ–onto→𝐴 ∧ 𝑥 ∈ 𝐵) → (∃𝑘 ∈ ℕ 𝑛 = (𝑓‘𝑘) → 𝑥 ∈ ∪ 𝑘 ∈ ℕ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵))
32	31	ex 413	. . . . . . . . . . . . 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 241	. . . . . . . . 9 ⊢ (𝑓:ℕ–onto→𝐴 → (𝑥 ∈ ∪ 𝑛 ∈ 𝐴 𝐵 → 𝑥 ∈ ∪ 𝑘 ∈ ℕ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵))
37	36	ssrdv 3988	. . . . . . . 8 ⊢ (𝑓:ℕ–onto→𝐴 → ∪ 𝑛 ∈ 𝐴 𝐵 ⊆ ∪ 𝑘 ∈ ℕ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵)
38	37	adantl 482	. . . . . . 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 6805	. . . . . . . . . . . . 13 ⊢ (𝑓:ℕ–onto→𝐴 → 𝑓:ℕ⟶𝐴)
40	39	adantl 482	. . . . . . . . . . . 12 ⊢ (((𝐴 ≼ ℕ ∧ ∀𝑛 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto→𝐴) → 𝑓:ℕ⟶𝐴)
41	40	ffvelcdmda 7086	. . . . . . . . . . 11 ⊢ ((((𝐴 ≼ ℕ ∧ ∀𝑛 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto→𝐴) ∧ 𝑘 ∈ ℕ) → (𝑓‘𝑘) ∈ 𝐴)
42		simpllr 774	. . . . . . . . . . 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 2903	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛ℝ
44	19, 43	nfss 3974	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛⦋(𝑓‘𝑘) / 𝑛⦌𝐵 ⊆ ℝ
45		nfcv 2903	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛vol*
46	45, 19	nffv 6901	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛(vol*‘⦋(𝑓‘𝑘) / 𝑛⦌𝐵)
47	46	nfeq1 2918	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛(vol*‘⦋(𝑓‘𝑘) / 𝑛⦌𝐵) = 0
48	44, 47	nfan 1902	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛(⦋(𝑓‘𝑘) / 𝑛⦌𝐵 ⊆ ℝ ∧ (vol*‘⦋(𝑓‘𝑘) / 𝑛⦌𝐵) = 0)
49	24	sseq1d 4013	. . . . . . . . . . . . 13 ⊢ (𝑛 = (𝑓‘𝑘) → (𝐵 ⊆ ℝ ↔ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵 ⊆ ℝ))
50	24	fveqeq2d 6899	. . . . . . . . . . . . 13 ⊢ (𝑛 = (𝑓‘𝑘) → ((vol*‘𝐵) = 0 ↔ (vol*‘⦋(𝑓‘𝑘) / 𝑛⦌𝐵) = 0))
51	49, 50	anbi12d 631	. . . . . . . . . . . 12 ⊢ (𝑛 = (𝑓‘𝑘) → ((𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ↔ (⦋(𝑓‘𝑘) / 𝑛⦌𝐵 ⊆ ℝ ∧ (vol*‘⦋(𝑓‘𝑘) / 𝑛⦌𝐵) = 0)))
52	48, 51	rspc 3600	. . . . . . . . . . 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 495	. . . . . . . . 9 ⊢ ((((𝐴 ≼ ℕ ∧ ∀𝑛 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto→𝐴) ∧ 𝑘 ∈ ℕ) → ⦋(𝑓‘𝑘) / 𝑛⦌𝐵 ⊆ ℝ)
55	54	ralrimiva 3146	. . . . . . . 8 ⊢ (((𝐴 ≼ ℕ ∧ ∀𝑛 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto→𝐴) → ∀𝑘 ∈ ℕ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵 ⊆ ℝ)
56		iunss 5048	. . . . . . . 8 ⊢ (∪ 𝑘 ∈ ℕ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵 ⊆ ℝ ↔ ∀𝑘 ∈ ℕ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵 ⊆ ℝ)
57	55, 56	sylibr 233	. . . . . . 7 ⊢ (((𝐴 ≼ ℕ ∧ ∀𝑛 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto→𝐴) → ∪ 𝑘 ∈ ℕ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵 ⊆ ℝ)
58		eqid 2732	. . . . . . . . . 10 ⊢ seq1( + , (𝑘 ∈ ℕ ↦ (vol*‘⦋(𝑓‘𝑘) / 𝑛⦌𝐵))) = seq1( + , (𝑘 ∈ ℕ ↦ (vol*‘⦋(𝑓‘𝑘) / 𝑛⦌𝐵)))
59		eqid 2732	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ ↦ (vol*‘⦋(𝑓‘𝑘) / 𝑛⦌𝐵)) = (𝑘 ∈ ℕ ↦ (vol*‘⦋(𝑓‘𝑘) / 𝑛⦌𝐵))
60	53	simprd 496	. . . . . . . . . . 11 ⊢ ((((𝐴 ≼ ℕ ∧ ∀𝑛 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto→𝐴) ∧ 𝑘 ∈ ℕ) → (vol*‘⦋(𝑓‘𝑘) / 𝑛⦌𝐵) = 0)
61		0re 11218	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ
62	60, 61	eqeltrdi 2841	. . . . . . . . . 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 5248	. . . . . . . . . . . . 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 5738	. . . . . . . . . . . . . 14 ⊢ (ℕ × {0}) = (𝑘 ∈ ℕ ↦ 0)
65		nnuz 12867	. . . . . . . . . . . . . . 15 ⊢ ℕ = (ℤ≥‘1)
66	65	xpeq1i 5702	. . . . . . . . . . . . . 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 13976	. . . . . . . . . . 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 12594	. . . . . . . . . . . 12 ⊢ 1 ∈ ℤ
71		serclim0 15523	. . . . . . . . . . . 12 ⊢ (1 ∈ ℤ → seq1( + , ((ℤ≥‘1) × {0})) ⇝ 0)
72		seqex 13970	. . . . . . . . . . . . 13 ⊢ seq1( + , ((ℤ≥‘1) × {0})) ∈ V
73		c0ex 11210	. . . . . . . . . . . . 13 ⊢ 0 ∈ V
74	72, 73	breldm 5908	. . . . . . . . . . . 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 2841	. . . . . . . . . 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 25030	. . . . . . . . 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 15651	. . . . . . . . . 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 4042	. . . . . . . . . . . 12 ⊢ ℕ ⊆ (ℤ≥‘1)
80	79	orci 863	. . . . . . . . . . 11 ⊢ (ℕ ⊆ (ℤ≥‘1) ∨ ℕ ∈ Fin)
81		sumz 15670	. . . . . . . . . . 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 5174	. . . . . . . 8 ⊢ (((𝐴 ≼ ℕ ∧ ∀𝑛 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto→𝐴) → (vol*‘∪ 𝑘 ∈ ℕ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵) ≤ 0)
85		ovolge0 25005	. . . . . . . . 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 25002	. . . . . . . . . 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 11263	. . . . . . . . 9 ⊢ 0 ∈ ℝ*
90		xrletri3 13135	. . . . . . . . 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 711	. . . . . . 7 ⊢ (((𝐴 ≼ ℕ ∧ ∀𝑛 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto→𝐴) → (vol*‘∪ 𝑘 ∈ ℕ ⦋(𝑓‘𝑘) / 𝑛⦌𝐵) = 0)
93		ovolssnul 25011	. . . . . . 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 1371	. . . . . 6 ⊢ (((𝐴 ≼ ℕ ∧ ∀𝑛 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto→𝐴) → (vol*‘∪ 𝑛 ∈ 𝐴 𝐵) = 0)
95	94	ex 413	. . . . 5 ⊢ ((𝐴 ≼ ℕ ∧ ∀𝑛 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) → (𝑓:ℕ–onto→𝐴 → (vol*‘∪ 𝑛 ∈ 𝐴 𝐵) = 0))
96	95	exlimdv 1936	. . . 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 259	. 2 ⊢ ((𝐴 ≼ ℕ ∧ ∀𝑛 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) → (𝐴 ≠ ∅ → (vol*‘∪ 𝑛 ∈ 𝐴 𝐵) = 0))
99	7, 98	pm2.61dne 3028	1 ⊢ ((𝐴 ≼ ℕ ∧ ∀𝑛 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) → (vol*‘∪ 𝑛 ∈ 𝐴 𝐵) = 0)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 845   = wceq 1541  ∃wex 1781   ∈ wcel 2106   ≠ wne 2940  ∀wral 3061  ∃wrex 3070  Vcvv 3474  ⦋csb 3893   ⊆ wss 3948  ∅c0 4322  {csn 4628  ∪ ciun 4997   class class class wbr 5148   ↦ cmpt 5231   × cxp 5674  dom cdm 5676  ⟶wf 6539  –onto→wfo 6541  ‘cfv 6543   ≼ cdom 8939   ≺ csdm 8940  Fincfn 8941  ℝcr 11111  0cc0 11112  1c1 11113   + caddc 11115  ℝ^*cxr 11249   ≤ cle 11251  ℕcn 12214  ℤcz 12560  ℤ_≥cuz 12824  seqcseq 13968   ⇝ cli 15430  Σcsu 15634  vol*covol 24986

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  ax-inf2 9638  ax-cc 10432  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189  ax-pre-sup 11190

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-of 7672  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-2o 8469  df-er 8705  df-map 8824  df-pm 8825  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-sup 9439  df-inf 9440  df-oi 9507  df-dju 9898  df-card 9936  df-pnf 11252  df-mnf 11253  df-xr 11254  df-ltxr 11255  df-le 11256  df-sub 11448  df-neg 11449  df-div 11874  df-nn 12215  df-2 12277  df-3 12278  df-n0 12475  df-z 12561  df-uz 12825  df-q 12935  df-rp 12977  df-xadd 13095  df-ioo 13330  df-ico 13332  df-icc 13333  df-fz 13487  df-fzo 13630  df-fl 13759  df-seq 13969  df-exp 14030  df-hash 14293  df-cj 15048  df-re 15049  df-im 15050  df-sqrt 15184  df-abs 15185  df-clim 15434  df-rlim 15435  df-sum 15635  df-xmet 20943  df-met 20944  df-ovol 24988

This theorem is referenced by: (None)