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Theorem lebnumii 24998

Description: Specialize the Lebesgue number lemma lebnum 24996 to the closed unit interval. (Contributed by Mario Carneiro, 14-Feb-2015.)

**Assertion**
Ref	Expression
lebnumii	⊢ ((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) → ∃𝑛 ∈ ℕ ∀𝑘 ∈ (1...𝑛)∃𝑢 ∈ 𝑈 (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛)) ⊆ 𝑢)

Distinct variable group: 𝑘,𝑛,𝑢,𝑈

Proof of Theorem lebnumii Dummy variables 𝑟 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ii 24903	. . 3 ⊢ II = (MetOpen‘((abs ∘ − ) ↾ ((0[,]1) × (0[,]1))))
2		cnmet 24792	. . . . 5 ⊢ (abs ∘ − ) ∈ (Met‘ℂ)
3		unitssre 13539	. . . . . 6 ⊢ (0[,]1) ⊆ ℝ
4		ax-resscn 11212	. . . . . 6 ⊢ ℝ ⊆ ℂ
5	3, 4	sstri 3993	. . . . 5 ⊢ (0[,]1) ⊆ ℂ
6		metres2 24373	. . . . 5 ⊢ (((abs ∘ − ) ∈ (Met‘ℂ) ∧ (0[,]1) ⊆ ℂ) → ((abs ∘ − ) ↾ ((0[,]1) × (0[,]1))) ∈ (Met‘(0[,]1)))
7	2, 5, 6	mp2an 692	. . . 4 ⊢ ((abs ∘ − ) ↾ ((0[,]1) × (0[,]1))) ∈ (Met‘(0[,]1))
8	7	a1i 11	. . 3 ⊢ ((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) → ((abs ∘ − ) ↾ ((0[,]1) × (0[,]1))) ∈ (Met‘(0[,]1)))
9		iicmp 24912	. . . 4 ⊢ II ∈ Comp
10	9	a1i 11	. . 3 ⊢ ((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) → II ∈ Comp)
11		simpl 482	. . 3 ⊢ ((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) → 𝑈 ⊆ II)
12		simpr 484	. . 3 ⊢ ((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) → (0[,]1) = ∪ 𝑈)
13	1, 8, 10, 11, 12	lebnum 24996	. 2 ⊢ ((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) → ∃𝑟 ∈ ℝ⁺ ∀𝑥 ∈ (0[,]1)∃𝑢 ∈ 𝑈 (𝑥(ball‘((abs ∘ − ) ↾ ((0[,]1) × (0[,]1))))𝑟) ⊆ 𝑢)
14		rpreccl 13061	. . . . . . . 8 ⊢ (𝑟 ∈ ℝ⁺ → (1 / 𝑟) ∈ ℝ⁺)
15	14	adantl 481	. . . . . . 7 ⊢ (((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) → (1 / 𝑟) ∈ ℝ⁺)
16	15	rpred 13077	. . . . . 6 ⊢ (((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) → (1 / 𝑟) ∈ ℝ)
17	15	rpge0d 13081	. . . . . 6 ⊢ (((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) → 0 ≤ (1 / 𝑟))
18		flge0nn0 13860	. . . . . 6 ⊢ (((1 / 𝑟) ∈ ℝ ∧ 0 ≤ (1 / 𝑟)) → (⌊‘(1 / 𝑟)) ∈ ℕ₀)
19	16, 17, 18	syl2anc 584	. . . . 5 ⊢ (((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) → (⌊‘(1 / 𝑟)) ∈ ℕ₀)
20		nn0p1nn 12565	. . . . 5 ⊢ ((⌊‘(1 / 𝑟)) ∈ ℕ₀ → ((⌊‘(1 / 𝑟)) + 1) ∈ ℕ)
21	19, 20	syl 17	. . . 4 ⊢ (((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) → ((⌊‘(1 / 𝑟)) + 1) ∈ ℕ)
22		elfznn 13593	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1)) → 𝑘 ∈ ℕ)
23	22	adantl 481	. . . . . . . . . . 11 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → 𝑘 ∈ ℕ)
24	23	nnrpd 13075	. . . . . . . . . 10 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → 𝑘 ∈ ℝ⁺)
25	21	adantr 480	. . . . . . . . . . 11 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → ((⌊‘(1 / 𝑟)) + 1) ∈ ℕ)
26	25	nnrpd 13075	. . . . . . . . . 10 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → ((⌊‘(1 / 𝑟)) + 1) ∈ ℝ⁺)
27	24, 26	rpdivcld 13094	. . . . . . . . 9 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → (𝑘 / ((⌊‘(1 / 𝑟)) + 1)) ∈ ℝ⁺)
28	27	rpred 13077	. . . . . . . 8 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → (𝑘 / ((⌊‘(1 / 𝑟)) + 1)) ∈ ℝ)
29	27	rpge0d 13081	. . . . . . . 8 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → 0 ≤ (𝑘 / ((⌊‘(1 / 𝑟)) + 1)))
30		elfzle2 13568	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1)) → 𝑘 ≤ ((⌊‘(1 / 𝑟)) + 1))
31	30	adantl 481	. . . . . . . . . 10 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → 𝑘 ≤ ((⌊‘(1 / 𝑟)) + 1))
32	25	nnred 12281	. . . . . . . . . . . 12 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → ((⌊‘(1 / 𝑟)) + 1) ∈ ℝ)
33	32	recnd 11289	. . . . . . . . . . 11 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → ((⌊‘(1 / 𝑟)) + 1) ∈ ℂ)
34	33	mulridd 11278	. . . . . . . . . 10 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → (((⌊‘(1 / 𝑟)) + 1) · 1) = ((⌊‘(1 / 𝑟)) + 1))
35	31, 34	breqtrrd 5171	. . . . . . . . 9 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → 𝑘 ≤ (((⌊‘(1 / 𝑟)) + 1) · 1))
36	23	nnred 12281	. . . . . . . . . 10 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → 𝑘 ∈ ℝ)
37		1red 11262	. . . . . . . . . 10 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → 1 ∈ ℝ)
38	25	nngt0d 12315	. . . . . . . . . 10 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → 0 < ((⌊‘(1 / 𝑟)) + 1))
39		ledivmul 12144	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ℝ ∧ 1 ∈ ℝ ∧ (((⌊‘(1 / 𝑟)) + 1) ∈ ℝ ∧ 0 < ((⌊‘(1 / 𝑟)) + 1))) → ((𝑘 / ((⌊‘(1 / 𝑟)) + 1)) ≤ 1 ↔ 𝑘 ≤ (((⌊‘(1 / 𝑟)) + 1) · 1)))
40	36, 37, 32, 38, 39	syl112anc 1376	. . . . . . . . 9 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → ((𝑘 / ((⌊‘(1 / 𝑟)) + 1)) ≤ 1 ↔ 𝑘 ≤ (((⌊‘(1 / 𝑟)) + 1) · 1)))
41	35, 40	mpbird 257	. . . . . . . 8 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → (𝑘 / ((⌊‘(1 / 𝑟)) + 1)) ≤ 1)
42		elicc01 13506	. . . . . . . 8 ⊢ ((𝑘 / ((⌊‘(1 / 𝑟)) + 1)) ∈ (0[,]1) ↔ ((𝑘 / ((⌊‘(1 / 𝑟)) + 1)) ∈ ℝ ∧ 0 ≤ (𝑘 / ((⌊‘(1 / 𝑟)) + 1)) ∧ (𝑘 / ((⌊‘(1 / 𝑟)) + 1)) ≤ 1))
43	28, 29, 41, 42	syl3anbrc 1344	. . . . . . 7 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → (𝑘 / ((⌊‘(1 / 𝑟)) + 1)) ∈ (0[,]1))
44		oveq1 7438	. . . . . . . . . 10 ⊢ (𝑥 = (𝑘 / ((⌊‘(1 / 𝑟)) + 1)) → (𝑥(ball‘((abs ∘ − ) ↾ ((0[,]1) × (0[,]1))))𝑟) = ((𝑘 / ((⌊‘(1 / 𝑟)) + 1))(ball‘((abs ∘ − ) ↾ ((0[,]1) × (0[,]1))))𝑟))
45	44	sseq1d 4015	. . . . . . . . 9 ⊢ (𝑥 = (𝑘 / ((⌊‘(1 / 𝑟)) + 1)) → ((𝑥(ball‘((abs ∘ − ) ↾ ((0[,]1) × (0[,]1))))𝑟) ⊆ 𝑢 ↔ ((𝑘 / ((⌊‘(1 / 𝑟)) + 1))(ball‘((abs ∘ − ) ↾ ((0[,]1) × (0[,]1))))𝑟) ⊆ 𝑢))
46	45	rexbidv 3179	. . . . . . . 8 ⊢ (𝑥 = (𝑘 / ((⌊‘(1 / 𝑟)) + 1)) → (∃𝑢 ∈ 𝑈 (𝑥(ball‘((abs ∘ − ) ↾ ((0[,]1) × (0[,]1))))𝑟) ⊆ 𝑢 ↔ ∃𝑢 ∈ 𝑈 ((𝑘 / ((⌊‘(1 / 𝑟)) + 1))(ball‘((abs ∘ − ) ↾ ((0[,]1) × (0[,]1))))𝑟) ⊆ 𝑢))
47	46	rspcv 3618	. . . . . . 7 ⊢ ((𝑘 / ((⌊‘(1 / 𝑟)) + 1)) ∈ (0[,]1) → (∀𝑥 ∈ (0[,]1)∃𝑢 ∈ 𝑈 (𝑥(ball‘((abs ∘ − ) ↾ ((0[,]1) × (0[,]1))))𝑟) ⊆ 𝑢 → ∃𝑢 ∈ 𝑈 ((𝑘 / ((⌊‘(1 / 𝑟)) + 1))(ball‘((abs ∘ − ) ↾ ((0[,]1) × (0[,]1))))𝑟) ⊆ 𝑢))
48	43, 47	syl 17	. . . . . 6 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → (∀𝑥 ∈ (0[,]1)∃𝑢 ∈ 𝑈 (𝑥(ball‘((abs ∘ − ) ↾ ((0[,]1) × (0[,]1))))𝑟) ⊆ 𝑢 → ∃𝑢 ∈ 𝑈 ((𝑘 / ((⌊‘(1 / 𝑟)) + 1))(ball‘((abs ∘ − ) ↾ ((0[,]1) × (0[,]1))))𝑟) ⊆ 𝑢))
49		simplr 769	. . . . . . . . . . . . . 14 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → 𝑟 ∈ ℝ⁺)
50	49	rpred 13077	. . . . . . . . . . . . 13 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → 𝑟 ∈ ℝ)
51	28, 50	resubcld 11691	. . . . . . . . . . . 12 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → ((𝑘 / ((⌊‘(1 / 𝑟)) + 1)) − 𝑟) ∈ ℝ)
52	51	rexrd 11311	. . . . . . . . . . 11 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → ((𝑘 / ((⌊‘(1 / 𝑟)) + 1)) − 𝑟) ∈ ℝ^*)
53	28, 50	readdcld 11290	. . . . . . . . . . . 12 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → ((𝑘 / ((⌊‘(1 / 𝑟)) + 1)) + 𝑟) ∈ ℝ)
54	53	rexrd 11311	. . . . . . . . . . 11 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → ((𝑘 / ((⌊‘(1 / 𝑟)) + 1)) + 𝑟) ∈ ℝ^*)
55		nnm1nn0 12567	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℕ → (𝑘 − 1) ∈ ℕ₀)
56	23, 55	syl 17	. . . . . . . . . . . . . 14 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → (𝑘 − 1) ∈ ℕ₀)
57	56	nn0red 12588	. . . . . . . . . . . . 13 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → (𝑘 − 1) ∈ ℝ)
58	57, 25	nndivred 12320	. . . . . . . . . . . 12 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → ((𝑘 − 1) / ((⌊‘(1 / 𝑟)) + 1)) ∈ ℝ)
59	36	recnd 11289	. . . . . . . . . . . . . . 15 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → 𝑘 ∈ ℂ)
60	57	recnd 11289	. . . . . . . . . . . . . . 15 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → (𝑘 − 1) ∈ ℂ)
61	25	nnne0d 12316	. . . . . . . . . . . . . . 15 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → ((⌊‘(1 / 𝑟)) + 1) ≠ 0)
62	59, 60, 33, 61	divsubdird 12082	. . . . . . . . . . . . . 14 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → ((𝑘 − (𝑘 − 1)) / ((⌊‘(1 / 𝑟)) + 1)) = ((𝑘 / ((⌊‘(1 / 𝑟)) + 1)) − ((𝑘 − 1) / ((⌊‘(1 / 𝑟)) + 1))))
63		ax-1cn 11213	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℂ
64		nncan 11538	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ ℂ ∧ 1 ∈ ℂ) → (𝑘 − (𝑘 − 1)) = 1)
65	59, 63, 64	sylancl 586	. . . . . . . . . . . . . . 15 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → (𝑘 − (𝑘 − 1)) = 1)
66	65	oveq1d 7446	. . . . . . . . . . . . . 14 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → ((𝑘 − (𝑘 − 1)) / ((⌊‘(1 / 𝑟)) + 1)) = (1 / ((⌊‘(1 / 𝑟)) + 1)))
67	62, 66	eqtr3d 2779	. . . . . . . . . . . . 13 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → ((𝑘 / ((⌊‘(1 / 𝑟)) + 1)) − ((𝑘 − 1) / ((⌊‘(1 / 𝑟)) + 1))) = (1 / ((⌊‘(1 / 𝑟)) + 1)))
68	49	rprecred 13088	. . . . . . . . . . . . . . 15 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → (1 / 𝑟) ∈ ℝ)
69		flltp1 13840	. . . . . . . . . . . . . . 15 ⊢ ((1 / 𝑟) ∈ ℝ → (1 / 𝑟) < ((⌊‘(1 / 𝑟)) + 1))
70	68, 69	syl 17	. . . . . . . . . . . . . 14 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → (1 / 𝑟) < ((⌊‘(1 / 𝑟)) + 1))
71		rpgt0 13047	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 ∈ ℝ⁺ → 0 < 𝑟)
72	71	ad2antlr 727	. . . . . . . . . . . . . . 15 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → 0 < 𝑟)
73		ltdiv23 12159	. . . . . . . . . . . . . . 15 ⊢ ((1 ∈ ℝ ∧ (𝑟 ∈ ℝ ∧ 0 < 𝑟) ∧ (((⌊‘(1 / 𝑟)) + 1) ∈ ℝ ∧ 0 < ((⌊‘(1 / 𝑟)) + 1))) → ((1 / 𝑟) < ((⌊‘(1 / 𝑟)) + 1) ↔ (1 / ((⌊‘(1 / 𝑟)) + 1)) < 𝑟))
74	37, 50, 72, 32, 38, 73	syl122anc 1381	. . . . . . . . . . . . . 14 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → ((1 / 𝑟) < ((⌊‘(1 / 𝑟)) + 1) ↔ (1 / ((⌊‘(1 / 𝑟)) + 1)) < 𝑟))
75	70, 74	mpbid 232	. . . . . . . . . . . . 13 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → (1 / ((⌊‘(1 / 𝑟)) + 1)) < 𝑟)
76	67, 75	eqbrtrd 5165	. . . . . . . . . . . 12 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → ((𝑘 / ((⌊‘(1 / 𝑟)) + 1)) − ((𝑘 − 1) / ((⌊‘(1 / 𝑟)) + 1))) < 𝑟)
77	28, 58, 50, 76	ltsub23d 11868	. . . . . . . . . . 11 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → ((𝑘 / ((⌊‘(1 / 𝑟)) + 1)) − 𝑟) < ((𝑘 − 1) / ((⌊‘(1 / 𝑟)) + 1)))
78	28, 49	ltaddrpd 13110	. . . . . . . . . . 11 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → (𝑘 / ((⌊‘(1 / 𝑟)) + 1)) < ((𝑘 / ((⌊‘(1 / 𝑟)) + 1)) + 𝑟))
79		iccssioo 13456	. . . . . . . . . . 11 ⊢ (((((𝑘 / ((⌊‘(1 / 𝑟)) + 1)) − 𝑟) ∈ ℝ^* ∧ ((𝑘 / ((⌊‘(1 / 𝑟)) + 1)) + 𝑟) ∈ ℝ^*) ∧ (((𝑘 / ((⌊‘(1 / 𝑟)) + 1)) − 𝑟) < ((𝑘 − 1) / ((⌊‘(1 / 𝑟)) + 1)) ∧ (𝑘 / ((⌊‘(1 / 𝑟)) + 1)) < ((𝑘 / ((⌊‘(1 / 𝑟)) + 1)) + 𝑟))) → (((𝑘 − 1) / ((⌊‘(1 / 𝑟)) + 1))[,](𝑘 / ((⌊‘(1 / 𝑟)) + 1))) ⊆ (((𝑘 / ((⌊‘(1 / 𝑟)) + 1)) − 𝑟)(,)((𝑘 / ((⌊‘(1 / 𝑟)) + 1)) + 𝑟)))
80	52, 54, 77, 78, 79	syl22anc 839	. . . . . . . . . 10 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → (((𝑘 − 1) / ((⌊‘(1 / 𝑟)) + 1))[,](𝑘 / ((⌊‘(1 / 𝑟)) + 1))) ⊆ (((𝑘 / ((⌊‘(1 / 𝑟)) + 1)) − 𝑟)(,)((𝑘 / ((⌊‘(1 / 𝑟)) + 1)) + 𝑟)))
81		0red 11264	. . . . . . . . . . 11 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → 0 ∈ ℝ)
82	56	nn0ge0d 12590	. . . . . . . . . . . 12 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → 0 ≤ (𝑘 − 1))
83		divge0 12137	. . . . . . . . . . . 12 ⊢ ((((𝑘 − 1) ∈ ℝ ∧ 0 ≤ (𝑘 − 1)) ∧ (((⌊‘(1 / 𝑟)) + 1) ∈ ℝ ∧ 0 < ((⌊‘(1 / 𝑟)) + 1))) → 0 ≤ ((𝑘 − 1) / ((⌊‘(1 / 𝑟)) + 1)))
84	57, 82, 32, 38, 83	syl22anc 839	. . . . . . . . . . 11 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → 0 ≤ ((𝑘 − 1) / ((⌊‘(1 / 𝑟)) + 1)))
85		iccss 13455	. . . . . . . . . . 11 ⊢ (((0 ∈ ℝ ∧ 1 ∈ ℝ) ∧ (0 ≤ ((𝑘 − 1) / ((⌊‘(1 / 𝑟)) + 1)) ∧ (𝑘 / ((⌊‘(1 / 𝑟)) + 1)) ≤ 1)) → (((𝑘 − 1) / ((⌊‘(1 / 𝑟)) + 1))[,](𝑘 / ((⌊‘(1 / 𝑟)) + 1))) ⊆ (0[,]1))
86	81, 37, 84, 41, 85	syl22anc 839	. . . . . . . . . 10 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → (((𝑘 − 1) / ((⌊‘(1 / 𝑟)) + 1))[,](𝑘 / ((⌊‘(1 / 𝑟)) + 1))) ⊆ (0[,]1))
87	80, 86	ssind 4241	. . . . . . . . 9 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → (((𝑘 − 1) / ((⌊‘(1 / 𝑟)) + 1))[,](𝑘 / ((⌊‘(1 / 𝑟)) + 1))) ⊆ ((((𝑘 / ((⌊‘(1 / 𝑟)) + 1)) − 𝑟)(,)((𝑘 / ((⌊‘(1 / 𝑟)) + 1)) + 𝑟)) ∩ (0[,]1)))
88		eqid 2737	. . . . . . . . . . . 12 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) = ((abs ∘ − ) ↾ (ℝ × ℝ))
89	88	rexmet 24812	. . . . . . . . . . 11 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) ∈ (∞Met‘ℝ)
90		sseqin2 4223	. . . . . . . . . . . . 13 ⊢ ((0[,]1) ⊆ ℝ ↔ (ℝ ∩ (0[,]1)) = (0[,]1))
91	3, 90	mpbi 230	. . . . . . . . . . . 12 ⊢ (ℝ ∩ (0[,]1)) = (0[,]1)
92	43, 91	eleqtrrdi 2852	. . . . . . . . . . 11 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → (𝑘 / ((⌊‘(1 / 𝑟)) + 1)) ∈ (ℝ ∩ (0[,]1)))
93		rpxr 13044	. . . . . . . . . . . 12 ⊢ (𝑟 ∈ ℝ⁺ → 𝑟 ∈ ℝ^*)
94	93	ad2antlr 727	. . . . . . . . . . 11 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → 𝑟 ∈ ℝ^*)
95		xpss12 5700	. . . . . . . . . . . . . . 15 ⊢ (((0[,]1) ⊆ ℝ ∧ (0[,]1) ⊆ ℝ) → ((0[,]1) × (0[,]1)) ⊆ (ℝ × ℝ))
96	3, 3, 95	mp2an 692	. . . . . . . . . . . . . 14 ⊢ ((0[,]1) × (0[,]1)) ⊆ (ℝ × ℝ)
97		resabs1 6024	. . . . . . . . . . . . . 14 ⊢ (((0[,]1) × (0[,]1)) ⊆ (ℝ × ℝ) → (((abs ∘ − ) ↾ (ℝ × ℝ)) ↾ ((0[,]1) × (0[,]1))) = ((abs ∘ − ) ↾ ((0[,]1) × (0[,]1))))
98	96, 97	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (((abs ∘ − ) ↾ (ℝ × ℝ)) ↾ ((0[,]1) × (0[,]1))) = ((abs ∘ − ) ↾ ((0[,]1) × (0[,]1)))
99	98	eqcomi 2746	. . . . . . . . . . . 12 ⊢ ((abs ∘ − ) ↾ ((0[,]1) × (0[,]1))) = (((abs ∘ − ) ↾ (ℝ × ℝ)) ↾ ((0[,]1) × (0[,]1)))
100	99	blres 24441	. . . . . . . . . . 11 ⊢ ((((abs ∘ − ) ↾ (ℝ × ℝ)) ∈ (∞Met‘ℝ) ∧ (𝑘 / ((⌊‘(1 / 𝑟)) + 1)) ∈ (ℝ ∩ (0[,]1)) ∧ 𝑟 ∈ ℝ^*) → ((𝑘 / ((⌊‘(1 / 𝑟)) + 1))(ball‘((abs ∘ − ) ↾ ((0[,]1) × (0[,]1))))𝑟) = (((𝑘 / ((⌊‘(1 / 𝑟)) + 1))(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ∩ (0[,]1)))
101	89, 92, 94, 100	mp3an2i 1468	. . . . . . . . . 10 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → ((𝑘 / ((⌊‘(1 / 𝑟)) + 1))(ball‘((abs ∘ − ) ↾ ((0[,]1) × (0[,]1))))𝑟) = (((𝑘 / ((⌊‘(1 / 𝑟)) + 1))(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ∩ (0[,]1)))
102	88	bl2ioo 24813	. . . . . . . . . . . 12 ⊢ (((𝑘 / ((⌊‘(1 / 𝑟)) + 1)) ∈ ℝ ∧ 𝑟 ∈ ℝ) → ((𝑘 / ((⌊‘(1 / 𝑟)) + 1))(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) = (((𝑘 / ((⌊‘(1 / 𝑟)) + 1)) − 𝑟)(,)((𝑘 / ((⌊‘(1 / 𝑟)) + 1)) + 𝑟)))
103	28, 50, 102	syl2anc 584	. . . . . . . . . . 11 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → ((𝑘 / ((⌊‘(1 / 𝑟)) + 1))(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) = (((𝑘 / ((⌊‘(1 / 𝑟)) + 1)) − 𝑟)(,)((𝑘 / ((⌊‘(1 / 𝑟)) + 1)) + 𝑟)))
104	103	ineq1d 4219	. . . . . . . . . 10 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → (((𝑘 / ((⌊‘(1 / 𝑟)) + 1))(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ∩ (0[,]1)) = ((((𝑘 / ((⌊‘(1 / 𝑟)) + 1)) − 𝑟)(,)((𝑘 / ((⌊‘(1 / 𝑟)) + 1)) + 𝑟)) ∩ (0[,]1)))
105	101, 104	eqtrd 2777	. . . . . . . . 9 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → ((𝑘 / ((⌊‘(1 / 𝑟)) + 1))(ball‘((abs ∘ − ) ↾ ((0[,]1) × (0[,]1))))𝑟) = ((((𝑘 / ((⌊‘(1 / 𝑟)) + 1)) − 𝑟)(,)((𝑘 / ((⌊‘(1 / 𝑟)) + 1)) + 𝑟)) ∩ (0[,]1)))
106	87, 105	sseqtrrd 4021	. . . . . . . 8 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → (((𝑘 − 1) / ((⌊‘(1 / 𝑟)) + 1))[,](𝑘 / ((⌊‘(1 / 𝑟)) + 1))) ⊆ ((𝑘 / ((⌊‘(1 / 𝑟)) + 1))(ball‘((abs ∘ − ) ↾ ((0[,]1) × (0[,]1))))𝑟))
107		sstr2 3990	. . . . . . . 8 ⊢ ((((𝑘 − 1) / ((⌊‘(1 / 𝑟)) + 1))[,](𝑘 / ((⌊‘(1 / 𝑟)) + 1))) ⊆ ((𝑘 / ((⌊‘(1 / 𝑟)) + 1))(ball‘((abs ∘ − ) ↾ ((0[,]1) × (0[,]1))))𝑟) → (((𝑘 / ((⌊‘(1 / 𝑟)) + 1))(ball‘((abs ∘ − ) ↾ ((0[,]1) × (0[,]1))))𝑟) ⊆ 𝑢 → (((𝑘 − 1) / ((⌊‘(1 / 𝑟)) + 1))[,](𝑘 / ((⌊‘(1 / 𝑟)) + 1))) ⊆ 𝑢))
108	106, 107	syl 17	. . . . . . 7 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → (((𝑘 / ((⌊‘(1 / 𝑟)) + 1))(ball‘((abs ∘ − ) ↾ ((0[,]1) × (0[,]1))))𝑟) ⊆ 𝑢 → (((𝑘 − 1) / ((⌊‘(1 / 𝑟)) + 1))[,](𝑘 / ((⌊‘(1 / 𝑟)) + 1))) ⊆ 𝑢))
109	108	reximdv 3170	. . . . . 6 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → (∃𝑢 ∈ 𝑈 ((𝑘 / ((⌊‘(1 / 𝑟)) + 1))(ball‘((abs ∘ − ) ↾ ((0[,]1) × (0[,]1))))𝑟) ⊆ 𝑢 → ∃𝑢 ∈ 𝑈 (((𝑘 − 1) / ((⌊‘(1 / 𝑟)) + 1))[,](𝑘 / ((⌊‘(1 / 𝑟)) + 1))) ⊆ 𝑢))
110	48, 109	syld 47	. . . . 5 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → (∀𝑥 ∈ (0[,]1)∃𝑢 ∈ 𝑈 (𝑥(ball‘((abs ∘ − ) ↾ ((0[,]1) × (0[,]1))))𝑟) ⊆ 𝑢 → ∃𝑢 ∈ 𝑈 (((𝑘 − 1) / ((⌊‘(1 / 𝑟)) + 1))[,](𝑘 / ((⌊‘(1 / 𝑟)) + 1))) ⊆ 𝑢))
111	110	ralrimdva 3154	. . . 4 ⊢ (((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) → (∀𝑥 ∈ (0[,]1)∃𝑢 ∈ 𝑈 (𝑥(ball‘((abs ∘ − ) ↾ ((0[,]1) × (0[,]1))))𝑟) ⊆ 𝑢 → ∀𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))∃𝑢 ∈ 𝑈 (((𝑘 − 1) / ((⌊‘(1 / 𝑟)) + 1))[,](𝑘 / ((⌊‘(1 / 𝑟)) + 1))) ⊆ 𝑢))
112		oveq2 7439	. . . . . 6 ⊢ (𝑛 = ((⌊‘(1 / 𝑟)) + 1) → (1...𝑛) = (1...((⌊‘(1 / 𝑟)) + 1)))
113		oveq2 7439	. . . . . . . . 9 ⊢ (𝑛 = ((⌊‘(1 / 𝑟)) + 1) → ((𝑘 − 1) / 𝑛) = ((𝑘 − 1) / ((⌊‘(1 / 𝑟)) + 1)))
114		oveq2 7439	. . . . . . . . 9 ⊢ (𝑛 = ((⌊‘(1 / 𝑟)) + 1) → (𝑘 / 𝑛) = (𝑘 / ((⌊‘(1 / 𝑟)) + 1)))
115	113, 114	oveq12d 7449	. . . . . . . 8 ⊢ (𝑛 = ((⌊‘(1 / 𝑟)) + 1) → (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛)) = (((𝑘 − 1) / ((⌊‘(1 / 𝑟)) + 1))[,](𝑘 / ((⌊‘(1 / 𝑟)) + 1))))
116	115	sseq1d 4015	. . . . . . 7 ⊢ (𝑛 = ((⌊‘(1 / 𝑟)) + 1) → ((((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛)) ⊆ 𝑢 ↔ (((𝑘 − 1) / ((⌊‘(1 / 𝑟)) + 1))[,](𝑘 / ((⌊‘(1 / 𝑟)) + 1))) ⊆ 𝑢))
117	116	rexbidv 3179	. . . . . 6 ⊢ (𝑛 = ((⌊‘(1 / 𝑟)) + 1) → (∃𝑢 ∈ 𝑈 (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛)) ⊆ 𝑢 ↔ ∃𝑢 ∈ 𝑈 (((𝑘 − 1) / ((⌊‘(1 / 𝑟)) + 1))[,](𝑘 / ((⌊‘(1 / 𝑟)) + 1))) ⊆ 𝑢))
118	112, 117	raleqbidv 3346	. . . . 5 ⊢ (𝑛 = ((⌊‘(1 / 𝑟)) + 1) → (∀𝑘 ∈ (1...𝑛)∃𝑢 ∈ 𝑈 (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛)) ⊆ 𝑢 ↔ ∀𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))∃𝑢 ∈ 𝑈 (((𝑘 − 1) / ((⌊‘(1 / 𝑟)) + 1))[,](𝑘 / ((⌊‘(1 / 𝑟)) + 1))) ⊆ 𝑢))
119	118	rspcev 3622	. . . 4 ⊢ ((((⌊‘(1 / 𝑟)) + 1) ∈ ℕ ∧ ∀𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))∃𝑢 ∈ 𝑈 (((𝑘 − 1) / ((⌊‘(1 / 𝑟)) + 1))[,](𝑘 / ((⌊‘(1 / 𝑟)) + 1))) ⊆ 𝑢) → ∃𝑛 ∈ ℕ ∀𝑘 ∈ (1...𝑛)∃𝑢 ∈ 𝑈 (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛)) ⊆ 𝑢)
120	21, 111, 119	syl6an 684	. . 3 ⊢ (((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) → (∀𝑥 ∈ (0[,]1)∃𝑢 ∈ 𝑈 (𝑥(ball‘((abs ∘ − ) ↾ ((0[,]1) × (0[,]1))))𝑟) ⊆ 𝑢 → ∃𝑛 ∈ ℕ ∀𝑘 ∈ (1...𝑛)∃𝑢 ∈ 𝑈 (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛)) ⊆ 𝑢))
121	120	rexlimdva 3155	. 2 ⊢ ((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) → (∃𝑟 ∈ ℝ⁺ ∀𝑥 ∈ (0[,]1)∃𝑢 ∈ 𝑈 (𝑥(ball‘((abs ∘ − ) ↾ ((0[,]1) × (0[,]1))))𝑟) ⊆ 𝑢 → ∃𝑛 ∈ ℕ ∀𝑘 ∈ (1...𝑛)∃𝑢 ∈ 𝑈 (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛)) ⊆ 𝑢))
122	13, 121	mpd 15	1 ⊢ ((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) → ∃𝑛 ∈ ℕ ∀𝑘 ∈ (1...𝑛)∃𝑢 ∈ 𝑈 (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛)) ⊆ 𝑢)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ∃wrex 3070 ∩ cin 3950 ⊆ wss 3951 ∪ cuni 4907 class class class wbr 5143 × cxp 5683 ↾ cres 5687 ∘ ccom 5689 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 ℝcr 11154 0cc0 11155 1c1 11156 + caddc 11158 · cmul 11160 ℝ^*cxr 11294 < clt 11295 ≤ cle 11296 − cmin 11492 / cdiv 11920 ℕcn 12266 ℕ₀cn0 12526 ℝ⁺crp 13034 (,)cioo 13387 [,]cicc 13390 ...cfz 13547 ⌊cfl 13830 abscabs 15273 ∞Metcxmet 21349 Metcmet 21350 ballcbl 21351 Compccmp 23394 IIcii 24901

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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-ec 8747 df-map 8868 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-fi 9451 df-sup 9482 df-inf 9483 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ioo 13391 df-ico 13393 df-icc 13394 df-fz 13548 df-fzo 13695 df-fl 13832 df-seq 14043 df-exp 14103 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-clim 15524 df-sum 15723 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-rest 17467 df-topn 17468 df-0g 17486 df-gsum 17487 df-topgen 17488 df-pt 17489 df-prds 17492 df-xrs 17547 df-qtop 17552 df-imas 17553 df-xps 17555 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-mulg 19086 df-cntz 19335 df-cmn 19800 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-cnfld 21365 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-cld 23027 df-ntr 23028 df-cls 23029 df-cn 23235 df-cnp 23236 df-cmp 23395 df-tx 23570 df-hmeo 23763 df-xms 24330 df-ms 24331 df-tms 24332 df-ii 24903

This theorem is referenced by: cvmliftlem15 35303

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