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Theorem lebnumii 25125

Description: Specialize the Lebesgue number lemma lebnum 25123 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 25036	. . 3 ⊢ II = (MetOpen‘((abs ∘ − ) ↾ ((0[,]1) × (0[,]1))))
2		cnmet 24928	. . . . 5 ⊢ (abs ∘ − ) ∈ (Met‘ℂ)
3		unitssre 13521	. . . . . 6 ⊢ (0[,]1) ⊆ ℝ
4		ax-resscn 11152	. . . . . 6 ⊢ ℝ ⊆ ℂ
5	3, 4	sstri 3946	. . . . 5 ⊢ (0[,]1) ⊆ ℂ
6		metres2 24520	. . . . 5 ⊢ (((abs ∘ − ) ∈ (Met‘ℂ) ∧ (0[,]1) ⊆ ℂ) → ((abs ∘ − ) ↾ ((0[,]1) × (0[,]1))) ∈ (Met‘(0[,]1)))
7	2, 5, 6	mp2an 704	. . . 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 25045	. . . 4 ⊢ II ∈ Comp
10	9	a1i 11	. . 3 ⊢ ((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) → II ∈ Comp)
11		simpl 487	. . 3 ⊢ ((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) → 𝑈 ⊆ II)
12		simpr 489	. . 3 ⊢ ((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) → (0[,]1) = ∪ 𝑈)
13	1, 8, 10, 11, 12	lebnum 25123	. 2 ⊢ ((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) → ∃𝑟 ∈ ℝ⁺ ∀𝑥 ∈ (0[,]1)∃𝑢 ∈ 𝑈 (𝑥(ball‘((abs ∘ − ) ↾ ((0[,]1) × (0[,]1))))𝑟) ⊆ 𝑢)
14		rpreccl 13039	. . . . . . . 8 ⊢ (𝑟 ∈ ℝ⁺ → (1 / 𝑟) ∈ ℝ⁺)
15	14	adantl 486	. . . . . . 7 ⊢ (((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) → (1 / 𝑟) ∈ ℝ⁺)
16	15	rpred 13055	. . . . . 6 ⊢ (((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) → (1 / 𝑟) ∈ ℝ)
17	15	rpge0d 13059	. . . . . 6 ⊢ (((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) → 0 ≤ (1 / 𝑟))
18		flge0nn0 13849	. . . . . 6 ⊢ (((1 / 𝑟) ∈ ℝ ∧ 0 ≤ (1 / 𝑟)) → (⌊‘(1 / 𝑟)) ∈ ℕ₀)
19	16, 17, 18	syl2anc 595	. . . . 5 ⊢ (((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) → (⌊‘(1 / 𝑟)) ∈ ℕ₀)
20		nn0p1nn 12538	. . . . 5 ⊢ ((⌊‘(1 / 𝑟)) ∈ ℕ₀ → ((⌊‘(1 / 𝑟)) + 1) ∈ ℕ)
21	19, 20	syl 18	. . . 4 ⊢ (((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) → ((⌊‘(1 / 𝑟)) + 1) ∈ ℕ)
22		elfznn 13577	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1)) → 𝑘 ∈ ℕ)
23	22	adantl 486	. . . . . . . . . . 11 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → 𝑘 ∈ ℕ)
24	23	nnrpd 13053	. . . . . . . . . 10 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → 𝑘 ∈ ℝ⁺)
25	21	adantr 485	. . . . . . . . . . 11 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → ((⌊‘(1 / 𝑟)) + 1) ∈ ℕ)
26	25	nnrpd 13053	. . . . . . . . . 10 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → ((⌊‘(1 / 𝑟)) + 1) ∈ ℝ⁺)
27	24, 26	rpdivcld 13072	. . . . . . . . 9 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → (𝑘 / ((⌊‘(1 / 𝑟)) + 1)) ∈ ℝ⁺)
28	27	rpred 13055	. . . . . . . 8 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → (𝑘 / ((⌊‘(1 / 𝑟)) + 1)) ∈ ℝ)
29	27	rpge0d 13059	. . . . . . . 8 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → 0 ≤ (𝑘 / ((⌊‘(1 / 𝑟)) + 1)))
30		elfzle2 13551	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1)) → 𝑘 ≤ ((⌊‘(1 / 𝑟)) + 1))
31	30	adantl 486	. . . . . . . . . 10 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → 𝑘 ≤ ((⌊‘(1 / 𝑟)) + 1))
32	25	nnred 12243	. . . . . . . . . . . 12 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → ((⌊‘(1 / 𝑟)) + 1) ∈ ℝ)
33	32	recnd 11232	. . . . . . . . . . 11 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → ((⌊‘(1 / 𝑟)) + 1) ∈ ℂ)
34	33	mulridd 11221	. . . . . . . . . 10 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → (((⌊‘(1 / 𝑟)) + 1) · 1) = ((⌊‘(1 / 𝑟)) + 1))
35	31, 34	breqtrrd 5139	. . . . . . . . 9 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → 𝑘 ≤ (((⌊‘(1 / 𝑟)) + 1) · 1))
36	23	nnred 12243	. . . . . . . . . 10 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → 𝑘 ∈ ℝ)
37		1red 11204	. . . . . . . . . 10 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → 1 ∈ ℝ)
38	25	nngt0d 12280	. . . . . . . . . 10 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → 0 < ((⌊‘(1 / 𝑟)) + 1))
39		ledivmul 12086	. . . . . . . . . 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 1401	. . . . . . . . 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 260	. . . . . . . 8 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → (𝑘 / ((⌊‘(1 / 𝑟)) + 1)) ≤ 1)
42		elicc01 13488	. . . . . . . 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 1362	. . . . . . 7 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → (𝑘 / ((⌊‘(1 / 𝑟)) + 1)) ∈ (0[,]1))
44		oveq1 7417	. . . . . . . . . 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 3968	. . . . . . . . 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 3189	. . . . . . . 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 3577	. . . . . . 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 18	. . . . . 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 780	. . . . . . . . . . . . . 14 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → 𝑟 ∈ ℝ⁺)
50	49	rpred 13055	. . . . . . . . . . . . 13 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → 𝑟 ∈ ℝ)
51	28, 50	resubcld 11637	. . . . . . . . . . . 12 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → ((𝑘 / ((⌊‘(1 / 𝑟)) + 1)) − 𝑟) ∈ ℝ)
52	51	rexrd 11254	. . . . . . . . . . 11 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → ((𝑘 / ((⌊‘(1 / 𝑟)) + 1)) − 𝑟) ∈ ℝ^*)
53	28, 50	readdcld 11233	. . . . . . . . . . . 12 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → ((𝑘 / ((⌊‘(1 / 𝑟)) + 1)) + 𝑟) ∈ ℝ)
54	53	rexrd 11254	. . . . . . . . . . 11 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → ((𝑘 / ((⌊‘(1 / 𝑟)) + 1)) + 𝑟) ∈ ℝ^*)
55		nnm1nn0 12540	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℕ → (𝑘 − 1) ∈ ℕ₀)
56	23, 55	syl 18	. . . . . . . . . . . . . 14 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → (𝑘 − 1) ∈ ℕ₀)
57	56	nn0red 12561	. . . . . . . . . . . . 13 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → (𝑘 − 1) ∈ ℝ)
58	57, 25	nndivred 12285	. . . . . . . . . . . 12 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → ((𝑘 − 1) / ((⌊‘(1 / 𝑟)) + 1)) ∈ ℝ)
59	36	recnd 11232	. . . . . . . . . . . . . . 15 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → 𝑘 ∈ ℂ)
60	57	recnd 11232	. . . . . . . . . . . . . . 15 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → (𝑘 − 1) ∈ ℂ)
61	25	nnne0d 12281	. . . . . . . . . . . . . . 15 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → ((⌊‘(1 / 𝑟)) + 1) ≠ 0)
62	59, 60, 33, 61	divsubdird 12025	. . . . . . . . . . . . . 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 11153	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℂ
64		nncan 11482	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ ℂ ∧ 1 ∈ ℂ) → (𝑘 − (𝑘 − 1)) = 1)
65	59, 63, 64	sylancl 597	. . . . . . . . . . . . . . 15 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → (𝑘 − (𝑘 − 1)) = 1)
66	65	oveq1d 7425	. . . . . . . . . . . . . 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 2800	. . . . . . . . . . . . 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 13066	. . . . . . . . . . . . . . 15 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → (1 / 𝑟) ∈ ℝ)
69		flltp1 13829	. . . . . . . . . . . . . . 15 ⊢ ((1 / 𝑟) ∈ ℝ → (1 / 𝑟) < ((⌊‘(1 / 𝑟)) + 1))
70	68, 69	syl 18	. . . . . . . . . . . . . 14 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → (1 / 𝑟) < ((⌊‘(1 / 𝑟)) + 1))
71		rpgt0 13024	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 ∈ ℝ⁺ → 0 < 𝑟)
72	71	ad2antlr 739	. . . . . . . . . . . . . . 15 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → 0 < 𝑟)
73		ltdiv23 12101	. . . . . . . . . . . . . . 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 1406	. . . . . . . . . . . . . 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 235	. . . . . . . . . . . . 13 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → (1 / ((⌊‘(1 / 𝑟)) + 1)) < 𝑟)
76	67, 75	eqbrtrd 5133	. . . . . . . . . . . 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 11814	. . . . . . . . . . 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 13088	. . . . . . . . . . 11 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → (𝑘 / ((⌊‘(1 / 𝑟)) + 1)) < ((𝑘 / ((⌊‘(1 / 𝑟)) + 1)) + 𝑟))
79		iccssioo 13437	. . . . . . . . . . 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 851	. . . . . . . . . 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 11206	. . . . . . . . . . 11 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → 0 ∈ ℝ)
82	56	nn0ge0d 12563	. . . . . . . . . . . 12 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → 0 ≤ (𝑘 − 1))
83		divge0 12079	. . . . . . . . . . . 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 851	. . . . . . . . . . 11 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → 0 ≤ ((𝑘 − 1) / ((⌊‘(1 / 𝑟)) + 1)))
85		iccss 13436	. . . . . . . . . . 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 851	. . . . . . . . . 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 4193	. . . . . . . . 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 2763	. . . . . . . . . . . 12 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) = ((abs ∘ − ) ↾ (ℝ × ℝ))
89	88	rexmet 24948	. . . . . . . . . . 11 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) ∈ (∞Met‘ℝ)
90		sseqin2 4176	. . . . . . . . . . . . 13 ⊢ ((0[,]1) ⊆ ℝ ↔ (ℝ ∩ (0[,]1)) = (0[,]1))
91	3, 90	mpbi 233	. . . . . . . . . . . 12 ⊢ (ℝ ∩ (0[,]1)) = (0[,]1)
92	43, 91	eleqtrrdi 2874	. . . . . . . . . . 11 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → (𝑘 / ((⌊‘(1 / 𝑟)) + 1)) ∈ (ℝ ∩ (0[,]1)))
93		rpxr 13021	. . . . . . . . . . . 12 ⊢ (𝑟 ∈ ℝ⁺ → 𝑟 ∈ ℝ^*)
94	93	ad2antlr 739	. . . . . . . . . . 11 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → 𝑟 ∈ ℝ^*)
95		xpss12 5676	. . . . . . . . . . . . . . 15 ⊢ (((0[,]1) ⊆ ℝ ∧ (0[,]1) ⊆ ℝ) → ((0[,]1) × (0[,]1)) ⊆ (ℝ × ℝ))
96	3, 3, 95	mp2an 704	. . . . . . . . . . . . . 14 ⊢ ((0[,]1) × (0[,]1)) ⊆ (ℝ × ℝ)
97		resabs1 6005	. . . . . . . . . . . . . 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 2772	. . . . . . . . . . . 12 ⊢ ((abs ∘ − ) ↾ ((0[,]1) × (0[,]1))) = (((abs ∘ − ) ↾ (ℝ × ℝ)) ↾ ((0[,]1) × (0[,]1)))
100	99	blres 24588	. . . . . . . . . . 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 1495	. . . . . . . . . 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 24949	. . . . . . . . . . . 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 595	. . . . . . . . . . 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 4172	. . . . . . . . . 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 2798	. . . . . . . . 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 3974	. . . . . . . 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 3944	. . . . . . . 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 18	. . . . . . 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 3180	. . . . . 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 48	. . . . 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 3165	. . . 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 7418	. . . . . 6 ⊢ (𝑛 = ((⌊‘(1 / 𝑟)) + 1) → (1...𝑛) = (1...((⌊‘(1 / 𝑟)) + 1)))
113		oveq2 7418	. . . . . . . . 9 ⊢ (𝑛 = ((⌊‘(1 / 𝑟)) + 1) → ((𝑘 − 1) / 𝑛) = ((𝑘 − 1) / ((⌊‘(1 / 𝑟)) + 1)))
114		oveq2 7418	. . . . . . . . 9 ⊢ (𝑛 = ((⌊‘(1 / 𝑟)) + 1) → (𝑘 / 𝑛) = (𝑘 / ((⌊‘(1 / 𝑟)) + 1)))
115	113, 114	oveq12d 7428	. . . . . . . 8 ⊢ (𝑛 = ((⌊‘(1 / 𝑟)) + 1) → (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛)) = (((𝑘 − 1) / ((⌊‘(1 / 𝑟)) + 1))[,](𝑘 / ((⌊‘(1 / 𝑟)) + 1))))
116	115	sseq1d 3968	. . . . . . 7 ⊢ (𝑛 = ((⌊‘(1 / 𝑟)) + 1) → ((((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛)) ⊆ 𝑢 ↔ (((𝑘 − 1) / ((⌊‘(1 / 𝑟)) + 1))[,](𝑘 / ((⌊‘(1 / 𝑟)) + 1))) ⊆ 𝑢))
117	116	rexbidv 3189	. . . . . 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 3338	. . . . 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 3581	. . . 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 696	. . 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 3166	. 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 16	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 209 ∧ wa 400 = wceq 1570 ∈ wcel 2143 ∀wral 3079 ∃wrex 3089 ∩ cin 3904 ⊆ wss 3905 ∪ cuni 4872 class class class wbr 5109 × cxp 5659 ↾ cres 5663 ∘ ccom 5665 ‘cfv 6536 (class class class)co 7410 ℂcc 11093 ℝcr 11094 0cc0 11095 1c1 11096 + caddc 11098 · cmul 11100 ℝ^*cxr 11237 < clt 11238 ≤ cle 11239 − cmin 11436 / cdiv 11866 ℕcn 12228 ℕ₀cn0 12499 ℝ⁺crp 13011 (,)cioo 13367 [,]cicc 13370 ...cfz 13530 ⌊cfl 13819 abscabs 15281 ∞Metcxmet 21507 Metcmet 21508 ballcbl 21509 Compccmp 23543 IIcii 25034

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5238 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-inf2 9606 ax-cnex 11151 ax-resscn 11152 ax-1cn 11153 ax-icn 11154 ax-addcl 11155 ax-addrcl 11156 ax-mulcl 11157 ax-mulrcl 11158 ax-mulcom 11159 ax-addass 11160 ax-mulass 11161 ax-distr 11162 ax-i2m1 11163 ax-1ne0 11164 ax-1rid 11165 ax-rnegex 11166 ax-rrecex 11167 ax-cnre 11168 ax-pre-lttri 11169 ax-pre-lttrn 11170 ax-pre-ltadd 11171 ax-pre-mulgt0 11172 ax-pre-sup 11173 ax-addf 11174

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4873 df-int 4913 df-iun 4958 df-iin 4959 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-se 5615 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-of 7674 df-om 7859 df-1st 7982 df-2nd 7983 df-supp 8153 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-2o 8450 df-er 8690 df-ec 8692 df-map 8822 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9318 df-fi 9367 df-sup 9398 df-inf 9399 df-oi 9468 df-card 9921 df-pnf 11240 df-mnf 11241 df-xr 11242 df-ltxr 11243 df-le 11244 df-sub 11438 df-neg 11439 df-div 11867 df-nn 12229 df-2 12298 df-3 12299 df-4 12300 df-5 12301 df-6 12302 df-7 12303 df-8 12304 df-9 12305 df-n0 12500 df-z 12587 df-dec 12707 df-uz 12858 df-q 12968 df-rp 13012 df-xneg 13132 df-xadd 13133 df-xmul 13134 df-ioo 13371 df-ico 13373 df-icc 13374 df-fz 13531 df-fzo 13679 df-fl 13821 df-seq 14034 df-exp 14094 df-hash 14363 df-cj 15146 df-re 15147 df-im 15148 df-sqrt 15282 df-abs 15283 df-clim 15535 df-sum 15734 df-struct 17202 df-sets 17219 df-slot 17237 df-ndx 17249 df-base 17265 df-ress 17286 df-plusg 17318 df-mulr 17319 df-starv 17320 df-sca 17321 df-vsca 17322 df-ip 17323 df-tset 17324 df-ple 17325 df-ds 17327 df-unif 17328 df-hom 17329 df-cco 17330 df-rest 17470 df-topn 17471 df-0g 17489 df-gsum 17490 df-topgen 17491 df-pt 17492 df-prds 17495 df-xrs 17551 df-qtop 17556 df-imas 17557 df-xps 17559 df-mre 17633 df-mrc 17634 df-acs 17636 df-mgm 18693 df-sgrp 18772 df-mnd 18788 df-submnd 18837 df-mulg 19129 df-cntz 19382 df-cmn 19847 df-psmet 21514 df-xmet 21515 df-met 21516 df-bl 21517 df-mopn 21518 df-cnfld 21523 df-top 23051 df-topon 23068 df-topsp 23090 df-bases 23103 df-cld 23176 df-ntr 23177 df-cls 23178 df-cn 23384 df-cnp 23385 df-cmp 23544 df-tx 23719 df-hmeo 23912 df-xms 24477 df-ms 24478 df-tms 24479 df-ii 25036

This theorem is referenced by: cvmliftlem15 35790

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