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Theorem lebnumii 24473

Description: Specialize the Lebesgue number lemma lebnum 24471 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 24384	. . 3 ⊢ II = (MetOpen‘((abs ∘ − ) ↾ ((0[,]1) × (0[,]1))))
2		cnmet 24279	. . . . 5 ⊢ (abs ∘ − ) ∈ (Met‘ℂ)
3		unitssre 13472	. . . . . 6 ⊢ (0[,]1) ⊆ ℝ
4		ax-resscn 11163	. . . . . 6 ⊢ ℝ ⊆ ℂ
5	3, 4	sstri 3990	. . . . 5 ⊢ (0[,]1) ⊆ ℂ
6		metres2 23860	. . . . 5 ⊢ (((abs ∘ − ) ∈ (Met‘ℂ) ∧ (0[,]1) ⊆ ℂ) → ((abs ∘ − ) ↾ ((0[,]1) × (0[,]1))) ∈ (Met‘(0[,]1)))
7	2, 5, 6	mp2an 690	. . . 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 24393	. . . 4 ⊢ II ∈ Comp
10	9	a1i 11	. . 3 ⊢ ((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) → II ∈ Comp)
11		simpl 483	. . 3 ⊢ ((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) → 𝑈 ⊆ II)
12		simpr 485	. . 3 ⊢ ((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) → (0[,]1) = ∪ 𝑈)
13	1, 8, 10, 11, 12	lebnum 24471	. 2 ⊢ ((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) → ∃𝑟 ∈ ℝ⁺ ∀𝑥 ∈ (0[,]1)∃𝑢 ∈ 𝑈 (𝑥(ball‘((abs ∘ − ) ↾ ((0[,]1) × (0[,]1))))𝑟) ⊆ 𝑢)
14		rpreccl 12996	. . . . . . . 8 ⊢ (𝑟 ∈ ℝ⁺ → (1 / 𝑟) ∈ ℝ⁺)
15	14	adantl 482	. . . . . . 7 ⊢ (((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) → (1 / 𝑟) ∈ ℝ⁺)
16	15	rpred 13012	. . . . . 6 ⊢ (((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) → (1 / 𝑟) ∈ ℝ)
17	15	rpge0d 13016	. . . . . 6 ⊢ (((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) → 0 ≤ (1 / 𝑟))
18		flge0nn0 13781	. . . . . 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 12507	. . . . 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 13526	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1)) → 𝑘 ∈ ℕ)
23	22	adantl 482	. . . . . . . . . . 11 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → 𝑘 ∈ ℕ)
24	23	nnrpd 13010	. . . . . . . . . 10 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → 𝑘 ∈ ℝ⁺)
25	21	adantr 481	. . . . . . . . . . 11 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → ((⌊‘(1 / 𝑟)) + 1) ∈ ℕ)
26	25	nnrpd 13010	. . . . . . . . . 10 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → ((⌊‘(1 / 𝑟)) + 1) ∈ ℝ⁺)
27	24, 26	rpdivcld 13029	. . . . . . . . 9 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → (𝑘 / ((⌊‘(1 / 𝑟)) + 1)) ∈ ℝ⁺)
28	27	rpred 13012	. . . . . . . 8 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → (𝑘 / ((⌊‘(1 / 𝑟)) + 1)) ∈ ℝ)
29	27	rpge0d 13016	. . . . . . . 8 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → 0 ≤ (𝑘 / ((⌊‘(1 / 𝑟)) + 1)))
30		elfzle2 13501	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1)) → 𝑘 ≤ ((⌊‘(1 / 𝑟)) + 1))
31	30	adantl 482	. . . . . . . . . 10 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → 𝑘 ≤ ((⌊‘(1 / 𝑟)) + 1))
32	25	nnred 12223	. . . . . . . . . . . 12 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → ((⌊‘(1 / 𝑟)) + 1) ∈ ℝ)
33	32	recnd 11238	. . . . . . . . . . 11 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → ((⌊‘(1 / 𝑟)) + 1) ∈ ℂ)
34	33	mulridd 11227	. . . . . . . . . 10 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → (((⌊‘(1 / 𝑟)) + 1) · 1) = ((⌊‘(1 / 𝑟)) + 1))
35	31, 34	breqtrrd 5175	. . . . . . . . 9 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → 𝑘 ≤ (((⌊‘(1 / 𝑟)) + 1) · 1))
36	23	nnred 12223	. . . . . . . . . 10 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → 𝑘 ∈ ℝ)
37		1red 11211	. . . . . . . . . 10 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → 1 ∈ ℝ)
38	25	nngt0d 12257	. . . . . . . . . 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 1374	. . . . . . . . 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 256	. . . . . . . 8 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → (𝑘 / ((⌊‘(1 / 𝑟)) + 1)) ≤ 1)
42		elicc01 13439	. . . . . . . 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 1343	. . . . . . 7 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → (𝑘 / ((⌊‘(1 / 𝑟)) + 1)) ∈ (0[,]1))
44		oveq1 7412	. . . . . . . . . 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 4012	. . . . . . . . 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 3178	. . . . . . . 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 3608	. . . . . . 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 767	. . . . . . . . . . . . . 14 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → 𝑟 ∈ ℝ⁺)
50	49	rpred 13012	. . . . . . . . . . . . 13 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → 𝑟 ∈ ℝ)
51	28, 50	resubcld 11638	. . . . . . . . . . . 12 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → ((𝑘 / ((⌊‘(1 / 𝑟)) + 1)) − 𝑟) ∈ ℝ)
52	51	rexrd 11260	. . . . . . . . . . 11 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → ((𝑘 / ((⌊‘(1 / 𝑟)) + 1)) − 𝑟) ∈ ℝ^*)
53	28, 50	readdcld 11239	. . . . . . . . . . . 12 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → ((𝑘 / ((⌊‘(1 / 𝑟)) + 1)) + 𝑟) ∈ ℝ)
54	53	rexrd 11260	. . . . . . . . . . 11 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → ((𝑘 / ((⌊‘(1 / 𝑟)) + 1)) + 𝑟) ∈ ℝ^*)
55		nnm1nn0 12509	. . . . . . . . . . . . . . 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 12529	. . . . . . . . . . . . 13 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → (𝑘 − 1) ∈ ℝ)
58	57, 25	nndivred 12262	. . . . . . . . . . . 12 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → ((𝑘 − 1) / ((⌊‘(1 / 𝑟)) + 1)) ∈ ℝ)
59	36	recnd 11238	. . . . . . . . . . . . . . 15 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → 𝑘 ∈ ℂ)
60	57	recnd 11238	. . . . . . . . . . . . . . 15 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → (𝑘 − 1) ∈ ℂ)
61	25	nnne0d 12258	. . . . . . . . . . . . . . 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 11164	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℂ
64		nncan 11485	. . . . . . . . . . . . . . . 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 7420	. . . . . . . . . . . . . 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 2774	. . . . . . . . . . . . 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 13023	. . . . . . . . . . . . . . 15 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → (1 / 𝑟) ∈ ℝ)
69		flltp1 13761	. . . . . . . . . . . . . . 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 12982	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 ∈ ℝ⁺ → 0 < 𝑟)
72	71	ad2antlr 725	. . . . . . . . . . . . . . 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 1379	. . . . . . . . . . . . . 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 231	. . . . . . . . . . . . 13 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → (1 / ((⌊‘(1 / 𝑟)) + 1)) < 𝑟)
76	67, 75	eqbrtrd 5169	. . . . . . . . . . . 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 11815	. . . . . . . . . . 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 13045	. . . . . . . . . . 11 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → (𝑘 / ((⌊‘(1 / 𝑟)) + 1)) < ((𝑘 / ((⌊‘(1 / 𝑟)) + 1)) + 𝑟))
79		iccssioo 13389	. . . . . . . . . . 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 837	. . . . . . . . . 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 11213	. . . . . . . . . . 11 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → 0 ∈ ℝ)
82	56	nn0ge0d 12531	. . . . . . . . . . . 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 837	. . . . . . . . . . 11 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → 0 ≤ ((𝑘 − 1) / ((⌊‘(1 / 𝑟)) + 1)))
85		iccss 13388	. . . . . . . . . . 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 837	. . . . . . . . . 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 4231	. . . . . . . . 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 2732	. . . . . . . . . . . 12 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) = ((abs ∘ − ) ↾ (ℝ × ℝ))
89	88	rexmet 24298	. . . . . . . . . . 11 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) ∈ (∞Met‘ℝ)
90		sseqin2 4214	. . . . . . . . . . . . 13 ⊢ ((0[,]1) ⊆ ℝ ↔ (ℝ ∩ (0[,]1)) = (0[,]1))
91	3, 90	mpbi 229	. . . . . . . . . . . 12 ⊢ (ℝ ∩ (0[,]1)) = (0[,]1)
92	43, 91	eleqtrrdi 2844	. . . . . . . . . . 11 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → (𝑘 / ((⌊‘(1 / 𝑟)) + 1)) ∈ (ℝ ∩ (0[,]1)))
93		rpxr 12979	. . . . . . . . . . . 12 ⊢ (𝑟 ∈ ℝ⁺ → 𝑟 ∈ ℝ^*)
94	93	ad2antlr 725	. . . . . . . . . . 11 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → 𝑟 ∈ ℝ^*)
95		xpss12 5690	. . . . . . . . . . . . . . 15 ⊢ (((0[,]1) ⊆ ℝ ∧ (0[,]1) ⊆ ℝ) → ((0[,]1) × (0[,]1)) ⊆ (ℝ × ℝ))
96	3, 3, 95	mp2an 690	. . . . . . . . . . . . . 14 ⊢ ((0[,]1) × (0[,]1)) ⊆ (ℝ × ℝ)
97		resabs1 6009	. . . . . . . . . . . . . 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 2741	. . . . . . . . . . . 12 ⊢ ((abs ∘ − ) ↾ ((0[,]1) × (0[,]1))) = (((abs ∘ − ) ↾ (ℝ × ℝ)) ↾ ((0[,]1) × (0[,]1)))
100	99	blres 23928	. . . . . . . . . . 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 1466	. . . . . . . . . 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 24299	. . . . . . . . . . . 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 4210	. . . . . . . . . 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 2772	. . . . . . . . 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 4022	. . . . . . . 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 3988	. . . . . . . 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 7413	. . . . . 6 ⊢ (𝑛 = ((⌊‘(1 / 𝑟)) + 1) → (1...𝑛) = (1...((⌊‘(1 / 𝑟)) + 1)))
113		oveq2 7413	. . . . . . . . 9 ⊢ (𝑛 = ((⌊‘(1 / 𝑟)) + 1) → ((𝑘 − 1) / 𝑛) = ((𝑘 − 1) / ((⌊‘(1 / 𝑟)) + 1)))
114		oveq2 7413	. . . . . . . . 9 ⊢ (𝑛 = ((⌊‘(1 / 𝑟)) + 1) → (𝑘 / 𝑛) = (𝑘 / ((⌊‘(1 / 𝑟)) + 1)))
115	113, 114	oveq12d 7423	. . . . . . . 8 ⊢ (𝑛 = ((⌊‘(1 / 𝑟)) + 1) → (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛)) = (((𝑘 − 1) / ((⌊‘(1 / 𝑟)) + 1))[,](𝑘 / ((⌊‘(1 / 𝑟)) + 1))))
116	115	sseq1d 4012	. . . . . . 7 ⊢ (𝑛 = ((⌊‘(1 / 𝑟)) + 1) → ((((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛)) ⊆ 𝑢 ↔ (((𝑘 − 1) / ((⌊‘(1 / 𝑟)) + 1))[,](𝑘 / ((⌊‘(1 / 𝑟)) + 1))) ⊆ 𝑢))
117	116	rexbidv 3178	. . . . . 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 3342	. . . . 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 3612	. . . 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 682	. . 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 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3061 ∃wrex 3070 ∩ cin 3946 ⊆ wss 3947 ∪ cuni 4907 class class class wbr 5147 × cxp 5673 ↾ cres 5677 ∘ ccom 5679 ‘cfv 6540 (class class class)co 7405 ℂcc 11104 ℝcr 11105 0cc0 11106 1c1 11107 + caddc 11109 · cmul 11111 ℝ^*cxr 11243 < clt 11244 ≤ cle 11245 − cmin 11440 / cdiv 11867 ℕcn 12208 ℕ₀cn0 12468 ℝ⁺crp 12970 (,)cioo 13320 [,]cicc 13323 ...cfz 13480 ⌊cfl 13751 abscabs 15177 ∞Metcxmet 20921 Metcmet 20922 ballcbl 20923 Compccmp 22881 IIcii 24382

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 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-inf2 9632 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185 ax-mulf 11186

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 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8143 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-er 8699 df-ec 8701 df-map 8818 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-fi 9402 df-sup 9433 df-inf 9434 df-oi 9501 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-ioo 13324 df-ico 13326 df-icc 13327 df-fz 13481 df-fzo 13624 df-fl 13753 df-seq 13963 df-exp 14024 df-hash 14287 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-clim 15428 df-sum 15629 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-starv 17208 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-unif 17216 df-hom 17217 df-cco 17218 df-rest 17364 df-topn 17365 df-0g 17383 df-gsum 17384 df-topgen 17385 df-pt 17386 df-prds 17389 df-xrs 17444 df-qtop 17449 df-imas 17450 df-xps 17452 df-mre 17526 df-mrc 17527 df-acs 17529 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-submnd 18668 df-mulg 18945 df-cntz 19175 df-cmn 19644 df-psmet 20928 df-xmet 20929 df-met 20930 df-bl 20931 df-mopn 20932 df-cnfld 20937 df-top 22387 df-topon 22404 df-topsp 22426 df-bases 22440 df-cld 22514 df-ntr 22515 df-cls 22516 df-cn 22722 df-cnp 22723 df-cmp 22882 df-tx 23057 df-hmeo 23250 df-xms 23817 df-ms 23818 df-tms 23819 df-ii 24384

This theorem is referenced by: cvmliftlem15 34277

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