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Theorem lebnumii 23184

Description: Specialize the Lebesgue number lemma lebnum 23182 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 23099	. . 3 ⊢ II = (MetOpen‘((abs ∘ − ) ↾ ((0[,]1) × (0[,]1))))
2		cnmet 22994	. . . . 5 ⊢ (abs ∘ − ) ∈ (Met‘ℂ)
3		unitssre 12641	. . . . . 6 ⊢ (0[,]1) ⊆ ℝ
4		ax-resscn 10331	. . . . . 6 ⊢ ℝ ⊆ ℂ
5	3, 4	sstri 3830	. . . . 5 ⊢ (0[,]1) ⊆ ℂ
6		metres2 22587	. . . . 5 ⊢ (((abs ∘ − ) ∈ (Met‘ℂ) ∧ (0[,]1) ⊆ ℂ) → ((abs ∘ − ) ↾ ((0[,]1) × (0[,]1))) ∈ (Met‘(0[,]1)))
7	2, 5, 6	mp2an 682	. . . 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 23108	. . . 4 ⊢ II ∈ Comp
10	9	a1i 11	. . 3 ⊢ ((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) → II ∈ Comp)
11		simpl 476	. . 3 ⊢ ((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) → 𝑈 ⊆ II)
12		simpr 479	. . 3 ⊢ ((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) → (0[,]1) = ∪ 𝑈)
13	1, 8, 10, 11, 12	lebnum 23182	. 2 ⊢ ((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) → ∃𝑟 ∈ ℝ⁺ ∀𝑥 ∈ (0[,]1)∃𝑢 ∈ 𝑈 (𝑥(ball‘((abs ∘ − ) ↾ ((0[,]1) × (0[,]1))))𝑟) ⊆ 𝑢)
14		rpreccl 12170	. . . . . . . 8 ⊢ (𝑟 ∈ ℝ⁺ → (1 / 𝑟) ∈ ℝ⁺)
15	14	adantl 475	. . . . . . 7 ⊢ (((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) → (1 / 𝑟) ∈ ℝ⁺)
16	15	rpred 12186	. . . . . 6 ⊢ (((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) → (1 / 𝑟) ∈ ℝ)
17	15	rpge0d 12190	. . . . . 6 ⊢ (((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) → 0 ≤ (1 / 𝑟))
18		flge0nn0 12945	. . . . . 6 ⊢ (((1 / 𝑟) ∈ ℝ ∧ 0 ≤ (1 / 𝑟)) → (⌊‘(1 / 𝑟)) ∈ ℕ₀)
19	16, 17, 18	syl2anc 579	. . . . 5 ⊢ (((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) → (⌊‘(1 / 𝑟)) ∈ ℕ₀)
20		nn0p1nn 11688	. . . . 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 12692	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1)) → 𝑘 ∈ ℕ)
23	22	adantl 475	. . . . . . . . . . 11 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → 𝑘 ∈ ℕ)
24	23	nnrpd 12184	. . . . . . . . . 10 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → 𝑘 ∈ ℝ⁺)
25	21	adantr 474	. . . . . . . . . . 11 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → ((⌊‘(1 / 𝑟)) + 1) ∈ ℕ)
26	25	nnrpd 12184	. . . . . . . . . 10 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → ((⌊‘(1 / 𝑟)) + 1) ∈ ℝ⁺)
27	24, 26	rpdivcld 12203	. . . . . . . . 9 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → (𝑘 / ((⌊‘(1 / 𝑟)) + 1)) ∈ ℝ⁺)
28	27	rpred 12186	. . . . . . . 8 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → (𝑘 / ((⌊‘(1 / 𝑟)) + 1)) ∈ ℝ)
29	27	rpge0d 12190	. . . . . . . 8 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → 0 ≤ (𝑘 / ((⌊‘(1 / 𝑟)) + 1)))
30		elfzle2 12667	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1)) → 𝑘 ≤ ((⌊‘(1 / 𝑟)) + 1))
31	30	adantl 475	. . . . . . . . . 10 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → 𝑘 ≤ ((⌊‘(1 / 𝑟)) + 1))
32	25	nnred 11396	. . . . . . . . . . . 12 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → ((⌊‘(1 / 𝑟)) + 1) ∈ ℝ)
33	32	recnd 10407	. . . . . . . . . . 11 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → ((⌊‘(1 / 𝑟)) + 1) ∈ ℂ)
34	33	mulid1d 10396	. . . . . . . . . 10 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → (((⌊‘(1 / 𝑟)) + 1) · 1) = ((⌊‘(1 / 𝑟)) + 1))
35	31, 34	breqtrrd 4916	. . . . . . . . 9 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → 𝑘 ≤ (((⌊‘(1 / 𝑟)) + 1) · 1))
36	23	nnred 11396	. . . . . . . . . 10 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → 𝑘 ∈ ℝ)
37		1red 10379	. . . . . . . . . 10 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → 1 ∈ ℝ)
38	25	nngt0d 11429	. . . . . . . . . 10 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → 0 < ((⌊‘(1 / 𝑟)) + 1))
39		ledivmul 11256	. . . . . . . . . 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 1442	. . . . . . . . 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 249	. . . . . . . 8 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → (𝑘 / ((⌊‘(1 / 𝑟)) + 1)) ≤ 1)
42		elicc01 12609	. . . . . . . 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 1400	. . . . . . 7 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → (𝑘 / ((⌊‘(1 / 𝑟)) + 1)) ∈ (0[,]1))
44		oveq1 6931	. . . . . . . . . 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 3851	. . . . . . . . 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 3237	. . . . . . . 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 3507	. . . . . . 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 759	. . . . . . . . . . . . . 14 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → 𝑟 ∈ ℝ⁺)
50	49	rpred 12186	. . . . . . . . . . . . 13 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → 𝑟 ∈ ℝ)
51	28, 50	resubcld 10806	. . . . . . . . . . . 12 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → ((𝑘 / ((⌊‘(1 / 𝑟)) + 1)) − 𝑟) ∈ ℝ)
52	51	rexrd 10428	. . . . . . . . . . 11 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → ((𝑘 / ((⌊‘(1 / 𝑟)) + 1)) − 𝑟) ∈ ℝ^*)
53	28, 50	readdcld 10408	. . . . . . . . . . . 12 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → ((𝑘 / ((⌊‘(1 / 𝑟)) + 1)) + 𝑟) ∈ ℝ)
54	53	rexrd 10428	. . . . . . . . . . 11 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → ((𝑘 / ((⌊‘(1 / 𝑟)) + 1)) + 𝑟) ∈ ℝ^*)
55		nnm1nn0 11690	. . . . . . . . . . . . . . 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 11708	. . . . . . . . . . . . 13 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → (𝑘 − 1) ∈ ℝ)
58	57, 25	nndivred 11434	. . . . . . . . . . . 12 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → ((𝑘 − 1) / ((⌊‘(1 / 𝑟)) + 1)) ∈ ℝ)
59	36	recnd 10407	. . . . . . . . . . . . . . 15 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → 𝑘 ∈ ℂ)
60	57	recnd 10407	. . . . . . . . . . . . . . 15 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → (𝑘 − 1) ∈ ℂ)
61	25	nnne0d 11430	. . . . . . . . . . . . . . 15 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → ((⌊‘(1 / 𝑟)) + 1) ≠ 0)
62	59, 60, 33, 61	divsubdird 11193	. . . . . . . . . . . . . 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 10332	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℂ
64		nncan 10654	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ ℂ ∧ 1 ∈ ℂ) → (𝑘 − (𝑘 − 1)) = 1)
65	59, 63, 64	sylancl 580	. . . . . . . . . . . . . . 15 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → (𝑘 − (𝑘 − 1)) = 1)
66	65	oveq1d 6939	. . . . . . . . . . . . . 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 2816	. . . . . . . . . . . . 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 12197	. . . . . . . . . . . . . . 15 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → (1 / 𝑟) ∈ ℝ)
69		flltp1 12925	. . . . . . . . . . . . . . 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 12156	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 ∈ ℝ⁺ → 0 < 𝑟)
72	71	ad2antlr 717	. . . . . . . . . . . . . . 15 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → 0 < 𝑟)
73		ltdiv23 11271	. . . . . . . . . . . . . . 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 1447	. . . . . . . . . . . . . 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 224	. . . . . . . . . . . . 13 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → (1 / ((⌊‘(1 / 𝑟)) + 1)) < 𝑟)
76	67, 75	eqbrtrd 4910	. . . . . . . . . . . 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 10983	. . . . . . . . . . 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 12219	. . . . . . . . . . 11 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → (𝑘 / ((⌊‘(1 / 𝑟)) + 1)) < ((𝑘 / ((⌊‘(1 / 𝑟)) + 1)) + 𝑟))
79		iccssioo 12559	. . . . . . . . . . 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 829	. . . . . . . . . 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 10382	. . . . . . . . . . 11 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → 0 ∈ ℝ)
82	56	nn0ge0d 11710	. . . . . . . . . . . 12 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → 0 ≤ (𝑘 − 1))
83		divge0 11249	. . . . . . . . . . . 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 829	. . . . . . . . . . 11 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → 0 ≤ ((𝑘 − 1) / ((⌊‘(1 / 𝑟)) + 1)))
85		iccss 12558	. . . . . . . . . . 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 829	. . . . . . . . . 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 4057	. . . . . . . . 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 2778	. . . . . . . . . . . 12 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) = ((abs ∘ − ) ↾ (ℝ × ℝ))
89	88	rexmet 23013	. . . . . . . . . . 11 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) ∈ (∞Met‘ℝ)
90		sseqin2 4040	. . . . . . . . . . . . 13 ⊢ ((0[,]1) ⊆ ℝ ↔ (ℝ ∩ (0[,]1)) = (0[,]1))
91	3, 90	mpbi 222	. . . . . . . . . . . 12 ⊢ (ℝ ∩ (0[,]1)) = (0[,]1)
92	43, 91	syl6eleqr 2870	. . . . . . . . . . 11 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → (𝑘 / ((⌊‘(1 / 𝑟)) + 1)) ∈ (ℝ ∩ (0[,]1)))
93		rpxr 12153	. . . . . . . . . . . 12 ⊢ (𝑟 ∈ ℝ⁺ → 𝑟 ∈ ℝ^*)
94	93	ad2antlr 717	. . . . . . . . . . 11 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → 𝑟 ∈ ℝ^*)
95		xpss12 5372	. . . . . . . . . . . . . . 15 ⊢ (((0[,]1) ⊆ ℝ ∧ (0[,]1) ⊆ ℝ) → ((0[,]1) × (0[,]1)) ⊆ (ℝ × ℝ))
96	3, 3, 95	mp2an 682	. . . . . . . . . . . . . 14 ⊢ ((0[,]1) × (0[,]1)) ⊆ (ℝ × ℝ)
97		resabs1 5678	. . . . . . . . . . . . . 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 2787	. . . . . . . . . . . 12 ⊢ ((abs ∘ − ) ↾ ((0[,]1) × (0[,]1))) = (((abs ∘ − ) ↾ (ℝ × ℝ)) ↾ ((0[,]1) × (0[,]1)))
100	99	blres 22655	. . . . . . . . . . 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 1539	. . . . . . . . . 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 23014	. . . . . . . . . . . 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 579	. . . . . . . . . . 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 4036	. . . . . . . . . 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 2814	. . . . . . . . 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	sseqtr4d 3861	. . . . . . . 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 3828	. . . . . . . 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 3197	. . . . . 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 3151	. . . 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 6932	. . . . . 6 ⊢ (𝑛 = ((⌊‘(1 / 𝑟)) + 1) → (1...𝑛) = (1...((⌊‘(1 / 𝑟)) + 1)))
113		oveq2 6932	. . . . . . . . 9 ⊢ (𝑛 = ((⌊‘(1 / 𝑟)) + 1) → ((𝑘 − 1) / 𝑛) = ((𝑘 − 1) / ((⌊‘(1 / 𝑟)) + 1)))
114		oveq2 6932	. . . . . . . . 9 ⊢ (𝑛 = ((⌊‘(1 / 𝑟)) + 1) → (𝑘 / 𝑛) = (𝑘 / ((⌊‘(1 / 𝑟)) + 1)))
115	113, 114	oveq12d 6942	. . . . . . . 8 ⊢ (𝑛 = ((⌊‘(1 / 𝑟)) + 1) → (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛)) = (((𝑘 − 1) / ((⌊‘(1 / 𝑟)) + 1))[,](𝑘 / ((⌊‘(1 / 𝑟)) + 1))))
116	115	sseq1d 3851	. . . . . . 7 ⊢ (𝑛 = ((⌊‘(1 / 𝑟)) + 1) → ((((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛)) ⊆ 𝑢 ↔ (((𝑘 − 1) / ((⌊‘(1 / 𝑟)) + 1))[,](𝑘 / ((⌊‘(1 / 𝑟)) + 1))) ⊆ 𝑢))
117	116	rexbidv 3237	. . . . . 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 3326	. . . . 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 3511	. . . 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 674	. . 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 3213	. 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 198 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ∀wral 3090 ∃wrex 3091 ∩ cin 3791 ⊆ wss 3792 ∪ cuni 4673 class class class wbr 4888 × cxp 5355 ↾ cres 5359 ∘ ccom 5361 ‘cfv 6137 (class class class)co 6924 ℂcc 10272 ℝcr 10273 0cc0 10274 1c1 10275 + caddc 10277 · cmul 10279 ℝ^*cxr 10412 < clt 10413 ≤ cle 10414 − cmin 10608 / cdiv 11035 ℕcn 11379 ℕ₀cn0 11647 ℝ⁺crp 12142 (,)cioo 12492 [,]cicc 12495 ...cfz 12648 ⌊cfl 12915 abscabs 14387 ∞Metcxmet 20138 Metcmet 20139 ballcbl 20140 Compccmp 21609 IIcii 23097

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-inf2 8837 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 ax-pre-sup 10352 ax-addf 10353 ax-mulf 10354

This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-int 4713 df-iun 4757 df-iin 4758 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-se 5317 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-isom 6146 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-of 7176 df-om 7346 df-1st 7447 df-2nd 7448 df-supp 7579 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-1o 7845 df-2o 7846 df-oadd 7849 df-er 8028 df-ec 8030 df-map 8144 df-ixp 8197 df-en 8244 df-dom 8245 df-sdom 8246 df-fin 8247 df-fsupp 8566 df-fi 8607 df-sup 8638 df-inf 8639 df-oi 8706 df-card 9100 df-cda 9327 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-div 11036 df-nn 11380 df-2 11443 df-3 11444 df-4 11445 df-5 11446 df-6 11447 df-7 11448 df-8 11449 df-9 11450 df-n0 11648 df-z 11734 df-dec 11851 df-uz 11998 df-q 12101 df-rp 12143 df-xneg 12262 df-xadd 12263 df-xmul 12264 df-ioo 12496 df-ico 12498 df-icc 12499 df-fz 12649 df-fzo 12790 df-fl 12917 df-seq 13125 df-exp 13184 df-hash 13442 df-cj 14252 df-re 14253 df-im 14254 df-sqrt 14388 df-abs 14389 df-clim 14636 df-sum 14834 df-struct 16268 df-ndx 16269 df-slot 16270 df-base 16272 df-sets 16273 df-ress 16274 df-plusg 16362 df-mulr 16363 df-starv 16364 df-sca 16365 df-vsca 16366 df-ip 16367 df-tset 16368 df-ple 16369 df-ds 16371 df-unif 16372 df-hom 16373 df-cco 16374 df-rest 16480 df-topn 16481 df-0g 16499 df-gsum 16500 df-topgen 16501 df-pt 16502 df-prds 16505 df-xrs 16559 df-qtop 16564 df-imas 16565 df-xps 16567 df-mre 16643 df-mrc 16644 df-acs 16646 df-mgm 17639 df-sgrp 17681 df-mnd 17692 df-submnd 17733 df-mulg 17939 df-cntz 18144 df-cmn 18592 df-psmet 20145 df-xmet 20146 df-met 20147 df-bl 20148 df-mopn 20149 df-cnfld 20154 df-top 21117 df-topon 21134 df-topsp 21156 df-bases 21169 df-cld 21242 df-ntr 21243 df-cls 21244 df-cn 21450 df-cnp 21451 df-cmp 21610 df-tx 21785 df-hmeo 21978 df-xms 22544 df-ms 22545 df-tms 22546 df-ii 23099

This theorem is referenced by: cvmliftlem15 31887

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