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Theorem lebnumii 25197

Description: Specialize the Lebesgue number lemma lebnum 25195 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 25108	. . 3 ⊢ II = (MetOpen‘((abs ∘ − ) ↾ ((0[,]1) × (0[,]1))))
2		cnmet 25000	. . . . 5 ⊢ (abs ∘ − ) ∈ (Met‘ℂ)
3		unitssre 13555	. . . . . 6 ⊢ (0[,]1) ⊆ ℝ
4		ax-resscn 11184	. . . . . 6 ⊢ ℝ ⊆ ℂ
5	3, 4	sstri 3940	. . . . 5 ⊢ (0[,]1) ⊆ ℂ
6		metres2 24592	. . . . 5 ⊢ (((abs ∘ − ) ∈ (Met‘ℂ) ∧ (0[,]1) ⊆ ℂ) → ((abs ∘ − ) ↾ ((0[,]1) × (0[,]1))) ∈ (Met‘(0[,]1)))
7	2, 5, 6	mp2an 705	. . . 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 25117	. . . 4 ⊢ II ∈ Comp
10	9	a1i 11	. . 3 ⊢ ((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) → II ∈ Comp)
11		simpl 488	. . 3 ⊢ ((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) → 𝑈 ⊆ II)
12		simpr 490	. . 3 ⊢ ((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) → (0[,]1) = ∪ 𝑈)
13	1, 8, 10, 11, 12	lebnum 25195	. 2 ⊢ ((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) → ∃𝑟 ∈ ℝ⁺ ∀𝑥 ∈ (0[,]1)∃𝑢 ∈ 𝑈 (𝑥(ball‘((abs ∘ − ) ↾ ((0[,]1) × (0[,]1))))𝑟) ⊆ 𝑢)
14		rpreccl 13073	. . . . . . . 8 ⊢ (𝑟 ∈ ℝ⁺ → (1 / 𝑟) ∈ ℝ⁺)
15	14	adantl 487	. . . . . . 7 ⊢ (((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) → (1 / 𝑟) ∈ ℝ⁺)
16	15	rpred 13089	. . . . . 6 ⊢ (((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) → (1 / 𝑟) ∈ ℝ)
17	15	rpge0d 13093	. . . . . 6 ⊢ (((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) → 0 ≤ (1 / 𝑟))
18		flge0nn0 13884	. . . . . 6 ⊢ (((1 / 𝑟) ∈ ℝ ∧ 0 ≤ (1 / 𝑟)) → (⌊‘(1 / 𝑟)) ∈ ℕ₀)
19	16, 17, 18	syl2anc 596	. . . . 5 ⊢ (((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) → (⌊‘(1 / 𝑟)) ∈ ℕ₀)
20		nn0p1nn 12570	. . . . 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 13611	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1)) → 𝑘 ∈ ℕ)
23	22	adantl 487	. . . . . . . . . . 11 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → 𝑘 ∈ ℕ)
24	23	nnrpd 13087	. . . . . . . . . 10 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → 𝑘 ∈ ℝ⁺)
25	21	adantr 486	. . . . . . . . . . 11 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → ((⌊‘(1 / 𝑟)) + 1) ∈ ℕ)
26	25	nnrpd 13087	. . . . . . . . . 10 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → ((⌊‘(1 / 𝑟)) + 1) ∈ ℝ⁺)
27	24, 26	rpdivcld 13106	. . . . . . . . 9 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → (𝑘 / ((⌊‘(1 / 𝑟)) + 1)) ∈ ℝ⁺)
28	27	rpred 13089	. . . . . . . 8 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → (𝑘 / ((⌊‘(1 / 𝑟)) + 1)) ∈ ℝ)
29	27	rpge0d 13093	. . . . . . . 8 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → 0 ≤ (𝑘 / ((⌊‘(1 / 𝑟)) + 1)))
30		elfzle2 13585	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1)) → 𝑘 ≤ ((⌊‘(1 / 𝑟)) + 1))
31	30	adantl 487	. . . . . . . . . 10 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → 𝑘 ≤ ((⌊‘(1 / 𝑟)) + 1))
32	25	nnred 12275	. . . . . . . . . . . 12 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → ((⌊‘(1 / 𝑟)) + 1) ∈ ℝ)
33	32	recnd 11264	. . . . . . . . . . 11 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → ((⌊‘(1 / 𝑟)) + 1) ∈ ℂ)
34	33	mulridd 11253	. . . . . . . . . 10 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → (((⌊‘(1 / 𝑟)) + 1) · 1) = ((⌊‘(1 / 𝑟)) + 1))
35	31, 34	breqtrrd 5133	. . . . . . . . 9 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → 𝑘 ≤ (((⌊‘(1 / 𝑟)) + 1) · 1))
36	23	nnred 12275	. . . . . . . . . 10 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → 𝑘 ∈ ℝ)
37		1red 11236	. . . . . . . . . 10 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → 1 ∈ ℝ)
38	25	nngt0d 12312	. . . . . . . . . 10 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → 0 < ((⌊‘(1 / 𝑟)) + 1))
39		ledivmul 12118	. . . . . . . . . 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 13522	. . . . . . . 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 7421	. . . . . . . . . 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 3962	. . . . . . . . 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 3186	. . . . . . . 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 3572	. . . . . . 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 781	. . . . . . . . . . . . . 14 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → 𝑟 ∈ ℝ⁺)
50	49	rpred 13089	. . . . . . . . . . . . 13 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → 𝑟 ∈ ℝ)
51	28, 50	resubcld 11669	. . . . . . . . . . . 12 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → ((𝑘 / ((⌊‘(1 / 𝑟)) + 1)) − 𝑟) ∈ ℝ)
52	51	rexrd 11286	. . . . . . . . . . 11 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → ((𝑘 / ((⌊‘(1 / 𝑟)) + 1)) − 𝑟) ∈ ℝ^*)
53	28, 50	readdcld 11265	. . . . . . . . . . . 12 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → ((𝑘 / ((⌊‘(1 / 𝑟)) + 1)) + 𝑟) ∈ ℝ)
54	53	rexrd 11286	. . . . . . . . . . 11 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → ((𝑘 / ((⌊‘(1 / 𝑟)) + 1)) + 𝑟) ∈ ℝ^*)
55		nnm1nn0 12572	. . . . . . . . . . . . . . 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 12593	. . . . . . . . . . . . 13 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → (𝑘 − 1) ∈ ℝ)
58	57, 25	nndivred 12317	. . . . . . . . . . . 12 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → ((𝑘 − 1) / ((⌊‘(1 / 𝑟)) + 1)) ∈ ℝ)
59	36	recnd 11264	. . . . . . . . . . . . . . 15 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → 𝑘 ∈ ℂ)
60	57	recnd 11264	. . . . . . . . . . . . . . 15 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → (𝑘 − 1) ∈ ℂ)
61	25	nnne0d 12313	. . . . . . . . . . . . . . 15 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → ((⌊‘(1 / 𝑟)) + 1) ≠ 0)
62	59, 60, 33, 61	divsubdird 12057	. . . . . . . . . . . . . 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 11185	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℂ
64		nncan 11514	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ ℂ ∧ 1 ∈ ℂ) → (𝑘 − (𝑘 − 1)) = 1)
65	59, 63, 64	sylancl 598	. . . . . . . . . . . . . . 15 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → (𝑘 − (𝑘 − 1)) = 1)
66	65	oveq1d 7429	. . . . . . . . . . . . . 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 2797	. . . . . . . . . . . . 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 13100	. . . . . . . . . . . . . . 15 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → (1 / 𝑟) ∈ ℝ)
69		flltp1 13864	. . . . . . . . . . . . . . 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 13058	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 ∈ ℝ⁺ → 0 < 𝑟)
72	71	ad2antlr 740	. . . . . . . . . . . . . . 15 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → 0 < 𝑟)
73		ltdiv23 12133	. . . . . . . . . . . . . . 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 5127	. . . . . . . . . . . 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 11846	. . . . . . . . . . 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 13122	. . . . . . . . . . 11 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → (𝑘 / ((⌊‘(1 / 𝑟)) + 1)) < ((𝑘 / ((⌊‘(1 / 𝑟)) + 1)) + 𝑟))
79		iccssioo 13471	. . . . . . . . . . 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 852	. . . . . . . . . 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 11238	. . . . . . . . . . 11 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → 0 ∈ ℝ)
82	56	nn0ge0d 12595	. . . . . . . . . . . 12 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → 0 ≤ (𝑘 − 1))
83		divge0 12111	. . . . . . . . . . . 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 852	. . . . . . . . . . 11 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → 0 ≤ ((𝑘 − 1) / ((⌊‘(1 / 𝑟)) + 1)))
85		iccss 13470	. . . . . . . . . . 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 852	. . . . . . . . . 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 4186	. . . . . . . . 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 2760	. . . . . . . . . . . 12 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) = ((abs ∘ − ) ↾ (ℝ × ℝ))
89	88	rexmet 25020	. . . . . . . . . . 11 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) ∈ (∞Met‘ℝ)
90		sseqin2 4169	. . . . . . . . . . . . 13 ⊢ ((0[,]1) ⊆ ℝ ↔ (ℝ ∩ (0[,]1)) = (0[,]1))
91	3, 90	mpbi 233	. . . . . . . . . . . 12 ⊢ (ℝ ∩ (0[,]1)) = (0[,]1)
92	43, 91	eleqtrrdi 2871	. . . . . . . . . . 11 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → (𝑘 / ((⌊‘(1 / 𝑟)) + 1)) ∈ (ℝ ∩ (0[,]1)))
93		rpxr 13055	. . . . . . . . . . . 12 ⊢ (𝑟 ∈ ℝ⁺ → 𝑟 ∈ ℝ^*)
94	93	ad2antlr 740	. . . . . . . . . . 11 ⊢ ((((𝑈 ⊆ II ∧ (0[,]1) = ∪ 𝑈) ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...((⌊‘(1 / 𝑟)) + 1))) → 𝑟 ∈ ℝ^*)
95		xpss12 5670	. . . . . . . . . . . . . . 15 ⊢ (((0[,]1) ⊆ ℝ ∧ (0[,]1) ⊆ ℝ) → ((0[,]1) × (0[,]1)) ⊆ (ℝ × ℝ))
96	3, 3, 95	mp2an 705	. . . . . . . . . . . . . 14 ⊢ ((0[,]1) × (0[,]1)) ⊆ (ℝ × ℝ)
97		resabs1 5999	. . . . . . . . . . . . . 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 2769	. . . . . . . . . . . 12 ⊢ ((abs ∘ − ) ↾ ((0[,]1) × (0[,]1))) = (((abs ∘ − ) ↾ (ℝ × ℝ)) ↾ ((0[,]1) × (0[,]1)))
100	99	blres 24660	. . . . . . . . . . 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 25021	. . . . . . . . . . . 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 596	. . . . . . . . . . 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 4165	. . . . . . . . . 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 2795	. . . . . . . . 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 3968	. . . . . . . 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 3938	. . . . . . . 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 3177	. . . . . 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 3162	. . . 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 7422	. . . . . 6 ⊢ (𝑛 = ((⌊‘(1 / 𝑟)) + 1) → (1...𝑛) = (1...((⌊‘(1 / 𝑟)) + 1)))
113		oveq2 7422	. . . . . . . . 9 ⊢ (𝑛 = ((⌊‘(1 / 𝑟)) + 1) → ((𝑘 − 1) / 𝑛) = ((𝑘 − 1) / ((⌊‘(1 / 𝑟)) + 1)))
114		oveq2 7422	. . . . . . . . 9 ⊢ (𝑛 = ((⌊‘(1 / 𝑟)) + 1) → (𝑘 / 𝑛) = (𝑘 / ((⌊‘(1 / 𝑟)) + 1)))
115	113, 114	oveq12d 7432	. . . . . . . 8 ⊢ (𝑛 = ((⌊‘(1 / 𝑟)) + 1) → (((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛)) = (((𝑘 − 1) / ((⌊‘(1 / 𝑟)) + 1))[,](𝑘 / ((⌊‘(1 / 𝑟)) + 1))))
116	115	sseq1d 3962	. . . . . . 7 ⊢ (𝑛 = ((⌊‘(1 / 𝑟)) + 1) → ((((𝑘 − 1) / 𝑛)[,](𝑘 / 𝑛)) ⊆ 𝑢 ↔ (((𝑘 − 1) / ((⌊‘(1 / 𝑟)) + 1))[,](𝑘 / ((⌊‘(1 / 𝑟)) + 1))) ⊆ 𝑢))
117	116	rexbidv 3186	. . . . . 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 3334	. . . . 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 3576	. . . 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 697	. . 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 3163	. 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

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ∀wral 3076 ∃wrex 3086 ∩ cin 3898 ⊆ wss 3899 ∪ cuni 4867 class class class wbr 5103 × cxp 5653 ↾ cres 5657 ∘ ccom 5659 ‘cfv 6533 (class class class)co 7414 ℂcc 11125 ℝcr 11126 0cc0 11127 1c1 11128 + caddc 11130 · cmul 11132 ℝ^*cxr 11269 < clt 11270 ≤ cle 11271 − cmin 11468 / cdiv 11898 ℕcn 12260 ℕ₀cn0 12531 ℝ⁺crp 13045 (,)cioo 13401 [,]cicc 13404 ...cfz 13564 ⌊cfl 13854 abscabs 15324 ∞Metcxmet 21573 Metcmet 21574 ballcbl 21575 Compccmp 23614 IIcii 25106

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7737 ax-inf2 9623 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204 ax-pre-sup 11205 ax-addf 11206

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7371 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7679 df-om 7864 df-1st 7987 df-2nd 7988 df-supp 8160 df-frecs 8281 df-wrecs 8312 df-recs 8361 df-rdg 8400 df-1o 8458 df-2o 8459 df-er 8699 df-ec 8701 df-map 8831 df-ixp 8908 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-fsupp 9335 df-fi 9384 df-sup 9415 df-inf 9416 df-oi 9485 df-card 9947 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11470 df-neg 11471 df-div 11899 df-nn 12261 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12532 df-z 12619 df-dec 12740 df-uz 12891 df-q 13001 df-rp 13046 df-xneg 13166 df-xadd 13167 df-xmul 13168 df-ioo 13405 df-ico 13407 df-icc 13408 df-fz 13565 df-fzo 13713 df-fl 13856 df-seq 14069 df-exp 14129 df-hash 14398 df-cj 15189 df-re 15190 df-im 15191 df-sqrt 15325 df-abs 15326 df-clim 15578 df-sum 15777 df-struct 17242 df-sets 17259 df-slot 17277 df-ndx 17289 df-base 17305 df-ress 17326 df-plusg 17358 df-mulr 17359 df-starv 17360 df-sca 17361 df-vsca 17362 df-ip 17363 df-tset 17364 df-ple 17365 df-ds 17367 df-unif 17368 df-hom 17369 df-cco 17370 df-rest 17510 df-topn 17511 df-0g 17529 df-gsum 17530 df-topgen 17531 df-pt 17532 df-prds 17535 df-xrs 17591 df-qtop 17596 df-imas 17597 df-xps 17599 df-mre 17673 df-mrc 17674 df-acs 17676 df-mgm 18733 df-sgrp 18824 df-mnd 18840 df-submnd 18895 df-mulg 19194 df-cntz 19447 df-cmn 19912 df-psmet 21580 df-xmet 21581 df-met 21582 df-bl 21583 df-mopn 21584 df-cnfld 21589 df-top 23122 df-topon 23139 df-topsp 23161 df-bases 23174 df-cld 23247 df-ntr 23248 df-cls 23249 df-cn 23455 df-cnp 23456 df-cmp 23615 df-tx 23791 df-hmeo 23984 df-xms 24549 df-ms 24550 df-tms 24551 df-ii 25108

This theorem is used by: cvmliftlem15 35880

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