opnmbllem0 - Mathbox for Brendan Leahy

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Theorem opnmbllem0 33445

Description: Lemma for ismblfin 33450; could also be used to shorten proof of opnmbllem 23369. (Contributed by Brendan Leahy, 13-Jul-2018.)

**Assertion**
Ref	Expression
opnmbllem0	⊢ (𝐴 ∈ (topGen‘ran (,)) → ∪ ([,] “ {𝑧 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ₀ ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑧) ⊆ 𝐴}) = 𝐴)

Distinct variable group: 𝑥,𝑦,𝑧,𝐴

Proof of Theorem opnmbllem0 Dummy variables 𝑛 𝑟 𝑠 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . . . . . 8 ⊢ (𝑧 = 𝑤 → ([,]‘𝑧) = ([,]‘𝑤))
2	1	sseq1d 3632	. . . . . . 7 ⊢ (𝑧 = 𝑤 → (([,]‘𝑧) ⊆ 𝐴 ↔ ([,]‘𝑤) ⊆ 𝐴))
3	2	elrab 3363	. . . . . 6 ⊢ (𝑤 ∈ {𝑧 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ₀ ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑧) ⊆ 𝐴} ↔ (𝑤 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ₀ ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∧ ([,]‘𝑤) ⊆ 𝐴))
4		simprr 796	. . . . . . 7 ⊢ ((𝐴 ∈ (topGen‘ran (,)) ∧ (𝑤 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ₀ ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∧ ([,]‘𝑤) ⊆ 𝐴)) → ([,]‘𝑤) ⊆ 𝐴)
5		fvex 6201	. . . . . . . 8 ⊢ ([,]‘𝑤) ∈ V
6	5	elpw 4164	. . . . . . 7 ⊢ (([,]‘𝑤) ∈ 𝒫 𝐴 ↔ ([,]‘𝑤) ⊆ 𝐴)
7	4, 6	sylibr 224	. . . . . 6 ⊢ ((𝐴 ∈ (topGen‘ran (,)) ∧ (𝑤 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ₀ ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∧ ([,]‘𝑤) ⊆ 𝐴)) → ([,]‘𝑤) ∈ 𝒫 𝐴)
8	3, 7	sylan2b 492	. . . . 5 ⊢ ((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ {𝑧 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ₀ ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑧) ⊆ 𝐴}) → ([,]‘𝑤) ∈ 𝒫 𝐴)
9	8	ralrimiva 2966	. . . 4 ⊢ (𝐴 ∈ (topGen‘ran (,)) → ∀𝑤 ∈ {𝑧 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ₀ ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑧) ⊆ 𝐴} ([,]‘𝑤) ∈ 𝒫 𝐴)
10		iccf 12272	. . . . . 6 ⊢ [,]:(ℝ^* × ℝ^)⟶𝒫 ℝ^
11		ffun 6048	. . . . . 6 ⊢ ([,]:(ℝ^* × ℝ^)⟶𝒫 ℝ^ → Fun [,])
12	10, 11	ax-mp 5	. . . . 5 ⊢ Fun [,]
13		ssrab2 3687	. . . . . . 7 ⊢ {𝑧 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ₀ ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑧) ⊆ 𝐴} ⊆ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ₀ ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩)
14		oveq1 6657	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑟 → (𝑥 / (2↑𝑦)) = (𝑟 / (2↑𝑦)))
15		oveq1 6657	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑟 → (𝑥 + 1) = (𝑟 + 1))
16	15	oveq1d 6665	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑟 → ((𝑥 + 1) / (2↑𝑦)) = ((𝑟 + 1) / (2↑𝑦)))
17	14, 16	opeq12d 4410	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑟 → ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩ = ⟨(𝑟 / (2↑𝑦)), ((𝑟 + 1) / (2↑𝑦))⟩)
18		oveq2 6658	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑠 → (2↑𝑦) = (2↑𝑠))
19	18	oveq2d 6666	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑠 → (𝑟 / (2↑𝑦)) = (𝑟 / (2↑𝑠)))
20	18	oveq2d 6666	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑠 → ((𝑟 + 1) / (2↑𝑦)) = ((𝑟 + 1) / (2↑𝑠)))
21	19, 20	opeq12d 4410	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑠 → ⟨(𝑟 / (2↑𝑦)), ((𝑟 + 1) / (2↑𝑦))⟩ = ⟨(𝑟 / (2↑𝑠)), ((𝑟 + 1) / (2↑𝑠))⟩)
22	17, 21	cbvmpt2v 6735	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℤ, 𝑦 ∈ ℕ₀ ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) = (𝑟 ∈ ℤ, 𝑠 ∈ ℕ₀ ↦ ⟨(𝑟 / (2↑𝑠)), ((𝑟 + 1) / (2↑𝑠))⟩)
23	22	dyadf 23359	. . . . . . . . 9 ⊢ (𝑥 ∈ ℤ, 𝑦 ∈ ℕ₀ ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩):(ℤ × ℕ₀)⟶( ≤ ∩ (ℝ × ℝ))
24		frn 6053	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℤ, 𝑦 ∈ ℕ₀ ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩):(ℤ × ℕ₀)⟶( ≤ ∩ (ℝ × ℝ)) → ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ₀ ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ⊆ ( ≤ ∩ (ℝ × ℝ)))
25	23, 24	ax-mp 5	. . . . . . . 8 ⊢ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ₀ ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ⊆ ( ≤ ∩ (ℝ × ℝ))
26		inss2 3834	. . . . . . . . 9 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
27		rexpssxrxp 10084	. . . . . . . . 9 ⊢ (ℝ × ℝ) ⊆ (ℝ^* × ℝ^*)
28	26, 27	sstri 3612	. . . . . . . 8 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ^* × ℝ^*)
29	25, 28	sstri 3612	. . . . . . 7 ⊢ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ₀ ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ⊆ (ℝ^* × ℝ^*)
30	13, 29	sstri 3612	. . . . . 6 ⊢ {𝑧 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ₀ ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑧) ⊆ 𝐴} ⊆ (ℝ^* × ℝ^*)
31	10	fdmi 6052	. . . . . 6 ⊢ dom [,] = (ℝ^* × ℝ^*)
32	30, 31	sseqtr4i 3638	. . . . 5 ⊢ {𝑧 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ₀ ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑧) ⊆ 𝐴} ⊆ dom [,]
33		funimass4 6247	. . . . 5 ⊢ ((Fun [,] ∧ {𝑧 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ₀ ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑧) ⊆ 𝐴} ⊆ dom [,]) → (([,] “ {𝑧 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ₀ ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑧) ⊆ 𝐴}) ⊆ 𝒫 𝐴 ↔ ∀𝑤 ∈ {𝑧 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ₀ ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑧) ⊆ 𝐴} ([,]‘𝑤) ∈ 𝒫 𝐴))
34	12, 32, 33	mp2an 708	. . . 4 ⊢ (([,] “ {𝑧 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ₀ ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑧) ⊆ 𝐴}) ⊆ 𝒫 𝐴 ↔ ∀𝑤 ∈ {𝑧 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ₀ ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑧) ⊆ 𝐴} ([,]‘𝑤) ∈ 𝒫 𝐴)
35	9, 34	sylibr 224	. . 3 ⊢ (𝐴 ∈ (topGen‘ran (,)) → ([,] “ {𝑧 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ₀ ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑧) ⊆ 𝐴}) ⊆ 𝒫 𝐴)
36		sspwuni 4611	. . 3 ⊢ (([,] “ {𝑧 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ₀ ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑧) ⊆ 𝐴}) ⊆ 𝒫 𝐴 ↔ ∪ ([,] “ {𝑧 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ₀ ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑧) ⊆ 𝐴}) ⊆ 𝐴)
37	35, 36	sylib 208	. 2 ⊢ (𝐴 ∈ (topGen‘ran (,)) → ∪ ([,] “ {𝑧 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ₀ ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑧) ⊆ 𝐴}) ⊆ 𝐴)
38		eqid 2622	. . . . . . 7 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) = ((abs ∘ − ) ↾ (ℝ × ℝ))
39	38	rexmet 22594	. . . . . 6 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) ∈ (∞Met‘ℝ)
40		eqid 2622	. . . . . . . 8 ⊢ (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ))) = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ)))
41	38, 40	tgioo 22599	. . . . . . 7 ⊢ (topGen‘ran (,)) = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ)))
42	41	mopni2 22298	. . . . . 6 ⊢ ((((abs ∘ − ) ↾ (ℝ × ℝ)) ∈ (∞Met‘ℝ) ∧ 𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) → ∃𝑟 ∈ ℝ⁺ (𝑤(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ⊆ 𝐴)
43	39, 42	mp3an1 1411	. . . . 5 ⊢ ((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) → ∃𝑟 ∈ ℝ⁺ (𝑤(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ⊆ 𝐴)
44		elssuni 4467	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (topGen‘ran (,)) → 𝐴 ⊆ ∪ (topGen‘ran (,)))
45		uniretop 22566	. . . . . . . . . . 11 ⊢ ℝ = ∪ (topGen‘ran (,))
46	44, 45	syl6sseqr 3652	. . . . . . . . . 10 ⊢ (𝐴 ∈ (topGen‘ran (,)) → 𝐴 ⊆ ℝ)
47	46	sselda 3603	. . . . . . . . 9 ⊢ ((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) → 𝑤 ∈ ℝ)
48		rpre 11839	. . . . . . . . 9 ⊢ (𝑟 ∈ ℝ⁺ → 𝑟 ∈ ℝ)
49	38	bl2ioo 22595	. . . . . . . . 9 ⊢ ((𝑤 ∈ ℝ ∧ 𝑟 ∈ ℝ) → (𝑤(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) = ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)))
50	47, 48, 49	syl2an 494	. . . . . . . 8 ⊢ (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ 𝑟 ∈ ℝ⁺) → (𝑤(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) = ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)))
51	50	sseq1d 3632	. . . . . . 7 ⊢ (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ 𝑟 ∈ ℝ⁺) → ((𝑤(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ⊆ 𝐴 ↔ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴))
52		2re 11090	. . . . . . . . . . 11 ⊢ 2 ∈ ℝ
53		1lt2 11194	. . . . . . . . . . 11 ⊢ 1 < 2
54		expnlbnd 12994	. . . . . . . . . . 11 ⊢ ((𝑟 ∈ ℝ⁺ ∧ 2 ∈ ℝ ∧ 1 < 2) → ∃𝑛 ∈ ℕ (1 / (2↑𝑛)) < 𝑟)
55	52, 53, 54	mp3an23 1416	. . . . . . . . . 10 ⊢ (𝑟 ∈ ℝ⁺ → ∃𝑛 ∈ ℕ (1 / (2↑𝑛)) < 𝑟)
56	55	ad2antrl 764	. . . . . . . . 9 ⊢ (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) → ∃𝑛 ∈ ℕ (1 / (2↑𝑛)) < 𝑟)
57	47	ad2antrr 762	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → 𝑤 ∈ ℝ)
58		2nn 11185	. . . . . . . . . . . . . . . . . 18 ⊢ 2 ∈ ℕ
59		nnnn0 11299	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ₀)
60	59	ad2antrl 764	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → 𝑛 ∈ ℕ₀)
61		nnexpcl 12873	. . . . . . . . . . . . . . . . . 18 ⊢ ((2 ∈ ℕ ∧ 𝑛 ∈ ℕ₀) → (2↑𝑛) ∈ ℕ)
62	58, 60, 61	sylancr 695	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → (2↑𝑛) ∈ ℕ)
63	62	nnred 11035	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → (2↑𝑛) ∈ ℝ)
64	57, 63	remulcld 10070	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → (𝑤 · (2↑𝑛)) ∈ ℝ)
65		fllelt 12598	. . . . . . . . . . . . . . 15 ⊢ ((𝑤 · (2↑𝑛)) ∈ ℝ → ((⌊‘(𝑤 · (2↑𝑛))) ≤ (𝑤 · (2↑𝑛)) ∧ (𝑤 · (2↑𝑛)) < ((⌊‘(𝑤 · (2↑𝑛))) + 1)))
66	64, 65	syl 17	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → ((⌊‘(𝑤 · (2↑𝑛))) ≤ (𝑤 · (2↑𝑛)) ∧ (𝑤 · (2↑𝑛)) < ((⌊‘(𝑤 · (2↑𝑛))) + 1)))
67	66	simpld 475	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → (⌊‘(𝑤 · (2↑𝑛))) ≤ (𝑤 · (2↑𝑛)))
68		reflcl 12597	. . . . . . . . . . . . . . 15 ⊢ ((𝑤 · (2↑𝑛)) ∈ ℝ → (⌊‘(𝑤 · (2↑𝑛))) ∈ ℝ)
69	64, 68	syl 17	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → (⌊‘(𝑤 · (2↑𝑛))) ∈ ℝ)
70	62	nngt0d 11064	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → 0 < (2↑𝑛))
71		ledivmul2 10902	. . . . . . . . . . . . . 14 ⊢ (((⌊‘(𝑤 · (2↑𝑛))) ∈ ℝ ∧ 𝑤 ∈ ℝ ∧ ((2↑𝑛) ∈ ℝ ∧ 0 < (2↑𝑛))) → (((⌊‘(𝑤 · (2↑𝑛))) / (2↑𝑛)) ≤ 𝑤 ↔ (⌊‘(𝑤 · (2↑𝑛))) ≤ (𝑤 · (2↑𝑛))))
72	69, 57, 63, 70, 71	syl112anc 1330	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → (((⌊‘(𝑤 · (2↑𝑛))) / (2↑𝑛)) ≤ 𝑤 ↔ (⌊‘(𝑤 · (2↑𝑛))) ≤ (𝑤 · (2↑𝑛))))
73	67, 72	mpbird 247	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → ((⌊‘(𝑤 · (2↑𝑛))) / (2↑𝑛)) ≤ 𝑤)
74		peano2re 10209	. . . . . . . . . . . . . . 15 ⊢ ((⌊‘(𝑤 · (2↑𝑛))) ∈ ℝ → ((⌊‘(𝑤 · (2↑𝑛))) + 1) ∈ ℝ)
75	69, 74	syl 17	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → ((⌊‘(𝑤 · (2↑𝑛))) + 1) ∈ ℝ)
76	75, 62	nndivred 11069	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → (((⌊‘(𝑤 · (2↑𝑛))) + 1) / (2↑𝑛)) ∈ ℝ)
77	66	simprd 479	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → (𝑤 · (2↑𝑛)) < ((⌊‘(𝑤 · (2↑𝑛))) + 1))
78		ltmuldiv 10896	. . . . . . . . . . . . . . 15 ⊢ ((𝑤 ∈ ℝ ∧ ((⌊‘(𝑤 · (2↑𝑛))) + 1) ∈ ℝ ∧ ((2↑𝑛) ∈ ℝ ∧ 0 < (2↑𝑛))) → ((𝑤 · (2↑𝑛)) < ((⌊‘(𝑤 · (2↑𝑛))) + 1) ↔ 𝑤 < (((⌊‘(𝑤 · (2↑𝑛))) + 1) / (2↑𝑛))))
79	57, 75, 63, 70, 78	syl112anc 1330	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → ((𝑤 · (2↑𝑛)) < ((⌊‘(𝑤 · (2↑𝑛))) + 1) ↔ 𝑤 < (((⌊‘(𝑤 · (2↑𝑛))) + 1) / (2↑𝑛))))
80	77, 79	mpbid 222	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → 𝑤 < (((⌊‘(𝑤 · (2↑𝑛))) + 1) / (2↑𝑛)))
81	57, 76, 80	ltled 10185	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → 𝑤 ≤ (((⌊‘(𝑤 · (2↑𝑛))) + 1) / (2↑𝑛)))
82	69, 62	nndivred 11069	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → ((⌊‘(𝑤 · (2↑𝑛))) / (2↑𝑛)) ∈ ℝ)
83		elicc2 12238	. . . . . . . . . . . . 13 ⊢ ((((⌊‘(𝑤 · (2↑𝑛))) / (2↑𝑛)) ∈ ℝ ∧ (((⌊‘(𝑤 · (2↑𝑛))) + 1) / (2↑𝑛)) ∈ ℝ) → (𝑤 ∈ (((⌊‘(𝑤 · (2↑𝑛))) / (2↑𝑛))[,](((⌊‘(𝑤 · (2↑𝑛))) + 1) / (2↑𝑛))) ↔ (𝑤 ∈ ℝ ∧ ((⌊‘(𝑤 · (2↑𝑛))) / (2↑𝑛)) ≤ 𝑤 ∧ 𝑤 ≤ (((⌊‘(𝑤 · (2↑𝑛))) + 1) / (2↑𝑛)))))
84	82, 76, 83	syl2anc 693	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → (𝑤 ∈ (((⌊‘(𝑤 · (2↑𝑛))) / (2↑𝑛))[,](((⌊‘(𝑤 · (2↑𝑛))) + 1) / (2↑𝑛))) ↔ (𝑤 ∈ ℝ ∧ ((⌊‘(𝑤 · (2↑𝑛))) / (2↑𝑛)) ≤ 𝑤 ∧ 𝑤 ≤ (((⌊‘(𝑤 · (2↑𝑛))) + 1) / (2↑𝑛)))))
85	57, 73, 81, 84	mpbir3and 1245	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → 𝑤 ∈ (((⌊‘(𝑤 · (2↑𝑛))) / (2↑𝑛))[,](((⌊‘(𝑤 · (2↑𝑛))) + 1) / (2↑𝑛))))
86	64	flcld 12599	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → (⌊‘(𝑤 · (2↑𝑛))) ∈ ℤ)
87	22	dyadval 23360	. . . . . . . . . . . . . 14 ⊢ (((⌊‘(𝑤 · (2↑𝑛))) ∈ ℤ ∧ 𝑛 ∈ ℕ₀) → ((⌊‘(𝑤 · (2↑𝑛)))(𝑥 ∈ ℤ, 𝑦 ∈ ℕ₀ ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩)𝑛) = ⟨((⌊‘(𝑤 · (2↑𝑛))) / (2↑𝑛)), (((⌊‘(𝑤 · (2↑𝑛))) + 1) / (2↑𝑛))⟩)
88	86, 60, 87	syl2anc 693	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → ((⌊‘(𝑤 · (2↑𝑛)))(𝑥 ∈ ℤ, 𝑦 ∈ ℕ₀ ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩)𝑛) = ⟨((⌊‘(𝑤 · (2↑𝑛))) / (2↑𝑛)), (((⌊‘(𝑤 · (2↑𝑛))) + 1) / (2↑𝑛))⟩)
89	88	fveq2d 6195	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → ([,]‘((⌊‘(𝑤 · (2↑𝑛)))(𝑥 ∈ ℤ, 𝑦 ∈ ℕ₀ ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩)𝑛)) = ([,]‘⟨((⌊‘(𝑤 · (2↑𝑛))) / (2↑𝑛)), (((⌊‘(𝑤 · (2↑𝑛))) + 1) / (2↑𝑛))⟩))
90		df-ov 6653	. . . . . . . . . . . 12 ⊢ (((⌊‘(𝑤 · (2↑𝑛))) / (2↑𝑛))[,](((⌊‘(𝑤 · (2↑𝑛))) + 1) / (2↑𝑛))) = ([,]‘⟨((⌊‘(𝑤 · (2↑𝑛))) / (2↑𝑛)), (((⌊‘(𝑤 · (2↑𝑛))) + 1) / (2↑𝑛))⟩)
91	89, 90	syl6eqr 2674	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → ([,]‘((⌊‘(𝑤 · (2↑𝑛)))(𝑥 ∈ ℤ, 𝑦 ∈ ℕ₀ ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩)𝑛)) = (((⌊‘(𝑤 · (2↑𝑛))) / (2↑𝑛))[,](((⌊‘(𝑤 · (2↑𝑛))) + 1) / (2↑𝑛))))
92	85, 91	eleqtrrd 2704	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → 𝑤 ∈ ([,]‘((⌊‘(𝑤 · (2↑𝑛)))(𝑥 ∈ ℤ, 𝑦 ∈ ℕ₀ ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩)𝑛)))
93		ffn 6045	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℤ, 𝑦 ∈ ℕ₀ ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩):(ℤ × ℕ₀)⟶( ≤ ∩ (ℝ × ℝ)) → (𝑥 ∈ ℤ, 𝑦 ∈ ℕ₀ ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) Fn (ℤ × ℕ₀))
94	23, 93	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℤ, 𝑦 ∈ ℕ₀ ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) Fn (ℤ × ℕ₀)
95		fnovrn 6809	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℤ, 𝑦 ∈ ℕ₀ ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) Fn (ℤ × ℕ₀) ∧ (⌊‘(𝑤 · (2↑𝑛))) ∈ ℤ ∧ 𝑛 ∈ ℕ₀) → ((⌊‘(𝑤 · (2↑𝑛)))(𝑥 ∈ ℤ, 𝑦 ∈ ℕ₀ ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩)𝑛) ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ₀ ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩))
96	94, 95	mp3an1 1411	. . . . . . . . . . . . 13 ⊢ (((⌊‘(𝑤 · (2↑𝑛))) ∈ ℤ ∧ 𝑛 ∈ ℕ₀) → ((⌊‘(𝑤 · (2↑𝑛)))(𝑥 ∈ ℤ, 𝑦 ∈ ℕ₀ ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩)𝑛) ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ₀ ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩))
97	86, 60, 96	syl2anc 693	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → ((⌊‘(𝑤 · (2↑𝑛)))(𝑥 ∈ ℤ, 𝑦 ∈ ℕ₀ ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩)𝑛) ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ₀ ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩))
98		simplrl 800	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → 𝑟 ∈ ℝ⁺)
99	98	rpred 11872	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → 𝑟 ∈ ℝ)
100	57, 99	resubcld 10458	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → (𝑤 − 𝑟) ∈ ℝ)
101	100	rexrd 10089	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → (𝑤 − 𝑟) ∈ ℝ^*)
102	57, 99	readdcld 10069	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → (𝑤 + 𝑟) ∈ ℝ)
103	102	rexrd 10089	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → (𝑤 + 𝑟) ∈ ℝ^*)
104	82, 99	readdcld 10069	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → (((⌊‘(𝑤 · (2↑𝑛))) / (2↑𝑛)) + 𝑟) ∈ ℝ)
105	69	recnd 10068	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → (⌊‘(𝑤 · (2↑𝑛))) ∈ ℂ)
106		1cnd 10056	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → 1 ∈ ℂ)
107	63	recnd 10068	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → (2↑𝑛) ∈ ℂ)
108	62	nnne0d 11065	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → (2↑𝑛) ≠ 0)
109	105, 106, 107, 108	divdird 10839	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → (((⌊‘(𝑤 · (2↑𝑛))) + 1) / (2↑𝑛)) = (((⌊‘(𝑤 · (2↑𝑛))) / (2↑𝑛)) + (1 / (2↑𝑛))))
110	62	nnrecred 11066	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → (1 / (2↑𝑛)) ∈ ℝ)
111		simprr 796	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → (1 / (2↑𝑛)) < 𝑟)
112	110, 99, 82, 111	ltadd2dd 10196	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → (((⌊‘(𝑤 · (2↑𝑛))) / (2↑𝑛)) + (1 / (2↑𝑛))) < (((⌊‘(𝑤 · (2↑𝑛))) / (2↑𝑛)) + 𝑟))
113	109, 112	eqbrtrd 4675	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → (((⌊‘(𝑤 · (2↑𝑛))) + 1) / (2↑𝑛)) < (((⌊‘(𝑤 · (2↑𝑛))) / (2↑𝑛)) + 𝑟))
114	57, 76, 104, 80, 113	lttrd 10198	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → 𝑤 < (((⌊‘(𝑤 · (2↑𝑛))) / (2↑𝑛)) + 𝑟))
115	57, 99, 82	ltsubaddd 10623	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → ((𝑤 − 𝑟) < ((⌊‘(𝑤 · (2↑𝑛))) / (2↑𝑛)) ↔ 𝑤 < (((⌊‘(𝑤 · (2↑𝑛))) / (2↑𝑛)) + 𝑟)))
116	114, 115	mpbird 247	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → (𝑤 − 𝑟) < ((⌊‘(𝑤 · (2↑𝑛))) / (2↑𝑛)))
117	57, 110	readdcld 10069	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → (𝑤 + (1 / (2↑𝑛))) ∈ ℝ)
118	82, 57, 110, 73	leadd1dd 10641	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → (((⌊‘(𝑤 · (2↑𝑛))) / (2↑𝑛)) + (1 / (2↑𝑛))) ≤ (𝑤 + (1 / (2↑𝑛))))
119	109, 118	eqbrtrd 4675	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → (((⌊‘(𝑤 · (2↑𝑛))) + 1) / (2↑𝑛)) ≤ (𝑤 + (1 / (2↑𝑛))))
120	110, 99, 57, 111	ltadd2dd 10196	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → (𝑤 + (1 / (2↑𝑛))) < (𝑤 + 𝑟))
121	76, 117, 102, 119, 120	lelttrd 10195	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → (((⌊‘(𝑤 · (2↑𝑛))) + 1) / (2↑𝑛)) < (𝑤 + 𝑟))
122		iccssioo 12242	. . . . . . . . . . . . . . 15 ⊢ ((((𝑤 − 𝑟) ∈ ℝ^* ∧ (𝑤 + 𝑟) ∈ ℝ^*) ∧ ((𝑤 − 𝑟) < ((⌊‘(𝑤 · (2↑𝑛))) / (2↑𝑛)) ∧ (((⌊‘(𝑤 · (2↑𝑛))) + 1) / (2↑𝑛)) < (𝑤 + 𝑟))) → (((⌊‘(𝑤 · (2↑𝑛))) / (2↑𝑛))[,](((⌊‘(𝑤 · (2↑𝑛))) + 1) / (2↑𝑛))) ⊆ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)))
123	101, 103, 116, 121, 122	syl22anc 1327	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → (((⌊‘(𝑤 · (2↑𝑛))) / (2↑𝑛))[,](((⌊‘(𝑤 · (2↑𝑛))) + 1) / (2↑𝑛))) ⊆ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)))
124	91, 123	eqsstrd 3639	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → ([,]‘((⌊‘(𝑤 · (2↑𝑛)))(𝑥 ∈ ℤ, 𝑦 ∈ ℕ₀ ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩)𝑛)) ⊆ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)))
125		simplrr 801	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)
126	124, 125	sstrd 3613	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → ([,]‘((⌊‘(𝑤 · (2↑𝑛)))(𝑥 ∈ ℤ, 𝑦 ∈ ℕ₀ ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩)𝑛)) ⊆ 𝐴)
127		fveq2 6191	. . . . . . . . . . . . . 14 ⊢ (𝑧 = ((⌊‘(𝑤 · (2↑𝑛)))(𝑥 ∈ ℤ, 𝑦 ∈ ℕ₀ ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩)𝑛) → ([,]‘𝑧) = ([,]‘((⌊‘(𝑤 · (2↑𝑛)))(𝑥 ∈ ℤ, 𝑦 ∈ ℕ₀ ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩)𝑛)))
128	127	sseq1d 3632	. . . . . . . . . . . . 13 ⊢ (𝑧 = ((⌊‘(𝑤 · (2↑𝑛)))(𝑥 ∈ ℤ, 𝑦 ∈ ℕ₀ ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩)𝑛) → (([,]‘𝑧) ⊆ 𝐴 ↔ ([,]‘((⌊‘(𝑤 · (2↑𝑛)))(𝑥 ∈ ℤ, 𝑦 ∈ ℕ₀ ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩)𝑛)) ⊆ 𝐴))
129	128	elrab 3363	. . . . . . . . . . . 12 ⊢ (((⌊‘(𝑤 · (2↑𝑛)))(𝑥 ∈ ℤ, 𝑦 ∈ ℕ₀ ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩)𝑛) ∈ {𝑧 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ₀ ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑧) ⊆ 𝐴} ↔ (((⌊‘(𝑤 · (2↑𝑛)))(𝑥 ∈ ℤ, 𝑦 ∈ ℕ₀ ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩)𝑛) ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ₀ ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∧ ([,]‘((⌊‘(𝑤 · (2↑𝑛)))(𝑥 ∈ ℤ, 𝑦 ∈ ℕ₀ ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩)𝑛)) ⊆ 𝐴))
130	97, 126, 129	sylanbrc 698	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → ((⌊‘(𝑤 · (2↑𝑛)))(𝑥 ∈ ℤ, 𝑦 ∈ ℕ₀ ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩)𝑛) ∈ {𝑧 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ₀ ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑧) ⊆ 𝐴})
131		funfvima2 6493	. . . . . . . . . . . 12 ⊢ ((Fun [,] ∧ {𝑧 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ₀ ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑧) ⊆ 𝐴} ⊆ dom [,]) → (((⌊‘(𝑤 · (2↑𝑛)))(𝑥 ∈ ℤ, 𝑦 ∈ ℕ₀ ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩)𝑛) ∈ {𝑧 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ₀ ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑧) ⊆ 𝐴} → ([,]‘((⌊‘(𝑤 · (2↑𝑛)))(𝑥 ∈ ℤ, 𝑦 ∈ ℕ₀ ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩)𝑛)) ∈ ([,] “ {𝑧 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ₀ ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑧) ⊆ 𝐴})))
132	12, 32, 131	mp2an 708	. . . . . . . . . . 11 ⊢ (((⌊‘(𝑤 · (2↑𝑛)))(𝑥 ∈ ℤ, 𝑦 ∈ ℕ₀ ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩)𝑛) ∈ {𝑧 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ₀ ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑧) ⊆ 𝐴} → ([,]‘((⌊‘(𝑤 · (2↑𝑛)))(𝑥 ∈ ℤ, 𝑦 ∈ ℕ₀ ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩)𝑛)) ∈ ([,] “ {𝑧 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ₀ ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑧) ⊆ 𝐴}))
133	130, 132	syl 17	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → ([,]‘((⌊‘(𝑤 · (2↑𝑛)))(𝑥 ∈ ℤ, 𝑦 ∈ ℕ₀ ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩)𝑛)) ∈ ([,] “ {𝑧 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ₀ ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑧) ⊆ 𝐴}))
134		elunii 4441	. . . . . . . . . 10 ⊢ ((𝑤 ∈ ([,]‘((⌊‘(𝑤 · (2↑𝑛)))(𝑥 ∈ ℤ, 𝑦 ∈ ℕ₀ ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩)𝑛)) ∧ ([,]‘((⌊‘(𝑤 · (2↑𝑛)))(𝑥 ∈ ℤ, 𝑦 ∈ ℕ₀ ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩)𝑛)) ∈ ([,] “ {𝑧 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ₀ ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑧) ⊆ 𝐴})) → 𝑤 ∈ ∪ ([,] “ {𝑧 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ₀ ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑧) ⊆ 𝐴}))
135	92, 133, 134	syl2anc 693	. . . . . . . . 9 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → 𝑤 ∈ ∪ ([,] “ {𝑧 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ₀ ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑧) ⊆ 𝐴}))
136	56, 135	rexlimddv 3035	. . . . . . . 8 ⊢ (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) → 𝑤 ∈ ∪ ([,] “ {𝑧 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ₀ ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑧) ⊆ 𝐴}))
137	136	expr 643	. . . . . . 7 ⊢ (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ 𝑟 ∈ ℝ⁺) → (((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴 → 𝑤 ∈ ∪ ([,] “ {𝑧 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ₀ ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑧) ⊆ 𝐴})))
138	51, 137	sylbid 230	. . . . . 6 ⊢ (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ 𝑟 ∈ ℝ⁺) → ((𝑤(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ⊆ 𝐴 → 𝑤 ∈ ∪ ([,] “ {𝑧 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ₀ ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑧) ⊆ 𝐴})))
139	138	rexlimdva 3031	. . . . 5 ⊢ ((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) → (∃𝑟 ∈ ℝ⁺ (𝑤(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ⊆ 𝐴 → 𝑤 ∈ ∪ ([,] “ {𝑧 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ₀ ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑧) ⊆ 𝐴})))
140	43, 139	mpd 15	. . . 4 ⊢ ((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) → 𝑤 ∈ ∪ ([,] “ {𝑧 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ₀ ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑧) ⊆ 𝐴}))
141	140	ex 450	. . 3 ⊢ (𝐴 ∈ (topGen‘ran (,)) → (𝑤 ∈ 𝐴 → 𝑤 ∈ ∪ ([,] “ {𝑧 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ₀ ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑧) ⊆ 𝐴})))
142	141	ssrdv 3609	. 2 ⊢ (𝐴 ∈ (topGen‘ran (,)) → 𝐴 ⊆ ∪ ([,] “ {𝑧 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ₀ ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑧) ⊆ 𝐴}))
143	37, 142	eqssd 3620	1 ⊢ (𝐴 ∈ (topGen‘ran (,)) → ∪ ([,] “ {𝑧 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ₀ ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑧) ⊆ 𝐴}) = 𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ∃wrex 2913 {crab 2916 ∩ cin 3573 ⊆ wss 3574 𝒫 cpw 4158 ⟨cop 4183 ∪ cuni 4436 class class class wbr 4653 × cxp 5112 dom cdm 5114 ran crn 5115 ↾ cres 5116 “ cima 5117 ∘ ccom 5118 Fun wfun 5882 Fn wfn 5883 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ↦ cmpt2 6652 ℝcr 9935 0cc0 9936 1c1 9937 + caddc 9939 · cmul 9941 ℝ^*cxr 10073 < clt 10074 ≤ cle 10075 − cmin 10266 / cdiv 10684 ℕcn 11020 2c2 11070 ℕ₀cn0 11292 ℤcz 11377 ℝ⁺crp 11832 (,)cioo 12175 [,]cicc 12178 ⌊cfl 12591 ↑cexp 12860 abscabs 13974 topGenctg 16098 ∞Metcxmt 19731 ballcbl 19733 MetOpencmopn 19736

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-inf 8349 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ioo 12179 df-icc 12182 df-fl 12593 df-seq 12802 df-exp 12861 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-topgen 16104 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-top 20699 df-topon 20716 df-bases 20750

This theorem is referenced by: mblfinlem1 33446 mblfinlem2 33447

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