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Theorem opnmbllem 25549

Description: Lemma for opnmbl 25550. (Contributed by Mario Carneiro, 26-Mar-2015.)

**Hypothesis**
Ref	Expression
dyadmbl.1	⊢ 𝐹 = (𝑥 ∈ ℤ, 𝑦 ∈ ℕ₀ ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩)

**Assertion**
Ref	Expression
opnmbllem	⊢ (𝐴 ∈ (topGen‘ran (,)) → 𝐴 ∈ dom vol)

Distinct variable groups: 𝑥,𝑦,𝐴 𝑥,𝐹,𝑦

Proof of Theorem opnmbllem Dummy variables 𝑐 𝑎 𝑏 𝑛 𝑤 𝑧 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6831	. . . . . . . . 9 ⊢ (𝑧 = 𝑤 → ([,]‘𝑧) = ([,]‘𝑤))
2	1	sseq1d 3962	. . . . . . . 8 ⊢ (𝑧 = 𝑤 → (([,]‘𝑧) ⊆ 𝐴 ↔ ([,]‘𝑤) ⊆ 𝐴))
3	2	elrab 3643	. . . . . . 7 ⊢ (𝑤 ∈ {𝑧 ∈ ran 𝐹 ∣ ([,]‘𝑧) ⊆ 𝐴} ↔ (𝑤 ∈ ran 𝐹 ∧ ([,]‘𝑤) ⊆ 𝐴))
4		simprr 772	. . . . . . . 8 ⊢ ((𝐴 ∈ (topGen‘ran (,)) ∧ (𝑤 ∈ ran 𝐹 ∧ ([,]‘𝑤) ⊆ 𝐴)) → ([,]‘𝑤) ⊆ 𝐴)
5		fvex 6844	. . . . . . . . 9 ⊢ ([,]‘𝑤) ∈ V
6	5	elpw 4555	. . . . . . . 8 ⊢ (([,]‘𝑤) ∈ 𝒫 𝐴 ↔ ([,]‘𝑤) ⊆ 𝐴)
7	4, 6	sylibr 234	. . . . . . 7 ⊢ ((𝐴 ∈ (topGen‘ran (,)) ∧ (𝑤 ∈ ran 𝐹 ∧ ([,]‘𝑤) ⊆ 𝐴)) → ([,]‘𝑤) ∈ 𝒫 𝐴)
8	3, 7	sylan2b 594	. . . . . 6 ⊢ ((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ {𝑧 ∈ ran 𝐹 ∣ ([,]‘𝑧) ⊆ 𝐴}) → ([,]‘𝑤) ∈ 𝒫 𝐴)
9	8	ralrimiva 3125	. . . . 5 ⊢ (𝐴 ∈ (topGen‘ran (,)) → ∀𝑤 ∈ {𝑧 ∈ ran 𝐹 ∣ ([,]‘𝑧) ⊆ 𝐴} ([,]‘𝑤) ∈ 𝒫 𝐴)
10		iccf 13355	. . . . . . 7 ⊢ [,]:(ℝ^* × ℝ^)⟶𝒫 ℝ^
11		ffun 6662	. . . . . . 7 ⊢ ([,]:(ℝ^* × ℝ^)⟶𝒫 ℝ^ → Fun [,])
12	10, 11	ax-mp 5	. . . . . 6 ⊢ Fun [,]
13		ssrab2 4029	. . . . . . . 8 ⊢ {𝑧 ∈ ran 𝐹 ∣ ([,]‘𝑧) ⊆ 𝐴} ⊆ ran 𝐹
14		dyadmbl.1	. . . . . . . . . . 11 ⊢ 𝐹 = (𝑥 ∈ ℤ, 𝑦 ∈ ℕ₀ ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩)
15	14	dyadf 25539	. . . . . . . . . 10 ⊢ 𝐹:(ℤ × ℕ₀)⟶( ≤ ∩ (ℝ × ℝ))
16		frn 6666	. . . . . . . . . 10 ⊢ (𝐹:(ℤ × ℕ₀)⟶( ≤ ∩ (ℝ × ℝ)) → ran 𝐹 ⊆ ( ≤ ∩ (ℝ × ℝ)))
17	15, 16	ax-mp 5	. . . . . . . . 9 ⊢ ran 𝐹 ⊆ ( ≤ ∩ (ℝ × ℝ))
18		inss2 4187	. . . . . . . . . 10 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
19		rexpssxrxp 11168	. . . . . . . . . 10 ⊢ (ℝ × ℝ) ⊆ (ℝ^* × ℝ^*)
20	18, 19	sstri 3940	. . . . . . . . 9 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ^* × ℝ^*)
21	17, 20	sstri 3940	. . . . . . . 8 ⊢ ran 𝐹 ⊆ (ℝ^* × ℝ^*)
22	13, 21	sstri 3940	. . . . . . 7 ⊢ {𝑧 ∈ ran 𝐹 ∣ ([,]‘𝑧) ⊆ 𝐴} ⊆ (ℝ^* × ℝ^*)
23	10	fdmi 6670	. . . . . . 7 ⊢ dom [,] = (ℝ^* × ℝ^*)
24	22, 23	sseqtrri 3980	. . . . . 6 ⊢ {𝑧 ∈ ran 𝐹 ∣ ([,]‘𝑧) ⊆ 𝐴} ⊆ dom [,]
25		funimass4 6895	. . . . . 6 ⊢ ((Fun [,] ∧ {𝑧 ∈ ran 𝐹 ∣ ([,]‘𝑧) ⊆ 𝐴} ⊆ dom [,]) → (([,] “ {𝑧 ∈ ran 𝐹 ∣ ([,]‘𝑧) ⊆ 𝐴}) ⊆ 𝒫 𝐴 ↔ ∀𝑤 ∈ {𝑧 ∈ ran 𝐹 ∣ ([,]‘𝑧) ⊆ 𝐴} ([,]‘𝑤) ∈ 𝒫 𝐴))
26	12, 24, 25	mp2an 692	. . . . 5 ⊢ (([,] “ {𝑧 ∈ ran 𝐹 ∣ ([,]‘𝑧) ⊆ 𝐴}) ⊆ 𝒫 𝐴 ↔ ∀𝑤 ∈ {𝑧 ∈ ran 𝐹 ∣ ([,]‘𝑧) ⊆ 𝐴} ([,]‘𝑤) ∈ 𝒫 𝐴)
27	9, 26	sylibr 234	. . . 4 ⊢ (𝐴 ∈ (topGen‘ran (,)) → ([,] “ {𝑧 ∈ ran 𝐹 ∣ ([,]‘𝑧) ⊆ 𝐴}) ⊆ 𝒫 𝐴)
28		sspwuni 5052	. . . 4 ⊢ (([,] “ {𝑧 ∈ ran 𝐹 ∣ ([,]‘𝑧) ⊆ 𝐴}) ⊆ 𝒫 𝐴 ↔ ∪ ([,] “ {𝑧 ∈ ran 𝐹 ∣ ([,]‘𝑧) ⊆ 𝐴}) ⊆ 𝐴)
29	27, 28	sylib 218	. . 3 ⊢ (𝐴 ∈ (topGen‘ran (,)) → ∪ ([,] “ {𝑧 ∈ ran 𝐹 ∣ ([,]‘𝑧) ⊆ 𝐴}) ⊆ 𝐴)
30		eqid 2733	. . . . . 6 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) = ((abs ∘ − ) ↾ (ℝ × ℝ))
31	30	rexmet 24726	. . . . 5 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) ∈ (∞Met‘ℝ)
32		eqid 2733	. . . . . . 7 ⊢ (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ))) = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ)))
33	30, 32	tgioo 24731	. . . . . 6 ⊢ (topGen‘ran (,)) = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ)))
34	33	mopni2 24428	. . . . 5 ⊢ ((((abs ∘ − ) ↾ (ℝ × ℝ)) ∈ (∞Met‘ℝ) ∧ 𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) → ∃𝑟 ∈ ℝ⁺ (𝑤(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ⊆ 𝐴)
35	31, 34	mp3an1 1450	. . . 4 ⊢ ((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) → ∃𝑟 ∈ ℝ⁺ (𝑤(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ⊆ 𝐴)
36		elssuni 4891	. . . . . . . . . 10 ⊢ (𝐴 ∈ (topGen‘ran (,)) → 𝐴 ⊆ ∪ (topGen‘ran (,)))
37		uniretop 24697	. . . . . . . . . 10 ⊢ ℝ = ∪ (topGen‘ran (,))
38	36, 37	sseqtrrdi 3972	. . . . . . . . 9 ⊢ (𝐴 ∈ (topGen‘ran (,)) → 𝐴 ⊆ ℝ)
39	38	sselda 3930	. . . . . . . 8 ⊢ ((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) → 𝑤 ∈ ℝ)
40		rpre 12905	. . . . . . . 8 ⊢ (𝑟 ∈ ℝ⁺ → 𝑟 ∈ ℝ)
41	30	bl2ioo 24727	. . . . . . . 8 ⊢ ((𝑤 ∈ ℝ ∧ 𝑟 ∈ ℝ) → (𝑤(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) = ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)))
42	39, 40, 41	syl2an 596	. . . . . . 7 ⊢ (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ 𝑟 ∈ ℝ⁺) → (𝑤(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) = ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)))
43	42	sseq1d 3962	. . . . . 6 ⊢ (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ 𝑟 ∈ ℝ⁺) → ((𝑤(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ⊆ 𝐴 ↔ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴))
44		2re 12210	. . . . . . . . . 10 ⊢ 2 ∈ ℝ
45		1lt2 12302	. . . . . . . . . 10 ⊢ 1 < 2
46		expnlbnd 14147	. . . . . . . . . 10 ⊢ ((𝑟 ∈ ℝ⁺ ∧ 2 ∈ ℝ ∧ 1 < 2) → ∃𝑛 ∈ ℕ (1 / (2↑𝑛)) < 𝑟)
47	44, 45, 46	mp3an23 1455	. . . . . . . . 9 ⊢ (𝑟 ∈ ℝ⁺ → ∃𝑛 ∈ ℕ (1 / (2↑𝑛)) < 𝑟)
48	47	ad2antrl 728	. . . . . . . 8 ⊢ (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) → ∃𝑛 ∈ ℕ (1 / (2↑𝑛)) < 𝑟)
49	39	ad2antrr 726	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → 𝑤 ∈ ℝ)
50		2nn 12209	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ ℕ
51		nnnn0 12399	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ₀)
52	51	ad2antrl 728	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → 𝑛 ∈ ℕ₀)
53		nnexpcl 13988	. . . . . . . . . . . . . . . . 17 ⊢ ((2 ∈ ℕ ∧ 𝑛 ∈ ℕ₀) → (2↑𝑛) ∈ ℕ)
54	50, 52, 53	sylancr 587	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → (2↑𝑛) ∈ ℕ)
55	54	nnred 12151	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → (2↑𝑛) ∈ ℝ)
56	49, 55	remulcld 11153	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → (𝑤 · (2↑𝑛)) ∈ ℝ)
57		fllelt 13708	. . . . . . . . . . . . . 14 ⊢ ((𝑤 · (2↑𝑛)) ∈ ℝ → ((⌊‘(𝑤 · (2↑𝑛))) ≤ (𝑤 · (2↑𝑛)) ∧ (𝑤 · (2↑𝑛)) < ((⌊‘(𝑤 · (2↑𝑛))) + 1)))
58	56, 57	syl 17	. . . . . . . . . . . . 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)) ∧ (𝑤 · (2↑𝑛)) < ((⌊‘(𝑤 · (2↑𝑛))) + 1)))
59	58	simpld 494	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → (⌊‘(𝑤 · (2↑𝑛))) ≤ (𝑤 · (2↑𝑛)))
60		reflcl 13707	. . . . . . . . . . . . . 14 ⊢ ((𝑤 · (2↑𝑛)) ∈ ℝ → (⌊‘(𝑤 · (2↑𝑛))) ∈ ℝ)
61	56, 60	syl 17	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → (⌊‘(𝑤 · (2↑𝑛))) ∈ ℝ)
62	54	nngt0d 12185	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → 0 < (2↑𝑛))
63		ledivmul2 12012	. . . . . . . . . . . . 13 ⊢ (((⌊‘(𝑤 · (2↑𝑛))) ∈ ℝ ∧ 𝑤 ∈ ℝ ∧ ((2↑𝑛) ∈ ℝ ∧ 0 < (2↑𝑛))) → (((⌊‘(𝑤 · (2↑𝑛))) / (2↑𝑛)) ≤ 𝑤 ↔ (⌊‘(𝑤 · (2↑𝑛))) ≤ (𝑤 · (2↑𝑛))))
64	61, 49, 55, 62, 63	syl112anc 1376	. . . . . . . . . . . 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 ↔ (⌊‘(𝑤 · (2↑𝑛))) ≤ (𝑤 · (2↑𝑛))))
65	59, 64	mpbird 257	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → ((⌊‘(𝑤 · (2↑𝑛))) / (2↑𝑛)) ≤ 𝑤)
66		peano2re 11297	. . . . . . . . . . . . . 14 ⊢ ((⌊‘(𝑤 · (2↑𝑛))) ∈ ℝ → ((⌊‘(𝑤 · (2↑𝑛))) + 1) ∈ ℝ)
67	61, 66	syl 17	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → ((⌊‘(𝑤 · (2↑𝑛))) + 1) ∈ ℝ)
68	67, 54	nndivred 12190	. . . . . . . . . . . 12 ⊢ ((((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)
69	58	simprd 495	. . . . . . . . . . . . 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))) + 1))
70		ltmuldiv 12006	. . . . . . . . . . . . . 14 ⊢ ((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))))
71	49, 67, 55, 62, 70	syl112anc 1376	. . . . . . . . . . . . 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))) + 1) ↔ 𝑤 < (((⌊‘(𝑤 · (2↑𝑛))) + 1) / (2↑𝑛))))
72	69, 71	mpbid 232	. . . . . . . . . . . 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)))
73	49, 68, 72	ltled 11272	. . . . . . . . . . 11 ⊢ ((((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)))
74	61, 54	nndivred 12190	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → ((⌊‘(𝑤 · (2↑𝑛))) / (2↑𝑛)) ∈ ℝ)
75		elicc2 13318	. . . . . . . . . . . 12 ⊢ ((((⌊‘(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)))))
76	74, 68, 75	syl2anc 584	. . . . . . . . . . 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))) ↔ (𝑤 ∈ ℝ ∧ ((⌊‘(𝑤 · (2↑𝑛))) / (2↑𝑛)) ≤ 𝑤 ∧ 𝑤 ≤ (((⌊‘(𝑤 · (2↑𝑛))) + 1) / (2↑𝑛)))))
77	49, 65, 73, 76	mpbir3and 1343	. . . . . . . . . 10 ⊢ ((((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))))
78	56	flcld 13709	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → (⌊‘(𝑤 · (2↑𝑛))) ∈ ℤ)
79	14	dyadval 25540	. . . . . . . . . . . . 13 ⊢ (((⌊‘(𝑤 · (2↑𝑛))) ∈ ℤ ∧ 𝑛 ∈ ℕ₀) → ((⌊‘(𝑤 · (2↑𝑛)))𝐹𝑛) = ⟨((⌊‘(𝑤 · (2↑𝑛))) / (2↑𝑛)), (((⌊‘(𝑤 · (2↑𝑛))) + 1) / (2↑𝑛))⟩)
80	78, 52, 79	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → ((⌊‘(𝑤 · (2↑𝑛)))𝐹𝑛) = ⟨((⌊‘(𝑤 · (2↑𝑛))) / (2↑𝑛)), (((⌊‘(𝑤 · (2↑𝑛))) + 1) / (2↑𝑛))⟩)
81	80	fveq2d 6835	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → ([,]‘((⌊‘(𝑤 · (2↑𝑛)))𝐹𝑛)) = ([,]‘⟨((⌊‘(𝑤 · (2↑𝑛))) / (2↑𝑛)), (((⌊‘(𝑤 · (2↑𝑛))) + 1) / (2↑𝑛))⟩))
82		df-ov 7358	. . . . . . . . . . 11 ⊢ (((⌊‘(𝑤 · (2↑𝑛))) / (2↑𝑛))[,](((⌊‘(𝑤 · (2↑𝑛))) + 1) / (2↑𝑛))) = ([,]‘⟨((⌊‘(𝑤 · (2↑𝑛))) / (2↑𝑛)), (((⌊‘(𝑤 · (2↑𝑛))) + 1) / (2↑𝑛))⟩)
83	81, 82	eqtr4di 2786	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → ([,]‘((⌊‘(𝑤 · (2↑𝑛)))𝐹𝑛)) = (((⌊‘(𝑤 · (2↑𝑛))) / (2↑𝑛))[,](((⌊‘(𝑤 · (2↑𝑛))) + 1) / (2↑𝑛))))
84	77, 83	eleqtrrd 2836	. . . . . . . . 9 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → 𝑤 ∈ ([,]‘((⌊‘(𝑤 · (2↑𝑛)))𝐹𝑛)))
85		fveq2 6831	. . . . . . . . . . . 12 ⊢ (𝑧 = ((⌊‘(𝑤 · (2↑𝑛)))𝐹𝑛) → ([,]‘𝑧) = ([,]‘((⌊‘(𝑤 · (2↑𝑛)))𝐹𝑛)))
86	85	sseq1d 3962	. . . . . . . . . . 11 ⊢ (𝑧 = ((⌊‘(𝑤 · (2↑𝑛)))𝐹𝑛) → (([,]‘𝑧) ⊆ 𝐴 ↔ ([,]‘((⌊‘(𝑤 · (2↑𝑛)))𝐹𝑛)) ⊆ 𝐴))
87		ffn 6659	. . . . . . . . . . . . 13 ⊢ (𝐹:(ℤ × ℕ₀)⟶( ≤ ∩ (ℝ × ℝ)) → 𝐹 Fn (ℤ × ℕ₀))
88	15, 87	ax-mp 5	. . . . . . . . . . . 12 ⊢ 𝐹 Fn (ℤ × ℕ₀)
89		fnovrn 7530	. . . . . . . . . . . 12 ⊢ ((𝐹 Fn (ℤ × ℕ₀) ∧ (⌊‘(𝑤 · (2↑𝑛))) ∈ ℤ ∧ 𝑛 ∈ ℕ₀) → ((⌊‘(𝑤 · (2↑𝑛)))𝐹𝑛) ∈ ran 𝐹)
90	88, 78, 52, 89	mp3an2i 1468	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → ((⌊‘(𝑤 · (2↑𝑛)))𝐹𝑛) ∈ ran 𝐹)
91		simplrl 776	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → 𝑟 ∈ ℝ⁺)
92	91	rpred 12940	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → 𝑟 ∈ ℝ)
93	49, 92	resubcld 11556	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → (𝑤 − 𝑟) ∈ ℝ)
94	93	rexrd 11173	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → (𝑤 − 𝑟) ∈ ℝ^*)
95	49, 92	readdcld 11152	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → (𝑤 + 𝑟) ∈ ℝ)
96	95	rexrd 11173	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → (𝑤 + 𝑟) ∈ ℝ^*)
97	74, 92	readdcld 11152	. . . . . . . . . . . . . . . 16 ⊢ ((((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)
98	61	recnd 11151	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → (⌊‘(𝑤 · (2↑𝑛))) ∈ ℂ)
99		1cnd 11118	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → 1 ∈ ℂ)
100	55	recnd 11151	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → (2↑𝑛) ∈ ℂ)
101	54	nnne0d 12186	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → (2↑𝑛) ≠ 0)
102	98, 99, 100, 101	divdird 11946	. . . . . . . . . . . . . . . . 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)) + (1 / (2↑𝑛))))
103	54	nnrecred 12187	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → (1 / (2↑𝑛)) ∈ ℝ)
104		simprr 772	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → (1 / (2↑𝑛)) < 𝑟)
105	103, 92, 74, 104	ltadd2dd 11283	. . . . . . . . . . . . . . . . 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 · (2↑𝑛))) / (2↑𝑛)) + 𝑟))
106	102, 105	eqbrtrd 5117	. . . . . . . . . . . . . . . 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 · (2↑𝑛))) / (2↑𝑛)) + 𝑟))
107	49, 68, 97, 72, 106	lttrd 11285	. . . . . . . . . . . . . . 15 ⊢ ((((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))
108	49, 92, 74	ltsubaddd 11724	. . . . . . . . . . . . . . 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)) ↔ 𝑤 < (((⌊‘(𝑤 · (2↑𝑛))) / (2↑𝑛)) + 𝑟)))
109	107, 108	mpbird 257	. . . . . . . . . . . . . 14 ⊢ ((((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)))
110	49, 103	readdcld 11152	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → (𝑤 + (1 / (2↑𝑛))) ∈ ℝ)
111	74, 49, 103, 65	leadd1dd 11742	. . . . . . . . . . . . . . . 16 ⊢ ((((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))))
112	102, 111	eqbrtrd 5117	. . . . . . . . . . . . . . 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 + (1 / (2↑𝑛))))
113	103, 92, 49, 104	ltadd2dd 11283	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → (𝑤 + (1 / (2↑𝑛))) < (𝑤 + 𝑟))
114	68, 110, 95, 112, 113	lelttrd 11282	. . . . . . . . . . . . . 14 ⊢ ((((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))
115		iccssioo 13322	. . . . . . . . . . . . . 14 ⊢ ((((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)))
116	94, 96, 109, 114, 115	syl22anc 838	. . . . . . . . . . . . 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 · (2↑𝑛))) + 1) / (2↑𝑛))) ⊆ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)))
117	83, 116	eqsstrd 3965	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → ([,]‘((⌊‘(𝑤 · (2↑𝑛)))𝐹𝑛)) ⊆ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)))
118		simplrr 777	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)
119	117, 118	sstrd 3941	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → ([,]‘((⌊‘(𝑤 · (2↑𝑛)))𝐹𝑛)) ⊆ 𝐴)
120	86, 90, 119	elrabd 3645	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → ((⌊‘(𝑤 · (2↑𝑛)))𝐹𝑛) ∈ {𝑧 ∈ ran 𝐹 ∣ ([,]‘𝑧) ⊆ 𝐴})
121		funfvima2 7174	. . . . . . . . . . 11 ⊢ ((Fun [,] ∧ {𝑧 ∈ ran 𝐹 ∣ ([,]‘𝑧) ⊆ 𝐴} ⊆ dom [,]) → (((⌊‘(𝑤 · (2↑𝑛)))𝐹𝑛) ∈ {𝑧 ∈ ran 𝐹 ∣ ([,]‘𝑧) ⊆ 𝐴} → ([,]‘((⌊‘(𝑤 · (2↑𝑛)))𝐹𝑛)) ∈ ([,] “ {𝑧 ∈ ran 𝐹 ∣ ([,]‘𝑧) ⊆ 𝐴})))
122	12, 24, 121	mp2an 692	. . . . . . . . . 10 ⊢ (((⌊‘(𝑤 · (2↑𝑛)))𝐹𝑛) ∈ {𝑧 ∈ ran 𝐹 ∣ ([,]‘𝑧) ⊆ 𝐴} → ([,]‘((⌊‘(𝑤 · (2↑𝑛)))𝐹𝑛)) ∈ ([,] “ {𝑧 ∈ ran 𝐹 ∣ ([,]‘𝑧) ⊆ 𝐴}))
123	120, 122	syl 17	. . . . . . . . 9 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → ([,]‘((⌊‘(𝑤 · (2↑𝑛)))𝐹𝑛)) ∈ ([,] “ {𝑧 ∈ ran 𝐹 ∣ ([,]‘𝑧) ⊆ 𝐴}))
124		elunii 4865	. . . . . . . . 9 ⊢ ((𝑤 ∈ ([,]‘((⌊‘(𝑤 · (2↑𝑛)))𝐹𝑛)) ∧ ([,]‘((⌊‘(𝑤 · (2↑𝑛)))𝐹𝑛)) ∈ ([,] “ {𝑧 ∈ ran 𝐹 ∣ ([,]‘𝑧) ⊆ 𝐴})) → 𝑤 ∈ ∪ ([,] “ {𝑧 ∈ ran 𝐹 ∣ ([,]‘𝑧) ⊆ 𝐴}))
125	84, 123, 124	syl2anc 584	. . . . . . . 8 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → 𝑤 ∈ ∪ ([,] “ {𝑧 ∈ ran 𝐹 ∣ ([,]‘𝑧) ⊆ 𝐴}))
126	48, 125	rexlimddv 3140	. . . . . . 7 ⊢ (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) → 𝑤 ∈ ∪ ([,] “ {𝑧 ∈ ran 𝐹 ∣ ([,]‘𝑧) ⊆ 𝐴}))
127	126	expr 456	. . . . . 6 ⊢ (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ 𝑟 ∈ ℝ⁺) → (((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴 → 𝑤 ∈ ∪ ([,] “ {𝑧 ∈ ran 𝐹 ∣ ([,]‘𝑧) ⊆ 𝐴})))
128	43, 127	sylbid 240	. . . . 5 ⊢ (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ 𝑟 ∈ ℝ⁺) → ((𝑤(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ⊆ 𝐴 → 𝑤 ∈ ∪ ([,] “ {𝑧 ∈ ran 𝐹 ∣ ([,]‘𝑧) ⊆ 𝐴})))
129	128	rexlimdva 3134	. . . 4 ⊢ ((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) → (∃𝑟 ∈ ℝ⁺ (𝑤(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ⊆ 𝐴 → 𝑤 ∈ ∪ ([,] “ {𝑧 ∈ ran 𝐹 ∣ ([,]‘𝑧) ⊆ 𝐴})))
130	35, 129	mpd 15	. . 3 ⊢ ((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) → 𝑤 ∈ ∪ ([,] “ {𝑧 ∈ ran 𝐹 ∣ ([,]‘𝑧) ⊆ 𝐴}))
131	29, 130	eqelssd 3952	. 2 ⊢ (𝐴 ∈ (topGen‘ran (,)) → ∪ ([,] “ {𝑧 ∈ ran 𝐹 ∣ ([,]‘𝑧) ⊆ 𝐴}) = 𝐴)
132		fveq2 6831	. . . . . . 7 ⊢ (𝑐 = 𝑎 → ([,]‘𝑐) = ([,]‘𝑎))
133	132	sseq1d 3962	. . . . . 6 ⊢ (𝑐 = 𝑎 → (([,]‘𝑐) ⊆ ([,]‘𝑏) ↔ ([,]‘𝑎) ⊆ ([,]‘𝑏)))
134		equequ1 2026	. . . . . 6 ⊢ (𝑐 = 𝑎 → (𝑐 = 𝑏 ↔ 𝑎 = 𝑏))
135	133, 134	imbi12d 344	. . . . 5 ⊢ (𝑐 = 𝑎 → ((([,]‘𝑐) ⊆ ([,]‘𝑏) → 𝑐 = 𝑏) ↔ (([,]‘𝑎) ⊆ ([,]‘𝑏) → 𝑎 = 𝑏)))
136	135	ralbidv 3156	. . . 4 ⊢ (𝑐 = 𝑎 → (∀𝑏 ∈ {𝑧 ∈ ran 𝐹 ∣ ([,]‘𝑧) ⊆ 𝐴} (([,]‘𝑐) ⊆ ([,]‘𝑏) → 𝑐 = 𝑏) ↔ ∀𝑏 ∈ {𝑧 ∈ ran 𝐹 ∣ ([,]‘𝑧) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑏) → 𝑎 = 𝑏)))
137	136	cbvrabv 3406	. . 3 ⊢ {𝑐 ∈ {𝑧 ∈ ran 𝐹 ∣ ([,]‘𝑧) ⊆ 𝐴} ∣ ∀𝑏 ∈ {𝑧 ∈ ran 𝐹 ∣ ([,]‘𝑧) ⊆ 𝐴} (([,]‘𝑐) ⊆ ([,]‘𝑏) → 𝑐 = 𝑏)} = {𝑎 ∈ {𝑧 ∈ ran 𝐹 ∣ ([,]‘𝑧) ⊆ 𝐴} ∣ ∀𝑏 ∈ {𝑧 ∈ ran 𝐹 ∣ ([,]‘𝑧) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑏) → 𝑎 = 𝑏)}
138	13	a1i 11	. . 3 ⊢ (𝐴 ∈ (topGen‘ran (,)) → {𝑧 ∈ ran 𝐹 ∣ ([,]‘𝑧) ⊆ 𝐴} ⊆ ran 𝐹)
139	14, 137, 138	dyadmbl 25548	. 2 ⊢ (𝐴 ∈ (topGen‘ran (,)) → ∪ ([,] “ {𝑧 ∈ ran 𝐹 ∣ ([,]‘𝑧) ⊆ 𝐴}) ∈ dom vol)
140	131, 139	eqeltrrd 2834	1 ⊢ (𝐴 ∈ (topGen‘ran (,)) → 𝐴 ∈ dom vol)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2113 ∀wral 3048 ∃wrex 3057 {crab 3396 ∩ cin 3897 ⊆ wss 3898 𝒫 cpw 4551 ⟨cop 4583 ∪ cuni 4860 class class class wbr 5095 × cxp 5619 dom cdm 5621 ran crn 5622 ↾ cres 5623 “ cima 5624 ∘ ccom 5625 Fun wfun 6483 Fn wfn 6484 ⟶wf 6485 ‘cfv 6489 (class class class)co 7355 ∈ cmpo 7357 ℝcr 11016 0cc0 11017 1c1 11018 + caddc 11020 · cmul 11022 ℝ^*cxr 11156 < clt 11157 ≤ cle 11158 − cmin 11355 / cdiv 11785 ℕcn 12136 2c2 12191 ℕ₀cn0 12392 ℤcz 12479 ℝ⁺crp 12896 (,)cioo 13252 [,]cicc 13255 ⌊cfl 13701 ↑cexp 13975 abscabs 15148 topGenctg 17348 ∞Metcxmet 21285 ballcbl 21287 MetOpencmopn 21290 volcvol 25411

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7677 ax-inf2 9542 ax-cnex 11073 ax-resscn 11074 ax-1cn 11075 ax-icn 11076 ax-addcl 11077 ax-addrcl 11078 ax-mulcl 11079 ax-mulrcl 11080 ax-mulcom 11081 ax-addass 11082 ax-mulass 11083 ax-distr 11084 ax-i2m1 11085 ax-1ne0 11086 ax-1rid 11087 ax-rnegex 11088 ax-rrecex 11089 ax-cnre 11090 ax-pre-lttri 11091 ax-pre-lttrn 11092 ax-pre-ltadd 11093 ax-pre-mulgt0 11094 ax-pre-sup 11095

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-rmo 3347 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4283 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4861 df-int 4900 df-iun 4945 df-disj 5063 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-se 5575 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-isom 6498 df-riota 7312 df-ov 7358 df-oprab 7359 df-mpo 7360 df-of 7619 df-om 7806 df-1st 7930 df-2nd 7931 df-frecs 8220 df-wrecs 8251 df-recs 8300 df-rdg 8338 df-1o 8394 df-2o 8395 df-oadd 8398 df-omul 8399 df-er 8631 df-map 8761 df-pm 8762 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-fi 9306 df-sup 9337 df-inf 9338 df-oi 9407 df-dju 9805 df-card 9843 df-acn 9846 df-pnf 11159 df-mnf 11160 df-xr 11161 df-ltxr 11162 df-le 11163 df-sub 11357 df-neg 11358 df-div 11786 df-nn 12137 df-2 12199 df-3 12200 df-4 12201 df-n0 12393 df-z 12480 df-uz 12743 df-q 12853 df-rp 12897 df-xneg 13017 df-xadd 13018 df-xmul 13019 df-ioo 13256 df-ico 13258 df-icc 13259 df-fz 13415 df-fzo 13562 df-fl 13703 df-seq 13916 df-exp 13976 df-hash 14245 df-cj 15013 df-re 15014 df-im 15015 df-sqrt 15149 df-abs 15150 df-clim 15402 df-rlim 15403 df-sum 15601 df-rest 17333 df-topgen 17354 df-psmet 21292 df-xmet 21293 df-met 21294 df-bl 21295 df-mopn 21296 df-top 22829 df-topon 22846 df-bases 22881 df-cmp 23322 df-ovol 25412 df-vol 25413

This theorem is referenced by: opnmbl 25550

W3C validator