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Theorem opnmbllem 25915

Description: Lemma for opnmbl 25916. (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 6883	. . . . . . . . 9 ⊢ (𝑧 = 𝑤 → ([,]‘𝑧) = ([,]‘𝑤))
2	1	sseq1d 3962	. . . . . . . 8 ⊢ (𝑧 = 𝑤 → (([,]‘𝑧) ⊆ 𝐴 ↔ ([,]‘𝑤) ⊆ 𝐴))
3	2	elrab 3645	. . . . . . 7 ⊢ (𝑤 ∈ {𝑧 ∈ ran 𝐹 ∣ ([,]‘𝑧) ⊆ 𝐴} ↔ (𝑤 ∈ ran 𝐹 ∧ ([,]‘𝑤) ⊆ 𝐴))
4		simprr 785	. . . . . . . 8 ⊢ ((𝐴 ∈ (topGen‘ran (,)) ∧ (𝑤 ∈ ran 𝐹 ∧ ([,]‘𝑤) ⊆ 𝐴)) → ([,]‘𝑤) ⊆ 𝐴)
5		fvex 6896	. . . . . . . . 9 ⊢ ([,]‘𝑤) ∈ V
6	5	elpw 4561	. . . . . . . 8 ⊢ (([,]‘𝑤) ∈ 𝒫 𝐴 ↔ ([,]‘𝑤) ⊆ 𝐴)
7	4, 6	sylibr 237	. . . . . . 7 ⊢ ((𝐴 ∈ (topGen‘ran (,)) ∧ (𝑤 ∈ ran 𝐹 ∧ ([,]‘𝑤) ⊆ 𝐴)) → ([,]‘𝑤) ∈ 𝒫 𝐴)
8	3, 7	sylan2b 606	. . . . . 6 ⊢ ((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ {𝑧 ∈ ran 𝐹 ∣ ([,]‘𝑧) ⊆ 𝐴}) → ([,]‘𝑤) ∈ 𝒫 𝐴)
9	8	ralrimiva 3155	. . . . 5 ⊢ (𝐴 ∈ (topGen‘ran (,)) → ∀𝑤 ∈ {𝑧 ∈ ran 𝐹 ∣ ([,]‘𝑧) ⊆ 𝐴} ([,]‘𝑤) ∈ 𝒫 𝐴)
10		iccf 13572	. . . . . . 7 ⊢ [,]:(ℝ^* × ℝ^)⟶𝒫 ℝ^
11		ffun 6710	. . . . . . 7 ⊢ ([,]:(ℝ^* × ℝ^)⟶𝒫 ℝ^ → Fun [,])
12	10, 11	ax-mp 5	. . . . . 6 ⊢ Fun [,]
13		ssrab2 4028	. . . . . . . 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 25905	. . . . . . . . . 10 ⊢ 𝐹:(ℤ × ℕ₀)⟶( ≤ ∩ (ℝ × ℝ))
16		frn 6715	. . . . . . . . . 10 ⊢ (𝐹:(ℤ × ℕ₀)⟶( ≤ ∩ (ℝ × ℝ)) → ran 𝐹 ⊆ ( ≤ ∩ (ℝ × ℝ)))
17	15, 16	ax-mp 5	. . . . . . . . 9 ⊢ ran 𝐹 ⊆ ( ≤ ∩ (ℝ × ℝ))
18		inss2 4183	. . . . . . . . . 10 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
19		rexpssxrxp 11347	. . . . . . . . . 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 6719	. . . . . . 7 ⊢ dom [,] = (ℝ^* × ℝ^*)
24	22, 23	sseqtrri 3980	. . . . . 6 ⊢ {𝑧 ∈ ran 𝐹 ∣ ([,]‘𝑧) ⊆ 𝐴} ⊆ dom [,]
25		funimass4 6947	. . . . . 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 705	. . . . 5 ⊢ (([,] “ {𝑧 ∈ ran 𝐹 ∣ ([,]‘𝑧) ⊆ 𝐴}) ⊆ 𝒫 𝐴 ↔ ∀𝑤 ∈ {𝑧 ∈ ran 𝐹 ∣ ([,]‘𝑧) ⊆ 𝐴} ([,]‘𝑤) ∈ 𝒫 𝐴)
27	9, 26	sylibr 237	. . . 4 ⊢ (𝐴 ∈ (topGen‘ran (,)) → ([,] “ {𝑧 ∈ ran 𝐹 ∣ ([,]‘𝑧) ⊆ 𝐴}) ⊆ 𝒫 𝐴)
28		sspwuni 5060	. . . 4 ⊢ (([,] “ {𝑧 ∈ ran 𝐹 ∣ ([,]‘𝑧) ⊆ 𝐴}) ⊆ 𝒫 𝐴 ↔ ∪ ([,] “ {𝑧 ∈ ran 𝐹 ∣ ([,]‘𝑧) ⊆ 𝐴}) ⊆ 𝐴)
29	27, 28	sylib 221	. . 3 ⊢ (𝐴 ∈ (topGen‘ran (,)) → ∪ ([,] “ {𝑧 ∈ ran 𝐹 ∣ ([,]‘𝑧) ⊆ 𝐴}) ⊆ 𝐴)
30		eqid 2761	. . . . . 6 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) = ((abs ∘ − ) ↾ (ℝ × ℝ))
31	30	rexmet 25103	. . . . 5 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) ∈ (∞Met‘ℝ)
32		eqid 2761	. . . . . . 7 ⊢ (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ))) = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ)))
33	30, 32	tgioo 25108	. . . . . 6 ⊢ (topGen‘ran (,)) = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ)))
34	33	mopni2 24805	. . . . 5 ⊢ ((((abs ∘ − ) ↾ (ℝ × ℝ)) ∈ (∞Met‘ℝ) ∧ 𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) → ∃𝑟 ∈ ℝ⁺ (𝑤(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ⊆ 𝐴)
35	31, 34	mp3an1 1477	. . . 4 ⊢ ((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) → ∃𝑟 ∈ ℝ⁺ (𝑤(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ⊆ 𝐴)
36		elssuni 4899	. . . . . . . . . 10 ⊢ (𝐴 ∈ (topGen‘ran (,)) → 𝐴 ⊆ ∪ (topGen‘ran (,)))
37		uniretop 25074	. . . . . . . . . 10 ⊢ ℝ = ∪ (topGen‘ran (,))
38	36, 37	sseqtrrdi 3972	. . . . . . . . 9 ⊢ (𝐴 ∈ (topGen‘ran (,)) → 𝐴 ⊆ ℝ)
39	38	sselda 3931	. . . . . . . 8 ⊢ ((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) → 𝑤 ∈ ℝ)
40		rpre 13122	. . . . . . . 8 ⊢ (𝑟 ∈ ℝ⁺ → 𝑟 ∈ ℝ)
41	30	bl2ioo 25104	. . . . . . . 8 ⊢ ((𝑤 ∈ ℝ ∧ 𝑟 ∈ ℝ) → (𝑤(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) = ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)))
42	39, 40, 41	syl2an 608	. . . . . . 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 12410	. . . . . . . . . 10 ⊢ 2 ∈ ℝ
45		1lt2 12508	. . . . . . . . . 10 ⊢ 1 < 2
46		expnlbnd 14370	. . . . . . . . . 10 ⊢ ((𝑟 ∈ ℝ⁺ ∧ 2 ∈ ℝ ∧ 1 < 2) → ∃𝑛 ∈ ℕ (1 / (2↑𝑛)) < 𝑟)
47	44, 45, 46	mp3an23 1482	. . . . . . . . 9 ⊢ (𝑟 ∈ ℝ⁺ → ∃𝑛 ∈ ℕ (1 / (2↑𝑛)) < 𝑟)
48	47	ad2antrl 741	. . . . . . . 8 ⊢ (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) → ∃𝑛 ∈ ℕ (1 / (2↑𝑛)) < 𝑟)
49	39	ad2antrr 739	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → 𝑤 ∈ ℝ)
50		2nn 12409	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ ℕ
51		nnnn0 12606	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ₀)
52	51	ad2antrl 741	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → 𝑛 ∈ ℕ₀)
53		nnexpcl 14210	. . . . . . . . . . . . . . . . 17 ⊢ ((2 ∈ ℕ ∧ 𝑛 ∈ ℕ₀) → (2↑𝑛) ∈ ℕ)
54	50, 52, 53	sylancr 599	. . . . . . . . . . . . . . . 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 12343	. . . . . . . . . . . . . . 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 11332	. . . . . . . . . . . . . 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 13930	. . . . . . . . . . . . . 14 ⊢ ((𝑤 · (2↑𝑛)) ∈ ℝ → ((⌊‘(𝑤 · (2↑𝑛))) ≤ (𝑤 · (2↑𝑛)) ∧ (𝑤 · (2↑𝑛)) < ((⌊‘(𝑤 · (2↑𝑛))) + 1)))
58	56, 57	syl 18	. . . . . . . . . . . . 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 500	. . . . . . . . . . . 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 13929	. . . . . . . . . . . . . 14 ⊢ ((𝑤 · (2↑𝑛)) ∈ ℝ → (⌊‘(𝑤 · (2↑𝑛))) ∈ ℝ)
61	56, 60	syl 18	. . . . . . . . . . . . 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 12380	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → 0 < (2↑𝑛))
63		ledivmul2 12189	. . . . . . . . . . . . 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 1401	. . . . . . . . . . . 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 260	. . . . . . . . . . 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 11476	. . . . . . . . . . . . . 14 ⊢ ((⌊‘(𝑤 · (2↑𝑛))) ∈ ℝ → ((⌊‘(𝑤 · (2↑𝑛))) + 1) ∈ ℝ)
67	61, 66	syl 18	. . . . . . . . . . . . 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 12385	. . . . . . . . . . . 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 501	. . . . . . . . . . . . 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 12183	. . . . . . . . . . . . . 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 1401	. . . . . . . . . . . . 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 235	. . . . . . . . . . . 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 11451	. . . . . . . . . . 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 12385	. . . . . . . . . . . 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 13535	. . . . . . . . . . . 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 596	. . . . . . . . . . 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 1361	. . . . . . . . . 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 13931	. . . . . . . . . . . . 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 25906	. . . . . . . . . . . . 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 596	. . . . . . . . . . . 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 6887	. . . . . . . . . . 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 7421	. . . . . . . . . . 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 2814	. . . . . . . . . 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 2864	. . . . . . . . 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 6883	. . . . . . . . . . . 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 6707	. . . . . . . . . . . . 13 ⊢ (𝐹:(ℤ × ℕ₀)⟶( ≤ ∩ (ℝ × ℝ)) → 𝐹 Fn (ℤ × ℕ₀))
88	15, 87	ax-mp 5	. . . . . . . . . . . 12 ⊢ 𝐹 Fn (ℤ × ℕ₀)
89		fnovrn 7594	. . . . . . . . . . . 12 ⊢ ((𝐹 Fn (ℤ × ℕ₀) ∧ (⌊‘(𝑤 · (2↑𝑛))) ∈ ℤ ∧ 𝑛 ∈ ℕ₀) → ((⌊‘(𝑤 · (2↑𝑛)))𝐹𝑛) ∈ ran 𝐹)
90	88, 78, 52, 89	mp3an2i 1495	. . . . . . . . . . 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 789	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → 𝑟 ∈ ℝ⁺)
92	91	rpred 13157	. . . . . . . . . . . . . . . 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 11737	. . . . . . . . . . . . . . 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 11352	. . . . . . . . . . . . . 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 11331	. . . . . . . . . . . . . . 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 11352	. . . . . . . . . . . . . 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 11331	. . . . . . . . . . . . . . . 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 11330	. . . . . . . . . . . . . . . . . 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 11295	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → 1 ∈ ℂ)
100	55	recnd 11330	. . . . . . . . . . . . . . . . . 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 12381	. . . . . . . . . . . . . . . . . 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 12124	. . . . . . . . . . . . . . . . 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 12382	. . . . . . . . . . . . . . . . . 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 785	. . . . . . . . . . . . . . . . . 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 11462	. . . . . . . . . . . . . . . . 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 5127	. . . . . . . . . . . . . . . 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 11464	. . . . . . . . . . . . . . 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 11905	. . . . . . . . . . . . . . 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 260	. . . . . . . . . . . . . 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 11331	. . . . . . . . . . . . . . 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 11923	. . . . . . . . . . . . . . . 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 5127	. . . . . . . . . . . . . . 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 11462	. . . . . . . . . . . . . . 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 11461	. . . . . . . . . . . . . 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 13539	. . . . . . . . . . . . . 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 852	. . . . . . . . . . . . 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 790	. . . . . . . . . . . 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 3647	. . . . . . . . . 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 7235	. . . . . . . . . . 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 705	. . . . . . . . . 10 ⊢ (((⌊‘(𝑤 · (2↑𝑛)))𝐹𝑛) ∈ {𝑧 ∈ ran 𝐹 ∣ ([,]‘𝑧) ⊆ 𝐴} → ([,]‘((⌊‘(𝑤 · (2↑𝑛)))𝐹𝑛)) ∈ ([,] “ {𝑧 ∈ ran 𝐹 ∣ ([,]‘𝑧) ⊆ 𝐴}))
123	120, 122	syl 18	. . . . . . . . 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 4872	. . . . . . . . 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 596	. . . . . . . 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 3170	. . . . . . 7 ⊢ (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) → 𝑤 ∈ ∪ ([,] “ {𝑧 ∈ ran 𝐹 ∣ ([,]‘𝑧) ⊆ 𝐴}))
127	126	expr 462	. . . . . 6 ⊢ (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ 𝑟 ∈ ℝ⁺) → (((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴 → 𝑤 ∈ ∪ ([,] “ {𝑧 ∈ ran 𝐹 ∣ ([,]‘𝑧) ⊆ 𝐴})))
128	43, 127	sylbid 243	. . . . 5 ⊢ (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ 𝑟 ∈ ℝ⁺) → ((𝑤(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ⊆ 𝐴 → 𝑤 ∈ ∪ ([,] “ {𝑧 ∈ ran 𝐹 ∣ ([,]‘𝑧) ⊆ 𝐴})))
129	128	rexlimdva 3164	. . . 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 16	. . 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 6883	. . . . . . 7 ⊢ (𝑐 = 𝑎 → ([,]‘𝑐) = ([,]‘𝑎))
133	132	sseq1d 3962	. . . . . 6 ⊢ (𝑐 = 𝑎 → (([,]‘𝑐) ⊆ ([,]‘𝑏) ↔ ([,]‘𝑎) ⊆ ([,]‘𝑏)))
134		equequ1 2058	. . . . . 6 ⊢ (𝑐 = 𝑎 → (𝑐 = 𝑏 ↔ 𝑎 = 𝑏))
135	133, 134	imbi12d 347	. . . . 5 ⊢ (𝑐 = 𝑎 → ((([,]‘𝑐) ⊆ ([,]‘𝑏) → 𝑐 = 𝑏) ↔ (([,]‘𝑎) ⊆ ([,]‘𝑏) → 𝑎 = 𝑏)))
136	135	ralbidv 3186	. . . 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 3423	. . 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 25914	. 2 ⊢ (𝐴 ∈ (topGen‘ran (,)) → ∪ ([,] “ {𝑧 ∈ ran 𝐹 ∣ ([,]‘𝑧) ⊆ 𝐴}) ∈ dom vol)
140	131, 139	eqeltrrd 2862	1 ⊢ (𝐴 ∈ (topGen‘ran (,)) → 𝐴 ∈ dom vol)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ∀wral 3077 ∃wrex 3087 {crab 3413 ∩ cin 3898 ⊆ wss 3899 𝒫 cpw 4557 ⟨cop 4590 ∪ cuni 4867 class class class wbr 5103 × cxp 5649 dom cdm 5651 ran crn 5652 ↾ cres 5653 “ cima 5654 ∘ ccom 5655 Fun wfun 6531 Fn wfn 6532 ⟶wf 6533 ‘cfv 6537 (class class class)co 7418 ∈ cmpo 7420 ℝcr 11192 0cc0 11193 1c1 11194 + caddc 11196 · cmul 11198 ℝ^*cxr 11335 < clt 11336 ≤ cle 11337 − cmin 11534 / cdiv 11966 ℕcn 12328 2c2 12390 ℕ₀cn0 12599 ℤcz 12686 ℝ⁺crp 13113 (,)cioo 13469 [,]cicc 13472 ⌊cfl 13923 ↑cexp 14197 abscabs 15394 topGenctg 17601 ∞Metcxmet 21656 ballcbl 21658 MetOpencmopn 21661 volcvol 25777

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 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7749 ax-inf2 9635 ax-cnex 11249 ax-resscn 11250 ax-1cn 11251 ax-icn 11252 ax-addcl 11253 ax-addrcl 11254 ax-mulcl 11255 ax-mulrcl 11256 ax-mulcom 11257 ax-addass 11258 ax-mulass 11259 ax-distr 11260 ax-i2m1 11261 ax-1ne0 11262 ax-1rid 11263 ax-rnegex 11264 ax-rrecex 11265 ax-cnre 11266 ax-pre-lttri 11267 ax-pre-lttrn 11268 ax-pre-ltadd 11269 ax-pre-mulgt0 11270 ax-pre-sup 11271

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 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 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-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-disj 5071 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7375 df-ov 7421 df-oprab 7422 df-mpo 7423 df-of 7691 df-om 7876 df-1st 7999 df-2nd 8000 df-frecs 8292 df-wrecs 8323 df-recs 8372 df-rdg 8411 df-1o 8469 df-2o 8470 df-oadd 8473 df-omul 8474 df-er 8710 df-map 8842 df-pm 8843 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-fi 9396 df-sup 9427 df-inf 9428 df-oi 9497 df-dju 9975 df-card 10013 df-acn 10016 df-pnf 11338 df-mnf 11339 df-xr 11340 df-ltxr 11341 df-le 11342 df-sub 11536 df-neg 11537 df-div 11967 df-nn 12329 df-2 12398 df-3 12399 df-4 12400 df-n0 12600 df-z 12687 df-uz 12959 df-q 13069 df-rp 13114 df-xneg 13234 df-xadd 13235 df-xmul 13236 df-ioo 13473 df-ico 13475 df-icc 13476 df-fz 13633 df-fzo 13782 df-fl 13925 df-seq 14138 df-exp 14198 df-hash 14468 df-cj 15259 df-re 15260 df-im 15261 df-sqrt 15395 df-abs 15396 df-clim 15648 df-rlim 15649 df-sum 15847 df-rest 17586 df-topgen 17607 df-psmet 21663 df-xmet 21664 df-met 21665 df-bl 21666 df-mopn 21667 df-top 23205 df-topon 23222 df-bases 23257 df-cmp 23698 df-ovol 25778 df-vol 25779

This theorem is used by: opnmbl 25916

W3C validator