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Theorem opnmbllem 25655

Description: Lemma for opnmbl 25656. (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 6920	. . . . . . . . 9 ⊢ (𝑧 = 𝑤 → ([,]‘𝑧) = ([,]‘𝑤))
2	1	sseq1d 4040	. . . . . . . 8 ⊢ (𝑧 = 𝑤 → (([,]‘𝑧) ⊆ 𝐴 ↔ ([,]‘𝑤) ⊆ 𝐴))
3	2	elrab 3708	. . . . . . 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 6933	. . . . . . . . 9 ⊢ ([,]‘𝑤) ∈ V
6	5	elpw 4626	. . . . . . . 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 593	. . . . . 6 ⊢ ((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ {𝑧 ∈ ran 𝐹 ∣ ([,]‘𝑧) ⊆ 𝐴}) → ([,]‘𝑤) ∈ 𝒫 𝐴)
9	8	ralrimiva 3152	. . . . 5 ⊢ (𝐴 ∈ (topGen‘ran (,)) → ∀𝑤 ∈ {𝑧 ∈ ran 𝐹 ∣ ([,]‘𝑧) ⊆ 𝐴} ([,]‘𝑤) ∈ 𝒫 𝐴)
10		iccf 13508	. . . . . . 7 ⊢ [,]:(ℝ^* × ℝ^)⟶𝒫 ℝ^
11		ffun 6750	. . . . . . 7 ⊢ ([,]:(ℝ^* × ℝ^)⟶𝒫 ℝ^ → Fun [,])
12	10, 11	ax-mp 5	. . . . . 6 ⊢ Fun [,]
13		ssrab2 4103	. . . . . . . 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 25645	. . . . . . . . . 10 ⊢ 𝐹:(ℤ × ℕ₀)⟶( ≤ ∩ (ℝ × ℝ))
16		frn 6754	. . . . . . . . . 10 ⊢ (𝐹:(ℤ × ℕ₀)⟶( ≤ ∩ (ℝ × ℝ)) → ran 𝐹 ⊆ ( ≤ ∩ (ℝ × ℝ)))
17	15, 16	ax-mp 5	. . . . . . . . 9 ⊢ ran 𝐹 ⊆ ( ≤ ∩ (ℝ × ℝ))
18		inss2 4259	. . . . . . . . . 10 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
19		rexpssxrxp 11335	. . . . . . . . . 10 ⊢ (ℝ × ℝ) ⊆ (ℝ^* × ℝ^*)
20	18, 19	sstri 4018	. . . . . . . . 9 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ^* × ℝ^*)
21	17, 20	sstri 4018	. . . . . . . 8 ⊢ ran 𝐹 ⊆ (ℝ^* × ℝ^*)
22	13, 21	sstri 4018	. . . . . . 7 ⊢ {𝑧 ∈ ran 𝐹 ∣ ([,]‘𝑧) ⊆ 𝐴} ⊆ (ℝ^* × ℝ^*)
23	10	fdmi 6758	. . . . . . 7 ⊢ dom [,] = (ℝ^* × ℝ^*)
24	22, 23	sseqtrri 4046	. . . . . 6 ⊢ {𝑧 ∈ ran 𝐹 ∣ ([,]‘𝑧) ⊆ 𝐴} ⊆ dom [,]
25		funimass4 6986	. . . . . 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 691	. . . . 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 5123	. . . 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 2740	. . . . . 6 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) = ((abs ∘ − ) ↾ (ℝ × ℝ))
31	30	rexmet 24832	. . . . 5 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) ∈ (∞Met‘ℝ)
32		eqid 2740	. . . . . . 7 ⊢ (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ))) = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ)))
33	30, 32	tgioo 24837	. . . . . 6 ⊢ (topGen‘ran (,)) = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ)))
34	33	mopni2 24527	. . . . 5 ⊢ ((((abs ∘ − ) ↾ (ℝ × ℝ)) ∈ (∞Met‘ℝ) ∧ 𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) → ∃𝑟 ∈ ℝ⁺ (𝑤(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ⊆ 𝐴)
35	31, 34	mp3an1 1448	. . . 4 ⊢ ((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) → ∃𝑟 ∈ ℝ⁺ (𝑤(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ⊆ 𝐴)
36		elssuni 4961	. . . . . . . . . 10 ⊢ (𝐴 ∈ (topGen‘ran (,)) → 𝐴 ⊆ ∪ (topGen‘ran (,)))
37		uniretop 24804	. . . . . . . . . 10 ⊢ ℝ = ∪ (topGen‘ran (,))
38	36, 37	sseqtrrdi 4060	. . . . . . . . 9 ⊢ (𝐴 ∈ (topGen‘ran (,)) → 𝐴 ⊆ ℝ)
39	38	sselda 4008	. . . . . . . 8 ⊢ ((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) → 𝑤 ∈ ℝ)
40		rpre 13065	. . . . . . . 8 ⊢ (𝑟 ∈ ℝ⁺ → 𝑟 ∈ ℝ)
41	30	bl2ioo 24833	. . . . . . . 8 ⊢ ((𝑤 ∈ ℝ ∧ 𝑟 ∈ ℝ) → (𝑤(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) = ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)))
42	39, 40, 41	syl2an 595	. . . . . . 7 ⊢ (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ 𝑟 ∈ ℝ⁺) → (𝑤(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) = ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)))
43	42	sseq1d 4040	. . . . . 6 ⊢ (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ 𝑟 ∈ ℝ⁺) → ((𝑤(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ⊆ 𝐴 ↔ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴))
44		2re 12367	. . . . . . . . . 10 ⊢ 2 ∈ ℝ
45		1lt2 12464	. . . . . . . . . 10 ⊢ 1 < 2
46		expnlbnd 14282	. . . . . . . . . 10 ⊢ ((𝑟 ∈ ℝ⁺ ∧ 2 ∈ ℝ ∧ 1 < 2) → ∃𝑛 ∈ ℕ (1 / (2↑𝑛)) < 𝑟)
47	44, 45, 46	mp3an23 1453	. . . . . . . . 9 ⊢ (𝑟 ∈ ℝ⁺ → ∃𝑛 ∈ ℕ (1 / (2↑𝑛)) < 𝑟)
48	47	ad2antrl 727	. . . . . . . 8 ⊢ (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) → ∃𝑛 ∈ ℕ (1 / (2↑𝑛)) < 𝑟)
49	39	ad2antrr 725	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → 𝑤 ∈ ℝ)
50		2nn 12366	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ ℕ
51		nnnn0 12560	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ₀)
52	51	ad2antrl 727	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → 𝑛 ∈ ℕ₀)
53		nnexpcl 14125	. . . . . . . . . . . . . . . . 17 ⊢ ((2 ∈ ℕ ∧ 𝑛 ∈ ℕ₀) → (2↑𝑛) ∈ ℕ)
54	50, 52, 53	sylancr 586	. . . . . . . . . . . . . . . 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 12308	. . . . . . . . . . . . . . 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 11320	. . . . . . . . . . . . . 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 13848	. . . . . . . . . . . . . 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 13847	. . . . . . . . . . . . . 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 12342	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → 0 < (2↑𝑛))
63		ledivmul2 12174	. . . . . . . . . . . . 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 1374	. . . . . . . . . . . 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 11463	. . . . . . . . . . . . . 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 12347	. . . . . . . . . . . 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 12168	. . . . . . . . . . . . . 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 1374	. . . . . . . . . . . . 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 11438	. . . . . . . . . . 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 12347	. . . . . . . . . . . 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 13472	. . . . . . . . . . . 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 583	. . . . . . . . . . 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 1342	. . . . . . . . . 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 13849	. . . . . . . . . . . . 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 25646	. . . . . . . . . . . . 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 583	. . . . . . . . . . . 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 6924	. . . . . . . . . . 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 7451	. . . . . . . . . . 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 2798	. . . . . . . . . 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 2847	. . . . . . . . 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 6920	. . . . . . . . . . . 12 ⊢ (𝑧 = ((⌊‘(𝑤 · (2↑𝑛)))𝐹𝑛) → ([,]‘𝑧) = ([,]‘((⌊‘(𝑤 · (2↑𝑛)))𝐹𝑛)))
86	85	sseq1d 4040	. . . . . . . . . . 11 ⊢ (𝑧 = ((⌊‘(𝑤 · (2↑𝑛)))𝐹𝑛) → (([,]‘𝑧) ⊆ 𝐴 ↔ ([,]‘((⌊‘(𝑤 · (2↑𝑛)))𝐹𝑛)) ⊆ 𝐴))
87		ffn 6747	. . . . . . . . . . . . 13 ⊢ (𝐹:(ℤ × ℕ₀)⟶( ≤ ∩ (ℝ × ℝ)) → 𝐹 Fn (ℤ × ℕ₀))
88	15, 87	ax-mp 5	. . . . . . . . . . . 12 ⊢ 𝐹 Fn (ℤ × ℕ₀)
89		fnovrn 7625	. . . . . . . . . . . 12 ⊢ ((𝐹 Fn (ℤ × ℕ₀) ∧ (⌊‘(𝑤 · (2↑𝑛))) ∈ ℤ ∧ 𝑛 ∈ ℕ₀) → ((⌊‘(𝑤 · (2↑𝑛)))𝐹𝑛) ∈ ran 𝐹)
90	88, 78, 52, 89	mp3an2i 1466	. . . . . . . . . . 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 13099	. . . . . . . . . . . . . . . 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 11718	. . . . . . . . . . . . . . 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 11340	. . . . . . . . . . . . . 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 11319	. . . . . . . . . . . . . . 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 11340	. . . . . . . . . . . . . 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 11319	. . . . . . . . . . . . . . . 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 11318	. . . . . . . . . . . . . . . . . 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 11285	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → 1 ∈ ℂ)
100	55	recnd 11318	. . . . . . . . . . . . . . . . . 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 12343	. . . . . . . . . . . . . . . . . 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 12108	. . . . . . . . . . . . . . . . 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 12344	. . . . . . . . . . . . . . . . . 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 11449	. . . . . . . . . . . . . . . . 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 5188	. . . . . . . . . . . . . . . 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 11451	. . . . . . . . . . . . . . 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 11886	. . . . . . . . . . . . . . 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 11319	. . . . . . . . . . . . . . 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 11904	. . . . . . . . . . . . . . . 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 5188	. . . . . . . . . . . . . . 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 11449	. . . . . . . . . . . . . . 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 11448	. . . . . . . . . . . . . 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 13476	. . . . . . . . . . . . . 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 4047	. . . . . . . . . . . 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 4019	. . . . . . . . . . 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 3710	. . . . . . . . . 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 7268	. . . . . . . . . . 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 691	. . . . . . . . . 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 4936	. . . . . . . . 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 583	. . . . . . . 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 3167	. . . . . . 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 3161	. . . 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 4030	. 2 ⊢ (𝐴 ∈ (topGen‘ran (,)) → ∪ ([,] “ {𝑧 ∈ ran 𝐹 ∣ ([,]‘𝑧) ⊆ 𝐴}) = 𝐴)
132		fveq2 6920	. . . . . . 7 ⊢ (𝑐 = 𝑎 → ([,]‘𝑐) = ([,]‘𝑎))
133	132	sseq1d 4040	. . . . . 6 ⊢ (𝑐 = 𝑎 → (([,]‘𝑐) ⊆ ([,]‘𝑏) ↔ ([,]‘𝑎) ⊆ ([,]‘𝑏)))
134		equequ1 2024	. . . . . 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 3184	. . . 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 3454	. . 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 25654	. 2 ⊢ (𝐴 ∈ (topGen‘ran (,)) → ∪ ([,] “ {𝑧 ∈ ran 𝐹 ∣ ([,]‘𝑧) ⊆ 𝐴}) ∈ dom vol)
140	131, 139	eqeltrrd 2845	1 ⊢ (𝐴 ∈ (topGen‘ran (,)) → 𝐴 ∈ dom vol)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1537 ∈ wcel 2108 ∀wral 3067 ∃wrex 3076 {crab 3443 ∩ cin 3975 ⊆ wss 3976 𝒫 cpw 4622 ⟨cop 4654 ∪ cuni 4931 class class class wbr 5166 × cxp 5698 dom cdm 5700 ran crn 5701 ↾ cres 5702 “ cima 5703 ∘ ccom 5704 Fun wfun 6567 Fn wfn 6568 ⟶wf 6569 ‘cfv 6573 (class class class)co 7448 ∈ cmpo 7450 ℝcr 11183 0cc0 11184 1c1 11185 + caddc 11187 · cmul 11189 ℝ^*cxr 11323 < clt 11324 ≤ cle 11325 − cmin 11520 / cdiv 11947 ℕcn 12293 2c2 12348 ℕ₀cn0 12553 ℤcz 12639 ℝ⁺crp 13057 (,)cioo 13407 [,]cicc 13410 ⌊cfl 13841 ↑cexp 14112 abscabs 15283 topGenctg 17497 ∞Metcxmet 21372 ballcbl 21374 MetOpencmopn 21377 volcvol 25517

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-inf2 9710 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-disj 5134 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-om 7904 df-1st 8030 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-oadd 8526 df-omul 8527 df-er 8763 df-map 8886 df-pm 8887 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fi 9480 df-sup 9511 df-inf 9512 df-oi 9579 df-dju 9970 df-card 10008 df-acn 10011 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-n0 12554 df-z 12640 df-uz 12904 df-q 13014 df-rp 13058 df-xneg 13175 df-xadd 13176 df-xmul 13177 df-ioo 13411 df-ico 13413 df-icc 13414 df-fz 13568 df-fzo 13712 df-fl 13843 df-seq 14053 df-exp 14113 df-hash 14380 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-clim 15534 df-rlim 15535 df-sum 15735 df-rest 17482 df-topgen 17503 df-psmet 21379 df-xmet 21380 df-met 21381 df-bl 21382 df-mopn 21383 df-top 22921 df-topon 22938 df-bases 22974 df-cmp 23416 df-ovol 25518 df-vol 25519

This theorem is referenced by: opnmbl 25656

W3C validator