opnmbllem - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > opnmbllem		Structured version Visualization version GIF version

Theorem opnmbllem 25524

Description: Lemma for opnmbl 25525. (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 6817	. . . . . . . . 9 ⊢ (𝑧 = 𝑤 → ([,]‘𝑧) = ([,]‘𝑤))
2	1	sseq1d 3961	. . . . . . . 8 ⊢ (𝑧 = 𝑤 → (([,]‘𝑧) ⊆ 𝐴 ↔ ([,]‘𝑤) ⊆ 𝐴))
3	2	elrab 3642	. . . . . . 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 6830	. . . . . . . . 9 ⊢ ([,]‘𝑤) ∈ V
6	5	elpw 4549	. . . . . . . 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 3124	. . . . 5 ⊢ (𝐴 ∈ (topGen‘ran (,)) → ∀𝑤 ∈ {𝑧 ∈ ran 𝐹 ∣ ([,]‘𝑧) ⊆ 𝐴} ([,]‘𝑤) ∈ 𝒫 𝐴)
10		iccf 13343	. . . . . . 7 ⊢ [,]:(ℝ^* × ℝ^)⟶𝒫 ℝ^
11		ffun 6649	. . . . . . 7 ⊢ ([,]:(ℝ^* × ℝ^)⟶𝒫 ℝ^ → Fun [,])
12	10, 11	ax-mp 5	. . . . . 6 ⊢ Fun [,]
13		ssrab2 4025	. . . . . . . 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 25514	. . . . . . . . . 10 ⊢ 𝐹:(ℤ × ℕ₀)⟶( ≤ ∩ (ℝ × ℝ))
16		frn 6653	. . . . . . . . . 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 11152	. . . . . . . . . 10 ⊢ (ℝ × ℝ) ⊆ (ℝ^* × ℝ^*)
20	18, 19	sstri 3939	. . . . . . . . 9 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ^* × ℝ^*)
21	17, 20	sstri 3939	. . . . . . . 8 ⊢ ran 𝐹 ⊆ (ℝ^* × ℝ^*)
22	13, 21	sstri 3939	. . . . . . 7 ⊢ {𝑧 ∈ ran 𝐹 ∣ ([,]‘𝑧) ⊆ 𝐴} ⊆ (ℝ^* × ℝ^*)
23	10	fdmi 6657	. . . . . . 7 ⊢ dom [,] = (ℝ^* × ℝ^*)
24	22, 23	sseqtrri 3979	. . . . . 6 ⊢ {𝑧 ∈ ran 𝐹 ∣ ([,]‘𝑧) ⊆ 𝐴} ⊆ dom [,]
25		funimass4 6881	. . . . . 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 5043	. . . 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 2731	. . . . . 6 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) = ((abs ∘ − ) ↾ (ℝ × ℝ))
31	30	rexmet 24701	. . . . 5 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) ∈ (∞Met‘ℝ)
32		eqid 2731	. . . . . . 7 ⊢ (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ))) = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ)))
33	30, 32	tgioo 24706	. . . . . 6 ⊢ (topGen‘ran (,)) = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ)))
34	33	mopni2 24403	. . . . 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 4884	. . . . . . . . . 10 ⊢ (𝐴 ∈ (topGen‘ran (,)) → 𝐴 ⊆ ∪ (topGen‘ran (,)))
37		uniretop 24672	. . . . . . . . . 10 ⊢ ℝ = ∪ (topGen‘ran (,))
38	36, 37	sseqtrrdi 3971	. . . . . . . . 9 ⊢ (𝐴 ∈ (topGen‘ran (,)) → 𝐴 ⊆ ℝ)
39	38	sselda 3929	. . . . . . . 8 ⊢ ((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) → 𝑤 ∈ ℝ)
40		rpre 12894	. . . . . . . 8 ⊢ (𝑟 ∈ ℝ⁺ → 𝑟 ∈ ℝ)
41	30	bl2ioo 24702	. . . . . . . 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 3961	. . . . . 6 ⊢ (((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ 𝑟 ∈ ℝ⁺) → ((𝑤(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ⊆ 𝐴 ↔ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴))
44		2re 12194	. . . . . . . . . 10 ⊢ 2 ∈ ℝ
45		1lt2 12286	. . . . . . . . . 10 ⊢ 1 < 2
46		expnlbnd 14135	. . . . . . . . . 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 12193	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ ℕ
51		nnnn0 12383	. . . . . . . . . . . . . . . . . 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 13976	. . . . . . . . . . . . . . . . 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 12135	. . . . . . . . . . . . . . 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 11137	. . . . . . . . . . . . . 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 13696	. . . . . . . . . . . . . 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 13695	. . . . . . . . . . . . . 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 12169	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → 0 < (2↑𝑛))
63		ledivmul2 11996	. . . . . . . . . . . . 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 11281	. . . . . . . . . . . . . 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 12174	. . . . . . . . . . . 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 11990	. . . . . . . . . . . . . 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 11256	. . . . . . . . . . 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 12174	. . . . . . . . . . . 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 13306	. . . . . . . . . . . 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 13697	. . . . . . . . . . . . 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 25515	. . . . . . . . . . . . 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 6821	. . . . . . . . . . 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 7344	. . . . . . . . . . 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 2784	. . . . . . . . . 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 2834	. . . . . . . . 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 6817	. . . . . . . . . . . 12 ⊢ (𝑧 = ((⌊‘(𝑤 · (2↑𝑛)))𝐹𝑛) → ([,]‘𝑧) = ([,]‘((⌊‘(𝑤 · (2↑𝑛)))𝐹𝑛)))
86	85	sseq1d 3961	. . . . . . . . . . 11 ⊢ (𝑧 = ((⌊‘(𝑤 · (2↑𝑛)))𝐹𝑛) → (([,]‘𝑧) ⊆ 𝐴 ↔ ([,]‘((⌊‘(𝑤 · (2↑𝑛)))𝐹𝑛)) ⊆ 𝐴))
87		ffn 6646	. . . . . . . . . . . . 13 ⊢ (𝐹:(ℤ × ℕ₀)⟶( ≤ ∩ (ℝ × ℝ)) → 𝐹 Fn (ℤ × ℕ₀))
88	15, 87	ax-mp 5	. . . . . . . . . . . 12 ⊢ 𝐹 Fn (ℤ × ℕ₀)
89		fnovrn 7516	. . . . . . . . . . . 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 12929	. . . . . . . . . . . . . . . 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 11540	. . . . . . . . . . . . . . 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 11157	. . . . . . . . . . . . . 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 11136	. . . . . . . . . . . . . . 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 11157	. . . . . . . . . . . . . 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 11136	. . . . . . . . . . . . . . . 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 11135	. . . . . . . . . . . . . . . . . 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 11102	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑤 ∈ 𝐴) ∧ (𝑟 ∈ ℝ⁺ ∧ ((𝑤 − 𝑟)(,)(𝑤 + 𝑟)) ⊆ 𝐴)) ∧ (𝑛 ∈ ℕ ∧ (1 / (2↑𝑛)) < 𝑟)) → 1 ∈ ℂ)
100	55	recnd 11135	. . . . . . . . . . . . . . . . . 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 12170	. . . . . . . . . . . . . . . . . 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 11930	. . . . . . . . . . . . . . . . 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 12171	. . . . . . . . . . . . . . . . . 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 11267	. . . . . . . . . . . . . . . . 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 5108	. . . . . . . . . . . . . . . 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 11269	. . . . . . . . . . . . . . 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 11708	. . . . . . . . . . . . . . 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 11136	. . . . . . . . . . . . . . 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 11726	. . . . . . . . . . . . . . . 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 5108	. . . . . . . . . . . . . . 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 11267	. . . . . . . . . . . . . . 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 11266	. . . . . . . . . . . . . 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 13310	. . . . . . . . . . . . . 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 3964	. . . . . . . . . . . 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 3940	. . . . . . . . . . 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 3644	. . . . . . . . . 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 7160	. . . . . . . . . . 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 4859	. . . . . . . . 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 3139	. . . . . . 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 3133	. . . 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 3951	. 2 ⊢ (𝐴 ∈ (topGen‘ran (,)) → ∪ ([,] “ {𝑧 ∈ ran 𝐹 ∣ ([,]‘𝑧) ⊆ 𝐴}) = 𝐴)
132		fveq2 6817	. . . . . . 7 ⊢ (𝑐 = 𝑎 → ([,]‘𝑐) = ([,]‘𝑎))
133	132	sseq1d 3961	. . . . . 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 3155	. . . 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 3405	. . 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 25523	. 2 ⊢ (𝐴 ∈ (topGen‘ran (,)) → ∪ ([,] “ {𝑧 ∈ ran 𝐹 ∣ ([,]‘𝑧) ⊆ 𝐴}) ∈ dom vol)
140	131, 139	eqeltrrd 2832	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 2111 ∀wral 3047 ∃wrex 3056 {crab 3395 ∩ cin 3896 ⊆ wss 3897 𝒫 cpw 4545 ⟨cop 4577 ∪ cuni 4854 class class class wbr 5086 × cxp 5609 dom cdm 5611 ran crn 5612 ↾ cres 5613 “ cima 5614 ∘ ccom 5615 Fun wfun 6470 Fn wfn 6471 ⟶wf 6472 ‘cfv 6476 (class class class)co 7341 ∈ cmpo 7343 ℝcr 11000 0cc0 11001 1c1 11002 + caddc 11004 · cmul 11006 ℝ^*cxr 11140 < clt 11141 ≤ cle 11142 − cmin 11339 / cdiv 11769 ℕcn 12120 2c2 12175 ℕ₀cn0 12376 ℤcz 12463 ℝ⁺crp 12885 (,)cioo 13240 [,]cicc 13243 ⌊cfl 13689 ↑cexp 13963 abscabs 15136 topGenctg 17336 ∞Metcxmet 21271 ballcbl 21273 MetOpencmopn 21276 volcvol 25386

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 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5212 ax-sep 5229 ax-nul 5239 ax-pow 5298 ax-pr 5365 ax-un 7663 ax-inf2 9526 ax-cnex 11057 ax-resscn 11058 ax-1cn 11059 ax-icn 11060 ax-addcl 11061 ax-addrcl 11062 ax-mulcl 11063 ax-mulrcl 11064 ax-mulcom 11065 ax-addass 11066 ax-mulass 11067 ax-distr 11068 ax-i2m1 11069 ax-1ne0 11070 ax-1rid 11071 ax-rnegex 11072 ax-rrecex 11073 ax-cnre 11074 ax-pre-lttri 11075 ax-pre-lttrn 11076 ax-pre-ltadd 11077 ax-pre-mulgt0 11078 ax-pre-sup 11079

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 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4279 df-if 4471 df-pw 4547 df-sn 4572 df-pr 4574 df-op 4578 df-uni 4855 df-int 4893 df-iun 4938 df-disj 5054 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5506 df-eprel 5511 df-po 5519 df-so 5520 df-fr 5564 df-se 5565 df-we 5566 df-xp 5617 df-rel 5618 df-cnv 5619 df-co 5620 df-dm 5621 df-rn 5622 df-res 5623 df-ima 5624 df-pred 6243 df-ord 6304 df-on 6305 df-lim 6306 df-suc 6307 df-iota 6432 df-fun 6478 df-fn 6479 df-f 6480 df-f1 6481 df-fo 6482 df-f1o 6483 df-fv 6484 df-isom 6485 df-riota 7298 df-ov 7344 df-oprab 7345 df-mpo 7346 df-of 7605 df-om 7792 df-1st 7916 df-2nd 7917 df-frecs 8206 df-wrecs 8237 df-recs 8286 df-rdg 8324 df-1o 8380 df-2o 8381 df-oadd 8384 df-omul 8385 df-er 8617 df-map 8747 df-pm 8748 df-en 8865 df-dom 8866 df-sdom 8867 df-fin 8868 df-fi 9290 df-sup 9321 df-inf 9322 df-oi 9391 df-dju 9789 df-card 9827 df-acn 9830 df-pnf 11143 df-mnf 11144 df-xr 11145 df-ltxr 11146 df-le 11147 df-sub 11341 df-neg 11342 df-div 11770 df-nn 12121 df-2 12183 df-3 12184 df-4 12185 df-n0 12377 df-z 12464 df-uz 12728 df-q 12842 df-rp 12886 df-xneg 13006 df-xadd 13007 df-xmul 13008 df-ioo 13244 df-ico 13246 df-icc 13247 df-fz 13403 df-fzo 13550 df-fl 13691 df-seq 13904 df-exp 13964 df-hash 14233 df-cj 15001 df-re 15002 df-im 15003 df-sqrt 15137 df-abs 15138 df-clim 15390 df-rlim 15391 df-sum 15589 df-rest 17321 df-topgen 17342 df-psmet 21278 df-xmet 21279 df-met 21280 df-bl 21281 df-mopn 21282 df-top 22804 df-topon 22821 df-bases 22856 df-cmp 23297 df-ovol 25387 df-vol 25388

This theorem is referenced by: opnmbl 25525

W3C validator