Theorem heiborlem8 38159

Description: Lemma for heibor 38162. The previous lemmas establish that the sequence 𝑀 is Cauchy, so using completeness we now consider the convergent point 𝑌. By assumption, 𝑈 is an open cover, so 𝑌 is an element of some 𝑍 ∈ 𝑈, and some ball centered at 𝑌 is contained in 𝑍. But the sequence contains arbitrarily small balls close to 𝑌, so some element ball(𝑀‘𝑛) of the sequence is contained in 𝑍. And finally we arrive at a contradiction, because {𝑍} is a finite subcover of 𝑈 that covers ball(𝑀‘𝑛), yet ball(𝑀‘𝑛) ∈ 𝐾. For convenience, we write this contradiction as 𝜑 → 𝜓 where 𝜑 is all the accumulated hypotheses and 𝜓 is anything at all. (Contributed by Jeff Madsen, 22-Jan-2014.)

Hypotheses

Ref	Expression
heibor.1	⊢ 𝐽 = (MetOpen‘𝐷)
heibor.3	⊢ 𝐾 = {𝑢 ∣ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 ⊆ ∪ 𝑣}
heibor.4	⊢ 𝐺 = {⟨𝑦, 𝑛⟩ ∣ (𝑛 ∈ ℕ0 ∧ 𝑦 ∈ (𝐹‘𝑛) ∧ (𝑦𝐵𝑛) ∈ 𝐾)}
heibor.5	⊢ 𝐵 = (𝑧 ∈ 𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))
heibor.6	⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋))
heibor.7	⊢ (𝜑 → 𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin))
heibor.8	⊢ (𝜑 → ∀𝑛 ∈ ℕ0 𝑋 = ∪ 𝑦 ∈ (𝐹‘𝑛)(𝑦𝐵𝑛))
heibor.9	⊢ (𝜑 → ∀𝑥 ∈ 𝐺 ((𝑇‘𝑥)𝐺((2nd ‘𝑥) + 1) ∧ ((𝐵‘𝑥) ∩ ((𝑇‘𝑥)𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾))
heibor.10	⊢ (𝜑 → 𝐶𝐺0)
heibor.11	⊢ 𝑆 = seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))))
heibor.12	⊢ 𝑀 = (𝑛 ∈ ℕ ↦ ⟨(𝑆‘𝑛), (3 / (2↑𝑛))⟩)
heibor.13	⊢ (𝜑 → 𝑈 ⊆ 𝐽)
heibor.14	⊢ 𝑌 ∈ V
heibor.15	⊢ (𝜑 → 𝑌 ∈ 𝑍)
heibor.16	⊢ (𝜑 → 𝑍 ∈ 𝑈)
heibor.17	⊢ (𝜑 → (1st ∘ 𝑀)(⇝𝑡‘𝐽)𝑌)

Assertion

Ref	Expression
heiborlem8	⊢ (𝜑 → 𝜓)

Distinct variable groups:   𝑥,𝑛,𝑦,𝑢,𝐹   𝑥,𝐺   𝜑,𝑥   𝑚,𝑛,𝑢,𝑣,𝑥,𝑦,𝑧,𝐷   𝑚,𝑀,𝑢,𝑥,𝑦,𝑧   𝑇,𝑚,𝑛,𝑥,𝑦,𝑧   𝐵,𝑛,𝑢,𝑣,𝑦   𝑚,𝐽,𝑛,𝑢,𝑣,𝑥,𝑦,𝑧   𝑈,𝑛,𝑢,𝑣,𝑥,𝑦,𝑧   𝜓,𝑦,𝑧   𝑆,𝑚,𝑛,𝑢,𝑣,𝑥,𝑦,𝑧   𝑚,𝑋,𝑛,𝑢,𝑣,𝑥,𝑦,𝑧   𝐶,𝑚,𝑛,𝑢,𝑣,𝑦   𝑛,𝐾,𝑥,𝑦,𝑧   𝑥,𝑌   𝑣,𝑍,𝑥   𝑥,𝐵

Allowed substitution hints:   𝜑(𝑦,𝑧,𝑣,𝑢,𝑚,𝑛)   𝜓(𝑥,𝑣,𝑢,𝑚,𝑛)   𝐵(𝑧,𝑚)   𝐶(𝑥,𝑧)   𝑇(𝑣,𝑢)   𝑈(𝑚)   𝐹(𝑧,𝑣,𝑚)   𝐺(𝑦,𝑧,𝑣,𝑢,𝑚,𝑛)   𝐾(𝑣,𝑢,𝑚)   𝑀(𝑣,𝑛)   𝑌(𝑦,𝑧,𝑣,𝑢,𝑚,𝑛)   𝑍(𝑦,𝑧,𝑢,𝑚,𝑛)

Proof of Theorem heiborlem8

Dummy variables 𝑡 𝑘 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		heibor.6	. . . 4 ⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋))
2		cmetmet 25267	. . . 4 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
3		metxmet 24313	. . . 4 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
4	1, 2, 3	3syl 18	. . 3 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋))
5		heibor.13	. . . 4 ⊢ (𝜑 → 𝑈 ⊆ 𝐽)
6		heibor.16	. . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝑈)
7	5, 6	sseldd 3923	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐽)
8		heibor.15	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑍)
9		heibor.1	. . . 4 ⊢ 𝐽 = (MetOpen‘𝐷)
10	9	mopni2 24472	. . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑍 ∈ 𝐽 ∧ 𝑌 ∈ 𝑍) → ∃𝑥 ∈ ℝ+ (𝑌(ball‘𝐷)𝑥) ⊆ 𝑍)
11	4, 7, 8, 10	syl3anc 1374	. 2 ⊢ (𝜑 → ∃𝑥 ∈ ℝ+ (𝑌(ball‘𝐷)𝑥) ⊆ 𝑍)
12		rphalfcl 12966	. . . . . 6 ⊢ (𝑥 ∈ ℝ+ → (𝑥 / 2) ∈ ℝ+)
13		breq2 5090	. . . . . . . 8 ⊢ (𝑟 = (𝑥 / 2) → ((2nd ‘(𝑀‘𝑘)) < 𝑟 ↔ (2nd ‘(𝑀‘𝑘)) < (𝑥 / 2)))
14	13	rexbidv 3162	. . . . . . 7 ⊢ (𝑟 = (𝑥 / 2) → (∃𝑘 ∈ ℕ (2nd ‘(𝑀‘𝑘)) < 𝑟 ↔ ∃𝑘 ∈ ℕ (2nd ‘(𝑀‘𝑘)) < (𝑥 / 2)))
15		heibor.3	. . . . . . . 8 ⊢ 𝐾 = {𝑢 ∣ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 ⊆ ∪ 𝑣}
16		heibor.4	. . . . . . . 8 ⊢ 𝐺 = {⟨𝑦, 𝑛⟩ ∣ (𝑛 ∈ ℕ0 ∧ 𝑦 ∈ (𝐹‘𝑛) ∧ (𝑦𝐵𝑛) ∈ 𝐾)}
17		heibor.5	. . . . . . . 8 ⊢ 𝐵 = (𝑧 ∈ 𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))
18		heibor.7	. . . . . . . 8 ⊢ (𝜑 → 𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin))
19		heibor.8	. . . . . . . 8 ⊢ (𝜑 → ∀𝑛 ∈ ℕ0 𝑋 = ∪ 𝑦 ∈ (𝐹‘𝑛)(𝑦𝐵𝑛))
20		heibor.9	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ 𝐺 ((𝑇‘𝑥)𝐺((2nd ‘𝑥) + 1) ∧ ((𝐵‘𝑥) ∩ ((𝑇‘𝑥)𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾))
21		heibor.10	. . . . . . . 8 ⊢ (𝜑 → 𝐶𝐺0)
22		heibor.11	. . . . . . . 8 ⊢ 𝑆 = seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))))
23		heibor.12	. . . . . . . 8 ⊢ 𝑀 = (𝑛 ∈ ℕ ↦ ⟨(𝑆‘𝑛), (3 / (2↑𝑛))⟩)
24	9, 15, 16, 17, 1, 18, 19, 20, 21, 22, 23	heiborlem7 38158	. . . . . . 7 ⊢ ∀𝑟 ∈ ℝ+ ∃𝑘 ∈ ℕ (2nd ‘(𝑀‘𝑘)) < 𝑟
25	14, 24	vtoclri 3533	. . . . . 6 ⊢ ((𝑥 / 2) ∈ ℝ+ → ∃𝑘 ∈ ℕ (2nd ‘(𝑀‘𝑘)) < (𝑥 / 2))
26	12, 25	syl 17	. . . . 5 ⊢ (𝑥 ∈ ℝ+ → ∃𝑘 ∈ ℕ (2nd ‘(𝑀‘𝑘)) < (𝑥 / 2))
27	26	adantl 481	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → ∃𝑘 ∈ ℕ (2nd ‘(𝑀‘𝑘)) < (𝑥 / 2))
28		nnnn0 12439	. . . . . . 7 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ0)
29	9, 15, 16, 17, 1, 18, 19, 20, 21, 22	heiborlem4 38155	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑆‘𝑘)𝐺𝑘)
30		fvex 6849	. . . . . . . . . 10 ⊢ (𝑆‘𝑘) ∈ V
31		vex 3434	. . . . . . . . . 10 ⊢ 𝑘 ∈ V
32	9, 15, 16, 30, 31	heiborlem2 38153	. . . . . . . . 9 ⊢ ((𝑆‘𝑘)𝐺𝑘 ↔ (𝑘 ∈ ℕ0 ∧ (𝑆‘𝑘) ∈ (𝐹‘𝑘) ∧ ((𝑆‘𝑘)𝐵𝑘) ∈ 𝐾))
33	32	simp3bi 1148	. . . . . . . 8 ⊢ ((𝑆‘𝑘)𝐺𝑘 → ((𝑆‘𝑘)𝐵𝑘) ∈ 𝐾)
34	29, 33	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝑆‘𝑘)𝐵𝑘) ∈ 𝐾)
35	28, 34	sylan2 594	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝑆‘𝑘)𝐵𝑘) ∈ 𝐾)
36	35	ad2ant2r 748	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀‘𝑘)) < (𝑥 / 2))) → ((𝑆‘𝑘)𝐵𝑘) ∈ 𝐾)
37	4	ad2antrr 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀‘𝑘)) < (𝑥 / 2))) → 𝐷 ∈ (∞Met‘𝑋))
38	9, 15, 16, 17, 1, 18, 19, 20, 21, 22, 23	heiborlem5 38156	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀:ℕ⟶(𝑋 × ℝ+))
39	38	ffvelcdmda 7032	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑀‘𝑘) ∈ (𝑋 × ℝ+))
40	39	ad2ant2r 748	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀‘𝑘)) < (𝑥 / 2))) → (𝑀‘𝑘) ∈ (𝑋 × ℝ+))
41		xp1st 7969	. . . . . . . . . . 11 ⊢ ((𝑀‘𝑘) ∈ (𝑋 × ℝ+) → (1st ‘(𝑀‘𝑘)) ∈ 𝑋)
42	40, 41	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀‘𝑘)) < (𝑥 / 2))) → (1st ‘(𝑀‘𝑘)) ∈ 𝑋)
43		2nn 12249	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℕ
44		nnexpcl 14031	. . . . . . . . . . . . . . 15 ⊢ ((2 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → (2↑𝑘) ∈ ℕ)
45	43, 28, 44	sylancr 588	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ → (2↑𝑘) ∈ ℕ)
46	45	nnrpd 12979	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ → (2↑𝑘) ∈ ℝ+)
47	46	rpreccld 12991	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → (1 / (2↑𝑘)) ∈ ℝ+)
48	47	ad2antrl 729	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀‘𝑘)) < (𝑥 / 2))) → (1 / (2↑𝑘)) ∈ ℝ+)
49	48	rpxrd 12982	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀‘𝑘)) < (𝑥 / 2))) → (1 / (2↑𝑘)) ∈ ℝ*)
50		xp2nd 7970	. . . . . . . . . . . 12 ⊢ ((𝑀‘𝑘) ∈ (𝑋 × ℝ+) → (2nd ‘(𝑀‘𝑘)) ∈ ℝ+)
51	40, 50	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀‘𝑘)) < (𝑥 / 2))) → (2nd ‘(𝑀‘𝑘)) ∈ ℝ+)
52	51	rpxrd 12982	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀‘𝑘)) < (𝑥 / 2))) → (2nd ‘(𝑀‘𝑘)) ∈ ℝ*)
53		1le3 12383	. . . . . . . . . . . . . 14 ⊢ 1 ≤ 3
54		elrp 12939	. . . . . . . . . . . . . . 15 ⊢ ((2↑𝑘) ∈ ℝ+ ↔ ((2↑𝑘) ∈ ℝ ∧ 0 < (2↑𝑘)))
55		1re 11139	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℝ
56		3re 12256	. . . . . . . . . . . . . . . 16 ⊢ 3 ∈ ℝ
57		lediv1 12016	. . . . . . . . . . . . . . . 16 ⊢ ((1 ∈ ℝ ∧ 3 ∈ ℝ ∧ ((2↑𝑘) ∈ ℝ ∧ 0 < (2↑𝑘))) → (1 ≤ 3 ↔ (1 / (2↑𝑘)) ≤ (3 / (2↑𝑘))))
58	55, 56, 57	mp3an12 1454	. . . . . . . . . . . . . . 15 ⊢ (((2↑𝑘) ∈ ℝ ∧ 0 < (2↑𝑘)) → (1 ≤ 3 ↔ (1 / (2↑𝑘)) ≤ (3 / (2↑𝑘))))
59	54, 58	sylbi 217	. . . . . . . . . . . . . 14 ⊢ ((2↑𝑘) ∈ ℝ+ → (1 ≤ 3 ↔ (1 / (2↑𝑘)) ≤ (3 / (2↑𝑘))))
60	53, 59	mpbii 233	. . . . . . . . . . . . 13 ⊢ ((2↑𝑘) ∈ ℝ+ → (1 / (2↑𝑘)) ≤ (3 / (2↑𝑘)))
61	46, 60	syl 17	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → (1 / (2↑𝑘)) ≤ (3 / (2↑𝑘)))
62	61	ad2antrl 729	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀‘𝑘)) < (𝑥 / 2))) → (1 / (2↑𝑘)) ≤ (3 / (2↑𝑘)))
63		fveq2 6836	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑘 → (𝑆‘𝑛) = (𝑆‘𝑘))
64		oveq2 7370	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑘 → (2↑𝑛) = (2↑𝑘))
65	64	oveq2d 7378	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑘 → (3 / (2↑𝑛)) = (3 / (2↑𝑘)))
66	63, 65	opeq12d 4825	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑘 → ⟨(𝑆‘𝑛), (3 / (2↑𝑛))⟩ = ⟨(𝑆‘𝑘), (3 / (2↑𝑘))⟩)
67		opex 5413	. . . . . . . . . . . . . . 15 ⊢ ⟨(𝑆‘𝑘), (3 / (2↑𝑘))⟩ ∈ V
68	66, 23, 67	fvmpt 6943	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ → (𝑀‘𝑘) = ⟨(𝑆‘𝑘), (3 / (2↑𝑘))⟩)
69	68	fveq2d 6840	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ → (2nd ‘(𝑀‘𝑘)) = (2nd ‘⟨(𝑆‘𝑘), (3 / (2↑𝑘))⟩))
70		ovex 7395	. . . . . . . . . . . . . 14 ⊢ (3 / (2↑𝑘)) ∈ V
71	30, 70	op2nd 7946	. . . . . . . . . . . . 13 ⊢ (2nd ‘⟨(𝑆‘𝑘), (3 / (2↑𝑘))⟩) = (3 / (2↑𝑘))
72	69, 71	eqtrdi 2788	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → (2nd ‘(𝑀‘𝑘)) = (3 / (2↑𝑘)))
73	72	ad2antrl 729	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀‘𝑘)) < (𝑥 / 2))) → (2nd ‘(𝑀‘𝑘)) = (3 / (2↑𝑘)))
74	62, 73	breqtrrd 5114	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀‘𝑘)) < (𝑥 / 2))) → (1 / (2↑𝑘)) ≤ (2nd ‘(𝑀‘𝑘)))
75		ssbl 24402	. . . . . . . . . 10 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ (1st ‘(𝑀‘𝑘)) ∈ 𝑋) ∧ ((1 / (2↑𝑘)) ∈ ℝ* ∧ (2nd ‘(𝑀‘𝑘)) ∈ ℝ*) ∧ (1 / (2↑𝑘)) ≤ (2nd ‘(𝑀‘𝑘))) → ((1st ‘(𝑀‘𝑘))(ball‘𝐷)(1 / (2↑𝑘))) ⊆ ((1st ‘(𝑀‘𝑘))(ball‘𝐷)(2nd ‘(𝑀‘𝑘))))
76	37, 42, 49, 52, 74, 75	syl221anc 1384	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀‘𝑘)) < (𝑥 / 2))) → ((1st ‘(𝑀‘𝑘))(ball‘𝐷)(1 / (2↑𝑘))) ⊆ ((1st ‘(𝑀‘𝑘))(ball‘𝐷)(2nd ‘(𝑀‘𝑘))))
77	28	ad2antrl 729	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀‘𝑘)) < (𝑥 / 2))) → 𝑘 ∈ ℕ0)
78		oveq1 7369	. . . . . . . . . . . 12 ⊢ (𝑧 = (1st ‘(𝑀‘𝑘)) → (𝑧(ball‘𝐷)(1 / (2↑𝑚))) = ((1st ‘(𝑀‘𝑘))(ball‘𝐷)(1 / (2↑𝑚))))
79		oveq2 7370	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑘 → (2↑𝑚) = (2↑𝑘))
80	79	oveq2d 7378	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑘 → (1 / (2↑𝑚)) = (1 / (2↑𝑘)))
81	80	oveq2d 7378	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑘 → ((1st ‘(𝑀‘𝑘))(ball‘𝐷)(1 / (2↑𝑚))) = ((1st ‘(𝑀‘𝑘))(ball‘𝐷)(1 / (2↑𝑘))))
82		ovex 7395	. . . . . . . . . . . 12 ⊢ ((1st ‘(𝑀‘𝑘))(ball‘𝐷)(1 / (2↑𝑘))) ∈ V
83	78, 81, 17, 82	ovmpo 7522	. . . . . . . . . . 11 ⊢ (((1st ‘(𝑀‘𝑘)) ∈ 𝑋 ∧ 𝑘 ∈ ℕ0) → ((1st ‘(𝑀‘𝑘))𝐵𝑘) = ((1st ‘(𝑀‘𝑘))(ball‘𝐷)(1 / (2↑𝑘))))
84	42, 77, 83	syl2anc 585	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀‘𝑘)) < (𝑥 / 2))) → ((1st ‘(𝑀‘𝑘))𝐵𝑘) = ((1st ‘(𝑀‘𝑘))(ball‘𝐷)(1 / (2↑𝑘))))
85	68	fveq2d 6840	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ → (1st ‘(𝑀‘𝑘)) = (1st ‘⟨(𝑆‘𝑘), (3 / (2↑𝑘))⟩))
86	30, 70	op1st 7945	. . . . . . . . . . . . 13 ⊢ (1st ‘⟨(𝑆‘𝑘), (3 / (2↑𝑘))⟩) = (𝑆‘𝑘)
87	85, 86	eqtrdi 2788	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → (1st ‘(𝑀‘𝑘)) = (𝑆‘𝑘))
88	87	ad2antrl 729	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀‘𝑘)) < (𝑥 / 2))) → (1st ‘(𝑀‘𝑘)) = (𝑆‘𝑘))
89	88	oveq1d 7377	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀‘𝑘)) < (𝑥 / 2))) → ((1st ‘(𝑀‘𝑘))𝐵𝑘) = ((𝑆‘𝑘)𝐵𝑘))
90	84, 89	eqtr3d 2774	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀‘𝑘)) < (𝑥 / 2))) → ((1st ‘(𝑀‘𝑘))(ball‘𝐷)(1 / (2↑𝑘))) = ((𝑆‘𝑘)𝐵𝑘))
91		df-ov 7365	. . . . . . . . . 10 ⊢ ((1st ‘(𝑀‘𝑘))(ball‘𝐷)(2nd ‘(𝑀‘𝑘))) = ((ball‘𝐷)‘⟨(1st ‘(𝑀‘𝑘)), (2nd ‘(𝑀‘𝑘))⟩)
92		1st2nd2 7976	. . . . . . . . . . . 12 ⊢ ((𝑀‘𝑘) ∈ (𝑋 × ℝ+) → (𝑀‘𝑘) = ⟨(1st ‘(𝑀‘𝑘)), (2nd ‘(𝑀‘𝑘))⟩)
93	40, 92	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀‘𝑘)) < (𝑥 / 2))) → (𝑀‘𝑘) = ⟨(1st ‘(𝑀‘𝑘)), (2nd ‘(𝑀‘𝑘))⟩)
94	93	fveq2d 6840	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀‘𝑘)) < (𝑥 / 2))) → ((ball‘𝐷)‘(𝑀‘𝑘)) = ((ball‘𝐷)‘⟨(1st ‘(𝑀‘𝑘)), (2nd ‘(𝑀‘𝑘))⟩))
95	91, 94	eqtr4id 2791	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀‘𝑘)) < (𝑥 / 2))) → ((1st ‘(𝑀‘𝑘))(ball‘𝐷)(2nd ‘(𝑀‘𝑘))) = ((ball‘𝐷)‘(𝑀‘𝑘)))
96	76, 90, 95	3sstr3d 3977	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀‘𝑘)) < (𝑥 / 2))) → ((𝑆‘𝑘)𝐵𝑘) ⊆ ((ball‘𝐷)‘(𝑀‘𝑘)))
97	9	mopntop 24419	. . . . . . . . . . 11 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top)
98	37, 97	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀‘𝑘)) < (𝑥 / 2))) → 𝐽 ∈ Top)
99		blssm 24397	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (1st ‘(𝑀‘𝑘)) ∈ 𝑋 ∧ (2nd ‘(𝑀‘𝑘)) ∈ ℝ*) → ((1st ‘(𝑀‘𝑘))(ball‘𝐷)(2nd ‘(𝑀‘𝑘))) ⊆ 𝑋)
100	37, 42, 52, 99	syl3anc 1374	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀‘𝑘)) < (𝑥 / 2))) → ((1st ‘(𝑀‘𝑘))(ball‘𝐷)(2nd ‘(𝑀‘𝑘))) ⊆ 𝑋)
101	9	mopnuni 24420	. . . . . . . . . . . 12 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = ∪ 𝐽)
102	37, 101	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀‘𝑘)) < (𝑥 / 2))) → 𝑋 = ∪ 𝐽)
103	100, 95, 102	3sstr3d 3977	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀‘𝑘)) < (𝑥 / 2))) → ((ball‘𝐷)‘(𝑀‘𝑘)) ⊆ ∪ 𝐽)
104		eqid 2737	. . . . . . . . . . 11 ⊢ ∪ 𝐽 = ∪ 𝐽
105	104	sscls 23035	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ ((ball‘𝐷)‘(𝑀‘𝑘)) ⊆ ∪ 𝐽) → ((ball‘𝐷)‘(𝑀‘𝑘)) ⊆ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑀‘𝑘))))
106	98, 103, 105	syl2anc 585	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀‘𝑘)) < (𝑥 / 2))) → ((ball‘𝐷)‘(𝑀‘𝑘)) ⊆ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑀‘𝑘))))
107	95	fveq2d 6840	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀‘𝑘)) < (𝑥 / 2))) → ((cls‘𝐽)‘((1st ‘(𝑀‘𝑘))(ball‘𝐷)(2nd ‘(𝑀‘𝑘)))) = ((cls‘𝐽)‘((ball‘𝐷)‘(𝑀‘𝑘))))
108	12	ad2antlr 728	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀‘𝑘)) < (𝑥 / 2))) → (𝑥 / 2) ∈ ℝ+)
109	108	rpxrd 12982	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀‘𝑘)) < (𝑥 / 2))) → (𝑥 / 2) ∈ ℝ*)
110		simprr 773	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀‘𝑘)) < (𝑥 / 2))) → (2nd ‘(𝑀‘𝑘)) < (𝑥 / 2))
111	9	blsscls 24486	. . . . . . . . . . . 12 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ (1st ‘(𝑀‘𝑘)) ∈ 𝑋) ∧ ((2nd ‘(𝑀‘𝑘)) ∈ ℝ* ∧ (𝑥 / 2) ∈ ℝ* ∧ (2nd ‘(𝑀‘𝑘)) < (𝑥 / 2))) → ((cls‘𝐽)‘((1st ‘(𝑀‘𝑘))(ball‘𝐷)(2nd ‘(𝑀‘𝑘)))) ⊆ ((1st ‘(𝑀‘𝑘))(ball‘𝐷)(𝑥 / 2)))
112	37, 42, 52, 109, 110, 111	syl23anc 1380	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀‘𝑘)) < (𝑥 / 2))) → ((cls‘𝐽)‘((1st ‘(𝑀‘𝑘))(ball‘𝐷)(2nd ‘(𝑀‘𝑘)))) ⊆ ((1st ‘(𝑀‘𝑘))(ball‘𝐷)(𝑥 / 2)))
113	107, 112	eqsstrrd 3958	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀‘𝑘)) < (𝑥 / 2))) → ((cls‘𝐽)‘((ball‘𝐷)‘(𝑀‘𝑘))) ⊆ ((1st ‘(𝑀‘𝑘))(ball‘𝐷)(𝑥 / 2)))
114		rpre 12946	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ+ → 𝑥 ∈ ℝ)
115	114	ad2antlr 728	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀‘𝑘)) < (𝑥 / 2))) → 𝑥 ∈ ℝ)
116		heibor.17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (1st ∘ 𝑀)(⇝𝑡‘𝐽)𝑌)
117	9, 15, 16, 17, 1, 18, 19, 20, 21, 22, 23	heiborlem6 38157	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ∀𝑡 ∈ ℕ ((ball‘𝐷)‘(𝑀‘(𝑡 + 1))) ⊆ ((ball‘𝐷)‘(𝑀‘𝑡)))
118	4, 38, 117, 9	caublcls 25290	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (1st ∘ 𝑀)(⇝𝑡‘𝐽)𝑌 ∧ 𝑘 ∈ ℕ) → 𝑌 ∈ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑀‘𝑘))))
119	118	3expia 1122	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (1st ∘ 𝑀)(⇝𝑡‘𝐽)𝑌) → (𝑘 ∈ ℕ → 𝑌 ∈ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑀‘𝑘)))))
120	116, 119	mpdan 688	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑘 ∈ ℕ → 𝑌 ∈ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑀‘𝑘)))))
121	120	imp 406	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑌 ∈ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑀‘𝑘))))
122	121	ad2ant2r 748	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀‘𝑘)) < (𝑥 / 2))) → 𝑌 ∈ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑀‘𝑘))))
123	113, 122	sseldd 3923	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀‘𝑘)) < (𝑥 / 2))) → 𝑌 ∈ ((1st ‘(𝑀‘𝑘))(ball‘𝐷)(𝑥 / 2)))
124		blhalf 24384	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ (1st ‘(𝑀‘𝑘)) ∈ 𝑋) ∧ (𝑥 ∈ ℝ ∧ 𝑌 ∈ ((1st ‘(𝑀‘𝑘))(ball‘𝐷)(𝑥 / 2)))) → ((1st ‘(𝑀‘𝑘))(ball‘𝐷)(𝑥 / 2)) ⊆ (𝑌(ball‘𝐷)𝑥))
125	37, 42, 115, 123, 124	syl22anc 839	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀‘𝑘)) < (𝑥 / 2))) → ((1st ‘(𝑀‘𝑘))(ball‘𝐷)(𝑥 / 2)) ⊆ (𝑌(ball‘𝐷)𝑥))
126	113, 125	sstrd 3933	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀‘𝑘)) < (𝑥 / 2))) → ((cls‘𝐽)‘((ball‘𝐷)‘(𝑀‘𝑘))) ⊆ (𝑌(ball‘𝐷)𝑥))
127	106, 126	sstrd 3933	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀‘𝑘)) < (𝑥 / 2))) → ((ball‘𝐷)‘(𝑀‘𝑘)) ⊆ (𝑌(ball‘𝐷)𝑥))
128	96, 127	sstrd 3933	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀‘𝑘)) < (𝑥 / 2))) → ((𝑆‘𝑘)𝐵𝑘) ⊆ (𝑌(ball‘𝐷)𝑥))
129		sstr2 3929	. . . . . . 7 ⊢ (((𝑆‘𝑘)𝐵𝑘) ⊆ (𝑌(ball‘𝐷)𝑥) → ((𝑌(ball‘𝐷)𝑥) ⊆ 𝑍 → ((𝑆‘𝑘)𝐵𝑘) ⊆ 𝑍))
130	128, 129	syl 17	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀‘𝑘)) < (𝑥 / 2))) → ((𝑌(ball‘𝐷)𝑥) ⊆ 𝑍 → ((𝑆‘𝑘)𝐵𝑘) ⊆ 𝑍))
131		unisng 4869	. . . . . . . . . . . . 13 ⊢ (𝑍 ∈ 𝑈 → ∪ {𝑍} = 𝑍)
132	6, 131	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → ∪ {𝑍} = 𝑍)
133	132	sseq2d 3955	. . . . . . . . . . 11 ⊢ (𝜑 → (((𝑆‘𝑘)𝐵𝑘) ⊆ ∪ {𝑍} ↔ ((𝑆‘𝑘)𝐵𝑘) ⊆ 𝑍))
134	133	biimpar 477	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑆‘𝑘)𝐵𝑘) ⊆ 𝑍) → ((𝑆‘𝑘)𝐵𝑘) ⊆ ∪ {𝑍})
135	6	snssd 4753	. . . . . . . . . . . . 13 ⊢ (𝜑 → {𝑍} ⊆ 𝑈)
136		snex 5378	. . . . . . . . . . . . . 14 ⊢ {𝑍} ∈ V
137	136	elpw 4546	. . . . . . . . . . . . 13 ⊢ ({𝑍} ∈ 𝒫 𝑈 ↔ {𝑍} ⊆ 𝑈)
138	135, 137	sylibr 234	. . . . . . . . . . . 12 ⊢ (𝜑 → {𝑍} ∈ 𝒫 𝑈)
139		snfi 8985	. . . . . . . . . . . . 13 ⊢ {𝑍} ∈ Fin
140	139	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → {𝑍} ∈ Fin)
141	138, 140	elind 4141	. . . . . . . . . . 11 ⊢ (𝜑 → {𝑍} ∈ (𝒫 𝑈 ∩ Fin))
142		unieq 4862	. . . . . . . . . . . . 13 ⊢ (𝑣 = {𝑍} → ∪ 𝑣 = ∪ {𝑍})
143	142	sseq2d 3955	. . . . . . . . . . . 12 ⊢ (𝑣 = {𝑍} → (((𝑆‘𝑘)𝐵𝑘) ⊆ ∪ 𝑣 ↔ ((𝑆‘𝑘)𝐵𝑘) ⊆ ∪ {𝑍}))
144	143	rspcev 3565	. . . . . . . . . . 11 ⊢ (({𝑍} ∈ (𝒫 𝑈 ∩ Fin) ∧ ((𝑆‘𝑘)𝐵𝑘) ⊆ ∪ {𝑍}) → ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)((𝑆‘𝑘)𝐵𝑘) ⊆ ∪ 𝑣)
145	141, 144	sylan 581	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑆‘𝑘)𝐵𝑘) ⊆ ∪ {𝑍}) → ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)((𝑆‘𝑘)𝐵𝑘) ⊆ ∪ 𝑣)
146	134, 145	syldan 592	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑆‘𝑘)𝐵𝑘) ⊆ 𝑍) → ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)((𝑆‘𝑘)𝐵𝑘) ⊆ ∪ 𝑣)
147		ovex 7395	. . . . . . . . . . 11 ⊢ ((𝑆‘𝑘)𝐵𝑘) ∈ V
148		sseq1 3948	. . . . . . . . . . . . 13 ⊢ (𝑢 = ((𝑆‘𝑘)𝐵𝑘) → (𝑢 ⊆ ∪ 𝑣 ↔ ((𝑆‘𝑘)𝐵𝑘) ⊆ ∪ 𝑣))
149	148	rexbidv 3162	. . . . . . . . . . . 12 ⊢ (𝑢 = ((𝑆‘𝑘)𝐵𝑘) → (∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 ⊆ ∪ 𝑣 ↔ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)((𝑆‘𝑘)𝐵𝑘) ⊆ ∪ 𝑣))
150	149	notbid 318	. . . . . . . . . . 11 ⊢ (𝑢 = ((𝑆‘𝑘)𝐵𝑘) → (¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 ⊆ ∪ 𝑣 ↔ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)((𝑆‘𝑘)𝐵𝑘) ⊆ ∪ 𝑣))
151	147, 150, 15	elab2 3626	. . . . . . . . . 10 ⊢ (((𝑆‘𝑘)𝐵𝑘) ∈ 𝐾 ↔ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)((𝑆‘𝑘)𝐵𝑘) ⊆ ∪ 𝑣)
152	151	con2bii 357	. . . . . . . . 9 ⊢ (∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)((𝑆‘𝑘)𝐵𝑘) ⊆ ∪ 𝑣 ↔ ¬ ((𝑆‘𝑘)𝐵𝑘) ∈ 𝐾)
153	146, 152	sylib 218	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑆‘𝑘)𝐵𝑘) ⊆ 𝑍) → ¬ ((𝑆‘𝑘)𝐵𝑘) ∈ 𝐾)
154	153	ex 412	. . . . . . 7 ⊢ (𝜑 → (((𝑆‘𝑘)𝐵𝑘) ⊆ 𝑍 → ¬ ((𝑆‘𝑘)𝐵𝑘) ∈ 𝐾))
155	154	ad2antrr 727	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀‘𝑘)) < (𝑥 / 2))) → (((𝑆‘𝑘)𝐵𝑘) ⊆ 𝑍 → ¬ ((𝑆‘𝑘)𝐵𝑘) ∈ 𝐾))
156	130, 155	syld 47	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀‘𝑘)) < (𝑥 / 2))) → ((𝑌(ball‘𝐷)𝑥) ⊆ 𝑍 → ¬ ((𝑆‘𝑘)𝐵𝑘) ∈ 𝐾))
157	36, 156	mt2d 136	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀‘𝑘)) < (𝑥 / 2))) → ¬ (𝑌(ball‘𝐷)𝑥) ⊆ 𝑍)
158	27, 157	rexlimddv 3145	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → ¬ (𝑌(ball‘𝐷)𝑥) ⊆ 𝑍)
159	158	nrexdv 3133	. 2 ⊢ (𝜑 → ¬ ∃𝑥 ∈ ℝ+ (𝑌(ball‘𝐷)𝑥) ⊆ 𝑍)
160	11, 159	pm2.21dd 195	1 ⊢ (𝜑 → 𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  {cab 2715  ∀wral 3052  ∃wrex 3062  Vcvv 3430   ∩ cin 3889   ⊆ wss 3890  ifcif 4467  𝒫 cpw 4542  {csn 4568  ⟨cop 4574  ∪ cuni 4851  ∪ ciun 4934   class class class wbr 5086  {copab 5148   ↦ cmpt 5167   × cxp 5624   ∘ ccom 5630  ⟶wf 6490  ‘cfv 6494  (class class class)co 7362   ∈ cmpo 7364  1^st c1st 7935  2^nd c2nd 7936  Fincfn 8888  ℝcr 11032  0cc0 11033  1c1 11034   + caddc 11036  ℝ^*cxr 11173   < clt 11174   ≤ cle 11175   − cmin 11372   / cdiv 11802  ℕcn 12169  2c2 12231  3c3 12232  ℕ₀cn0 12432  ℝ⁺crp 12937  seqcseq 13958  ↑cexp 14018  ∞Metcxmet 21333  Metcmet 21334  ballcbl 21335  MetOpencmopn 21338  Topctop 22872  clsccl 22997  ⇝_𝑡clm 23205  CMetccmet 25235

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5304  ax-pr 5372  ax-un 7684  ax-cnex 11089  ax-resscn 11090  ax-1cn 11091  ax-icn 11092  ax-addcl 11093  ax-addrcl 11094  ax-mulcl 11095  ax-mulrcl 11096  ax-mulcom 11097  ax-addass 11098  ax-mulass 11099  ax-distr 11100  ax-i2m1 11101  ax-1ne0 11102  ax-1rid 11103  ax-rnegex 11104  ax-rrecex 11105  ax-cnre 11106  ax-pre-lttri 11107  ax-pre-lttrn 11108  ax-pre-ltadd 11109  ax-pre-mulgt0 11110  ax-pre-sup 11111

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-iin 4937  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5521  df-eprel 5526  df-po 5534  df-so 5535  df-fr 5579  df-we 5581  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-res 5638  df-ima 5639  df-pred 6261  df-ord 6322  df-on 6323  df-lim 6324  df-suc 6325  df-iota 6450  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-riota 7319  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7813  df-1st 7937  df-2nd 7938  df-frecs 8226  df-wrecs 8257  df-recs 8306  df-rdg 8344  df-1o 8400  df-er 8638  df-map 8770  df-pm 8771  df-en 8889  df-dom 8890  df-sdom 8891  df-fin 8892  df-sup 9350  df-inf 9351  df-pnf 11176  df-mnf 11177  df-xr 11178  df-ltxr 11179  df-le 11180  df-sub 11374  df-neg 11375  df-div 11803  df-nn 12170  df-2 12239  df-3 12240  df-n0 12433  df-z 12520  df-uz 12784  df-q 12894  df-rp 12938  df-xneg 13058  df-xadd 13059  df-xmul 13060  df-fl 13746  df-seq 13959  df-exp 14019  df-topgen 17401  df-psmet 21340  df-xmet 21341  df-met 21342  df-bl 21343  df-mopn 21344  df-top 22873  df-topon 22890  df-bases 22925  df-cld 22998  df-ntr 22999  df-cls 23000  df-lm 23208  df-cmet 25238

This theorem is referenced by:  heiborlem9  38160