Theorem heiborlem6 38941

Description: Lemma for heibor 38946. Since the sequence of balls connected by the function 𝑇 ensures that each ball nontrivially intersects with the next (since the empty set has a finite subcover, the intersection of any two successive balls in the sequence is nonempty), and each ball is half the size of the previous one, the distance between the centers is at most 3 / 2 times the size of the larger, and so if we expand each ball by a factor of 3 we get a nested sequence of balls. (Contributed by Jeff Madsen, 23-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))⟩)

Assertion

Ref	Expression
heiborlem6	⊢ (𝜑 → ∀𝑘 ∈ ℕ ((ball‘𝐷)‘(𝑀‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝑀‘𝑘)))

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

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

Proof of Theorem heiborlem6

Step	Hyp	Ref	Expression
1		nnnn0 12637	. . . 4 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ0)
2		heibor.6	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋))
3		cmetmet 25634	. . . . . . . 8 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
4	2, 3	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋))
5		metxmet 24680	. . . . . . 7 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
6	4, 5	syl 18	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋))
7	6	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝐷 ∈ (∞Met‘𝑋))
8		heibor.7	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin))
9		inss1 4136	. . . . . . . . 9 ⊢ (𝒫 𝑋 ∩ Fin) ⊆ 𝒫 𝑋
10		fss 6745	. . . . . . . . 9 ⊢ ((𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin) ∧ (𝒫 𝑋 ∩ Fin) ⊆ 𝒫 𝑋) → 𝐹:ℕ0⟶𝒫 𝑋)
11	8, 9, 10	sylancl 598	. . . . . . . 8 ⊢ (𝜑 → 𝐹:ℕ0⟶𝒫 𝑋)
12		peano2nn0 12670	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ0 → (𝑘 + 1) ∈ ℕ0)
13		ffvelcdm 7102	. . . . . . . 8 ⊢ ((𝐹:ℕ0⟶𝒫 𝑋 ∧ (𝑘 + 1) ∈ ℕ0) → (𝐹‘(𝑘 + 1)) ∈ 𝒫 𝑋)
14	11, 12, 13	syl2an 608	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘(𝑘 + 1)) ∈ 𝒫 𝑋)
15	14	elpwid 4569	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘(𝑘 + 1)) ⊆ 𝑋)
16		heibor.1	. . . . . . . . 9 ⊢ 𝐽 = (MetOpen‘𝐷)
17		heibor.3	. . . . . . . . 9 ⊢ 𝐾 = {𝑢 ∣ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 ⊆ ∪ 𝑣}
18		heibor.4	. . . . . . . . 9 ⊢ 𝐺 = {⟨𝑦, 𝑛⟩ ∣ (𝑛 ∈ ℕ0 ∧ 𝑦 ∈ (𝐹‘𝑛) ∧ (𝑦𝐵𝑛) ∈ 𝐾)}
19		heibor.5	. . . . . . . . 9 ⊢ 𝐵 = (𝑧 ∈ 𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))
20		heibor.8	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑛 ∈ ℕ0 𝑋 = ∪ 𝑦 ∈ (𝐹‘𝑛)(𝑦𝐵𝑛))
21		heibor.9	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑥 ∈ 𝐺 ((𝑇‘𝑥)𝐺((2nd ‘𝑥) + 1) ∧ ((𝐵‘𝑥) ∩ ((𝑇‘𝑥)𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾))
22		heibor.10	. . . . . . . . 9 ⊢ (𝜑 → 𝐶𝐺0)
23		heibor.11	. . . . . . . . 9 ⊢ 𝑆 = seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))))
24	16, 17, 18, 19, 2, 8, 20, 21, 22, 23	heiborlem4 38939	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 + 1) ∈ ℕ0) → (𝑆‘(𝑘 + 1))𝐺(𝑘 + 1))
25	12, 24	sylan2 605	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑆‘(𝑘 + 1))𝐺(𝑘 + 1))
26		fvex 6918	. . . . . . . . 9 ⊢ (𝑆‘(𝑘 + 1)) ∈ V
27		ovex 7473	. . . . . . . . 9 ⊢ (𝑘 + 1) ∈ V
28	16, 17, 18, 26, 27	heiborlem2 38937	. . . . . . . 8 ⊢ ((𝑆‘(𝑘 + 1))𝐺(𝑘 + 1) ↔ ((𝑘 + 1) ∈ ℕ0 ∧ (𝑆‘(𝑘 + 1)) ∈ (𝐹‘(𝑘 + 1)) ∧ ((𝑆‘(𝑘 + 1))𝐵(𝑘 + 1)) ∈ 𝐾))
29	28	simp2bi 1164	. . . . . . 7 ⊢ ((𝑆‘(𝑘 + 1))𝐺(𝑘 + 1) → (𝑆‘(𝑘 + 1)) ∈ (𝐹‘(𝑘 + 1)))
30	25, 29	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑆‘(𝑘 + 1)) ∈ (𝐹‘(𝑘 + 1)))
31	15, 30	sseldd 3914	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑆‘(𝑘 + 1)) ∈ 𝑋)
32	11	ffvelcdmda 7105	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘𝑘) ∈ 𝒫 𝑋)
33	32	elpwid 4569	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘𝑘) ⊆ 𝑋)
34	16, 17, 18, 19, 2, 8, 20, 21, 22, 23	heiborlem4 38939	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑆‘𝑘)𝐺𝑘)
35		fvex 6918	. . . . . . . . 9 ⊢ (𝑆‘𝑘) ∈ V
36		vex 3458	. . . . . . . . 9 ⊢ 𝑘 ∈ V
37	16, 17, 18, 35, 36	heiborlem2 38937	. . . . . . . 8 ⊢ ((𝑆‘𝑘)𝐺𝑘 ↔ (𝑘 ∈ ℕ0 ∧ (𝑆‘𝑘) ∈ (𝐹‘𝑘) ∧ ((𝑆‘𝑘)𝐵𝑘) ∈ 𝐾))
38	37	simp2bi 1164	. . . . . . 7 ⊢ ((𝑆‘𝑘)𝐺𝑘 → (𝑆‘𝑘) ∈ (𝐹‘𝑘))
39	34, 38	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑆‘𝑘) ∈ (𝐹‘𝑘))
40	33, 39	sseldd 3914	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑆‘𝑘) ∈ 𝑋)
41		3re 12447	. . . . . 6 ⊢ 3 ∈ ℝ
42		2nn 12440	. . . . . . . . 9 ⊢ 2 ∈ ℕ
43		nnexpcl 14241	. . . . . . . . 9 ⊢ ((2 ∈ ℕ ∧ (𝑘 + 1) ∈ ℕ0) → (2↑(𝑘 + 1)) ∈ ℕ)
44	42, 12, 43	sylancr 599	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ0 → (2↑(𝑘 + 1)) ∈ ℕ)
45	44	nnrpd 13186	. . . . . . 7 ⊢ (𝑘 ∈ ℕ0 → (2↑(𝑘 + 1)) ∈ ℝ+)
46	45	adantl 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (2↑(𝑘 + 1)) ∈ ℝ+)
47		rerpdivcl 13176	. . . . . 6 ⊢ ((3 ∈ ℝ ∧ (2↑(𝑘 + 1)) ∈ ℝ+) → (3 / (2↑(𝑘 + 1))) ∈ ℝ)
48	41, 46, 47	sylancr 599	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (3 / (2↑(𝑘 + 1))) ∈ ℝ)
49		nnexpcl 14241	. . . . . . . . 9 ⊢ ((2 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → (2↑𝑘) ∈ ℕ)
50	42, 49	mpan 703	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ0 → (2↑𝑘) ∈ ℕ)
51	50	nnrpd 13186	. . . . . . 7 ⊢ (𝑘 ∈ ℕ0 → (2↑𝑘) ∈ ℝ+)
52	51	adantl 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (2↑𝑘) ∈ ℝ+)
53		rerpdivcl 13176	. . . . . 6 ⊢ ((3 ∈ ℝ ∧ (2↑𝑘) ∈ ℝ+) → (3 / (2↑𝑘)) ∈ ℝ)
54	41, 52, 53	sylancr 599	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (3 / (2↑𝑘)) ∈ ℝ)
55		oveq1 7447	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝑆‘𝑘) → (𝑧(ball‘𝐷)(1 / (2↑𝑚))) = ((𝑆‘𝑘)(ball‘𝐷)(1 / (2↑𝑚))))
56		oveq2 7448	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑘 → (2↑𝑚) = (2↑𝑘))
57	56	oveq2d 7456	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑘 → (1 / (2↑𝑚)) = (1 / (2↑𝑘)))
58	57	oveq2d 7456	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑘 → ((𝑆‘𝑘)(ball‘𝐷)(1 / (2↑𝑚))) = ((𝑆‘𝑘)(ball‘𝐷)(1 / (2↑𝑘))))
59		ovex 7473	. . . . . . . . . . . 12 ⊢ ((𝑆‘𝑘)(ball‘𝐷)(1 / (2↑𝑘))) ∈ V
60	55, 58, 19, 59	ovmpo 7601	. . . . . . . . . . 11 ⊢ (((𝑆‘𝑘) ∈ 𝑋 ∧ 𝑘 ∈ ℕ0) → ((𝑆‘𝑘)𝐵𝑘) = ((𝑆‘𝑘)(ball‘𝐷)(1 / (2↑𝑘))))
61	40, 60	sylancom 600	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝑆‘𝑘)𝐵𝑘) = ((𝑆‘𝑘)(ball‘𝐷)(1 / (2↑𝑘))))
62		df-br 5107	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑆‘𝑘)𝐺𝑘 ↔ ⟨(𝑆‘𝑘), 𝑘⟩ ∈ 𝐺)
63		fveq2 6905	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = ⟨(𝑆‘𝑘), 𝑘⟩ → (𝑇‘𝑥) = (𝑇‘⟨(𝑆‘𝑘), 𝑘⟩))
64		df-ov 7443	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑆‘𝑘)𝑇𝑘) = (𝑇‘⟨(𝑆‘𝑘), 𝑘⟩)
65	63, 64	eqtr4di 2816	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = ⟨(𝑆‘𝑘), 𝑘⟩ → (𝑇‘𝑥) = ((𝑆‘𝑘)𝑇𝑘))
66	35, 36	op2ndd 8033	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = ⟨(𝑆‘𝑘), 𝑘⟩ → (2nd ‘𝑥) = 𝑘)
67	66	oveq1d 7455	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = ⟨(𝑆‘𝑘), 𝑘⟩ → ((2nd ‘𝑥) + 1) = (𝑘 + 1))
68	65, 67	breq12d 5119	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = ⟨(𝑆‘𝑘), 𝑘⟩ → ((𝑇‘𝑥)𝐺((2nd ‘𝑥) + 1) ↔ ((𝑆‘𝑘)𝑇𝑘)𝐺(𝑘 + 1)))
69		fveq2 6905	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = ⟨(𝑆‘𝑘), 𝑘⟩ → (𝐵‘𝑥) = (𝐵‘⟨(𝑆‘𝑘), 𝑘⟩))
70		df-ov 7443	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑆‘𝑘)𝐵𝑘) = (𝐵‘⟨(𝑆‘𝑘), 𝑘⟩)
71	69, 70	eqtr4di 2816	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = ⟨(𝑆‘𝑘), 𝑘⟩ → (𝐵‘𝑥) = ((𝑆‘𝑘)𝐵𝑘))
72	65, 67	oveq12d 7458	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = ⟨(𝑆‘𝑘), 𝑘⟩ → ((𝑇‘𝑥)𝐵((2nd ‘𝑥) + 1)) = (((𝑆‘𝑘)𝑇𝑘)𝐵(𝑘 + 1)))
73	71, 72	ineq12d 4121	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = ⟨(𝑆‘𝑘), 𝑘⟩ → ((𝐵‘𝑥) ∩ ((𝑇‘𝑥)𝐵((2nd ‘𝑥) + 1))) = (((𝑆‘𝑘)𝐵𝑘) ∩ (((𝑆‘𝑘)𝑇𝑘)𝐵(𝑘 + 1))))
74	73	eleq1d 2849	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = ⟨(𝑆‘𝑘), 𝑘⟩ → (((𝐵‘𝑥) ∩ ((𝑇‘𝑥)𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾 ↔ (((𝑆‘𝑘)𝐵𝑘) ∩ (((𝑆‘𝑘)𝑇𝑘)𝐵(𝑘 + 1))) ∈ 𝐾))
75	68, 74	anbi12d 644	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = ⟨(𝑆‘𝑘), 𝑘⟩ → (((𝑇‘𝑥)𝐺((2nd ‘𝑥) + 1) ∧ ((𝐵‘𝑥) ∩ ((𝑇‘𝑥)𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾) ↔ (((𝑆‘𝑘)𝑇𝑘)𝐺(𝑘 + 1) ∧ (((𝑆‘𝑘)𝐵𝑘) ∩ (((𝑆‘𝑘)𝑇𝑘)𝐵(𝑘 + 1))) ∈ 𝐾)))
76	75	rspccv 3577	. . . . . . . . . . . . . . . . . 18 ⊢ (∀𝑥 ∈ 𝐺 ((𝑇‘𝑥)𝐺((2nd ‘𝑥) + 1) ∧ ((𝐵‘𝑥) ∩ ((𝑇‘𝑥)𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾) → (⟨(𝑆‘𝑘), 𝑘⟩ ∈ 𝐺 → (((𝑆‘𝑘)𝑇𝑘)𝐺(𝑘 + 1) ∧ (((𝑆‘𝑘)𝐵𝑘) ∩ (((𝑆‘𝑘)𝑇𝑘)𝐵(𝑘 + 1))) ∈ 𝐾)))
77	21, 76	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (⟨(𝑆‘𝑘), 𝑘⟩ ∈ 𝐺 → (((𝑆‘𝑘)𝑇𝑘)𝐺(𝑘 + 1) ∧ (((𝑆‘𝑘)𝐵𝑘) ∩ (((𝑆‘𝑘)𝑇𝑘)𝐵(𝑘 + 1))) ∈ 𝐾)))
78	62, 77	biimtrid 245	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝑆‘𝑘)𝐺𝑘 → (((𝑆‘𝑘)𝑇𝑘)𝐺(𝑘 + 1) ∧ (((𝑆‘𝑘)𝐵𝑘) ∩ (((𝑆‘𝑘)𝑇𝑘)𝐵(𝑘 + 1))) ∈ 𝐾)))
79	78	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝑆‘𝑘)𝐺𝑘 → (((𝑆‘𝑘)𝑇𝑘)𝐺(𝑘 + 1) ∧ (((𝑆‘𝑘)𝐵𝑘) ∩ (((𝑆‘𝑘)𝑇𝑘)𝐵(𝑘 + 1))) ∈ 𝐾)))
80	34, 79	mpd 16	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (((𝑆‘𝑘)𝑇𝑘)𝐺(𝑘 + 1) ∧ (((𝑆‘𝑘)𝐵𝑘) ∩ (((𝑆‘𝑘)𝑇𝑘)𝐵(𝑘 + 1))) ∈ 𝐾))
81	80	simpld 500	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝑆‘𝑘)𝑇𝑘)𝐺(𝑘 + 1))
82		ovex 7473	. . . . . . . . . . . . . . 15 ⊢ ((𝑆‘𝑘)𝑇𝑘) ∈ V
83	16, 17, 18, 82, 27	heiborlem2 38937	. . . . . . . . . . . . . 14 ⊢ (((𝑆‘𝑘)𝑇𝑘)𝐺(𝑘 + 1) ↔ ((𝑘 + 1) ∈ ℕ0 ∧ ((𝑆‘𝑘)𝑇𝑘) ∈ (𝐹‘(𝑘 + 1)) ∧ (((𝑆‘𝑘)𝑇𝑘)𝐵(𝑘 + 1)) ∈ 𝐾))
84	83	simp2bi 1164	. . . . . . . . . . . . 13 ⊢ (((𝑆‘𝑘)𝑇𝑘)𝐺(𝑘 + 1) → ((𝑆‘𝑘)𝑇𝑘) ∈ (𝐹‘(𝑘 + 1)))
85	81, 84	syl 18	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝑆‘𝑘)𝑇𝑘) ∈ (𝐹‘(𝑘 + 1)))
86	15, 85	sseldd 3914	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝑆‘𝑘)𝑇𝑘) ∈ 𝑋)
87	12	adantl 487	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑘 + 1) ∈ ℕ0)
88		oveq1 7447	. . . . . . . . . . . 12 ⊢ (𝑧 = ((𝑆‘𝑘)𝑇𝑘) → (𝑧(ball‘𝐷)(1 / (2↑𝑚))) = (((𝑆‘𝑘)𝑇𝑘)(ball‘𝐷)(1 / (2↑𝑚))))
89		oveq2 7448	. . . . . . . . . . . . . 14 ⊢ (𝑚 = (𝑘 + 1) → (2↑𝑚) = (2↑(𝑘 + 1)))
90	89	oveq2d 7456	. . . . . . . . . . . . 13 ⊢ (𝑚 = (𝑘 + 1) → (1 / (2↑𝑚)) = (1 / (2↑(𝑘 + 1))))
91	90	oveq2d 7456	. . . . . . . . . . . 12 ⊢ (𝑚 = (𝑘 + 1) → (((𝑆‘𝑘)𝑇𝑘)(ball‘𝐷)(1 / (2↑𝑚))) = (((𝑆‘𝑘)𝑇𝑘)(ball‘𝐷)(1 / (2↑(𝑘 + 1)))))
92		ovex 7473	. . . . . . . . . . . 12 ⊢ (((𝑆‘𝑘)𝑇𝑘)(ball‘𝐷)(1 / (2↑(𝑘 + 1)))) ∈ V
93	88, 91, 19, 92	ovmpo 7601	. . . . . . . . . . 11 ⊢ ((((𝑆‘𝑘)𝑇𝑘) ∈ 𝑋 ∧ (𝑘 + 1) ∈ ℕ0) → (((𝑆‘𝑘)𝑇𝑘)𝐵(𝑘 + 1)) = (((𝑆‘𝑘)𝑇𝑘)(ball‘𝐷)(1 / (2↑(𝑘 + 1)))))
94	86, 87, 93	syl2anc 596	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (((𝑆‘𝑘)𝑇𝑘)𝐵(𝑘 + 1)) = (((𝑆‘𝑘)𝑇𝑘)(ball‘𝐷)(1 / (2↑(𝑘 + 1)))))
95	61, 94	ineq12d 4121	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (((𝑆‘𝑘)𝐵𝑘) ∩ (((𝑆‘𝑘)𝑇𝑘)𝐵(𝑘 + 1))) = (((𝑆‘𝑘)(ball‘𝐷)(1 / (2↑𝑘))) ∩ (((𝑆‘𝑘)𝑇𝑘)(ball‘𝐷)(1 / (2↑(𝑘 + 1))))))
96	80	simprd 501	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (((𝑆‘𝑘)𝐵𝑘) ∩ (((𝑆‘𝑘)𝑇𝑘)𝐵(𝑘 + 1))) ∈ 𝐾)
97		0elpw 5321	. . . . . . . . . . . . 13 ⊢ ∅ ∈ 𝒫 𝑈
98		0fi 9095	. . . . . . . . . . . . 13 ⊢ ∅ ∈ Fin
99		elin 4096	. . . . . . . . . . . . 13 ⊢ (∅ ∈ (𝒫 𝑈 ∩ Fin) ↔ (∅ ∈ 𝒫 𝑈 ∧ ∅ ∈ Fin))
100	97, 98, 99	mpbir2an 724	. . . . . . . . . . . 12 ⊢ ∅ ∈ (𝒫 𝑈 ∩ Fin)
101		0ss 4353	. . . . . . . . . . . 12 ⊢ ∅ ⊆ ∪ ∅
102		unieq 4881	. . . . . . . . . . . . . 14 ⊢ (𝑣 = ∅ → ∪ 𝑣 = ∪ ∅)
103	102	sseq2d 3945	. . . . . . . . . . . . 13 ⊢ (𝑣 = ∅ → (∅ ⊆ ∪ 𝑣 ↔ ∅ ⊆ ∪ ∅))
104	103	rspcev 3580	. . . . . . . . . . . 12 ⊢ ((∅ ∈ (𝒫 𝑈 ∩ Fin) ∧ ∅ ⊆ ∪ ∅) → ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)∅ ⊆ ∪ 𝑣)
105	100, 101, 104	mp2an 705	. . . . . . . . . . 11 ⊢ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)∅ ⊆ ∪ 𝑣
106		0ex 5264	. . . . . . . . . . . . 13 ⊢ ∅ ∈ V
107		sseq1 3938	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = ∅ → (𝑢 ⊆ ∪ 𝑣 ↔ ∅ ⊆ ∪ 𝑣))
108	107	rexbidv 3190	. . . . . . . . . . . . . 14 ⊢ (𝑢 = ∅ → (∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 ⊆ ∪ 𝑣 ↔ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)∅ ⊆ ∪ 𝑣))
109	108	notbid 321	. . . . . . . . . . . . 13 ⊢ (𝑢 = ∅ → (¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 ⊆ ∪ 𝑣 ↔ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)∅ ⊆ ∪ 𝑣))
110	106, 109, 17	elab2 3639	. . . . . . . . . . . 12 ⊢ (∅ ∈ 𝐾 ↔ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)∅ ⊆ ∪ 𝑣)
111	110	con2bii 360	. . . . . . . . . . 11 ⊢ (∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)∅ ⊆ ∪ 𝑣 ↔ ¬ ∅ ∈ 𝐾)
112	105, 111	mpbi 233	. . . . . . . . . 10 ⊢ ¬ ∅ ∈ 𝐾
113		nelne2 3057	. . . . . . . . . 10 ⊢ (((((𝑆‘𝑘)𝐵𝑘) ∩ (((𝑆‘𝑘)𝑇𝑘)𝐵(𝑘 + 1))) ∈ 𝐾 ∧ ¬ ∅ ∈ 𝐾) → (((𝑆‘𝑘)𝐵𝑘) ∩ (((𝑆‘𝑘)𝑇𝑘)𝐵(𝑘 + 1))) ≠ ∅)
114	96, 112, 113	sylancl 598	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (((𝑆‘𝑘)𝐵𝑘) ∩ (((𝑆‘𝑘)𝑇𝑘)𝐵(𝑘 + 1))) ≠ ∅)
115	95, 114	eqnetrrd 3027	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (((𝑆‘𝑘)(ball‘𝐷)(1 / (2↑𝑘))) ∩ (((𝑆‘𝑘)𝑇𝑘)(ball‘𝐷)(1 / (2↑(𝑘 + 1))))) ≠ ∅)
116	51	rpreccld 13198	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ0 → (1 / (2↑𝑘)) ∈ ℝ+)
117	116	adantl 487	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (1 / (2↑𝑘)) ∈ ℝ+)
118	117	rpred 13188	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (1 / (2↑𝑘)) ∈ ℝ)
119	45	rpreccld 13198	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ0 → (1 / (2↑(𝑘 + 1))) ∈ ℝ+)
120	119	adantl 487	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (1 / (2↑(𝑘 + 1))) ∈ ℝ+)
121	120	rpred 13188	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (1 / (2↑(𝑘 + 1))) ∈ ℝ)
122		rexadd 13386	. . . . . . . . . . . 12 ⊢ (((1 / (2↑𝑘)) ∈ ℝ ∧ (1 / (2↑(𝑘 + 1))) ∈ ℝ) → ((1 / (2↑𝑘)) +e (1 / (2↑(𝑘 + 1)))) = ((1 / (2↑𝑘)) + (1 / (2↑(𝑘 + 1)))))
123	118, 121, 122	syl2anc 596	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((1 / (2↑𝑘)) +e (1 / (2↑(𝑘 + 1)))) = ((1 / (2↑𝑘)) + (1 / (2↑(𝑘 + 1)))))
124	123	breq1d 5116	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (((1 / (2↑𝑘)) +e (1 / (2↑(𝑘 + 1)))) ≤ ((𝑆‘𝑘)𝐷((𝑆‘𝑘)𝑇𝑘)) ↔ ((1 / (2↑𝑘)) + (1 / (2↑(𝑘 + 1)))) ≤ ((𝑆‘𝑘)𝐷((𝑆‘𝑘)𝑇𝑘))))
125	117	rpxrd 13189	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (1 / (2↑𝑘)) ∈ ℝ*)
126	120	rpxrd 13189	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (1 / (2↑(𝑘 + 1))) ∈ ℝ*)
127		bldisj 24744	. . . . . . . . . . . . 13 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑆‘𝑘) ∈ 𝑋 ∧ ((𝑆‘𝑘)𝑇𝑘) ∈ 𝑋) ∧ ((1 / (2↑𝑘)) ∈ ℝ* ∧ (1 / (2↑(𝑘 + 1))) ∈ ℝ* ∧ ((1 / (2↑𝑘)) +e (1 / (2↑(𝑘 + 1)))) ≤ ((𝑆‘𝑘)𝐷((𝑆‘𝑘)𝑇𝑘)))) → (((𝑆‘𝑘)(ball‘𝐷)(1 / (2↑𝑘))) ∩ (((𝑆‘𝑘)𝑇𝑘)(ball‘𝐷)(1 / (2↑(𝑘 + 1))))) = ∅)
128	127	3exp2 1373	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑆‘𝑘) ∈ 𝑋 ∧ ((𝑆‘𝑘)𝑇𝑘) ∈ 𝑋) → ((1 / (2↑𝑘)) ∈ ℝ* → ((1 / (2↑(𝑘 + 1))) ∈ ℝ* → (((1 / (2↑𝑘)) +e (1 / (2↑(𝑘 + 1)))) ≤ ((𝑆‘𝑘)𝐷((𝑆‘𝑘)𝑇𝑘)) → (((𝑆‘𝑘)(ball‘𝐷)(1 / (2↑𝑘))) ∩ (((𝑆‘𝑘)𝑇𝑘)(ball‘𝐷)(1 / (2↑(𝑘 + 1))))) = ∅))))
129	128	imp32 424	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑆‘𝑘) ∈ 𝑋 ∧ ((𝑆‘𝑘)𝑇𝑘) ∈ 𝑋) ∧ ((1 / (2↑𝑘)) ∈ ℝ* ∧ (1 / (2↑(𝑘 + 1))) ∈ ℝ*)) → (((1 / (2↑𝑘)) +e (1 / (2↑(𝑘 + 1)))) ≤ ((𝑆‘𝑘)𝐷((𝑆‘𝑘)𝑇𝑘)) → (((𝑆‘𝑘)(ball‘𝐷)(1 / (2↑𝑘))) ∩ (((𝑆‘𝑘)𝑇𝑘)(ball‘𝐷)(1 / (2↑(𝑘 + 1))))) = ∅))
130	7, 40, 86, 125, 126, 129	syl32anc 1406	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (((1 / (2↑𝑘)) +e (1 / (2↑(𝑘 + 1)))) ≤ ((𝑆‘𝑘)𝐷((𝑆‘𝑘)𝑇𝑘)) → (((𝑆‘𝑘)(ball‘𝐷)(1 / (2↑𝑘))) ∩ (((𝑆‘𝑘)𝑇𝑘)(ball‘𝐷)(1 / (2↑(𝑘 + 1))))) = ∅))
131	124, 130	sylbird 263	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (((1 / (2↑𝑘)) + (1 / (2↑(𝑘 + 1)))) ≤ ((𝑆‘𝑘)𝐷((𝑆‘𝑘)𝑇𝑘)) → (((𝑆‘𝑘)(ball‘𝐷)(1 / (2↑𝑘))) ∩ (((𝑆‘𝑘)𝑇𝑘)(ball‘𝐷)(1 / (2↑(𝑘 + 1))))) = ∅))
132	131	necon3ad 2972	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((((𝑆‘𝑘)(ball‘𝐷)(1 / (2↑𝑘))) ∩ (((𝑆‘𝑘)𝑇𝑘)(ball‘𝐷)(1 / (2↑(𝑘 + 1))))) ≠ ∅ → ¬ ((1 / (2↑𝑘)) + (1 / (2↑(𝑘 + 1)))) ≤ ((𝑆‘𝑘)𝐷((𝑆‘𝑘)𝑇𝑘))))
133	115, 132	mpd 16	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ¬ ((1 / (2↑𝑘)) + (1 / (2↑(𝑘 + 1)))) ≤ ((𝑆‘𝑘)𝐷((𝑆‘𝑘)𝑇𝑘)))
134	117, 120	rpaddcld 13203	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((1 / (2↑𝑘)) + (1 / (2↑(𝑘 + 1)))) ∈ ℝ+)
135	134	rpred 13188	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((1 / (2↑𝑘)) + (1 / (2↑(𝑘 + 1)))) ∈ ℝ)
136	4	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝐷 ∈ (Met‘𝑋))
137		metcl 24678	. . . . . . . . . 10 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ (𝑆‘𝑘) ∈ 𝑋 ∧ ((𝑆‘𝑘)𝑇𝑘) ∈ 𝑋) → ((𝑆‘𝑘)𝐷((𝑆‘𝑘)𝑇𝑘)) ∈ ℝ)
138	136, 40, 86, 137	syl3anc 1399	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝑆‘𝑘)𝐷((𝑆‘𝑘)𝑇𝑘)) ∈ ℝ)
139	135, 138	letrid 11486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (((1 / (2↑𝑘)) + (1 / (2↑(𝑘 + 1)))) ≤ ((𝑆‘𝑘)𝐷((𝑆‘𝑘)𝑇𝑘)) ∨ ((𝑆‘𝑘)𝐷((𝑆‘𝑘)𝑇𝑘)) ≤ ((1 / (2↑𝑘)) + (1 / (2↑(𝑘 + 1))))))
140	139	ord 878	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (¬ ((1 / (2↑𝑘)) + (1 / (2↑(𝑘 + 1)))) ≤ ((𝑆‘𝑘)𝐷((𝑆‘𝑘)𝑇𝑘)) → ((𝑆‘𝑘)𝐷((𝑆‘𝑘)𝑇𝑘)) ≤ ((1 / (2↑𝑘)) + (1 / (2↑(𝑘 + 1))))))
141	133, 140	mpd 16	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝑆‘𝑘)𝐷((𝑆‘𝑘)𝑇𝑘)) ≤ ((1 / (2↑𝑘)) + (1 / (2↑(𝑘 + 1)))))
142		seqp1 14183	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (ℤ≥‘0) → (seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))))‘(𝑘 + 1)) = ((seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))))‘𝑘)𝑇((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))‘(𝑘 + 1))))
143		nn0uz 13027	. . . . . . . . . . . 12 ⊢ ℕ0 = (ℤ≥‘0)
144	142, 143	eleq2s 2882	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ0 → (seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))))‘(𝑘 + 1)) = ((seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))))‘𝑘)𝑇((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))‘(𝑘 + 1))))
145	23	fveq1i 6906	. . . . . . . . . . 11 ⊢ (𝑆‘(𝑘 + 1)) = (seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))))‘(𝑘 + 1))
146	23	fveq1i 6906	. . . . . . . . . . . 12 ⊢ (𝑆‘𝑘) = (seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))))‘𝑘)
147	146	oveq1i 7450	. . . . . . . . . . 11 ⊢ ((𝑆‘𝑘)𝑇((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))‘(𝑘 + 1))) = ((seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))))‘𝑘)𝑇((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))‘(𝑘 + 1)))
148	144, 145, 147	3eqtr4g 2823	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ0 → (𝑆‘(𝑘 + 1)) = ((𝑆‘𝑘)𝑇((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))‘(𝑘 + 1))))
149		eqid 2763	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))) = (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))
150		eqeq1 2767	. . . . . . . . . . . . . 14 ⊢ (𝑚 = (𝑘 + 1) → (𝑚 = 0 ↔ (𝑘 + 1) = 0))
151		oveq1 7447	. . . . . . . . . . . . . 14 ⊢ (𝑚 = (𝑘 + 1) → (𝑚 − 1) = ((𝑘 + 1) − 1))
152	150, 151	ifbieq2d 4512	. . . . . . . . . . . . 13 ⊢ (𝑚 = (𝑘 + 1) → if(𝑚 = 0, 𝐶, (𝑚 − 1)) = if((𝑘 + 1) = 0, 𝐶, ((𝑘 + 1) − 1)))
153		nn0p1nn 12669	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ ℕ0 → (𝑘 + 1) ∈ ℕ)
154		nnne0 12396	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘 + 1) ∈ ℕ → (𝑘 + 1) ≠ 0)
155	154	neneqd 2964	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 + 1) ∈ ℕ → ¬ (𝑘 + 1) = 0)
156	153, 155	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℕ0 → ¬ (𝑘 + 1) = 0)
157	156	iffalsed 4496	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ0 → if((𝑘 + 1) = 0, 𝐶, ((𝑘 + 1) − 1)) = ((𝑘 + 1) − 1))
158		ovex 7473	. . . . . . . . . . . . . 14 ⊢ ((𝑘 + 1) − 1) ∈ V
159	157, 158	eqeltrdi 2872	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ0 → if((𝑘 + 1) = 0, 𝐶, ((𝑘 + 1) − 1)) ∈ V)
160	149, 152, 12, 159	fvmptd3 7037	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ0 → ((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))‘(𝑘 + 1)) = if((𝑘 + 1) = 0, 𝐶, ((𝑘 + 1) − 1)))
161		nn0cn 12640	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ0 → 𝑘 ∈ ℂ)
162		ax-1cn 11282	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℂ
163		pncan 11587	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑘 + 1) − 1) = 𝑘)
164	161, 162, 163	sylancl 598	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ0 → ((𝑘 + 1) − 1) = 𝑘)
165	160, 157, 164	3eqtrd 2802	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ0 → ((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))‘(𝑘 + 1)) = 𝑘)
166	165	oveq2d 7456	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ0 → ((𝑆‘𝑘)𝑇((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))‘(𝑘 + 1))) = ((𝑆‘𝑘)𝑇𝑘))
167	148, 166	eqtrd 2798	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ0 → (𝑆‘(𝑘 + 1)) = ((𝑆‘𝑘)𝑇𝑘))
168	167	adantl 487	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑆‘(𝑘 + 1)) = ((𝑆‘𝑘)𝑇𝑘))
169	168	oveq1d 7455	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝑆‘(𝑘 + 1))𝐷(𝑆‘𝑘)) = (((𝑆‘𝑘)𝑇𝑘)𝐷(𝑆‘𝑘)))
170		metsym 24696	. . . . . . . 8 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ ((𝑆‘𝑘)𝑇𝑘) ∈ 𝑋 ∧ (𝑆‘𝑘) ∈ 𝑋) → (((𝑆‘𝑘)𝑇𝑘)𝐷(𝑆‘𝑘)) = ((𝑆‘𝑘)𝐷((𝑆‘𝑘)𝑇𝑘)))
171	136, 86, 40, 170	syl3anc 1399	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (((𝑆‘𝑘)𝑇𝑘)𝐷(𝑆‘𝑘)) = ((𝑆‘𝑘)𝐷((𝑆‘𝑘)𝑇𝑘)))
172	169, 171	eqtrd 2798	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝑆‘(𝑘 + 1))𝐷(𝑆‘𝑘)) = ((𝑆‘𝑘)𝐷((𝑆‘𝑘)𝑇𝑘)))
173		3cn 12448	. . . . . . . . . . . . 13 ⊢ 3 ∈ ℂ
174	173	2timesi 12504	. . . . . . . . . . . 12 ⊢ (2 · 3) = (3 + 3)
175	174	oveq1i 7450	. . . . . . . . . . 11 ⊢ ((2 · 3) − 3) = ((3 + 3) − 3)
176	173, 173	pncan3oi 11597	. . . . . . . . . . 11 ⊢ ((3 + 3) − 3) = 3
177		df-3 12430	. . . . . . . . . . 11 ⊢ 3 = (2 + 1)
178	175, 176, 177	3eqtri 2790	. . . . . . . . . 10 ⊢ ((2 · 3) − 3) = (2 + 1)
179	178	oveq1i 7450	. . . . . . . . 9 ⊢ (((2 · 3) − 3) / (2↑(𝑘 + 1))) = ((2 + 1) / (2↑(𝑘 + 1)))
180		rpcn 13155	. . . . . . . . . . 11 ⊢ ((2↑(𝑘 + 1)) ∈ ℝ+ → (2↑(𝑘 + 1)) ∈ ℂ)
181		rpne0 13161	. . . . . . . . . . 11 ⊢ ((2↑(𝑘 + 1)) ∈ ℝ+ → (2↑(𝑘 + 1)) ≠ 0)
182		2cn 12442	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℂ
183	182, 173	mulcli 11340	. . . . . . . . . . . 12 ⊢ (2 · 3) ∈ ℂ
184		divsubdir 12034	. . . . . . . . . . . 12 ⊢ (((2 · 3) ∈ ℂ ∧ 3 ∈ ℂ ∧ ((2↑(𝑘 + 1)) ∈ ℂ ∧ (2↑(𝑘 + 1)) ≠ 0)) → (((2 · 3) − 3) / (2↑(𝑘 + 1))) = (((2 · 3) / (2↑(𝑘 + 1))) − (3 / (2↑(𝑘 + 1)))))
185	183, 173, 184	mp3an12 1481	. . . . . . . . . . 11 ⊢ (((2↑(𝑘 + 1)) ∈ ℂ ∧ (2↑(𝑘 + 1)) ≠ 0) → (((2 · 3) − 3) / (2↑(𝑘 + 1))) = (((2 · 3) / (2↑(𝑘 + 1))) − (3 / (2↑(𝑘 + 1)))))
186	180, 181, 185	syl2anc 596	. . . . . . . . . 10 ⊢ ((2↑(𝑘 + 1)) ∈ ℝ+ → (((2 · 3) − 3) / (2↑(𝑘 + 1))) = (((2 · 3) / (2↑(𝑘 + 1))) − (3 / (2↑(𝑘 + 1)))))
187	45, 186	syl 18	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ0 → (((2 · 3) − 3) / (2↑(𝑘 + 1))) = (((2 · 3) / (2↑(𝑘 + 1))) − (3 / (2↑(𝑘 + 1)))))
188		divdir 12023	. . . . . . . . . . . 12 ⊢ ((2 ∈ ℂ ∧ 1 ∈ ℂ ∧ ((2↑(𝑘 + 1)) ∈ ℂ ∧ (2↑(𝑘 + 1)) ≠ 0)) → ((2 + 1) / (2↑(𝑘 + 1))) = ((2 / (2↑(𝑘 + 1))) + (1 / (2↑(𝑘 + 1)))))
189	182, 162, 188	mp3an12 1481	. . . . . . . . . . 11 ⊢ (((2↑(𝑘 + 1)) ∈ ℂ ∧ (2↑(𝑘 + 1)) ≠ 0) → ((2 + 1) / (2↑(𝑘 + 1))) = ((2 / (2↑(𝑘 + 1))) + (1 / (2↑(𝑘 + 1)))))
190	180, 181, 189	syl2anc 596	. . . . . . . . . 10 ⊢ ((2↑(𝑘 + 1)) ∈ ℝ+ → ((2 + 1) / (2↑(𝑘 + 1))) = ((2 / (2↑(𝑘 + 1))) + (1 / (2↑(𝑘 + 1)))))
191	45, 190	syl 18	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ0 → ((2 + 1) / (2↑(𝑘 + 1))) = ((2 / (2↑(𝑘 + 1))) + (1 / (2↑(𝑘 + 1)))))
192	179, 187, 191	3eqtr3a 2822	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ0 → (((2 · 3) / (2↑(𝑘 + 1))) − (3 / (2↑(𝑘 + 1)))) = ((2 / (2↑(𝑘 + 1))) + (1 / (2↑(𝑘 + 1)))))
193		rpcn 13155	. . . . . . . . . . . 12 ⊢ ((2↑𝑘) ∈ ℝ+ → (2↑𝑘) ∈ ℂ)
194		rpne0 13161	. . . . . . . . . . . 12 ⊢ ((2↑𝑘) ∈ ℝ+ → (2↑𝑘) ≠ 0)
195		2cnne0 12579	. . . . . . . . . . . . 13 ⊢ (2 ∈ ℂ ∧ 2 ≠ 0)
196		divcan5 12043	. . . . . . . . . . . . 13 ⊢ ((3 ∈ ℂ ∧ ((2↑𝑘) ∈ ℂ ∧ (2↑𝑘) ≠ 0) ∧ (2 ∈ ℂ ∧ 2 ≠ 0)) → ((2 · 3) / (2 · (2↑𝑘))) = (3 / (2↑𝑘)))
197	173, 195, 196	mp3an13 1482	. . . . . . . . . . . 12 ⊢ (((2↑𝑘) ∈ ℂ ∧ (2↑𝑘) ≠ 0) → ((2 · 3) / (2 · (2↑𝑘))) = (3 / (2↑𝑘)))
198	193, 194, 197	syl2anc 596	. . . . . . . . . . 11 ⊢ ((2↑𝑘) ∈ ℝ+ → ((2 · 3) / (2 · (2↑𝑘))) = (3 / (2↑𝑘)))
199	51, 198	syl 18	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ0 → ((2 · 3) / (2 · (2↑𝑘))) = (3 / (2↑𝑘)))
200	51, 193	syl 18	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ0 → (2↑𝑘) ∈ ℂ)
201		mulcom 11310	. . . . . . . . . . . . 13 ⊢ ((2 ∈ ℂ ∧ (2↑𝑘) ∈ ℂ) → (2 · (2↑𝑘)) = ((2↑𝑘) · 2))
202	182, 200, 201	sylancr 599	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ0 → (2 · (2↑𝑘)) = ((2↑𝑘) · 2))
203		expp1 14235	. . . . . . . . . . . . 13 ⊢ ((2 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (2↑(𝑘 + 1)) = ((2↑𝑘) · 2))
204	182, 203	mpan 703	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ0 → (2↑(𝑘 + 1)) = ((2↑𝑘) · 2))
205	202, 204	eqtr4d 2801	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ0 → (2 · (2↑𝑘)) = (2↑(𝑘 + 1)))
206	205	oveq2d 7456	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ0 → ((2 · 3) / (2 · (2↑𝑘))) = ((2 · 3) / (2↑(𝑘 + 1))))
207	199, 206	eqtr3d 2800	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ0 → (3 / (2↑𝑘)) = ((2 · 3) / (2↑(𝑘 + 1))))
208	207	oveq1d 7455	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ0 → ((3 / (2↑𝑘)) − (3 / (2↑(𝑘 + 1)))) = (((2 · 3) / (2↑(𝑘 + 1))) − (3 / (2↑(𝑘 + 1)))))
209		divcan5 12043	. . . . . . . . . . . . 13 ⊢ ((1 ∈ ℂ ∧ ((2↑𝑘) ∈ ℂ ∧ (2↑𝑘) ≠ 0) ∧ (2 ∈ ℂ ∧ 2 ≠ 0)) → ((2 · 1) / (2 · (2↑𝑘))) = (1 / (2↑𝑘)))
210	162, 195, 209	mp3an13 1482	. . . . . . . . . . . 12 ⊢ (((2↑𝑘) ∈ ℂ ∧ (2↑𝑘) ≠ 0) → ((2 · 1) / (2 · (2↑𝑘))) = (1 / (2↑𝑘)))
211	193, 194, 210	syl2anc 596	. . . . . . . . . . 11 ⊢ ((2↑𝑘) ∈ ℝ+ → ((2 · 1) / (2 · (2↑𝑘))) = (1 / (2↑𝑘)))
212	51, 211	syl 18	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ0 → ((2 · 1) / (2 · (2↑𝑘))) = (1 / (2↑𝑘)))
213		2t1e2 12529	. . . . . . . . . . . 12 ⊢ (2 · 1) = 2
214	213	a1i 11	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ0 → (2 · 1) = 2)
215	214, 205	oveq12d 7458	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ0 → ((2 · 1) / (2 · (2↑𝑘))) = (2 / (2↑(𝑘 + 1))))
216	212, 215	eqtr3d 2800	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ0 → (1 / (2↑𝑘)) = (2 / (2↑(𝑘 + 1))))
217	216	oveq1d 7455	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ0 → ((1 / (2↑𝑘)) + (1 / (2↑(𝑘 + 1)))) = ((2 / (2↑(𝑘 + 1))) + (1 / (2↑(𝑘 + 1)))))
218	192, 208, 217	3eqtr4d 2808	. . . . . . 7 ⊢ (𝑘 ∈ ℕ0 → ((3 / (2↑𝑘)) − (3 / (2↑(𝑘 + 1)))) = ((1 / (2↑𝑘)) + (1 / (2↑(𝑘 + 1)))))
219	218	adantl 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((3 / (2↑𝑘)) − (3 / (2↑(𝑘 + 1)))) = ((1 / (2↑𝑘)) + (1 / (2↑(𝑘 + 1)))))
220	141, 172, 219	3brtr4d 5140	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝑆‘(𝑘 + 1))𝐷(𝑆‘𝑘)) ≤ ((3 / (2↑𝑘)) − (3 / (2↑(𝑘 + 1)))))
221		blss2 24750	. . . . 5 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ (𝑆‘(𝑘 + 1)) ∈ 𝑋 ∧ (𝑆‘𝑘) ∈ 𝑋) ∧ ((3 / (2↑(𝑘 + 1))) ∈ ℝ ∧ (3 / (2↑𝑘)) ∈ ℝ ∧ ((𝑆‘(𝑘 + 1))𝐷(𝑆‘𝑘)) ≤ ((3 / (2↑𝑘)) − (3 / (2↑(𝑘 + 1)))))) → ((𝑆‘(𝑘 + 1))(ball‘𝐷)(3 / (2↑(𝑘 + 1)))) ⊆ ((𝑆‘𝑘)(ball‘𝐷)(3 / (2↑𝑘))))
222	7, 31, 40, 48, 54, 220, 221	syl33anc 1413	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝑆‘(𝑘 + 1))(ball‘𝐷)(3 / (2↑(𝑘 + 1)))) ⊆ ((𝑆‘𝑘)(ball‘𝐷)(3 / (2↑𝑘))))
223	1, 222	sylan2 605	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝑆‘(𝑘 + 1))(ball‘𝐷)(3 / (2↑(𝑘 + 1)))) ⊆ ((𝑆‘𝑘)(ball‘𝐷)(3 / (2↑𝑘))))
224		peano2nn 12371	. . . . . . 7 ⊢ (𝑘 ∈ ℕ → (𝑘 + 1) ∈ ℕ)
225		fveq2 6905	. . . . . . . . 9 ⊢ (𝑛 = (𝑘 + 1) → (𝑆‘𝑛) = (𝑆‘(𝑘 + 1)))
226		oveq2 7448	. . . . . . . . . 10 ⊢ (𝑛 = (𝑘 + 1) → (2↑𝑛) = (2↑(𝑘 + 1)))
227	226	oveq2d 7456	. . . . . . . . 9 ⊢ (𝑛 = (𝑘 + 1) → (3 / (2↑𝑛)) = (3 / (2↑(𝑘 + 1))))
228	225, 227	opeq12d 4851	. . . . . . . 8 ⊢ (𝑛 = (𝑘 + 1) → ⟨(𝑆‘𝑛), (3 / (2↑𝑛))⟩ = ⟨(𝑆‘(𝑘 + 1)), (3 / (2↑(𝑘 + 1)))⟩)
229		heibor.12	. . . . . . . 8 ⊢ 𝑀 = (𝑛 ∈ ℕ ↦ ⟨(𝑆‘𝑛), (3 / (2↑𝑛))⟩)
230		opex 5436	. . . . . . . 8 ⊢ ⟨(𝑆‘(𝑘 + 1)), (3 / (2↑(𝑘 + 1)))⟩ ∈ V
231	228, 229, 230	fvmpt 7013	. . . . . . 7 ⊢ ((𝑘 + 1) ∈ ℕ → (𝑀‘(𝑘 + 1)) = ⟨(𝑆‘(𝑘 + 1)), (3 / (2↑(𝑘 + 1)))⟩)
232	224, 231	syl 18	. . . . . 6 ⊢ (𝑘 ∈ ℕ → (𝑀‘(𝑘 + 1)) = ⟨(𝑆‘(𝑘 + 1)), (3 / (2↑(𝑘 + 1)))⟩)
233	232	adantl 487	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑀‘(𝑘 + 1)) = ⟨(𝑆‘(𝑘 + 1)), (3 / (2↑(𝑘 + 1)))⟩)
234	233	fveq2d 6909	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((ball‘𝐷)‘(𝑀‘(𝑘 + 1))) = ((ball‘𝐷)‘⟨(𝑆‘(𝑘 + 1)), (3 / (2↑(𝑘 + 1)))⟩))
235		df-ov 7443	. . . 4 ⊢ ((𝑆‘(𝑘 + 1))(ball‘𝐷)(3 / (2↑(𝑘 + 1)))) = ((ball‘𝐷)‘⟨(𝑆‘(𝑘 + 1)), (3 / (2↑(𝑘 + 1)))⟩)
236	234, 235	eqtr4di 2816	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((ball‘𝐷)‘(𝑀‘(𝑘 + 1))) = ((𝑆‘(𝑘 + 1))(ball‘𝐷)(3 / (2↑(𝑘 + 1)))))
237		fveq2 6905	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → (𝑆‘𝑛) = (𝑆‘𝑘))
238		oveq2 7448	. . . . . . . . 9 ⊢ (𝑛 = 𝑘 → (2↑𝑛) = (2↑𝑘))
239	238	oveq2d 7456	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → (3 / (2↑𝑛)) = (3 / (2↑𝑘)))
240	237, 239	opeq12d 4851	. . . . . . 7 ⊢ (𝑛 = 𝑘 → ⟨(𝑆‘𝑛), (3 / (2↑𝑛))⟩ = ⟨(𝑆‘𝑘), (3 / (2↑𝑘))⟩)
241		opex 5436	. . . . . . 7 ⊢ ⟨(𝑆‘𝑘), (3 / (2↑𝑘))⟩ ∈ V
242	240, 229, 241	fvmpt 7013	. . . . . 6 ⊢ (𝑘 ∈ ℕ → (𝑀‘𝑘) = ⟨(𝑆‘𝑘), (3 / (2↑𝑘))⟩)
243	242	fveq2d 6909	. . . . 5 ⊢ (𝑘 ∈ ℕ → ((ball‘𝐷)‘(𝑀‘𝑘)) = ((ball‘𝐷)‘⟨(𝑆‘𝑘), (3 / (2↑𝑘))⟩))
244		df-ov 7443	. . . . 5 ⊢ ((𝑆‘𝑘)(ball‘𝐷)(3 / (2↑𝑘))) = ((ball‘𝐷)‘⟨(𝑆‘𝑘), (3 / (2↑𝑘))⟩)
245	243, 244	eqtr4di 2816	. . . 4 ⊢ (𝑘 ∈ ℕ → ((ball‘𝐷)‘(𝑀‘𝑘)) = ((𝑆‘𝑘)(ball‘𝐷)(3 / (2↑𝑘))))
246	245	adantl 487	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((ball‘𝐷)‘(𝑀‘𝑘)) = ((𝑆‘𝑘)(ball‘𝐷)(3 / (2↑𝑘))))
247	223, 236, 246	3sstr4d 3968	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((ball‘𝐷)‘(𝑀‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝑀‘𝑘)))
248	247	ralrimiva 3158	1 ⊢ (𝜑 → ∀𝑘 ∈ ℕ ((ball‘𝐷)‘(𝑀‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝑀‘𝑘)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  {cab 2741   ≠ wne 2959  ∀wral 3080  ∃wrex 3090  Vcvv 3454   ⊆ wss 3900   ∩ cin 4092  ∅c0 4281  ifcif 4485  𝒫 cpw 4560  ⟨cop 4838  ∪ cuni 4870  ∪ ciun 4954   class class class wbr 5106  {copab 5170   ↦ cmpt 5189  ⟶wf 6551  ‘cfv 6555  (class class class)co 7440   ∈ cmpo 7442  2^nd c2nd 8021  Fincfn 8998  ℂcc 11222  ℝcr 11223  0cc0 11224  1c1 11225   + caddc 11227   · cmul 11229  ℝ^*cxr 11366   ≤ cle 11368   − cmin 11565   / cdiv 11997  ℕcn 12359  2c2 12421  3c3 12422  ℕ₀cn0 12630  ℤ_≥cuz 12989  ℝ⁺crp 13144   +_e cxad 13263  seqcseq 14168  ↑cexp 14228  ∞Metcxmet 21690  Metcmet 21691  ballcbl 21692  MetOpencmopn 21695  CMetccmet 25602

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772  ax-cnex 11280  ax-resscn 11281  ax-1cn 11282  ax-icn 11283  ax-addcl 11284  ax-addrcl 11285  ax-mulcl 11286  ax-mulrcl 11287  ax-mulcom 11288  ax-addass 11289  ax-mulass 11290  ax-distr 11291  ax-i2m1 11292  ax-1ne0 11293  ax-1rid 11294  ax-rnegex 11295  ax-rrecex 11296  ax-cnre 11297  ax-pre-lttri 11298  ax-pre-lttrn 11299  ax-pre-ltadd 11300  ax-pre-mulgt0 11301

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6319  df-ord 6380  df-on 6381  df-lim 6382  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-riota 7397  df-ov 7443  df-oprab 7444  df-mpo 7445  df-om 7899  df-1st 8022  df-2nd 8023  df-frecs 8324  df-wrecs 8355  df-recs 8404  df-rdg 8443  df-er 8742  df-map 8874  df-en 8999  df-dom 9000  df-sdom 9001  df-fin 9002  df-pnf 11369  df-mnf 11370  df-xr 11371  df-ltxr 11372  df-le 11373  df-sub 11567  df-neg 11568  df-div 11998  df-nn 12360  df-2 12429  df-3 12430  df-n0 12631  df-z 12718  df-uz 12990  df-rp 13145  df-xneg 13265  df-xadd 13266  df-xmul 13267  df-seq 14169  df-exp 14229  df-psmet 21697  df-xmet 21698  df-met 21699  df-bl 21700  df-cmet 25605

This theorem is used by:  heiborlem8  38943  heiborlem9  38944