Theorem heiborlem6 36348

Description: Lemma for heibor 36353. 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 12429	. . . 4 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ0)
2		heibor.6	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋))
3		cmetmet 24687	. . . . . . . 8 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
4	2, 3	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋))
5		metxmet 23724	. . . . . . 7 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
6	4, 5	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋))
7	6	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝐷 ∈ (∞Met‘𝑋))
8		heibor.7	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin))
9		inss1 4193	. . . . . . . . 9 ⊢ (𝒫 𝑋 ∩ Fin) ⊆ 𝒫 𝑋
10		fss 6690	. . . . . . . . 9 ⊢ ((𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin) ∧ (𝒫 𝑋 ∩ Fin) ⊆ 𝒫 𝑋) → 𝐹:ℕ0⟶𝒫 𝑋)
11	8, 9, 10	sylancl 586	. . . . . . . 8 ⊢ (𝜑 → 𝐹:ℕ0⟶𝒫 𝑋)
12		peano2nn0 12462	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ0 → (𝑘 + 1) ∈ ℕ0)
13		ffvelcdm 7037	. . . . . . . 8 ⊢ ((𝐹:ℕ0⟶𝒫 𝑋 ∧ (𝑘 + 1) ∈ ℕ0) → (𝐹‘(𝑘 + 1)) ∈ 𝒫 𝑋)
14	11, 12, 13	syl2an 596	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘(𝑘 + 1)) ∈ 𝒫 𝑋)
15	14	elpwid 4574	. . . . . 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 36346	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 + 1) ∈ ℕ0) → (𝑆‘(𝑘 + 1))𝐺(𝑘 + 1))
25	12, 24	sylan2 593	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑆‘(𝑘 + 1))𝐺(𝑘 + 1))
26		fvex 6860	. . . . . . . . 9 ⊢ (𝑆‘(𝑘 + 1)) ∈ V
27		ovex 7395	. . . . . . . . 9 ⊢ (𝑘 + 1) ∈ V
28	16, 17, 18, 26, 27	heiborlem2 36344	. . . . . . . 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 1146	. . . . . . 7 ⊢ ((𝑆‘(𝑘 + 1))𝐺(𝑘 + 1) → (𝑆‘(𝑘 + 1)) ∈ (𝐹‘(𝑘 + 1)))
30	25, 29	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑆‘(𝑘 + 1)) ∈ (𝐹‘(𝑘 + 1)))
31	15, 30	sseldd 3948	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑆‘(𝑘 + 1)) ∈ 𝑋)
32	11	ffvelcdmda 7040	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘𝑘) ∈ 𝒫 𝑋)
33	32	elpwid 4574	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘𝑘) ⊆ 𝑋)
34	16, 17, 18, 19, 2, 8, 20, 21, 22, 23	heiborlem4 36346	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑆‘𝑘)𝐺𝑘)
35		fvex 6860	. . . . . . . . 9 ⊢ (𝑆‘𝑘) ∈ V
36		vex 3450	. . . . . . . . 9 ⊢ 𝑘 ∈ V
37	16, 17, 18, 35, 36	heiborlem2 36344	. . . . . . . 8 ⊢ ((𝑆‘𝑘)𝐺𝑘 ↔ (𝑘 ∈ ℕ0 ∧ (𝑆‘𝑘) ∈ (𝐹‘𝑘) ∧ ((𝑆‘𝑘)𝐵𝑘) ∈ 𝐾))
38	37	simp2bi 1146	. . . . . . 7 ⊢ ((𝑆‘𝑘)𝐺𝑘 → (𝑆‘𝑘) ∈ (𝐹‘𝑘))
39	34, 38	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑆‘𝑘) ∈ (𝐹‘𝑘))
40	33, 39	sseldd 3948	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑆‘𝑘) ∈ 𝑋)
41		3re 12242	. . . . . 6 ⊢ 3 ∈ ℝ
42		2nn 12235	. . . . . . . . 9 ⊢ 2 ∈ ℕ
43		nnexpcl 13990	. . . . . . . . 9 ⊢ ((2 ∈ ℕ ∧ (𝑘 + 1) ∈ ℕ0) → (2↑(𝑘 + 1)) ∈ ℕ)
44	42, 12, 43	sylancr 587	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ0 → (2↑(𝑘 + 1)) ∈ ℕ)
45	44	nnrpd 12964	. . . . . . 7 ⊢ (𝑘 ∈ ℕ0 → (2↑(𝑘 + 1)) ∈ ℝ+)
46	45	adantl 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (2↑(𝑘 + 1)) ∈ ℝ+)
47		rerpdivcl 12954	. . . . . 6 ⊢ ((3 ∈ ℝ ∧ (2↑(𝑘 + 1)) ∈ ℝ+) → (3 / (2↑(𝑘 + 1))) ∈ ℝ)
48	41, 46, 47	sylancr 587	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (3 / (2↑(𝑘 + 1))) ∈ ℝ)
49		nnexpcl 13990	. . . . . . . . 9 ⊢ ((2 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → (2↑𝑘) ∈ ℕ)
50	42, 49	mpan 688	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ0 → (2↑𝑘) ∈ ℕ)
51	50	nnrpd 12964	. . . . . . 7 ⊢ (𝑘 ∈ ℕ0 → (2↑𝑘) ∈ ℝ+)
52	51	adantl 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (2↑𝑘) ∈ ℝ+)
53		rerpdivcl 12954	. . . . . 6 ⊢ ((3 ∈ ℝ ∧ (2↑𝑘) ∈ ℝ+) → (3 / (2↑𝑘)) ∈ ℝ)
54	41, 52, 53	sylancr 587	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (3 / (2↑𝑘)) ∈ ℝ)
55		oveq1 7369	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝑆‘𝑘) → (𝑧(ball‘𝐷)(1 / (2↑𝑚))) = ((𝑆‘𝑘)(ball‘𝐷)(1 / (2↑𝑚))))
56		oveq2 7370	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑘 → (2↑𝑚) = (2↑𝑘))
57	56	oveq2d 7378	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑘 → (1 / (2↑𝑚)) = (1 / (2↑𝑘)))
58	57	oveq2d 7378	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑘 → ((𝑆‘𝑘)(ball‘𝐷)(1 / (2↑𝑚))) = ((𝑆‘𝑘)(ball‘𝐷)(1 / (2↑𝑘))))
59		ovex 7395	. . . . . . . . . . . 12 ⊢ ((𝑆‘𝑘)(ball‘𝐷)(1 / (2↑𝑘))) ∈ V
60	55, 58, 19, 59	ovmpo 7520	. . . . . . . . . . 11 ⊢ (((𝑆‘𝑘) ∈ 𝑋 ∧ 𝑘 ∈ ℕ0) → ((𝑆‘𝑘)𝐵𝑘) = ((𝑆‘𝑘)(ball‘𝐷)(1 / (2↑𝑘))))
61	40, 60	sylancom 588	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝑆‘𝑘)𝐵𝑘) = ((𝑆‘𝑘)(ball‘𝐷)(1 / (2↑𝑘))))
62		df-br 5111	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑆‘𝑘)𝐺𝑘 ↔ ⟨(𝑆‘𝑘), 𝑘⟩ ∈ 𝐺)
63		fveq2 6847	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = ⟨(𝑆‘𝑘), 𝑘⟩ → (𝑇‘𝑥) = (𝑇‘⟨(𝑆‘𝑘), 𝑘⟩))
64		df-ov 7365	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑆‘𝑘)𝑇𝑘) = (𝑇‘⟨(𝑆‘𝑘), 𝑘⟩)
65	63, 64	eqtr4di 2789	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = ⟨(𝑆‘𝑘), 𝑘⟩ → (𝑇‘𝑥) = ((𝑆‘𝑘)𝑇𝑘))
66	35, 36	op2ndd 7937	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = ⟨(𝑆‘𝑘), 𝑘⟩ → (2nd ‘𝑥) = 𝑘)
67	66	oveq1d 7377	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = ⟨(𝑆‘𝑘), 𝑘⟩ → ((2nd ‘𝑥) + 1) = (𝑘 + 1))
68	65, 67	breq12d 5123	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = ⟨(𝑆‘𝑘), 𝑘⟩ → ((𝑇‘𝑥)𝐺((2nd ‘𝑥) + 1) ↔ ((𝑆‘𝑘)𝑇𝑘)𝐺(𝑘 + 1)))
69		fveq2 6847	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = ⟨(𝑆‘𝑘), 𝑘⟩ → (𝐵‘𝑥) = (𝐵‘⟨(𝑆‘𝑘), 𝑘⟩))
70		df-ov 7365	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑆‘𝑘)𝐵𝑘) = (𝐵‘⟨(𝑆‘𝑘), 𝑘⟩)
71	69, 70	eqtr4di 2789	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = ⟨(𝑆‘𝑘), 𝑘⟩ → (𝐵‘𝑥) = ((𝑆‘𝑘)𝐵𝑘))
72	65, 67	oveq12d 7380	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = ⟨(𝑆‘𝑘), 𝑘⟩ → ((𝑇‘𝑥)𝐵((2nd ‘𝑥) + 1)) = (((𝑆‘𝑘)𝑇𝑘)𝐵(𝑘 + 1)))
73	71, 72	ineq12d 4178	. . . . . . . . . . . . . . . . . . . . 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 2817	. . . . . . . . . . . . . . . . . . . 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 631	. . . . . . . . . . . . . . . . . . 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 3579	. . . . . . . . . . . . . . . . . 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 17	. . . . . . . . . . . . . . . . 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 241	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝑆‘𝑘)𝐺𝑘 → (((𝑆‘𝑘)𝑇𝑘)𝐺(𝑘 + 1) ∧ (((𝑆‘𝑘)𝐵𝑘) ∩ (((𝑆‘𝑘)𝑇𝑘)𝐵(𝑘 + 1))) ∈ 𝐾)))
79	78	adantr 481	. . . . . . . . . . . . . . 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 15	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (((𝑆‘𝑘)𝑇𝑘)𝐺(𝑘 + 1) ∧ (((𝑆‘𝑘)𝐵𝑘) ∩ (((𝑆‘𝑘)𝑇𝑘)𝐵(𝑘 + 1))) ∈ 𝐾))
81	80	simpld 495	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝑆‘𝑘)𝑇𝑘)𝐺(𝑘 + 1))
82		ovex 7395	. . . . . . . . . . . . . . 15 ⊢ ((𝑆‘𝑘)𝑇𝑘) ∈ V
83	16, 17, 18, 82, 27	heiborlem2 36344	. . . . . . . . . . . . . 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 1146	. . . . . . . . . . . . 13 ⊢ (((𝑆‘𝑘)𝑇𝑘)𝐺(𝑘 + 1) → ((𝑆‘𝑘)𝑇𝑘) ∈ (𝐹‘(𝑘 + 1)))
85	81, 84	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝑆‘𝑘)𝑇𝑘) ∈ (𝐹‘(𝑘 + 1)))
86	15, 85	sseldd 3948	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝑆‘𝑘)𝑇𝑘) ∈ 𝑋)
87	12	adantl 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑘 + 1) ∈ ℕ0)
88		oveq1 7369	. . . . . . . . . . . 12 ⊢ (𝑧 = ((𝑆‘𝑘)𝑇𝑘) → (𝑧(ball‘𝐷)(1 / (2↑𝑚))) = (((𝑆‘𝑘)𝑇𝑘)(ball‘𝐷)(1 / (2↑𝑚))))
89		oveq2 7370	. . . . . . . . . . . . . 14 ⊢ (𝑚 = (𝑘 + 1) → (2↑𝑚) = (2↑(𝑘 + 1)))
90	89	oveq2d 7378	. . . . . . . . . . . . 13 ⊢ (𝑚 = (𝑘 + 1) → (1 / (2↑𝑚)) = (1 / (2↑(𝑘 + 1))))
91	90	oveq2d 7378	. . . . . . . . . . . 12 ⊢ (𝑚 = (𝑘 + 1) → (((𝑆‘𝑘)𝑇𝑘)(ball‘𝐷)(1 / (2↑𝑚))) = (((𝑆‘𝑘)𝑇𝑘)(ball‘𝐷)(1 / (2↑(𝑘 + 1)))))
92		ovex 7395	. . . . . . . . . . . 12 ⊢ (((𝑆‘𝑘)𝑇𝑘)(ball‘𝐷)(1 / (2↑(𝑘 + 1)))) ∈ V
93	88, 91, 19, 92	ovmpo 7520	. . . . . . . . . . 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 584	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (((𝑆‘𝑘)𝑇𝑘)𝐵(𝑘 + 1)) = (((𝑆‘𝑘)𝑇𝑘)(ball‘𝐷)(1 / (2↑(𝑘 + 1)))))
95	61, 94	ineq12d 4178	. . . . . . . . 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 496	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (((𝑆‘𝑘)𝐵𝑘) ∩ (((𝑆‘𝑘)𝑇𝑘)𝐵(𝑘 + 1))) ∈ 𝐾)
97		0elpw 5316	. . . . . . . . . . . . 13 ⊢ ∅ ∈ 𝒫 𝑈
98		0fin 9122	. . . . . . . . . . . . 13 ⊢ ∅ ∈ Fin
99		elin 3929	. . . . . . . . . . . . 13 ⊢ (∅ ∈ (𝒫 𝑈 ∩ Fin) ↔ (∅ ∈ 𝒫 𝑈 ∧ ∅ ∈ Fin))
100	97, 98, 99	mpbir2an 709	. . . . . . . . . . . 12 ⊢ ∅ ∈ (𝒫 𝑈 ∩ Fin)
101		0ss 4361	. . . . . . . . . . . 12 ⊢ ∅ ⊆ ∪ ∅
102		unieq 4881	. . . . . . . . . . . . . 14 ⊢ (𝑣 = ∅ → ∪ 𝑣 = ∪ ∅)
103	102	sseq2d 3979	. . . . . . . . . . . . 13 ⊢ (𝑣 = ∅ → (∅ ⊆ ∪ 𝑣 ↔ ∅ ⊆ ∪ ∅))
104	103	rspcev 3582	. . . . . . . . . . . 12 ⊢ ((∅ ∈ (𝒫 𝑈 ∩ Fin) ∧ ∅ ⊆ ∪ ∅) → ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)∅ ⊆ ∪ 𝑣)
105	100, 101, 104	mp2an 690	. . . . . . . . . . 11 ⊢ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)∅ ⊆ ∪ 𝑣
106		0ex 5269	. . . . . . . . . . . . 13 ⊢ ∅ ∈ V
107		sseq1 3972	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = ∅ → (𝑢 ⊆ ∪ 𝑣 ↔ ∅ ⊆ ∪ 𝑣))
108	107	rexbidv 3171	. . . . . . . . . . . . . 14 ⊢ (𝑢 = ∅ → (∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 ⊆ ∪ 𝑣 ↔ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)∅ ⊆ ∪ 𝑣))
109	108	notbid 317	. . . . . . . . . . . . 13 ⊢ (𝑢 = ∅ → (¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 ⊆ ∪ 𝑣 ↔ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)∅ ⊆ ∪ 𝑣))
110	106, 109, 17	elab2 3637	. . . . . . . . . . . 12 ⊢ (∅ ∈ 𝐾 ↔ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)∅ ⊆ ∪ 𝑣)
111	110	con2bii 357	. . . . . . . . . . 11 ⊢ (∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)∅ ⊆ ∪ 𝑣 ↔ ¬ ∅ ∈ 𝐾)
112	105, 111	mpbi 229	. . . . . . . . . 10 ⊢ ¬ ∅ ∈ 𝐾
113		nelne2 3039	. . . . . . . . . 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 586	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (((𝑆‘𝑘)𝐵𝑘) ∩ (((𝑆‘𝑘)𝑇𝑘)𝐵(𝑘 + 1))) ≠ ∅)
115	95, 114	eqnetrrd 3008	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (((𝑆‘𝑘)(ball‘𝐷)(1 / (2↑𝑘))) ∩ (((𝑆‘𝑘)𝑇𝑘)(ball‘𝐷)(1 / (2↑(𝑘 + 1))))) ≠ ∅)
116	51	rpreccld 12976	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ0 → (1 / (2↑𝑘)) ∈ ℝ+)
117	116	adantl 482	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (1 / (2↑𝑘)) ∈ ℝ+)
118	117	rpred 12966	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (1 / (2↑𝑘)) ∈ ℝ)
119	45	rpreccld 12976	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ0 → (1 / (2↑(𝑘 + 1))) ∈ ℝ+)
120	119	adantl 482	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (1 / (2↑(𝑘 + 1))) ∈ ℝ+)
121	120	rpred 12966	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (1 / (2↑(𝑘 + 1))) ∈ ℝ)
122		rexadd 13161	. . . . . . . . . . . 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 584	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((1 / (2↑𝑘)) +𝑒 (1 / (2↑(𝑘 + 1)))) = ((1 / (2↑𝑘)) + (1 / (2↑(𝑘 + 1)))))
124	123	breq1d 5120	. . . . . . . . . 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 12967	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (1 / (2↑𝑘)) ∈ ℝ*)
126	120	rpxrd 12967	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (1 / (2↑(𝑘 + 1))) ∈ ℝ*)
127		bldisj 23788	. . . . . . . . . . . . 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 1354	. . . . . . . . . . . 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 419	. . . . . . . . . . 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 1378	. . . . . . . . . 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 259	. . . . . . . . 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 2952	. . . . . . . 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 15	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ¬ ((1 / (2↑𝑘)) + (1 / (2↑(𝑘 + 1)))) ≤ ((𝑆‘𝑘)𝐷((𝑆‘𝑘)𝑇𝑘)))
134	117, 120	rpaddcld 12981	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((1 / (2↑𝑘)) + (1 / (2↑(𝑘 + 1)))) ∈ ℝ+)
135	134	rpred 12966	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((1 / (2↑𝑘)) + (1 / (2↑(𝑘 + 1)))) ∈ ℝ)
136	4	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝐷 ∈ (Met‘𝑋))
137		metcl 23722	. . . . . . . . . 10 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ (𝑆‘𝑘) ∈ 𝑋 ∧ ((𝑆‘𝑘)𝑇𝑘) ∈ 𝑋) → ((𝑆‘𝑘)𝐷((𝑆‘𝑘)𝑇𝑘)) ∈ ℝ)
138	136, 40, 86, 137	syl3anc 1371	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝑆‘𝑘)𝐷((𝑆‘𝑘)𝑇𝑘)) ∈ ℝ)
139	135, 138	letrid 11316	. . . . . . . 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 862	. . . . . . 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 15	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝑆‘𝑘)𝐷((𝑆‘𝑘)𝑇𝑘)) ≤ ((1 / (2↑𝑘)) + (1 / (2↑(𝑘 + 1)))))
142		seqp1 13931	. . . . . . . . . . . 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 12814	. . . . . . . . . . . 12 ⊢ ℕ0 = (ℤ≥‘0)
144	142, 143	eleq2s 2850	. . . . . . . . . . 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 6848	. . . . . . . . . . 11 ⊢ (𝑆‘(𝑘 + 1)) = (seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))))‘(𝑘 + 1))
146	23	fveq1i 6848	. . . . . . . . . . . 12 ⊢ (𝑆‘𝑘) = (seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))))‘𝑘)
147	146	oveq1i 7372	. . . . . . . . . . 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 2796	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ0 → (𝑆‘(𝑘 + 1)) = ((𝑆‘𝑘)𝑇((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))‘(𝑘 + 1))))
149		eqid 2731	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))) = (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))
150		eqeq1 2735	. . . . . . . . . . . . . 14 ⊢ (𝑚 = (𝑘 + 1) → (𝑚 = 0 ↔ (𝑘 + 1) = 0))
151		oveq1 7369	. . . . . . . . . . . . . 14 ⊢ (𝑚 = (𝑘 + 1) → (𝑚 − 1) = ((𝑘 + 1) − 1))
152	150, 151	ifbieq2d 4517	. . . . . . . . . . . . 13 ⊢ (𝑚 = (𝑘 + 1) → if(𝑚 = 0, 𝐶, (𝑚 − 1)) = if((𝑘 + 1) = 0, 𝐶, ((𝑘 + 1) − 1)))
153		nn0p1nn 12461	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ ℕ0 → (𝑘 + 1) ∈ ℕ)
154		nnne0 12196	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘 + 1) ∈ ℕ → (𝑘 + 1) ≠ 0)
155	154	neneqd 2944	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 + 1) ∈ ℕ → ¬ (𝑘 + 1) = 0)
156	153, 155	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℕ0 → ¬ (𝑘 + 1) = 0)
157	156	iffalsed 4502	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ0 → if((𝑘 + 1) = 0, 𝐶, ((𝑘 + 1) − 1)) = ((𝑘 + 1) − 1))
158		ovex 7395	. . . . . . . . . . . . . 14 ⊢ ((𝑘 + 1) − 1) ∈ V
159	157, 158	eqeltrdi 2840	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ0 → if((𝑘 + 1) = 0, 𝐶, ((𝑘 + 1) − 1)) ∈ V)
160	149, 152, 12, 159	fvmptd3 6976	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ0 → ((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))‘(𝑘 + 1)) = if((𝑘 + 1) = 0, 𝐶, ((𝑘 + 1) − 1)))
161		nn0cn 12432	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ0 → 𝑘 ∈ ℂ)
162		ax-1cn 11118	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℂ
163		pncan 11416	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑘 + 1) − 1) = 𝑘)
164	161, 162, 163	sylancl 586	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ0 → ((𝑘 + 1) − 1) = 𝑘)
165	160, 157, 164	3eqtrd 2775	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ0 → ((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))‘(𝑘 + 1)) = 𝑘)
166	165	oveq2d 7378	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ0 → ((𝑆‘𝑘)𝑇((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))‘(𝑘 + 1))) = ((𝑆‘𝑘)𝑇𝑘))
167	148, 166	eqtrd 2771	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ0 → (𝑆‘(𝑘 + 1)) = ((𝑆‘𝑘)𝑇𝑘))
168	167	adantl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑆‘(𝑘 + 1)) = ((𝑆‘𝑘)𝑇𝑘))
169	168	oveq1d 7377	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝑆‘(𝑘 + 1))𝐷(𝑆‘𝑘)) = (((𝑆‘𝑘)𝑇𝑘)𝐷(𝑆‘𝑘)))
170		metsym 23740	. . . . . . . 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 1371	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (((𝑆‘𝑘)𝑇𝑘)𝐷(𝑆‘𝑘)) = ((𝑆‘𝑘)𝐷((𝑆‘𝑘)𝑇𝑘)))
172	169, 171	eqtrd 2771	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝑆‘(𝑘 + 1))𝐷(𝑆‘𝑘)) = ((𝑆‘𝑘)𝐷((𝑆‘𝑘)𝑇𝑘)))
173		3cn 12243	. . . . . . . . . . . . 13 ⊢ 3 ∈ ℂ
174	173	2timesi 12300	. . . . . . . . . . . 12 ⊢ (2 · 3) = (3 + 3)
175	174	oveq1i 7372	. . . . . . . . . . 11 ⊢ ((2 · 3) − 3) = ((3 + 3) − 3)
176	173, 173	pncan3oi 11426	. . . . . . . . . . 11 ⊢ ((3 + 3) − 3) = 3
177		df-3 12226	. . . . . . . . . . 11 ⊢ 3 = (2 + 1)
178	175, 176, 177	3eqtri 2763	. . . . . . . . . 10 ⊢ ((2 · 3) − 3) = (2 + 1)
179	178	oveq1i 7372	. . . . . . . . 9 ⊢ (((2 · 3) − 3) / (2↑(𝑘 + 1))) = ((2 + 1) / (2↑(𝑘 + 1)))
180		rpcn 12934	. . . . . . . . . . 11 ⊢ ((2↑(𝑘 + 1)) ∈ ℝ+ → (2↑(𝑘 + 1)) ∈ ℂ)
181		rpne0 12940	. . . . . . . . . . 11 ⊢ ((2↑(𝑘 + 1)) ∈ ℝ+ → (2↑(𝑘 + 1)) ≠ 0)
182		2cn 12237	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℂ
183	182, 173	mulcli 11171	. . . . . . . . . . . 12 ⊢ (2 · 3) ∈ ℂ
184		divsubdir 11858	. . . . . . . . . . . 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 1451	. . . . . . . . . . 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 584	. . . . . . . . . 10 ⊢ ((2↑(𝑘 + 1)) ∈ ℝ+ → (((2 · 3) − 3) / (2↑(𝑘 + 1))) = (((2 · 3) / (2↑(𝑘 + 1))) − (3 / (2↑(𝑘 + 1)))))
187	45, 186	syl 17	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ0 → (((2 · 3) − 3) / (2↑(𝑘 + 1))) = (((2 · 3) / (2↑(𝑘 + 1))) − (3 / (2↑(𝑘 + 1)))))
188		divdir 11847	. . . . . . . . . . . 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 1451	. . . . . . . . . . 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 584	. . . . . . . . . 10 ⊢ ((2↑(𝑘 + 1)) ∈ ℝ+ → ((2 + 1) / (2↑(𝑘 + 1))) = ((2 / (2↑(𝑘 + 1))) + (1 / (2↑(𝑘 + 1)))))
191	45, 190	syl 17	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ0 → ((2 + 1) / (2↑(𝑘 + 1))) = ((2 / (2↑(𝑘 + 1))) + (1 / (2↑(𝑘 + 1)))))
192	179, 187, 191	3eqtr3a 2795	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ0 → (((2 · 3) / (2↑(𝑘 + 1))) − (3 / (2↑(𝑘 + 1)))) = ((2 / (2↑(𝑘 + 1))) + (1 / (2↑(𝑘 + 1)))))
193		rpcn 12934	. . . . . . . . . . . 12 ⊢ ((2↑𝑘) ∈ ℝ+ → (2↑𝑘) ∈ ℂ)
194		rpne0 12940	. . . . . . . . . . . 12 ⊢ ((2↑𝑘) ∈ ℝ+ → (2↑𝑘) ≠ 0)
195		2cnne0 12372	. . . . . . . . . . . . 13 ⊢ (2 ∈ ℂ ∧ 2 ≠ 0)
196		divcan5 11866	. . . . . . . . . . . . 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 1452	. . . . . . . . . . . 12 ⊢ (((2↑𝑘) ∈ ℂ ∧ (2↑𝑘) ≠ 0) → ((2 · 3) / (2 · (2↑𝑘))) = (3 / (2↑𝑘)))
198	193, 194, 197	syl2anc 584	. . . . . . . . . . 11 ⊢ ((2↑𝑘) ∈ ℝ+ → ((2 · 3) / (2 · (2↑𝑘))) = (3 / (2↑𝑘)))
199	51, 198	syl 17	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ0 → ((2 · 3) / (2 · (2↑𝑘))) = (3 / (2↑𝑘)))
200	51, 193	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ0 → (2↑𝑘) ∈ ℂ)
201		mulcom 11146	. . . . . . . . . . . . 13 ⊢ ((2 ∈ ℂ ∧ (2↑𝑘) ∈ ℂ) → (2 · (2↑𝑘)) = ((2↑𝑘) · 2))
202	182, 200, 201	sylancr 587	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ0 → (2 · (2↑𝑘)) = ((2↑𝑘) · 2))
203		expp1 13984	. . . . . . . . . . . . 13 ⊢ ((2 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (2↑(𝑘 + 1)) = ((2↑𝑘) · 2))
204	182, 203	mpan 688	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ0 → (2↑(𝑘 + 1)) = ((2↑𝑘) · 2))
205	202, 204	eqtr4d 2774	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ0 → (2 · (2↑𝑘)) = (2↑(𝑘 + 1)))
206	205	oveq2d 7378	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ0 → ((2 · 3) / (2 · (2↑𝑘))) = ((2 · 3) / (2↑(𝑘 + 1))))
207	199, 206	eqtr3d 2773	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ0 → (3 / (2↑𝑘)) = ((2 · 3) / (2↑(𝑘 + 1))))
208	207	oveq1d 7377	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ0 → ((3 / (2↑𝑘)) − (3 / (2↑(𝑘 + 1)))) = (((2 · 3) / (2↑(𝑘 + 1))) − (3 / (2↑(𝑘 + 1)))))
209		divcan5 11866	. . . . . . . . . . . . 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 1452	. . . . . . . . . . . 12 ⊢ (((2↑𝑘) ∈ ℂ ∧ (2↑𝑘) ≠ 0) → ((2 · 1) / (2 · (2↑𝑘))) = (1 / (2↑𝑘)))
211	193, 194, 210	syl2anc 584	. . . . . . . . . . 11 ⊢ ((2↑𝑘) ∈ ℝ+ → ((2 · 1) / (2 · (2↑𝑘))) = (1 / (2↑𝑘)))
212	51, 211	syl 17	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ0 → ((2 · 1) / (2 · (2↑𝑘))) = (1 / (2↑𝑘)))
213		2t1e2 12325	. . . . . . . . . . . 12 ⊢ (2 · 1) = 2
214	213	a1i 11	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ0 → (2 · 1) = 2)
215	214, 205	oveq12d 7380	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ0 → ((2 · 1) / (2 · (2↑𝑘))) = (2 / (2↑(𝑘 + 1))))
216	212, 215	eqtr3d 2773	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ0 → (1 / (2↑𝑘)) = (2 / (2↑(𝑘 + 1))))
217	216	oveq1d 7377	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ0 → ((1 / (2↑𝑘)) + (1 / (2↑(𝑘 + 1)))) = ((2 / (2↑(𝑘 + 1))) + (1 / (2↑(𝑘 + 1)))))
218	192, 208, 217	3eqtr4d 2781	. . . . . . 7 ⊢ (𝑘 ∈ ℕ0 → ((3 / (2↑𝑘)) − (3 / (2↑(𝑘 + 1)))) = ((1 / (2↑𝑘)) + (1 / (2↑(𝑘 + 1)))))
219	218	adantl 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((3 / (2↑𝑘)) − (3 / (2↑(𝑘 + 1)))) = ((1 / (2↑𝑘)) + (1 / (2↑(𝑘 + 1)))))
220	141, 172, 219	3brtr4d 5142	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝑆‘(𝑘 + 1))𝐷(𝑆‘𝑘)) ≤ ((3 / (2↑𝑘)) − (3 / (2↑(𝑘 + 1)))))
221		blss2 23794	. . . . 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 1385	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝑆‘(𝑘 + 1))(ball‘𝐷)(3 / (2↑(𝑘 + 1)))) ⊆ ((𝑆‘𝑘)(ball‘𝐷)(3 / (2↑𝑘))))
223	1, 222	sylan2 593	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝑆‘(𝑘 + 1))(ball‘𝐷)(3 / (2↑(𝑘 + 1)))) ⊆ ((𝑆‘𝑘)(ball‘𝐷)(3 / (2↑𝑘))))
224		peano2nn 12174	. . . . . . 7 ⊢ (𝑘 ∈ ℕ → (𝑘 + 1) ∈ ℕ)
225		fveq2 6847	. . . . . . . . 9 ⊢ (𝑛 = (𝑘 + 1) → (𝑆‘𝑛) = (𝑆‘(𝑘 + 1)))
226		oveq2 7370	. . . . . . . . . 10 ⊢ (𝑛 = (𝑘 + 1) → (2↑𝑛) = (2↑(𝑘 + 1)))
227	226	oveq2d 7378	. . . . . . . . 9 ⊢ (𝑛 = (𝑘 + 1) → (3 / (2↑𝑛)) = (3 / (2↑(𝑘 + 1))))
228	225, 227	opeq12d 4843	. . . . . . . 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 5426	. . . . . . . 8 ⊢ ⟨(𝑆‘(𝑘 + 1)), (3 / (2↑(𝑘 + 1)))⟩ ∈ V
231	228, 229, 230	fvmpt 6953	. . . . . . 7 ⊢ ((𝑘 + 1) ∈ ℕ → (𝑀‘(𝑘 + 1)) = ⟨(𝑆‘(𝑘 + 1)), (3 / (2↑(𝑘 + 1)))⟩)
232	224, 231	syl 17	. . . . . 6 ⊢ (𝑘 ∈ ℕ → (𝑀‘(𝑘 + 1)) = ⟨(𝑆‘(𝑘 + 1)), (3 / (2↑(𝑘 + 1)))⟩)
233	232	adantl 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑀‘(𝑘 + 1)) = ⟨(𝑆‘(𝑘 + 1)), (3 / (2↑(𝑘 + 1)))⟩)
234	233	fveq2d 6851	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((ball‘𝐷)‘(𝑀‘(𝑘 + 1))) = ((ball‘𝐷)‘⟨(𝑆‘(𝑘 + 1)), (3 / (2↑(𝑘 + 1)))⟩))
235		df-ov 7365	. . . 4 ⊢ ((𝑆‘(𝑘 + 1))(ball‘𝐷)(3 / (2↑(𝑘 + 1)))) = ((ball‘𝐷)‘⟨(𝑆‘(𝑘 + 1)), (3 / (2↑(𝑘 + 1)))⟩)
236	234, 235	eqtr4di 2789	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((ball‘𝐷)‘(𝑀‘(𝑘 + 1))) = ((𝑆‘(𝑘 + 1))(ball‘𝐷)(3 / (2↑(𝑘 + 1)))))
237		fveq2 6847	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → (𝑆‘𝑛) = (𝑆‘𝑘))
238		oveq2 7370	. . . . . . . . 9 ⊢ (𝑛 = 𝑘 → (2↑𝑛) = (2↑𝑘))
239	238	oveq2d 7378	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → (3 / (2↑𝑛)) = (3 / (2↑𝑘)))
240	237, 239	opeq12d 4843	. . . . . . 7 ⊢ (𝑛 = 𝑘 → ⟨(𝑆‘𝑛), (3 / (2↑𝑛))⟩ = ⟨(𝑆‘𝑘), (3 / (2↑𝑘))⟩)
241		opex 5426	. . . . . . 7 ⊢ ⟨(𝑆‘𝑘), (3 / (2↑𝑘))⟩ ∈ V
242	240, 229, 241	fvmpt 6953	. . . . . 6 ⊢ (𝑘 ∈ ℕ → (𝑀‘𝑘) = ⟨(𝑆‘𝑘), (3 / (2↑𝑘))⟩)
243	242	fveq2d 6851	. . . . 5 ⊢ (𝑘 ∈ ℕ → ((ball‘𝐷)‘(𝑀‘𝑘)) = ((ball‘𝐷)‘⟨(𝑆‘𝑘), (3 / (2↑𝑘))⟩))
244		df-ov 7365	. . . . 5 ⊢ ((𝑆‘𝑘)(ball‘𝐷)(3 / (2↑𝑘))) = ((ball‘𝐷)‘⟨(𝑆‘𝑘), (3 / (2↑𝑘))⟩)
245	243, 244	eqtr4di 2789	. . . 4 ⊢ (𝑘 ∈ ℕ → ((ball‘𝐷)‘(𝑀‘𝑘)) = ((𝑆‘𝑘)(ball‘𝐷)(3 / (2↑𝑘))))
246	245	adantl 482	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((ball‘𝐷)‘(𝑀‘𝑘)) = ((𝑆‘𝑘)(ball‘𝐷)(3 / (2↑𝑘))))
247	223, 236, 246	3sstr4d 3994	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((ball‘𝐷)‘(𝑀‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝑀‘𝑘)))
248	247	ralrimiva 3139	1 ⊢ (𝜑 → ∀𝑘 ∈ ℕ ((ball‘𝐷)‘(𝑀‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝑀‘𝑘)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  {cab 2708   ≠ wne 2939  ∀wral 3060  ∃wrex 3069  Vcvv 3446   ∩ cin 3912   ⊆ wss 3913  ∅c0 4287  ifcif 4491  𝒫 cpw 4565  ⟨cop 4597  ∪ cuni 4870  ∪ ciun 4959   class class class wbr 5110  {copab 5172   ↦ cmpt 5193  ⟶wf 6497  ‘cfv 6501  (class class class)co 7362   ∈ cmpo 7364  2^nd c2nd 7925  Fincfn 8890  ℂcc 11058  ℝcr 11059  0cc0 11060  1c1 11061   + caddc 11063   · cmul 11065  ℝ^*cxr 11197   ≤ cle 11199   − cmin 11394   / cdiv 11821  ℕcn 12162  2c2 12217  3c3 12218  ℕ₀cn0 12422  ℤ_≥cuz 12772  ℝ⁺crp 12924   +_𝑒 cxad 13040  seqcseq 13916  ↑cexp 13977  ∞Metcxmet 20818  Metcmet 20819  ballcbl 20820  MetOpencmopn 20823  CMetccmet 24655

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-cnex 11116  ax-resscn 11117  ax-1cn 11118  ax-icn 11119  ax-addcl 11120  ax-addrcl 11121  ax-mulcl 11122  ax-mulrcl 11123  ax-mulcom 11124  ax-addass 11125  ax-mulass 11126  ax-distr 11127  ax-i2m1 11128  ax-1ne0 11129  ax-1rid 11130  ax-rnegex 11131  ax-rrecex 11132  ax-cnre 11133  ax-pre-lttri 11134  ax-pre-lttrn 11135  ax-pre-ltadd 11136  ax-pre-mulgt0 11137

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3351  df-reu 3352  df-rab 3406  df-v 3448  df-sbc 3743  df-csb 3859  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-er 8655  df-map 8774  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-pnf 11200  df-mnf 11201  df-xr 11202  df-ltxr 11203  df-le 11204  df-sub 11396  df-neg 11397  df-div 11822  df-nn 12163  df-2 12225  df-3 12226  df-n0 12423  df-z 12509  df-uz 12773  df-rp 12925  df-xneg 13042  df-xadd 13043  df-xmul 13044  df-seq 13917  df-exp 13978  df-psmet 20825  df-xmet 20826  df-met 20827  df-bl 20828  df-cmet 24658

This theorem is referenced by:  heiborlem8  36350  heiborlem9  36351