Theorem heiborlem6 36275

Description: Lemma for heibor 36280. 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 12420	. . . 4 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ0)
2		heibor.6	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋))
3		cmetmet 24650	. . . . . . . 8 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
4	2, 3	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋))
5		metxmet 23687	. . . . . . 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 4188	. . . . . . . . 9 ⊢ (𝒫 𝑋 ∩ Fin) ⊆ 𝒫 𝑋
10		fss 6685	. . . . . . . . 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 12453	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ0 → (𝑘 + 1) ∈ ℕ0)
13		ffvelcdm 7032	. . . . . . . 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 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 36273	. . . . . . . 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 6855	. . . . . . . . 9 ⊢ (𝑆‘(𝑘 + 1)) ∈ V
27		ovex 7390	. . . . . . . . 9 ⊢ (𝑘 + 1) ∈ V
28	16, 17, 18, 26, 27	heiborlem2 36271	. . . . . . . 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 3945	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑆‘(𝑘 + 1)) ∈ 𝑋)
32	11	ffvelcdmda 7035	. . . . . . 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 36273	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑆‘𝑘)𝐺𝑘)
35		fvex 6855	. . . . . . . . 9 ⊢ (𝑆‘𝑘) ∈ V
36		vex 3449	. . . . . . . . 9 ⊢ 𝑘 ∈ V
37	16, 17, 18, 35, 36	heiborlem2 36271	. . . . . . . 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 3945	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑆‘𝑘) ∈ 𝑋)
41		3re 12233	. . . . . 6 ⊢ 3 ∈ ℝ
42		2nn 12226	. . . . . . . . 9 ⊢ 2 ∈ ℕ
43		nnexpcl 13980	. . . . . . . . 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 12955	. . . . . . 7 ⊢ (𝑘 ∈ ℕ0 → (2↑(𝑘 + 1)) ∈ ℝ+)
46	45	adantl 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (2↑(𝑘 + 1)) ∈ ℝ+)
47		rerpdivcl 12945	. . . . . 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 13980	. . . . . . . . 9 ⊢ ((2 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → (2↑𝑘) ∈ ℕ)
50	42, 49	mpan 688	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ0 → (2↑𝑘) ∈ ℕ)
51	50	nnrpd 12955	. . . . . . 7 ⊢ (𝑘 ∈ ℕ0 → (2↑𝑘) ∈ ℝ+)
52	51	adantl 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (2↑𝑘) ∈ ℝ+)
53		rerpdivcl 12945	. . . . . 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 7364	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝑆‘𝑘) → (𝑧(ball‘𝐷)(1 / (2↑𝑚))) = ((𝑆‘𝑘)(ball‘𝐷)(1 / (2↑𝑚))))
56		oveq2 7365	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑘 → (2↑𝑚) = (2↑𝑘))
57	56	oveq2d 7373	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑘 → (1 / (2↑𝑚)) = (1 / (2↑𝑘)))
58	57	oveq2d 7373	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑘 → ((𝑆‘𝑘)(ball‘𝐷)(1 / (2↑𝑚))) = ((𝑆‘𝑘)(ball‘𝐷)(1 / (2↑𝑘))))
59		ovex 7390	. . . . . . . . . . . 12 ⊢ ((𝑆‘𝑘)(ball‘𝐷)(1 / (2↑𝑘))) ∈ V
60	55, 58, 19, 59	ovmpo 7515	. . . . . . . . . . 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 5106	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑆‘𝑘)𝐺𝑘 ↔ ⟨(𝑆‘𝑘), 𝑘⟩ ∈ 𝐺)
63		fveq2 6842	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = ⟨(𝑆‘𝑘), 𝑘⟩ → (𝑇‘𝑥) = (𝑇‘⟨(𝑆‘𝑘), 𝑘⟩))
64		df-ov 7360	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑆‘𝑘)𝑇𝑘) = (𝑇‘⟨(𝑆‘𝑘), 𝑘⟩)
65	63, 64	eqtr4di 2794	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = ⟨(𝑆‘𝑘), 𝑘⟩ → (𝑇‘𝑥) = ((𝑆‘𝑘)𝑇𝑘))
66	35, 36	op2ndd 7932	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = ⟨(𝑆‘𝑘), 𝑘⟩ → (2nd ‘𝑥) = 𝑘)
67	66	oveq1d 7372	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = ⟨(𝑆‘𝑘), 𝑘⟩ → ((2nd ‘𝑥) + 1) = (𝑘 + 1))
68	65, 67	breq12d 5118	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = ⟨(𝑆‘𝑘), 𝑘⟩ → ((𝑇‘𝑥)𝐺((2nd ‘𝑥) + 1) ↔ ((𝑆‘𝑘)𝑇𝑘)𝐺(𝑘 + 1)))
69		fveq2 6842	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = ⟨(𝑆‘𝑘), 𝑘⟩ → (𝐵‘𝑥) = (𝐵‘⟨(𝑆‘𝑘), 𝑘⟩))
70		df-ov 7360	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑆‘𝑘)𝐵𝑘) = (𝐵‘⟨(𝑆‘𝑘), 𝑘⟩)
71	69, 70	eqtr4di 2794	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = ⟨(𝑆‘𝑘), 𝑘⟩ → (𝐵‘𝑥) = ((𝑆‘𝑘)𝐵𝑘))
72	65, 67	oveq12d 7375	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = ⟨(𝑆‘𝑘), 𝑘⟩ → ((𝑇‘𝑥)𝐵((2nd ‘𝑥) + 1)) = (((𝑆‘𝑘)𝑇𝑘)𝐵(𝑘 + 1)))
73	71, 72	ineq12d 4173	. . . . . . . . . . . . . . . . . . . . 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 2822	. . . . . . . . . . . . . . . . . . . 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 3578	. . . . . . . . . . . . . . . . . 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 7390	. . . . . . . . . . . . . . 15 ⊢ ((𝑆‘𝑘)𝑇𝑘) ∈ V
83	16, 17, 18, 82, 27	heiborlem2 36271	. . . . . . . . . . . . . 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 3945	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝑆‘𝑘)𝑇𝑘) ∈ 𝑋)
87	12	adantl 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑘 + 1) ∈ ℕ0)
88		oveq1 7364	. . . . . . . . . . . 12 ⊢ (𝑧 = ((𝑆‘𝑘)𝑇𝑘) → (𝑧(ball‘𝐷)(1 / (2↑𝑚))) = (((𝑆‘𝑘)𝑇𝑘)(ball‘𝐷)(1 / (2↑𝑚))))
89		oveq2 7365	. . . . . . . . . . . . . 14 ⊢ (𝑚 = (𝑘 + 1) → (2↑𝑚) = (2↑(𝑘 + 1)))
90	89	oveq2d 7373	. . . . . . . . . . . . 13 ⊢ (𝑚 = (𝑘 + 1) → (1 / (2↑𝑚)) = (1 / (2↑(𝑘 + 1))))
91	90	oveq2d 7373	. . . . . . . . . . . 12 ⊢ (𝑚 = (𝑘 + 1) → (((𝑆‘𝑘)𝑇𝑘)(ball‘𝐷)(1 / (2↑𝑚))) = (((𝑆‘𝑘)𝑇𝑘)(ball‘𝐷)(1 / (2↑(𝑘 + 1)))))
92		ovex 7390	. . . . . . . . . . . 12 ⊢ (((𝑆‘𝑘)𝑇𝑘)(ball‘𝐷)(1 / (2↑(𝑘 + 1)))) ∈ V
93	88, 91, 19, 92	ovmpo 7515	. . . . . . . . . . 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 4173	. . . . . . . . 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 5311	. . . . . . . . . . . . 13 ⊢ ∅ ∈ 𝒫 𝑈
98		0fin 9115	. . . . . . . . . . . . 13 ⊢ ∅ ∈ Fin
99		elin 3926	. . . . . . . . . . . . 13 ⊢ (∅ ∈ (𝒫 𝑈 ∩ Fin) ↔ (∅ ∈ 𝒫 𝑈 ∧ ∅ ∈ Fin))
100	97, 98, 99	mpbir2an 709	. . . . . . . . . . . 12 ⊢ ∅ ∈ (𝒫 𝑈 ∩ Fin)
101		0ss 4356	. . . . . . . . . . . 12 ⊢ ∅ ⊆ ∪ ∅
102		unieq 4876	. . . . . . . . . . . . . 14 ⊢ (𝑣 = ∅ → ∪ 𝑣 = ∪ ∅)
103	102	sseq2d 3976	. . . . . . . . . . . . 13 ⊢ (𝑣 = ∅ → (∅ ⊆ ∪ 𝑣 ↔ ∅ ⊆ ∪ ∅))
104	103	rspcev 3581	. . . . . . . . . . . 12 ⊢ ((∅ ∈ (𝒫 𝑈 ∩ Fin) ∧ ∅ ⊆ ∪ ∅) → ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)∅ ⊆ ∪ 𝑣)
105	100, 101, 104	mp2an 690	. . . . . . . . . . 11 ⊢ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)∅ ⊆ ∪ 𝑣
106		0ex 5264	. . . . . . . . . . . . 13 ⊢ ∅ ∈ V
107		sseq1 3969	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = ∅ → (𝑢 ⊆ ∪ 𝑣 ↔ ∅ ⊆ ∪ 𝑣))
108	107	rexbidv 3175	. . . . . . . . . . . . . 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 3634	. . . . . . . . . . . 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 3042	. . . . . . . . . 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 3012	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (((𝑆‘𝑘)(ball‘𝐷)(1 / (2↑𝑘))) ∩ (((𝑆‘𝑘)𝑇𝑘)(ball‘𝐷)(1 / (2↑(𝑘 + 1))))) ≠ ∅)
116	51	rpreccld 12967	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ0 → (1 / (2↑𝑘)) ∈ ℝ+)
117	116	adantl 482	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (1 / (2↑𝑘)) ∈ ℝ+)
118	117	rpred 12957	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (1 / (2↑𝑘)) ∈ ℝ)
119	45	rpreccld 12967	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ0 → (1 / (2↑(𝑘 + 1))) ∈ ℝ+)
120	119	adantl 482	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (1 / (2↑(𝑘 + 1))) ∈ ℝ+)
121	120	rpred 12957	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (1 / (2↑(𝑘 + 1))) ∈ ℝ)
122		rexadd 13151	. . . . . . . . . . . 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 5115	. . . . . . . . . 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 12958	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (1 / (2↑𝑘)) ∈ ℝ*)
126	120	rpxrd 12958	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (1 / (2↑(𝑘 + 1))) ∈ ℝ*)
127		bldisj 23751	. . . . . . . . . . . . 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 2956	. . . . . . . 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 12972	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((1 / (2↑𝑘)) + (1 / (2↑(𝑘 + 1)))) ∈ ℝ+)
135	134	rpred 12957	. . . . . . . . 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 23685	. . . . . . . . . 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 11307	. . . . . . . 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 13921	. . . . . . . . . . . 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 12805	. . . . . . . . . . . 12 ⊢ ℕ0 = (ℤ≥‘0)
144	142, 143	eleq2s 2856	. . . . . . . . . . 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 6843	. . . . . . . . . . 11 ⊢ (𝑆‘(𝑘 + 1)) = (seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))))‘(𝑘 + 1))
146	23	fveq1i 6843	. . . . . . . . . . . 12 ⊢ (𝑆‘𝑘) = (seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))))‘𝑘)
147	146	oveq1i 7367	. . . . . . . . . . 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 2801	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ0 → (𝑆‘(𝑘 + 1)) = ((𝑆‘𝑘)𝑇((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))‘(𝑘 + 1))))
149		eqid 2736	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))) = (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))
150		eqeq1 2740	. . . . . . . . . . . . . 14 ⊢ (𝑚 = (𝑘 + 1) → (𝑚 = 0 ↔ (𝑘 + 1) = 0))
151		oveq1 7364	. . . . . . . . . . . . . 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 12452	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ ℕ0 → (𝑘 + 1) ∈ ℕ)
154		nnne0 12187	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘 + 1) ∈ ℕ → (𝑘 + 1) ≠ 0)
155	154	neneqd 2948	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 + 1) ∈ ℕ → ¬ (𝑘 + 1) = 0)
156	153, 155	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℕ0 → ¬ (𝑘 + 1) = 0)
157	156	iffalsed 4497	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ0 → if((𝑘 + 1) = 0, 𝐶, ((𝑘 + 1) − 1)) = ((𝑘 + 1) − 1))
158		ovex 7390	. . . . . . . . . . . . . 14 ⊢ ((𝑘 + 1) − 1) ∈ V
159	157, 158	eqeltrdi 2846	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ0 → if((𝑘 + 1) = 0, 𝐶, ((𝑘 + 1) − 1)) ∈ V)
160	149, 152, 12, 159	fvmptd3 6971	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ0 → ((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))‘(𝑘 + 1)) = if((𝑘 + 1) = 0, 𝐶, ((𝑘 + 1) − 1)))
161		nn0cn 12423	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ0 → 𝑘 ∈ ℂ)
162		ax-1cn 11109	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℂ
163		pncan 11407	. . . . . . . . . . . . 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 2780	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ0 → ((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))‘(𝑘 + 1)) = 𝑘)
166	165	oveq2d 7373	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ0 → ((𝑆‘𝑘)𝑇((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))‘(𝑘 + 1))) = ((𝑆‘𝑘)𝑇𝑘))
167	148, 166	eqtrd 2776	. . . . . . . . 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 7372	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝑆‘(𝑘 + 1))𝐷(𝑆‘𝑘)) = (((𝑆‘𝑘)𝑇𝑘)𝐷(𝑆‘𝑘)))
170		metsym 23703	. . . . . . . 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 2776	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝑆‘(𝑘 + 1))𝐷(𝑆‘𝑘)) = ((𝑆‘𝑘)𝐷((𝑆‘𝑘)𝑇𝑘)))
173		3cn 12234	. . . . . . . . . . . . 13 ⊢ 3 ∈ ℂ
174	173	2timesi 12291	. . . . . . . . . . . 12 ⊢ (2 · 3) = (3 + 3)
175	174	oveq1i 7367	. . . . . . . . . . 11 ⊢ ((2 · 3) − 3) = ((3 + 3) − 3)
176	173, 173	pncan3oi 11417	. . . . . . . . . . 11 ⊢ ((3 + 3) − 3) = 3
177		df-3 12217	. . . . . . . . . . 11 ⊢ 3 = (2 + 1)
178	175, 176, 177	3eqtri 2768	. . . . . . . . . 10 ⊢ ((2 · 3) − 3) = (2 + 1)
179	178	oveq1i 7367	. . . . . . . . 9 ⊢ (((2 · 3) − 3) / (2↑(𝑘 + 1))) = ((2 + 1) / (2↑(𝑘 + 1)))
180		rpcn 12925	. . . . . . . . . . 11 ⊢ ((2↑(𝑘 + 1)) ∈ ℝ+ → (2↑(𝑘 + 1)) ∈ ℂ)
181		rpne0 12931	. . . . . . . . . . 11 ⊢ ((2↑(𝑘 + 1)) ∈ ℝ+ → (2↑(𝑘 + 1)) ≠ 0)
182		2cn 12228	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℂ
183	182, 173	mulcli 11162	. . . . . . . . . . . 12 ⊢ (2 · 3) ∈ ℂ
184		divsubdir 11849	. . . . . . . . . . . 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 11838	. . . . . . . . . . . 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 2800	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ0 → (((2 · 3) / (2↑(𝑘 + 1))) − (3 / (2↑(𝑘 + 1)))) = ((2 / (2↑(𝑘 + 1))) + (1 / (2↑(𝑘 + 1)))))
193		rpcn 12925	. . . . . . . . . . . 12 ⊢ ((2↑𝑘) ∈ ℝ+ → (2↑𝑘) ∈ ℂ)
194		rpne0 12931	. . . . . . . . . . . 12 ⊢ ((2↑𝑘) ∈ ℝ+ → (2↑𝑘) ≠ 0)
195		2cnne0 12363	. . . . . . . . . . . . 13 ⊢ (2 ∈ ℂ ∧ 2 ≠ 0)
196		divcan5 11857	. . . . . . . . . . . . 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 11137	. . . . . . . . . . . . 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 13974	. . . . . . . . . . . . 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 2779	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ0 → (2 · (2↑𝑘)) = (2↑(𝑘 + 1)))
206	205	oveq2d 7373	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ0 → ((2 · 3) / (2 · (2↑𝑘))) = ((2 · 3) / (2↑(𝑘 + 1))))
207	199, 206	eqtr3d 2778	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ0 → (3 / (2↑𝑘)) = ((2 · 3) / (2↑(𝑘 + 1))))
208	207	oveq1d 7372	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ0 → ((3 / (2↑𝑘)) − (3 / (2↑(𝑘 + 1)))) = (((2 · 3) / (2↑(𝑘 + 1))) − (3 / (2↑(𝑘 + 1)))))
209		divcan5 11857	. . . . . . . . . . . . 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 12316	. . . . . . . . . . . 12 ⊢ (2 · 1) = 2
214	213	a1i 11	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ0 → (2 · 1) = 2)
215	214, 205	oveq12d 7375	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ0 → ((2 · 1) / (2 · (2↑𝑘))) = (2 / (2↑(𝑘 + 1))))
216	212, 215	eqtr3d 2778	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ0 → (1 / (2↑𝑘)) = (2 / (2↑(𝑘 + 1))))
217	216	oveq1d 7372	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ0 → ((1 / (2↑𝑘)) + (1 / (2↑(𝑘 + 1)))) = ((2 / (2↑(𝑘 + 1))) + (1 / (2↑(𝑘 + 1)))))
218	192, 208, 217	3eqtr4d 2786	. . . . . . 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 5137	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝑆‘(𝑘 + 1))𝐷(𝑆‘𝑘)) ≤ ((3 / (2↑𝑘)) − (3 / (2↑(𝑘 + 1)))))
221		blss2 23757	. . . . 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 12165	. . . . . . 7 ⊢ (𝑘 ∈ ℕ → (𝑘 + 1) ∈ ℕ)
225		fveq2 6842	. . . . . . . . 9 ⊢ (𝑛 = (𝑘 + 1) → (𝑆‘𝑛) = (𝑆‘(𝑘 + 1)))
226		oveq2 7365	. . . . . . . . . 10 ⊢ (𝑛 = (𝑘 + 1) → (2↑𝑛) = (2↑(𝑘 + 1)))
227	226	oveq2d 7373	. . . . . . . . 9 ⊢ (𝑛 = (𝑘 + 1) → (3 / (2↑𝑛)) = (3 / (2↑(𝑘 + 1))))
228	225, 227	opeq12d 4838	. . . . . . . 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 5421	. . . . . . . 8 ⊢ ⟨(𝑆‘(𝑘 + 1)), (3 / (2↑(𝑘 + 1)))⟩ ∈ V
231	228, 229, 230	fvmpt 6948	. . . . . . 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 6846	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((ball‘𝐷)‘(𝑀‘(𝑘 + 1))) = ((ball‘𝐷)‘⟨(𝑆‘(𝑘 + 1)), (3 / (2↑(𝑘 + 1)))⟩))
235		df-ov 7360	. . . 4 ⊢ ((𝑆‘(𝑘 + 1))(ball‘𝐷)(3 / (2↑(𝑘 + 1)))) = ((ball‘𝐷)‘⟨(𝑆‘(𝑘 + 1)), (3 / (2↑(𝑘 + 1)))⟩)
236	234, 235	eqtr4di 2794	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((ball‘𝐷)‘(𝑀‘(𝑘 + 1))) = ((𝑆‘(𝑘 + 1))(ball‘𝐷)(3 / (2↑(𝑘 + 1)))))
237		fveq2 6842	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → (𝑆‘𝑛) = (𝑆‘𝑘))
238		oveq2 7365	. . . . . . . . 9 ⊢ (𝑛 = 𝑘 → (2↑𝑛) = (2↑𝑘))
239	238	oveq2d 7373	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → (3 / (2↑𝑛)) = (3 / (2↑𝑘)))
240	237, 239	opeq12d 4838	. . . . . . 7 ⊢ (𝑛 = 𝑘 → ⟨(𝑆‘𝑛), (3 / (2↑𝑛))⟩ = ⟨(𝑆‘𝑘), (3 / (2↑𝑘))⟩)
241		opex 5421	. . . . . . 7 ⊢ ⟨(𝑆‘𝑘), (3 / (2↑𝑘))⟩ ∈ V
242	240, 229, 241	fvmpt 6948	. . . . . 6 ⊢ (𝑘 ∈ ℕ → (𝑀‘𝑘) = ⟨(𝑆‘𝑘), (3 / (2↑𝑘))⟩)
243	242	fveq2d 6846	. . . . 5 ⊢ (𝑘 ∈ ℕ → ((ball‘𝐷)‘(𝑀‘𝑘)) = ((ball‘𝐷)‘⟨(𝑆‘𝑘), (3 / (2↑𝑘))⟩))
244		df-ov 7360	. . . . 5 ⊢ ((𝑆‘𝑘)(ball‘𝐷)(3 / (2↑𝑘))) = ((ball‘𝐷)‘⟨(𝑆‘𝑘), (3 / (2↑𝑘))⟩)
245	243, 244	eqtr4di 2794	. . . 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 3991	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((ball‘𝐷)‘(𝑀‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝑀‘𝑘)))
248	247	ralrimiva 3143	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 2713   ≠ wne 2943  ∀wral 3064  ∃wrex 3073  Vcvv 3445   ∩ cin 3909   ⊆ wss 3910  ∅c0 4282  ifcif 4486  𝒫 cpw 4560  ⟨cop 4592  ∪ cuni 4865  ∪ ciun 4954   class class class wbr 5105  {copab 5167   ↦ cmpt 5188  ⟶wf 6492  ‘cfv 6496  (class class class)co 7357   ∈ cmpo 7359  2^nd c2nd 7920  Fincfn 8883  ℂcc 11049  ℝcr 11050  0cc0 11051  1c1 11052   + caddc 11054   · cmul 11056  ℝ^*cxr 11188   ≤ cle 11190   − cmin 11385   / cdiv 11812  ℕcn 12153  2c2 12208  3c3 12209  ℕ₀cn0 12413  ℤ_≥cuz 12763  ℝ⁺crp 12915   +_𝑒 cxad 13031  seqcseq 13906  ↑cexp 13967  ∞Metcxmet 20781  Metcmet 20782  ballcbl 20783  MetOpencmopn 20786  CMetccmet 24618

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 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128

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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-n0 12414  df-z 12500  df-uz 12764  df-rp 12916  df-xneg 13033  df-xadd 13034  df-xmul 13035  df-seq 13907  df-exp 13968  df-psmet 20788  df-xmet 20789  df-met 20790  df-bl 20791  df-cmet 24621

This theorem is referenced by:  heiborlem8  36277  heiborlem9  36278