Theorem heiborlem6 35711

Description: Lemma for heibor 35716. 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 12097	. . . 4 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ0)
2		heibor.6	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋))
3		cmetmet 24183	. . . . . . . 8 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
4	2, 3	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋))
5		metxmet 23232	. . . . . . 7 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
6	4, 5	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋))
7	6	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝐷 ∈ (∞Met‘𝑋))
8		heibor.7	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin))
9		inss1 4143	. . . . . . . . 9 ⊢ (𝒫 𝑋 ∩ Fin) ⊆ 𝒫 𝑋
10		fss 6562	. . . . . . . . 9 ⊢ ((𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin) ∧ (𝒫 𝑋 ∩ Fin) ⊆ 𝒫 𝑋) → 𝐹:ℕ0⟶𝒫 𝑋)
11	8, 9, 10	sylancl 589	. . . . . . . 8 ⊢ (𝜑 → 𝐹:ℕ0⟶𝒫 𝑋)
12		peano2nn0 12130	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ0 → (𝑘 + 1) ∈ ℕ0)
13		ffvelrn 6902	. . . . . . . 8 ⊢ ((𝐹:ℕ0⟶𝒫 𝑋 ∧ (𝑘 + 1) ∈ ℕ0) → (𝐹‘(𝑘 + 1)) ∈ 𝒫 𝑋)
14	11, 12, 13	syl2an 599	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘(𝑘 + 1)) ∈ 𝒫 𝑋)
15	14	elpwid 4524	. . . . . 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 35709	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 + 1) ∈ ℕ0) → (𝑆‘(𝑘 + 1))𝐺(𝑘 + 1))
25	12, 24	sylan2 596	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑆‘(𝑘 + 1))𝐺(𝑘 + 1))
26		fvex 6730	. . . . . . . . 9 ⊢ (𝑆‘(𝑘 + 1)) ∈ V
27		ovex 7246	. . . . . . . . 9 ⊢ (𝑘 + 1) ∈ V
28	16, 17, 18, 26, 27	heiborlem2 35707	. . . . . . . 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 1148	. . . . . . 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 3902	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑆‘(𝑘 + 1)) ∈ 𝑋)
32	11	ffvelrnda 6904	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘𝑘) ∈ 𝒫 𝑋)
33	32	elpwid 4524	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘𝑘) ⊆ 𝑋)
34	16, 17, 18, 19, 2, 8, 20, 21, 22, 23	heiborlem4 35709	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑆‘𝑘)𝐺𝑘)
35		fvex 6730	. . . . . . . . 9 ⊢ (𝑆‘𝑘) ∈ V
36		vex 3412	. . . . . . . . 9 ⊢ 𝑘 ∈ V
37	16, 17, 18, 35, 36	heiborlem2 35707	. . . . . . . 8 ⊢ ((𝑆‘𝑘)𝐺𝑘 ↔ (𝑘 ∈ ℕ0 ∧ (𝑆‘𝑘) ∈ (𝐹‘𝑘) ∧ ((𝑆‘𝑘)𝐵𝑘) ∈ 𝐾))
38	37	simp2bi 1148	. . . . . . 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 3902	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑆‘𝑘) ∈ 𝑋)
41		3re 11910	. . . . . 6 ⊢ 3 ∈ ℝ
42		2nn 11903	. . . . . . . . 9 ⊢ 2 ∈ ℕ
43		nnexpcl 13648	. . . . . . . . 9 ⊢ ((2 ∈ ℕ ∧ (𝑘 + 1) ∈ ℕ0) → (2↑(𝑘 + 1)) ∈ ℕ)
44	42, 12, 43	sylancr 590	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ0 → (2↑(𝑘 + 1)) ∈ ℕ)
45	44	nnrpd 12626	. . . . . . 7 ⊢ (𝑘 ∈ ℕ0 → (2↑(𝑘 + 1)) ∈ ℝ+)
46	45	adantl 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (2↑(𝑘 + 1)) ∈ ℝ+)
47		rerpdivcl 12616	. . . . . 6 ⊢ ((3 ∈ ℝ ∧ (2↑(𝑘 + 1)) ∈ ℝ+) → (3 / (2↑(𝑘 + 1))) ∈ ℝ)
48	41, 46, 47	sylancr 590	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (3 / (2↑(𝑘 + 1))) ∈ ℝ)
49		nnexpcl 13648	. . . . . . . . 9 ⊢ ((2 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → (2↑𝑘) ∈ ℕ)
50	42, 49	mpan 690	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ0 → (2↑𝑘) ∈ ℕ)
51	50	nnrpd 12626	. . . . . . 7 ⊢ (𝑘 ∈ ℕ0 → (2↑𝑘) ∈ ℝ+)
52	51	adantl 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (2↑𝑘) ∈ ℝ+)
53		rerpdivcl 12616	. . . . . 6 ⊢ ((3 ∈ ℝ ∧ (2↑𝑘) ∈ ℝ+) → (3 / (2↑𝑘)) ∈ ℝ)
54	41, 52, 53	sylancr 590	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (3 / (2↑𝑘)) ∈ ℝ)
55		oveq1 7220	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝑆‘𝑘) → (𝑧(ball‘𝐷)(1 / (2↑𝑚))) = ((𝑆‘𝑘)(ball‘𝐷)(1 / (2↑𝑚))))
56		oveq2 7221	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑘 → (2↑𝑚) = (2↑𝑘))
57	56	oveq2d 7229	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑘 → (1 / (2↑𝑚)) = (1 / (2↑𝑘)))
58	57	oveq2d 7229	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑘 → ((𝑆‘𝑘)(ball‘𝐷)(1 / (2↑𝑚))) = ((𝑆‘𝑘)(ball‘𝐷)(1 / (2↑𝑘))))
59		ovex 7246	. . . . . . . . . . . 12 ⊢ ((𝑆‘𝑘)(ball‘𝐷)(1 / (2↑𝑘))) ∈ V
60	55, 58, 19, 59	ovmpo 7369	. . . . . . . . . . 11 ⊢ (((𝑆‘𝑘) ∈ 𝑋 ∧ 𝑘 ∈ ℕ0) → ((𝑆‘𝑘)𝐵𝑘) = ((𝑆‘𝑘)(ball‘𝐷)(1 / (2↑𝑘))))
61	40, 60	sylancom 591	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝑆‘𝑘)𝐵𝑘) = ((𝑆‘𝑘)(ball‘𝐷)(1 / (2↑𝑘))))
62		df-br 5054	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑆‘𝑘)𝐺𝑘 ↔ ⟨(𝑆‘𝑘), 𝑘⟩ ∈ 𝐺)
63		fveq2 6717	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = ⟨(𝑆‘𝑘), 𝑘⟩ → (𝑇‘𝑥) = (𝑇‘⟨(𝑆‘𝑘), 𝑘⟩))
64		df-ov 7216	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑆‘𝑘)𝑇𝑘) = (𝑇‘⟨(𝑆‘𝑘), 𝑘⟩)
65	63, 64	eqtr4di 2796	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = ⟨(𝑆‘𝑘), 𝑘⟩ → (𝑇‘𝑥) = ((𝑆‘𝑘)𝑇𝑘))
66	35, 36	op2ndd 7772	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = ⟨(𝑆‘𝑘), 𝑘⟩ → (2nd ‘𝑥) = 𝑘)
67	66	oveq1d 7228	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = ⟨(𝑆‘𝑘), 𝑘⟩ → ((2nd ‘𝑥) + 1) = (𝑘 + 1))
68	65, 67	breq12d 5066	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = ⟨(𝑆‘𝑘), 𝑘⟩ → ((𝑇‘𝑥)𝐺((2nd ‘𝑥) + 1) ↔ ((𝑆‘𝑘)𝑇𝑘)𝐺(𝑘 + 1)))
69		fveq2 6717	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = ⟨(𝑆‘𝑘), 𝑘⟩ → (𝐵‘𝑥) = (𝐵‘⟨(𝑆‘𝑘), 𝑘⟩))
70		df-ov 7216	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑆‘𝑘)𝐵𝑘) = (𝐵‘⟨(𝑆‘𝑘), 𝑘⟩)
71	69, 70	eqtr4di 2796	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = ⟨(𝑆‘𝑘), 𝑘⟩ → (𝐵‘𝑥) = ((𝑆‘𝑘)𝐵𝑘))
72	65, 67	oveq12d 7231	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = ⟨(𝑆‘𝑘), 𝑘⟩ → ((𝑇‘𝑥)𝐵((2nd ‘𝑥) + 1)) = (((𝑆‘𝑘)𝑇𝑘)𝐵(𝑘 + 1)))
73	71, 72	ineq12d 4128	. . . . . . . . . . . . . . . . . . . . 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 634	. . . . . . . . . . . . . . . . . . 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 3534	. . . . . . . . . . . . . . . . . 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	syl5bi 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 484	. . . . . . . . . . . . . . 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 498	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝑆‘𝑘)𝑇𝑘)𝐺(𝑘 + 1))
82		ovex 7246	. . . . . . . . . . . . . . 15 ⊢ ((𝑆‘𝑘)𝑇𝑘) ∈ V
83	16, 17, 18, 82, 27	heiborlem2 35707	. . . . . . . . . . . . . 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 1148	. . . . . . . . . . . . 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 3902	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝑆‘𝑘)𝑇𝑘) ∈ 𝑋)
87	12	adantl 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑘 + 1) ∈ ℕ0)
88		oveq1 7220	. . . . . . . . . . . 12 ⊢ (𝑧 = ((𝑆‘𝑘)𝑇𝑘) → (𝑧(ball‘𝐷)(1 / (2↑𝑚))) = (((𝑆‘𝑘)𝑇𝑘)(ball‘𝐷)(1 / (2↑𝑚))))
89		oveq2 7221	. . . . . . . . . . . . . 14 ⊢ (𝑚 = (𝑘 + 1) → (2↑𝑚) = (2↑(𝑘 + 1)))
90	89	oveq2d 7229	. . . . . . . . . . . . 13 ⊢ (𝑚 = (𝑘 + 1) → (1 / (2↑𝑚)) = (1 / (2↑(𝑘 + 1))))
91	90	oveq2d 7229	. . . . . . . . . . . 12 ⊢ (𝑚 = (𝑘 + 1) → (((𝑆‘𝑘)𝑇𝑘)(ball‘𝐷)(1 / (2↑𝑚))) = (((𝑆‘𝑘)𝑇𝑘)(ball‘𝐷)(1 / (2↑(𝑘 + 1)))))
92		ovex 7246	. . . . . . . . . . . 12 ⊢ (((𝑆‘𝑘)𝑇𝑘)(ball‘𝐷)(1 / (2↑(𝑘 + 1)))) ∈ V
93	88, 91, 19, 92	ovmpo 7369	. . . . . . . . . . 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 587	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (((𝑆‘𝑘)𝑇𝑘)𝐵(𝑘 + 1)) = (((𝑆‘𝑘)𝑇𝑘)(ball‘𝐷)(1 / (2↑(𝑘 + 1)))))
95	61, 94	ineq12d 4128	. . . . . . . . 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 499	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (((𝑆‘𝑘)𝐵𝑘) ∩ (((𝑆‘𝑘)𝑇𝑘)𝐵(𝑘 + 1))) ∈ 𝐾)
97		0elpw 5247	. . . . . . . . . . . . 13 ⊢ ∅ ∈ 𝒫 𝑈
98		0fin 8849	. . . . . . . . . . . . 13 ⊢ ∅ ∈ Fin
99		elin 3882	. . . . . . . . . . . . 13 ⊢ (∅ ∈ (𝒫 𝑈 ∩ Fin) ↔ (∅ ∈ 𝒫 𝑈 ∧ ∅ ∈ Fin))
100	97, 98, 99	mpbir2an 711	. . . . . . . . . . . 12 ⊢ ∅ ∈ (𝒫 𝑈 ∩ Fin)
101		0ss 4311	. . . . . . . . . . . 12 ⊢ ∅ ⊆ ∪ ∅
102		unieq 4830	. . . . . . . . . . . . . 14 ⊢ (𝑣 = ∅ → ∪ 𝑣 = ∪ ∅)
103	102	sseq2d 3933	. . . . . . . . . . . . 13 ⊢ (𝑣 = ∅ → (∅ ⊆ ∪ 𝑣 ↔ ∅ ⊆ ∪ ∅))
104	103	rspcev 3537	. . . . . . . . . . . 12 ⊢ ((∅ ∈ (𝒫 𝑈 ∩ Fin) ∧ ∅ ⊆ ∪ ∅) → ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)∅ ⊆ ∪ 𝑣)
105	100, 101, 104	mp2an 692	. . . . . . . . . . 11 ⊢ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)∅ ⊆ ∪ 𝑣
106		0ex 5200	. . . . . . . . . . . . 13 ⊢ ∅ ∈ V
107		sseq1 3926	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = ∅ → (𝑢 ⊆ ∪ 𝑣 ↔ ∅ ⊆ ∪ 𝑣))
108	107	rexbidv 3216	. . . . . . . . . . . . . 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 3591	. . . . . . . . . . . 12 ⊢ (∅ ∈ 𝐾 ↔ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)∅ ⊆ ∪ 𝑣)
111	110	con2bii 361	. . . . . . . . . . 11 ⊢ (∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)∅ ⊆ ∪ 𝑣 ↔ ¬ ∅ ∈ 𝐾)
112	105, 111	mpbi 233	. . . . . . . . . 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 589	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (((𝑆‘𝑘)𝐵𝑘) ∩ (((𝑆‘𝑘)𝑇𝑘)𝐵(𝑘 + 1))) ≠ ∅)
115	95, 114	eqnetrrd 3009	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (((𝑆‘𝑘)(ball‘𝐷)(1 / (2↑𝑘))) ∩ (((𝑆‘𝑘)𝑇𝑘)(ball‘𝐷)(1 / (2↑(𝑘 + 1))))) ≠ ∅)
116	51	rpreccld 12638	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ0 → (1 / (2↑𝑘)) ∈ ℝ+)
117	116	adantl 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (1 / (2↑𝑘)) ∈ ℝ+)
118	117	rpred 12628	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (1 / (2↑𝑘)) ∈ ℝ)
119	45	rpreccld 12638	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ0 → (1 / (2↑(𝑘 + 1))) ∈ ℝ+)
120	119	adantl 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (1 / (2↑(𝑘 + 1))) ∈ ℝ+)
121	120	rpred 12628	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (1 / (2↑(𝑘 + 1))) ∈ ℝ)
122		rexadd 12822	. . . . . . . . . . . 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 587	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((1 / (2↑𝑘)) +𝑒 (1 / (2↑(𝑘 + 1)))) = ((1 / (2↑𝑘)) + (1 / (2↑(𝑘 + 1)))))
124	123	breq1d 5063	. . . . . . . . . 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 12629	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (1 / (2↑𝑘)) ∈ ℝ*)
126	120	rpxrd 12629	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (1 / (2↑(𝑘 + 1))) ∈ ℝ*)
127		bldisj 23296	. . . . . . . . . . . . 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 1356	. . . . . . . . . . . 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 422	. . . . . . . . . . 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 1380	. . . . . . . . . 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 2953	. . . . . . . 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 12643	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((1 / (2↑𝑘)) + (1 / (2↑(𝑘 + 1)))) ∈ ℝ+)
135	134	rpred 12628	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((1 / (2↑𝑘)) + (1 / (2↑(𝑘 + 1)))) ∈ ℝ)
136	4	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝐷 ∈ (Met‘𝑋))
137		metcl 23230	. . . . . . . . . 10 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ (𝑆‘𝑘) ∈ 𝑋 ∧ ((𝑆‘𝑘)𝑇𝑘) ∈ 𝑋) → ((𝑆‘𝑘)𝐷((𝑆‘𝑘)𝑇𝑘)) ∈ ℝ)
138	136, 40, 86, 137	syl3anc 1373	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝑆‘𝑘)𝐷((𝑆‘𝑘)𝑇𝑘)) ∈ ℝ)
139	135, 138	letrid 10984	. . . . . . . 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 864	. . . . . . 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 13589	. . . . . . . . . . . 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 12476	. . . . . . . . . . . 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 6718	. . . . . . . . . . 11 ⊢ (𝑆‘(𝑘 + 1)) = (seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))))‘(𝑘 + 1))
146	23	fveq1i 6718	. . . . . . . . . . . 12 ⊢ (𝑆‘𝑘) = (seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))))‘𝑘)
147	146	oveq1i 7223	. . . . . . . . . . 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 2803	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ0 → (𝑆‘(𝑘 + 1)) = ((𝑆‘𝑘)𝑇((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))‘(𝑘 + 1))))
149		eqid 2737	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))) = (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))
150		eqeq1 2741	. . . . . . . . . . . . . 14 ⊢ (𝑚 = (𝑘 + 1) → (𝑚 = 0 ↔ (𝑘 + 1) = 0))
151		oveq1 7220	. . . . . . . . . . . . . 14 ⊢ (𝑚 = (𝑘 + 1) → (𝑚 − 1) = ((𝑘 + 1) − 1))
152	150, 151	ifbieq2d 4465	. . . . . . . . . . . . 13 ⊢ (𝑚 = (𝑘 + 1) → if(𝑚 = 0, 𝐶, (𝑚 − 1)) = if((𝑘 + 1) = 0, 𝐶, ((𝑘 + 1) − 1)))
153		nn0p1nn 12129	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ ℕ0 → (𝑘 + 1) ∈ ℕ)
154		nnne0 11864	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘 + 1) ∈ ℕ → (𝑘 + 1) ≠ 0)
155	154	neneqd 2945	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 + 1) ∈ ℕ → ¬ (𝑘 + 1) = 0)
156	153, 155	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℕ0 → ¬ (𝑘 + 1) = 0)
157	156	iffalsed 4450	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ0 → if((𝑘 + 1) = 0, 𝐶, ((𝑘 + 1) − 1)) = ((𝑘 + 1) − 1))
158		ovex 7246	. . . . . . . . . . . . . 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 6841	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ0 → ((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))‘(𝑘 + 1)) = if((𝑘 + 1) = 0, 𝐶, ((𝑘 + 1) − 1)))
161		nn0cn 12100	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ0 → 𝑘 ∈ ℂ)
162		ax-1cn 10787	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℂ
163		pncan 11084	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑘 + 1) − 1) = 𝑘)
164	161, 162, 163	sylancl 589	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ0 → ((𝑘 + 1) − 1) = 𝑘)
165	160, 157, 164	3eqtrd 2781	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ0 → ((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))‘(𝑘 + 1)) = 𝑘)
166	165	oveq2d 7229	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ0 → ((𝑆‘𝑘)𝑇((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))‘(𝑘 + 1))) = ((𝑆‘𝑘)𝑇𝑘))
167	148, 166	eqtrd 2777	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ0 → (𝑆‘(𝑘 + 1)) = ((𝑆‘𝑘)𝑇𝑘))
168	167	adantl 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑆‘(𝑘 + 1)) = ((𝑆‘𝑘)𝑇𝑘))
169	168	oveq1d 7228	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝑆‘(𝑘 + 1))𝐷(𝑆‘𝑘)) = (((𝑆‘𝑘)𝑇𝑘)𝐷(𝑆‘𝑘)))
170		metsym 23248	. . . . . . . 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 1373	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (((𝑆‘𝑘)𝑇𝑘)𝐷(𝑆‘𝑘)) = ((𝑆‘𝑘)𝐷((𝑆‘𝑘)𝑇𝑘)))
172	169, 171	eqtrd 2777	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝑆‘(𝑘 + 1))𝐷(𝑆‘𝑘)) = ((𝑆‘𝑘)𝐷((𝑆‘𝑘)𝑇𝑘)))
173		3cn 11911	. . . . . . . . . . . . 13 ⊢ 3 ∈ ℂ
174	173	2timesi 11968	. . . . . . . . . . . 12 ⊢ (2 · 3) = (3 + 3)
175	174	oveq1i 7223	. . . . . . . . . . 11 ⊢ ((2 · 3) − 3) = ((3 + 3) − 3)
176	173, 173	pncan3oi 11094	. . . . . . . . . . 11 ⊢ ((3 + 3) − 3) = 3
177		df-3 11894	. . . . . . . . . . 11 ⊢ 3 = (2 + 1)
178	175, 176, 177	3eqtri 2769	. . . . . . . . . 10 ⊢ ((2 · 3) − 3) = (2 + 1)
179	178	oveq1i 7223	. . . . . . . . 9 ⊢ (((2 · 3) − 3) / (2↑(𝑘 + 1))) = ((2 + 1) / (2↑(𝑘 + 1)))
180		rpcn 12596	. . . . . . . . . . 11 ⊢ ((2↑(𝑘 + 1)) ∈ ℝ+ → (2↑(𝑘 + 1)) ∈ ℂ)
181		rpne0 12602	. . . . . . . . . . 11 ⊢ ((2↑(𝑘 + 1)) ∈ ℝ+ → (2↑(𝑘 + 1)) ≠ 0)
182		2cn 11905	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℂ
183	182, 173	mulcli 10840	. . . . . . . . . . . 12 ⊢ (2 · 3) ∈ ℂ
184		divsubdir 11526	. . . . . . . . . . . 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 1453	. . . . . . . . . . 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 587	. . . . . . . . . 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 11515	. . . . . . . . . . . 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 1453	. . . . . . . . . . 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 587	. . . . . . . . . 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 2802	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ0 → (((2 · 3) / (2↑(𝑘 + 1))) − (3 / (2↑(𝑘 + 1)))) = ((2 / (2↑(𝑘 + 1))) + (1 / (2↑(𝑘 + 1)))))
193		rpcn 12596	. . . . . . . . . . . 12 ⊢ ((2↑𝑘) ∈ ℝ+ → (2↑𝑘) ∈ ℂ)
194		rpne0 12602	. . . . . . . . . . . 12 ⊢ ((2↑𝑘) ∈ ℝ+ → (2↑𝑘) ≠ 0)
195		2cnne0 12040	. . . . . . . . . . . . 13 ⊢ (2 ∈ ℂ ∧ 2 ≠ 0)
196		divcan5 11534	. . . . . . . . . . . . 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 1454	. . . . . . . . . . . 12 ⊢ (((2↑𝑘) ∈ ℂ ∧ (2↑𝑘) ≠ 0) → ((2 · 3) / (2 · (2↑𝑘))) = (3 / (2↑𝑘)))
198	193, 194, 197	syl2anc 587	. . . . . . . . . . 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 10815	. . . . . . . . . . . . 13 ⊢ ((2 ∈ ℂ ∧ (2↑𝑘) ∈ ℂ) → (2 · (2↑𝑘)) = ((2↑𝑘) · 2))
202	182, 200, 201	sylancr 590	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ0 → (2 · (2↑𝑘)) = ((2↑𝑘) · 2))
203		expp1 13642	. . . . . . . . . . . . 13 ⊢ ((2 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (2↑(𝑘 + 1)) = ((2↑𝑘) · 2))
204	182, 203	mpan 690	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ0 → (2↑(𝑘 + 1)) = ((2↑𝑘) · 2))
205	202, 204	eqtr4d 2780	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ0 → (2 · (2↑𝑘)) = (2↑(𝑘 + 1)))
206	205	oveq2d 7229	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ0 → ((2 · 3) / (2 · (2↑𝑘))) = ((2 · 3) / (2↑(𝑘 + 1))))
207	199, 206	eqtr3d 2779	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ0 → (3 / (2↑𝑘)) = ((2 · 3) / (2↑(𝑘 + 1))))
208	207	oveq1d 7228	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ0 → ((3 / (2↑𝑘)) − (3 / (2↑(𝑘 + 1)))) = (((2 · 3) / (2↑(𝑘 + 1))) − (3 / (2↑(𝑘 + 1)))))
209		divcan5 11534	. . . . . . . . . . . . 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 1454	. . . . . . . . . . . 12 ⊢ (((2↑𝑘) ∈ ℂ ∧ (2↑𝑘) ≠ 0) → ((2 · 1) / (2 · (2↑𝑘))) = (1 / (2↑𝑘)))
211	193, 194, 210	syl2anc 587	. . . . . . . . . . 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 11993	. . . . . . . . . . . 12 ⊢ (2 · 1) = 2
214	213	a1i 11	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ0 → (2 · 1) = 2)
215	214, 205	oveq12d 7231	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ0 → ((2 · 1) / (2 · (2↑𝑘))) = (2 / (2↑(𝑘 + 1))))
216	212, 215	eqtr3d 2779	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ0 → (1 / (2↑𝑘)) = (2 / (2↑(𝑘 + 1))))
217	216	oveq1d 7228	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ0 → ((1 / (2↑𝑘)) + (1 / (2↑(𝑘 + 1)))) = ((2 / (2↑(𝑘 + 1))) + (1 / (2↑(𝑘 + 1)))))
218	192, 208, 217	3eqtr4d 2787	. . . . . . 7 ⊢ (𝑘 ∈ ℕ0 → ((3 / (2↑𝑘)) − (3 / (2↑(𝑘 + 1)))) = ((1 / (2↑𝑘)) + (1 / (2↑(𝑘 + 1)))))
219	218	adantl 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((3 / (2↑𝑘)) − (3 / (2↑(𝑘 + 1)))) = ((1 / (2↑𝑘)) + (1 / (2↑(𝑘 + 1)))))
220	141, 172, 219	3brtr4d 5085	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝑆‘(𝑘 + 1))𝐷(𝑆‘𝑘)) ≤ ((3 / (2↑𝑘)) − (3 / (2↑(𝑘 + 1)))))
221		blss2 23302	. . . . 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 1387	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝑆‘(𝑘 + 1))(ball‘𝐷)(3 / (2↑(𝑘 + 1)))) ⊆ ((𝑆‘𝑘)(ball‘𝐷)(3 / (2↑𝑘))))
223	1, 222	sylan2 596	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝑆‘(𝑘 + 1))(ball‘𝐷)(3 / (2↑(𝑘 + 1)))) ⊆ ((𝑆‘𝑘)(ball‘𝐷)(3 / (2↑𝑘))))
224		peano2nn 11842	. . . . . . 7 ⊢ (𝑘 ∈ ℕ → (𝑘 + 1) ∈ ℕ)
225		fveq2 6717	. . . . . . . . 9 ⊢ (𝑛 = (𝑘 + 1) → (𝑆‘𝑛) = (𝑆‘(𝑘 + 1)))
226		oveq2 7221	. . . . . . . . . 10 ⊢ (𝑛 = (𝑘 + 1) → (2↑𝑛) = (2↑(𝑘 + 1)))
227	226	oveq2d 7229	. . . . . . . . 9 ⊢ (𝑛 = (𝑘 + 1) → (3 / (2↑𝑛)) = (3 / (2↑(𝑘 + 1))))
228	225, 227	opeq12d 4792	. . . . . . . 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 5348	. . . . . . . 8 ⊢ ⟨(𝑆‘(𝑘 + 1)), (3 / (2↑(𝑘 + 1)))⟩ ∈ V
231	228, 229, 230	fvmpt 6818	. . . . . . 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 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑀‘(𝑘 + 1)) = ⟨(𝑆‘(𝑘 + 1)), (3 / (2↑(𝑘 + 1)))⟩)
234	233	fveq2d 6721	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((ball‘𝐷)‘(𝑀‘(𝑘 + 1))) = ((ball‘𝐷)‘⟨(𝑆‘(𝑘 + 1)), (3 / (2↑(𝑘 + 1)))⟩))
235		df-ov 7216	. . . 4 ⊢ ((𝑆‘(𝑘 + 1))(ball‘𝐷)(3 / (2↑(𝑘 + 1)))) = ((ball‘𝐷)‘⟨(𝑆‘(𝑘 + 1)), (3 / (2↑(𝑘 + 1)))⟩)
236	234, 235	eqtr4di 2796	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((ball‘𝐷)‘(𝑀‘(𝑘 + 1))) = ((𝑆‘(𝑘 + 1))(ball‘𝐷)(3 / (2↑(𝑘 + 1)))))
237		fveq2 6717	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → (𝑆‘𝑛) = (𝑆‘𝑘))
238		oveq2 7221	. . . . . . . . 9 ⊢ (𝑛 = 𝑘 → (2↑𝑛) = (2↑𝑘))
239	238	oveq2d 7229	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → (3 / (2↑𝑛)) = (3 / (2↑𝑘)))
240	237, 239	opeq12d 4792	. . . . . . 7 ⊢ (𝑛 = 𝑘 → ⟨(𝑆‘𝑛), (3 / (2↑𝑛))⟩ = ⟨(𝑆‘𝑘), (3 / (2↑𝑘))⟩)
241		opex 5348	. . . . . . 7 ⊢ ⟨(𝑆‘𝑘), (3 / (2↑𝑘))⟩ ∈ V
242	240, 229, 241	fvmpt 6818	. . . . . 6 ⊢ (𝑘 ∈ ℕ → (𝑀‘𝑘) = ⟨(𝑆‘𝑘), (3 / (2↑𝑘))⟩)
243	242	fveq2d 6721	. . . . 5 ⊢ (𝑘 ∈ ℕ → ((ball‘𝐷)‘(𝑀‘𝑘)) = ((ball‘𝐷)‘⟨(𝑆‘𝑘), (3 / (2↑𝑘))⟩))
244		df-ov 7216	. . . . 5 ⊢ ((𝑆‘𝑘)(ball‘𝐷)(3 / (2↑𝑘))) = ((ball‘𝐷)‘⟨(𝑆‘𝑘), (3 / (2↑𝑘))⟩)
245	243, 244	eqtr4di 2796	. . . 4 ⊢ (𝑘 ∈ ℕ → ((ball‘𝐷)‘(𝑀‘𝑘)) = ((𝑆‘𝑘)(ball‘𝐷)(3 / (2↑𝑘))))
246	245	adantl 485	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((ball‘𝐷)‘(𝑀‘𝑘)) = ((𝑆‘𝑘)(ball‘𝐷)(3 / (2↑𝑘))))
247	223, 236, 246	3sstr4d 3948	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((ball‘𝐷)‘(𝑀‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝑀‘𝑘)))
248	247	ralrimiva 3105	1 ⊢ (𝜑 → ∀𝑘 ∈ ℕ ((ball‘𝐷)‘(𝑀‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝑀‘𝑘)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∧ w3a 1089   = wceq 1543   ∈ wcel 2110  {cab 2714   ≠ wne 2940  ∀wral 3061  ∃wrex 3062  Vcvv 3408   ∩ cin 3865   ⊆ wss 3866  ∅c0 4237  ifcif 4439  𝒫 cpw 4513  ⟨cop 4547  ∪ cuni 4819  ∪ ciun 4904   class class class wbr 5053  {copab 5115   ↦ cmpt 5135  ⟶wf 6376  ‘cfv 6380  (class class class)co 7213   ∈ cmpo 7215  2^nd c2nd 7760  Fincfn 8626  ℂcc 10727  ℝcr 10728  0cc0 10729  1c1 10730   + caddc 10732   · cmul 10734  ℝ^*cxr 10866   ≤ cle 10868   − cmin 11062   / cdiv 11489  ℕcn 11830  2c2 11885  3c3 11886  ℕ₀cn0 12090  ℤ_≥cuz 12438  ℝ⁺crp 12586   +_𝑒 cxad 12702  seqcseq 13574  ↑cexp 13635  ∞Metcxmet 20348  Metcmet 20349  ballcbl 20350  MetOpencmopn 20353  CMetccmet 24151

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-cnex 10785  ax-resscn 10786  ax-1cn 10787  ax-icn 10788  ax-addcl 10789  ax-addrcl 10790  ax-mulcl 10791  ax-mulrcl 10792  ax-mulcom 10793  ax-addass 10794  ax-mulass 10795  ax-distr 10796  ax-i2m1 10797  ax-1ne0 10798  ax-1rid 10799  ax-rnegex 10800  ax-rrecex 10801  ax-cnre 10802  ax-pre-lttri 10803  ax-pre-lttrn 10804  ax-pre-ltadd 10805  ax-pre-mulgt0 10806

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-1st 7761  df-2nd 7762  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-er 8391  df-map 8510  df-en 8627  df-dom 8628  df-sdom 8629  df-fin 8630  df-pnf 10869  df-mnf 10870  df-xr 10871  df-ltxr 10872  df-le 10873  df-sub 11064  df-neg 11065  df-div 11490  df-nn 11831  df-2 11893  df-3 11894  df-n0 12091  df-z 12177  df-uz 12439  df-rp 12587  df-xneg 12704  df-xadd 12705  df-xmul 12706  df-seq 13575  df-exp 13636  df-psmet 20355  df-xmet 20356  df-met 20357  df-bl 20358  df-cmet 24154

This theorem is referenced by:  heiborlem8  35713  heiborlem9  35714